Introduction

Two-dimensional transition metal dichalcogenides (TMDs) exhibit remarkable electronic and optical properties, making them exciting platforms for exploring excited-state physics. In monolayer TMDs, the combination of quantum confinement and reduced dielectric screening enhances Coulomb interactions, leading to the existence of strongly-bound excitons, with binding energies of around 0.5 eV, as well as other multiparticle excitations1,2,3,4,5,6,7. These excitons dominate the linear and nonlinear8 optical properties of monolayer TMDs and facilitate strong light-matter interactions that enable the concurrent manipulation of charge, spin, and valley degrees of freedom9,10,11,12,13.

When two monolayer TMDs, or other layered 2D materials, with a small lattice mismatch or twist angle are vertically stacked, a moiré pattern that can span several nanometers emerges. This moiré pattern creates a long-range spatially varying potential that can be leveraged to engineer the electronic band structure and explore correlated electron states, such as Mott-insulators14,15,16,17,18,19, Wigner crystals14,15,17,20,21,22 and various topological states23,24,25,26,27,28,29,30,31,32,33,34. Often these strongly correlated ground states are studied via probing excitons14,17,29,35,36, necessitating a detailed understanding of the exciton spectra. However, the exciton spectra of heterobilayer samples are complicated and can be difficult to assign correctly. One complication is lattice reconstruction. Unlike in twisted bilayer graphene37,38, at small twist angles (large moiré unit cells), TMDs undergo substantial lattice relaxation leading to an atomic reconstruction with strained regions where the stacking is nearly commensurate39,40,41,42,43,44,45. These strained regions host a large variety of tunable, localized exciton states, including interlayer excitons46, inter- and intralayer moiré excitons47,48, where the electron and hole sit at different moiré lattice sites, and hybrid states, where inter- and intralayer excitons hybridize49. Moreover, the moiré-induced localization enables access to excitons with a distribution of finite momenta, whose spatial confinement has the potential to be exploited as quantum dots and single-photon emitters50.

The exciton spectrum of MoSe2/WS2 heterobilayers is particularly complicated. While heterobilayers of MX2 TMDs (M = Mo, W; X = S, Se) mostly exhibit type II band alignment, in which the conduction band minimum (CBM) and valence band maximum (VBM) lie in different layers51,52,53,54, MoSe2/WS2 is believed to exhibit near degeneracy of the conduction bands within the two layers. Consequently, the band alignment and the character of excitonic states in MoSe2/WS2 heterobilayers remain the subject of considerable debate. Early first-principles density functional theory (DFT) and many-body GW calculations indicated that MoSe2/WS2 exhibits a type II band alignment, with the VBM in the MoSe2 layer and the CBM in the WS2 layer51,52,53,54. Assumptions about the type II band alignment combined with the apparent near degeneracy of the conduction band in both layers also lead to the hypothesis that three distinct low energy peaks in the optical absorption spectrum arise from the hybridization of interlayer and intralayer exciton states55. However, more recent experiments and corresponding model calculations indicate that MoSe2/WS2 actually has a type I band alignment with both the VBM and CBM in the MoSe2 layer. Experimentally, angle-resolved photoemission spectroscopy (ARPES) measurements incorporating electrostatic doping to populate the conduction band show some evidence for the CBM being in MoSe256, and reflectance contrast spectra as a function of doping density reveal that charged excitons only form in the MoSe2 layer in large twist-angle MoSe2/WS2 heterostructures, consistent with a type I band alignment57,58. Theoretically, a phenomenological model for intra- and interlayer excitons, fit to the experimental differential reflectance signal of MoSe2/WS2 as a function of the out-of-plane electric field, suggests that the MoSe2 CBM is lower than the WS2 CBM by about 50 meV59. The hybrid exciton hypothesis49,55 for the splitting of exciton peaks near the MoSe2 A resonance has also been challenged, as reflectance spectroscopy and photoluminescence (PL) measurements show no evidence of out-of-plane dipoles near the MoSe2 A peak57,59,60. The controversy over this basic property of a widely-studied TMD heterobilayer highlights the need for synergistic first principles theory that can capture the complex interplay of crystals structure and many-body effects with experimental techniques that can track exciton dynamics in momentum space.

In this work, we employ a combination of first-principles GW and GW plus Bethe Salpeter equation (GW-BSE) calculations and time- and angle-resolved photoemission spectroscopy (tr-ARPES) to investigate the quasiparticle (QP) band alignment of MoSe2/WS2 heterobilayer and the character of the low-energy excitons near the MoSe2 A resonance. We find that in the absence of atomic reconstruction, consistent with large twist angles (small moiré unit cell), the MoSe2/WS2 heterobilayer exhibits a uniform type Ⅰ band alignment after many-electron interaction effects are included at the GW level. At small twist angles (large moiré unit cell), there is a large atomic reconstruction resulting in strained high-symmetry stacking regions, where the strain in each layer considerably alters the electronic structure, consistent with studies in other TMD bilayers43,44. This large reconstruction results in the simultaneous emergence of alternating domains of local type Ⅰ band alignment, hosting bright intralayer excitons, and local type Ⅱ band alignment, hosting interlayer excitons, in the moiré supercell of MoSe2/WS2, though the global band alignment remains type I. We compare these calculations to tr-ARPES measurements of the excitons, which measure the momentum distributions of the participating electrons and holes61,62,63, the binding energy of the excitons64, and also the layer occupation of the electron via the extreme surface sensitivity of ARPES. In our tr-ARPES measurements, we observe long-lived intralayer excitons, with electron population only in the MoSe2 layer, for samples with large twist angle, while for samples with small twist angles and large moiré unit cells, we observe both long-lived intra- and interlayer exciton species with different binding energies, consistent with our calculations. Finally, our GW-BSE calculations on moiré-free heterobilayers that are strained to match the moiré-driven reconstruction reveal that contrary to previous hypotheses of the existence of hybrid excitons49,55, the lowest-energy excitons in the MoSe2/WS2 heterostructure are purely intralayer excitons, with negligible interlayer character. We note that the strain induced by the lattice reconstruction shifts absorption peaks in different high-symmetry stacking regions, and we determine that the co-existence of the three distinct stacking regions contributes to the formation of the three low-energy exciton peaks seen in the optical spectrum at small twist angles49,57,59 and the two distinct lowest energy excitons in our tr-ARPES measurements.

Results

Band alignment at different twist angles

We start by calculating the absolute band alignment of individual isolated monolayers with respect to the vacuum level51,52,53,54. Figure 1a, d shows the vacuum-aligned DFT band structure of the individual monolayers using the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)65, which exhibits type Ⅱ band alignment with the valence band in the MoSe2 layer and the conduction band in the WS2 layer. Spin-orbit coupling splits the conduction band of WS2 by 30 meV, and the CBM of MoSe2 is nearly degenerate with WS2, lying only 27 meV above the WS2 CBM, between the spin-orbit split bands, consistent with previous DFT calculations52. However, when we consider many-electron interactions and correct the band energies by the GW approximation, the MoSe2 CBM becomes lower than the WS2 CBM by 73 meV resulting in a type I band alignment (Fig. 1b, e). Of course, in a real heterostructure, even if wavefunctions of the individual monolayers do not hybridize at the band edge, the electronic states are still screened by both layers. To isolate the role of bilayer screening on the band alignment, we then perform GW calculations for each monolayer using screening that includes the polarizability of both layers in the random phase approximation (RPA)66,67, essentially treating the second layer as a substrate whose only contribution is to the polarizability3,68,69,70. The bilayer screening renormalizes the QP bandgap, reducing the gap in WS2 from 2.48 eV to 2.22 eV and the gap in MoSe2 from 2.15 eV to 1.99 eV, but the overall type I GW band alignment remains unchanged (Fig. 1c, f).

Fig. 1: Band structure and alignment of monolayer MoSe2 and WS2 across different computational methods.
Fig. 1: Band structure and alignment of monolayer MoSe2 and WS2 across different computational methods.
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a–c Band structures of monolayer MoSe2 (red) and WS2 (blue) calculated by DFT (with the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation), GW with monolayer and bilayer dielectric screening. Calculations are performed individually for each monolayer, and the bands are shifted so that the vacuum level is set to zero to obtain the band alignment. The bilayer dielectric screening is obtained from the heterobilayer MoSe2/WS2. d–f Corresponding band alignments of MoSe2/WS2 at the K/K’ point. Solid (dashed) lines indicate spin up (down) bands. DFT calculations show a type Ⅱ band alignment, while one-shot G0W0 calculations (both monolayer and bilayer screening) indicate type Ⅰ band alignment.

In a real heterobilayer, the band alignment may also be influenced by atomic reconstruction43,44,45,71,72. Next, we explore the effect of atomic reconstruction on the band alignment at both large twist angles, where the bandstructure at the GW level can be explicitly calculated, and small twist angles, where we approximate large-area reconstructions as regions of nearly uniform stacking (see the Section IV in Supplemental Material). A detailed discussion of the crossover from large to small twist angles (defined as near 0° or 60°) can be found in the Supplemental Material Section VII. We start by exploring the effect of twist angle on atomic reconstruction by constructing large supercells with a MoSe2 monolayer stacked on a WS2 monolayer at 8 different twist angles (1.8°, 11.2°, 19.1°, 27.8°, 35.5°, 43.9°, 51.7°, 60°). Then, we relax the supercells using molecular dynamics (MD) following previous workflows44. The size of the moiré superlattice and the maximum strain due to reconstruction in each layer after relaxation are reported in Table 1. The smallest moiré supercell appears at 43.9° with only 75 atoms and a moiré lattice constant of 1.1 nm. In the small moiré cell, the change in lattice constants due to atomic strain is ±0.006 Å, indicating that there is almost no reconstruction. This small amount of strain has a negligible effect on the band energies (see Supplementary Fig. 2 in Supplemental Material). When the twist angle is near 0° or 60°, the moiré supercell is much larger, extending 6–7 nm, and includes thousands of atoms. At these twist angles, there is a large reconstruction. For example, in the case of a 60° twist angle, the lattice constant of the both the MoSe2 and WS2 layers experience a maximum strain of 0.9%. At this level of strain, band energies shift by hundreds of milli-electronvolts (see Supplementary Fig. 2 in the Supplemental Material), which is considerably larger than the CBM offset between MoSe2 and WS2 in the absence of moiré reconstruction, suggesting the possibility of a twist-angle dependent change in band alignment.

Table 1 Molecular dynamics (MD) forcefield relaxation of MoSe2/WS2 moiré supercells at different twist angles
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We start by analyzing the small moiré supercell case, for a twist angle of 43.9° where the reconstruction is small. Figure 2a, b shows the crystal structure and corresponding Brillouin Zone (BZ). DFT and GW calculations are performed directly on the relaxed supercell. Figure 2c shows the DFT band structure colored according to the projected density of states on each layer. The band edges of the MoSe2 and WS2 layers are at the \(\gamma\) and \(\kappa\) points of the moiré BZ, respectively, as expected from the unfolding of the moiré BZ shown in Fig. 2b. At the DFT level, MoSe2/WS2 has type Ⅱ band alignment, which is identical to that of the isolated monolayers. At the GW level (Fig. 2d), the MoSe2 CBM lies 106 meV below the WS2 CBM, so the heterostructure has type Ⅰ band alignment, which is again consistent with calculations on the isolated monolayers. Therefore, at 43.9°, or more generally, at large twist angles when the moiré lattice is small, the moiré potential does not change the band alignment of MoSe2/WS2, and the heterostructure has a type I band alignment across the entire moiré lattice.

Fig. 2: Supercell calculations for the MoSe2/WS2 heterostructure with a 43.9° twist angle.
Fig. 2: Supercell calculations for the MoSe2/WS2 heterostructure with a 43.9° twist angle.
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a Atomic structure of 43.9° twist-angle MoSe2/WS2 (top view). b Brillouin Zones (BZ) of MoSe2(red), WS2(blue), and moiré heterostructure (black). The MoSe2 K point (KMo) is folded to the \(\gamma\) point of the moiré BZ, while the WS2 K point (Kw) is folded to the \(\kappa\) point of the moiré BZ. c DFT band structure colored according to the projected density of states on each layer. The blue dashed line marks the position of the WS2 CBM, which is lower than the MoSe2 CBM by 34 meV, highlighting the type Ⅱ band alignment at the DFT level. d Quasiparticle (QP) band edge at \(\gamma\) and \(\kappa\) of the moiré BZ obtained from GW calculations. The MoSe2 CBM (red), located at \(\gamma\), is lower than WS2 CBM (blue), located at \(\kappa\), by 102 meV. At the GW level, MoSe2/WS2 has a type Ⅰ band alignment.

We then investigate the case where the twist angle is near 60°, and the moiré superlattice is large. Figure 3a shows the moiré superlattice. Three distinct stacking regions that form after the relaxation are labeled \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\), denoting a configuration where the central atom is a metal atom stacked on a chalcogen atom in the neighboring layer, \({{{{\rm{H}}}}}_{{{{\rm{M}}}}}^{{{{\rm{M}}}}}\), denoting a central metal atom stacked on a metal atom, and \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\), denoting a central chalcogen in one layer stacked with a chalcogen in the other layer. Figure 3b shows the strain distribution in the same moiré supercell. Positive (negative) percentages mean that the lattice is under compressive (tensile) strain. We see that the MoSe2 layer is under tensile strain in the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) region and compressive strain in the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) region, while WS2 experiences the opposite strain. In the \({{{{\rm{H}}}}}_{{{{\rm{M}}}}}^{{{{\rm{M}}}}}\) region both materials are relatively unstrained. Within a radius of 1.2 nm, which roughly corresponds to the radius of the exciton (see Supplemental Material Section VIII), the standard deviation of the lattice constants in \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\), \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) and \({{{{\rm{H}}}}}_{{{{\rm{M}}}}}^{{{{\rm{M}}}}}\) are 0.002 Å, 0.005 Å, and 0.006 Å, respectively (see the Section IV in Supplemental Material).

Fig. 3: Strain distribution and band alignment in various high-symmetry regions of the MoSe2/WS2 moiré supercell at a 60° twist angle.
Fig. 3: Strain distribution and band alignment in various high-symmetry regions of the MoSe2/WS2 moiré supercell at a 60° twist angle.
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a Moiré lattice for 60° twist angle MoSe2/WS2 (top). \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\), \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\), and \({{{{\rm{H}}}}}_{{{{\rm{M}}}}}^{{{{\rm{M}}}}}\) (where M is the transition metal atom and X is the chalcogen atom) are three high-symmetry regions in the unit cell. Their stacking configurations are shown in the black dashed box (bottom). b Strain distribution in the same moiré lattice plotted as a variation in local lattice constants from equilibrium. Red (positive) indicates compressive strain; blue (negative) indicates tensile strain. In the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) region, MoSe2 is under compressive strain while WS2 is under tensile strain. The \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) region shows the opposite. c GW band edges (conduction band minimum (CBM) and valence band maximum (VBM)) of strained monolayer MoSe2 and WS2 with bilayer screening. All energies are aligned relative to the vacuum level. In each region, WS2 and MoSe2 are strained in opposite directions, so MoSe2 is plotted with increasing strain, and WS2 is plotted with decreasing strain. Circles are unstrained lattice constant in \({{{{\rm{H}}}}}_{{{{\rm{M}}}}}^{{{{\rm{M}}}}}\). Squares are average length of the lattice constants in high-symmetry regions \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) and \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\). Triangles are maximum or minimum length of the lattice constant, appearing at the boundary of regions \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) and \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\). Results indicate type Ⅰ band alignment in the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) region and type Ⅱ band alignment in the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) region.

Since the radius of each stacking region for the large moiré superlattice is comparable to the exciton radius2, and the strain is relatively uniform within each region, it is reasonable to approximate each region of the moiré superlattice as a heterobilayer with a quasi-uniform strain. We start by determining the change in the band alignment due to the strain in each stacking region at the GW level. Since each stacking is not perfectly commensurate, it is not possible to construct a unit cell with the correct average strain. Instead, to obtain the correct band alignment, we start by calculating the absolute VBM and CBM energies with respect to the vacuum level of monolayer MoSe2 and WS2 and incorporate screening taken from the bilayer with the correct stacking configuration. Figure 3c shows the CBM and VBM of MoSe2 and WS2 with lattice constants under different strains. Circles indicate unstrained lattice constants in the \({{{{\rm{H}}}}}_{{{{\rm{M}}}}}^{{{{\rm{M}}}}}\) region. Squares mark the average lattice constant in the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) and \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) regions. Triangles mark maximum or minimum lattice constants appearing at the boundary of \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) and \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) regions. Our results are consistent with the widely recognized fact that tensile strain decreases both the band gap and the absolute energy of the band edges at the K point in TMDs, whereas compressive strain increases them73,74. In the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) region, MoSe2 is stretched while WS2 is compressed, so the MoSe2 CBM decreases and the WS2 CBM increases, resulting in local type Ⅰ band alignment with a band gap of 1.92 eV compared to the bandgap of 1.99 eV found in the unstrained bilayer. The opposite trend is observed in the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) region, leading to local type Ⅱ band alignment with a bandgap of 1.86 eV.

Our GW calculations show that MoSe2/WS2 has type Ⅰ band alignment at large twist angles, where the moiré lattice and reconstruction are small, and both local type Ⅰ and Ⅱ band alignment in different high-symmetry regions at small twist angles, where the moiré lattice and reconstruction are large. The coexistence of two types of band alignment introduces the possibility of hosting both intralayer and interlayer excitons in different regions of space within the same heterobilayer.

Excitons and absorption spectra

Next, we investigate excitons that can arise in these different regions within the ab initio GW plus Bethe Salpeter equation (GW-BSE) approach75,76. To avoid the computational demand of performing calculations on the full moiré supercell, we again make the reasonable approximation that each stacking region is well-described as a uniform heterobilayer. To account for both interlayer and intralayer excitons, we construct a strained commensurate unit cell of MoSe2/WS2. Because the commensurate cell has slightly different strain than the high symmetry stacking regions in the reconstructed moiré superlattice, we then treat the change in energy due to strain from the moiré lattice as a first order perturbation on the energies of the commensurate bilayer. This is reasonable since the strain is small and the wavefunctions do not hybridize near the K point.

Figure 4a, c, e shows the calculated absorption spectrum corresponding to each high- symmetry stacking region, and (b), (d), and (f) show the contribution of different QP bands projected onto the MoSe2 or WS2 layer to the exciton wavefunction at each energy. In the Tamm-Dancoff approximation76,77, the exciton can be written as a linear combination of electron-hole pairs \({|S}\rangle={\Sigma }_{{vc}{{{\bf{k}}}}}{A}_{{vc}{{{\bf{k}}}}}^{S}{|vc}{{{\bf{k}}}}\rangle\), where \({A}_{{vc}{{{\bf{k}}}}}^{S}\) denotes the electron-hole amplitude at the \({{{\bf{k}}}}\) point in the reciprocal space. v and c are indices for the valence and conduction bands respectively and \(S\) is the principle quantum number of the exciton. In Fig. 4b, d, f, every exciton is depicted as a column of points, where the size of each dot corresponds to the square of the electron-hole amplitude, \({\sum }_{c{{{\bf{k}}}}}{|{A}_{{vc}{{{\bf{k}}}}}^{S}|}^{2}\) for valence-band states, and \({\sum }_{v{{{\bf{k}}}}}{|{A}_{{vc}{{{\bf{k}}}}}^{S}|}^{2}\) for conduction-band states.

Fig. 4: Absorption spectra and excitonic compositions at three distinct high-symmetry regions.
Fig. 4: Absorption spectra and excitonic compositions at three distinct high-symmetry regions.
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a, c, e Absorption spectra of MoSe2/WS2 in different high-symmetry regions with a constant broadening of 0.05 eV. BSE calculations are performed in a patch of 0.193 Å−1 around the K point in the BZ. The highest peak corresponds to the MoSe2 A exciton and its center is marked with a black dashed line. The exciton peak position shifts due to different strain in the three high-symmetry regions. b, d, f Contributions of different quasiparticle (QP) bands to each exciton state. MoSe2 valence band maximum (VBM), conduction band minimum(CBM), and higher conduction band (CB+n) contributions are in red, while WS2 conduction band contributions are in blue. The exciton state can be written as a linear combination of electron-hole pairs \({|S}\rangle={\Sigma }_{{vc}{{{\bf{k}}}}}{A}_{{vc}{{{\bf{k}}}}}^{S}{|vc}{{{\bf{k}}}}\rangle\), where \({A}_{{vc}{{{\bf{k}}}}}^{S}\) is the electron-hole amplitude for the valence band v and conduction band c at k-point k. The size of each dot is proportional to \({\sum }_{c{{{\bf{k}}}}}{|{A}_{{vc}{{{\bf{k}}}}}^{S}|}^{2}\) for valence-band states, and \({\sum }_{v{{{\bf{k}}}}}{|{A}_{{vc}{{{\bf{k}}}}}^{S}|}^{2}\) for conduction-band states. Intralayer and interlayer excitons are clearly distinguishable, indicating that no hybrid excitons contribute to the main peak in the absorption spectrum in each stacking region.

From this decomposition, the lowest energy exciton appears in the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) region where the bandgap is smallest. It is a purely intralayer exciton corresponding to the MoSe2 A exciton. We note that contrary to previous hypotheses49,55, the MoSe2 A exciton does not include any interlayer hybridization and comes purely from the MoSe2 layer. In our calculations, the GW correction lifts the near degeneracy of the MoSe2 and WS2 conduction band edges, which was the basis of previous hypotheses of inter- and intralayer hybridization55. However, even when the two conduction bands are nearly degenerate, we still do not see hybrid excitons in our BSE calculation. Some higher energy interlayer excitons (above or near the continuum) do exhibit a degree of hybridization due to interaction with the continuum of lower energy intralayer excitons. (see details in Supplementary Material VI).

Next, we turn to understanding the absorption features in different stacking regions. The absorption spectrum of MoSe2/WS2 with a near 60-degree twist is dominated by three distinct exciton peaks at 1.39 eV, 1.46 eV, and 1.49 eV. As previously noted, the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) region is dominated by a peak corresponding to the intralayer MoSe2 A exciton at 1.39 eV, which has a binding energy of 0.53 eV. For convenience, we will refer to this as the \({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) exciton. In the \({{{{\rm{H}}}}}_{{{{\rm{M}}}}}^{{{{\rm{M}}}}}\) region, the band alignment remains type I, and the lowest energy exciton is still the intralayer MoSe2 A exciton (\({{{{\rm{A}}}}}_{{{{\rm{M}}}}}^{{{{\rm{M}}}}}\)), now shifted to a higher energy of 1.46 eV. It has a similar binding energy of 0.53 eV. In the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) region, since there is type II band alignment, the lowest energy exciton is an interlayer exciton (\({{{{\rm{I}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\)) with the hole in the MoSe2 layer and the electron in the WS2 layer. The interlayer exciton is nearly dark, and the absorption spectrum is dominated by the bright intralayer exciton in MoSe2 (\({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\)) at 1.49 eV. In the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) region, the binding energy of the interlayer exciton is 0.43 eV, while the binding energy of the intralayer exciton is 0.53 eV. We note that the experiment is performed on an hBN substrate, which may renormalize the exciton binding energy to 200–300 meV59. Accounting for the renormalization may bring the experimental results in closer alignment with theory. Although it is hard to observe the interlayer exciton directly in reflectance contrast experiment because of its weak oscillator strength, it can be brightened by applying out-of-plane electric field. The existence of the interlayer exciton in the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) region at an energy between \({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) and \({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) is consistent with experimental observation of a weak dispersion of the second and third absorption peaks under a perpendicular electric field57,59. The energy difference between the \({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) and \({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) peaks is about 100 meV, which is similar to the difference in energy between the first and third absorption peaks observed in previous optical measurements57,59. Thus, we attribute the three low-energy peaks in the experimental spectrum to intralayer excitons in the three different high-symmetry regions, consistent with prior phenomenological work based on continuum model without explicit atomic reconstructions59. We note that the energy ordering we calculate for small-twist MoSe2/WS2, with the bright \({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) state being the lowest energy exction, also explains why photoluminescence is not observed from the interlayer excitons in MoSe2/WS2 at zero electrical bias60. This energy ordering is atypical for TMD heterostructures, where the long-lived interlayer exciton is usually the lowest energy state, and has an observable PL signature.

It is worth noting that the energy of the middle peak has more uncertainty in the calculation than the other two peaks. The reason is because \({{{{\rm{H}}}}}_{{{{\rm{M}}}}}^{{{{\rm{M}}}}}\) regions form the boundary between the other two stacking regions. As a result, it is nearly unstrained, but unlike the other stacking regions, the small strain within a single layer is not uniform but has opposite sign depending on whether it is moving toward the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) or \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) region. Consequently, there is a larger error in the calculated position introduced by the uniform strain approximation. Also, while this calculation does not account for the possibility of intralayer charge transfer excitons71, it sufficiently captures interlayer and intralayer excitons localized at each moiré site, which are expected to be much brighter in the absorption spectrum than charge transfer excitons.

TR-ARPES

To experimentally determine the nature of the excitons produced in different heterostructures and also compare dark excitons, we apply tr-ARPES to MoSe2/WS2 samples with different twist angles and different layer stacking orders. For samples with large twist angles, the K valleys of the two layers are easily discernible in momentum-space maps and tr-ARPES directly reports on the layer occupation of the electron component of the exciton wave function after photoexcitation. For samples with small twist angles, where the K valleys of the two layers coincide in momentum space, tr-ARPES can still delineate between inter- and intralayer excitons via the extreme surface sensitivity of the technique. Photoelectron signals from the lower layer are heavily attenuated due to the very small electron mean free path at ~20 eV energy56,78,79, such that our measurement predominantly measures the electron population in the top-most layer of the heterostructure. The Stony Brook time-resolved ARPES beamline and its application to 2D materials have been described previously61,80,81,82. Heterostructure regions of the samples are selected using an aperture in a real-space image plane of the momentum microscope83. The high data rate of the Stony Brook beamline enables us to perform a systematic series of measurements on many samples. In all experiments, the samples are excited with 2.4 eV pump photon energy, which initially populates many exciton states in both layers. In this study, we focus on the long-lived excitons persisting after initial relaxation.

Figure 5a–f shows 2D cuts of the 3D ARPES data along the K-\(\Gamma\)-K direction for 6 different samples, for pump-probe delays greater than 5 ps, recorded with 25.2 eV photon energy. Figure 5a, b show signals from monolayer MoSe2 and WS2, Fig. 5c, d are for heterostructures with twist angles near 60° (the small twist/large moiré supercell condition) and Fig. 5e, f are for heterostructures near 40° (the large twist/small moiré supercell condition). For the monolayers, the energy scale is referenced to the global VBM at K. For the bilayers, the precise maximum at K is not clearly visible due to the photoemission matrix element effect, therefore we reference energies to the local VBM at \(\Gamma ({E}_{\Gamma })\). The monolayer signals show the VBM at K, while the heterobilayer signals show that the VBM at \(\Gamma\) is nearly degenerate with the MoSe2 VBM at K, with the WS2 VBM at K substantially lower in energy, consistent with our calculations and previous work84. Similar to previous ARPES work56 and also consistent with our GW-BSE calculations, in the bilayers we observe a 490–590 meV splitting of the valence bands at \(\Gamma\) due to strong hybridization of the chalcogen pz orbitals.

Fig. 5: Tr-ARPES of monolayer WS2(ML WS2), monolayer MoSe2 (ML MoSe2), and twisted heterobilayers.
Fig. 5: Tr-ARPES of monolayer WS2(ML WS2), monolayer MoSe2 (ML MoSe2), and twisted heterobilayers.
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a–f K-\(\Gamma\)-K slice constructed from tr-ARPES data set by summing all pump/probe delays 5 ps and longer. The signal above 1.3 eV is multiplied by the constant displayed to show the exciton signal within the band gap. A nonlinear color scale is used to make all features visible. Logarithmic and linear scale versions appear in the Supplemental Materials. Normalized intensity traces comparing the integrated exciton decay of the monolayer samples to heterobilayers with (g) small and (h) large relative twist angles. i Energy distribution curves (EDCs) of the exciton signal from the small twist angle bilayers with Gaussian fits. The fitted peak position shifts by 110\(\pm\)13 (statistical error) \(\pm\)42 (systematic error) meV upon permutation of the layer stack ordering.

Exciton signals are observed above the VBM but below the CBM due to the exciton binding energy63,85,86,87,88,89. Figure 5g, h shows the pump-probe delay dependence of the exciton signals at K for the different samples. Monolayer exciton signals (shown in both Fig. 5g, h) appear at energies consistent with PL measurements90,91,92, and decay on the 10−100 ps timescale, consistent with our previous work61. For the bilayers, signals recorded from MoSe2 on WS2, which are sensitive to the electron population in the MoSe2 layer, show the same dynamics as the intralayer excitons in monolayer MoSe2 regardless of the twist angle. The exciton signal appears at 1.59 \(\pm\) 0.03 eV above the \({E}_{\Gamma }\), which is in reasonable agreement with the calculated excitation energy of the \({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) exciton of 1.39 eV. In contrast, signals recorded from WS2 on MoSe2, which are sensitive to the electron population in the WS2 layer, show a strong twist-angle dependence. In large-twist WS2 on MoSe2 samples, no long-lived signals are observed in ARPES. This suggests that the long-lived excitons in large-twist-angle samples are intralayer excitons with both the hole and the electron residing in the MoSe2 layer, consistent with the type I band alignment predicted in our calculations.

However, for samples with near 60° twist angle, we do observe a long-lived electron population in the WS2 layer, which indicates the presence of long-lived interlayer excitons coexisting with the intralayer excitons. Furthermore, the interlayer excitons appear at a different photoelectron energy. Figure 5i shows the energy distribution curves (EDCs) of the long-lived exciton signal from the near 60° bilayers, along with the Gaussian fits. The signal observed from interlayer excitons in the small-twist-angle WS2-on-MoSe2 sample appears at 110 \(\pm\) 13 (statistical error) \(\pm\)42 (systematic error) meV higher photoelectron energies than the photoelectron signals from intralayer excitons in the small-twist-angle MoSe2-on-WS2 sample. This energy shift is consistent with the expected signature of an interlayer exciton \({{{{\rm{I}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) in the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) stacking region. The calculated excitation energy of \({{{{\rm{I}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) is 40 meV higher than the energy of the \({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) exciton. In addition, the valence band edge in the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) region is 47 meV higher than the \({{{{\rm{H}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) region, which means that photoionization of the \({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\) exciton leaves the hole unrelaxed by 47 meV with a corresponding reduction in the photoelectron energy referenced to the global VBM, leading to an overall energy shift of 87 meV for the tr-ARPES signal of \({{{{\rm{I}}}}}_{{{{\rm{X}}}}}^{{{{\rm{M}}}}}\) compared to \({{{{\rm{A}}}}}_{{{{\rm{X}}}}}^{{{{\rm{X}}}}}\).

Discussion

In summary, we have performed first-principles GW-BSE calculations on a MoSe2/WS2 heterobilayer and show that the heterobilayer has a type I band alignment when the twist angle is large and the moiré length is small. At small twist angles, when the moiré length is large, we find that the large atomic reconstruction introduces significant strain, resulting in regions of both type I band alignment, hosting bright intralayer excitons, and type II band alignment, hosting long-lived interlayer excitons, at different moiré lattice sites. This spatial modulation between intra- and interlayer excitons could be an interesting platform for exploring novel phases of electrons and charged excitons93. Moreover, the system would form a natural lateral heterojunction with the energy landscape directing electron flow to the lower energy region, giving rise to the possibility of enhanced emission and population inversion with long-lived interlayer excitons acting as an exciton reservoir. This work also clarifies the assignment of optical spectra in MoSe2/WS2. Our BSE calculations show that the absorption spectrum is dominated by three low-energy exciton peaks corresponding to the MoSe2 A exciton in different stacking regions, with energy splitting in good agreement with experiment57,59. We observe no low-energy hybrid excitons, and the absorption peaks appear to be adequately explained by the lattice reconstruction. Our tr-ARPES measurements, which see a long-lived exciton population only in the MoSe2 layer at large twist angles and at both the energy of the intralayer exciton in MoSe2 and the interlayer exciton in WS2 at small twist angles validates our calculations. Our work provides a comprehensive picture of the complex electronic and excitonic landscape in MoSe2/WS2 heterobilayers, where the band alignment and optical properties can be precisely tuned through twist angles. The discovery of coexisting type I and type II band alignments within a single supercell opens pathways for moiré-engineered optoelectronic devices with intrinsic lateral heterojunctions and tunable light-matter interactions.

Methods

GW-BSE calculations and molecular dynamics relaxation

We start by performing DFT calculations as implemented in the Quantum ESPRESSO code94 with the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)65 for the exchange-correlation energy. Then, we calculate the QP band alignment and exciton states within the GW and GW-BSE formalism on top of the DFT mean field. The GW and GW-BSE calculations, with a fully relativistic spinor formalism, are performed with the BerkeleyGW package75,76,95,96. We construct twisted bilayer TMDs with the help of the TWISTER code97, relax atomic structures using the Stillinger−Weber (SW) force field98 and the parametrized Kolmogorov−Crespi potential99,100,101 as implemented in the LAMMPS package102, and visualize atomic strain using OVITO103 and VESTA104,105. Additional computational details can be found in the Supplemental Material.

Sample fabrication and characterization

The WS2/MoSe2/hBN heterostructures were fabricated using mechanical exfoliation and dry transfer method. First, hexagonal boron nitride (hBN) was mechanically exfoliated onto a highly doped n-type silicon substrate (0.001–0.005 Ω·cm) using Scotch tape. An hBN flake with a thickness of 20–30 nm was selected via optical contrast to construct the heterostructure. Then, monolayer WS2 and MoSe2 were mechanically exfoliated onto a polydimethylsiloxane (PDMS) substrate. Second harmonic generation (SHG) spectroscopy was further performed to control the twist angle between the layers. Finally, the monolayer MoSe2 and WS2 were sequentially transferred onto the hBN flake using the dry transfer method. Some part of the exfoliated MoSe2 or WS2 flake was draped over the edge of the hBN to ensure a conductive pathway exists and sample charging does not occur during ARPES measurements.

The heterostructures with MoSe2 on WS2 have twist angles 57.7° ± 0.2° and 40.1° ± 0.3°. The photoluminescence and reflection contrast measurements from the monolayer and heterobilayer regions, the SHG characterization of the twist angle, the optical image, and the photoemission electron microscopy (PEEM) image are shown in Supplementary Figs. 7 and 8. The photoluminescence and reflection contrast measurements were acquired at 90 K.

The heterostructures with WS2 on MoSe2 have twist angles 59.7° ± 0.5° and 40.2° ± 0.4°. The photoluminescence from monolayer and heterobilayer regions, the SHG characterization of the twist angle, the optical image, and the PEEM image are shown in Supplementary Figs. 9 and 10. These measurements were acquired at room temperature.

Time-resolved ARPES measurements and data analysis

Tr-ARPES measurements were performed at the Stony Brook tr-ARPES beamline facility61,79,80,81. All measurements were done with vacuum pressure at or below \(9\times {10}^{-10}\) torr and were measured with 25.2 eV probe photon energy. All measurements are performed at room temperature and with a pump wavelength of 517 nm (2.4 eV). The monolayer MoSe2 (ML-MoSe2), monolayer WS2 (ML-WS2), and 40° MoSe2 on WS2 bilayer were excited with 101 \({{\mathrm{\mu J}}}/{{{\mathrm{cm}}}}^{2}\) pump fluence. The 58° MoSe2 on WS2 bilayer was excited with 193 \({{\mathrm{\mu J}}}/{{{\mathrm{cm}}}}^{2}\). The pump pulses were s-polarized during these measurements. The 59° WS2 on MoSe2 bilayer was excited with 43 \({{\mathrm{\mu J}}}/{{{\mathrm{cm}}}}^{2}\) and the 40° WS2 on MoSe2 bilayer was excited with 45 \({{\mathrm{\mu J}}}/{{{\mathrm{cm}}}}^{2}\). The pump pulses were p-polarized during these measurements. For the bilayer samples, data are collected from a ~10 μm region of interest (ROI) selected using a field aperture in a real-space image plane of the microscope83.

The data obtained from the instrument is post processed to eliminate image distortions and for image calibration. First, the surface photovoltage (SPV) produced from the pump pulse exciting the Si substrate is corrected for and energy axes referenced to the valence band maxima (VBMs) are generated. The procedure for this correction has been described previously61,82. In ML-WS2, the VBM is determined by fitting an EDC of the spin-orbit split bands at K with a double Gaussian and using the fitted peak position as the VBM. For ML-MoSe2, the ARPES signal at the VBM at K is very weak due to the photoemission matrix element, so the VBM is determined by fitting an EDC from the \(\Gamma\) band with a Gaussian, then adding previously reported 380 meV106 to determine the true VBM at K. For the bilayer samples, the signal at K is very weak in general so all energies are referenced to the local valence band maximum at \(\Gamma\), \({E}_{\Gamma }\). Since the electronic states at \(\Gamma\) have contributions from both layers, we do not expect the position of \({E}_{\Gamma }\) observed in ARPES to depend on the layer orientation. Within the experimental error, \(\Gamma\) and K are energetically degenerate in the bilayer samples.

Once the energy axes are determined, each 3D image \(({k}_{x},{k}_{y},E)\) is normalized to the counts within a large ROI around \(\Gamma\). This normalization accounts for any drift in the photoelectron yield at different pump-probe time delays. A background signal \(({k}_{x},{k}_{y},E)\) is constructed from the average of data taken with negative pump-probe time delays and is subtracted from the remaining signal \(({k}_{x},{k}_{y},E,t)\).

To produce the time traces shown in Fig. 5g, h, the exciton signal at the K-point is integrated using a \({0.28\mathring{\rm A} }^{-1}\times 0.28{\mathring{\rm A} }^{-1}\) kx-ky bin size and an energy bin tuned to capture the entire exciton signal. Then, electron distribution curves (EDCs) are extracted from the same regions of interest in the background-subtracted exciton signal, summing data taken at or after 5 ps pump-probe delay. The EDCs are fit with Gaussians and the extracted peak positions are used for comparison between samples. To produce the K-\(\Gamma\)-K cuts, the data is distortion-corrected within the kx-ky plane using thin-spline transformations obtained from symmetry guided image registration107. Image shear along the E-kx and E-ky axes are corrected using affine transformations. The background subtracted data is stitched to data without background subtraction to show the exciton signal and the valence band signal together. After the shear is eliminated, the data is 6-fold symmetrized in the kx-ky plane. Then, a \({0.10\mathring{\rm A} }^{-1}\) bin is used to generate the K-\(\Gamma\)-K cut displayed and the exciton signal is multiplied by a constant so that it can be seen. Data taken at and after 5 ps pump-probe time delay is summed to show the final state after the initial dynamics have decayed. Supplementary Figs. 11 and 12 show the K-\(\Gamma\)-K cuts on linear and natural logarithmic scales.

The systematic error is dominated by uncertainties in the calibration function that maps time-of-flight in the momentum microscope to energy. Typically features within a spectrum are compared to other features within the same spectrum. Therefore, any systematic offset from the absolute energy would be eliminated when comparing differences within that spectrum. However, here features across different datasets where the systematic errors might differ are compared. Therefore, a cancellation of the systematic error is not expected in this case.