Introduction

Monolayer transition-metal-dichalcogenide (TMD) semiconductors have a direct bandgap and exhibit optically accessible bright excitons1,2,3,4. In addition, they also show a variety of optically forbidden dark excitons, either due to spin-flip or momentum transfer5,6,7. For example, both K-K bright and momentum-forbidden K-Q excitons emerge in monolayer WSe2, as confirmed by time- and angle-resolved photoemission spectroscopy6. These momentum-forbidden indirect excitons can also be realized in van der Waals (vdW) heterostructures, where bound electrons and holes are localized in two different layers, which is called interlayer excitons7,8,9,10,11,12,13,14. Emergence of these indirect interlayer excitons strongly depends on the stacking angle between the layers8,10,12 and is independent of stacking sequential order10,15. Despite this, a few recent reports have accidentally observed optical anomalies in heterostructures that rely on the sequential order of the layers15,16. However, the underlying mechanisms that give rise to these anomalies are not yet fully realized. This lack of understanding may be attributed in part to the hypothesis that altering the stacking sequence between layers would not significantly affect the intrinsic properties of heterostructures. Therefore, it is important to explore stacking sequential order dependence to further use of real device applications.

In this work, we tune dark excitons by changing the stacking sequence between TMD layers, whereas other experimental parameters remain constant. Such emergence or disappearance of dark excitons depending upon stacking sequence has not been reported to date. Moreover, the emergence of intra-layer dark exciton in TMDs heterostructure also remains elusive. This phenomenon is investigated by analyzing the photoluminescence (PL) spectra of 2D TMD heterostructures in different stacking configurations with a series of experiments to reveal such an unusual phenomenon. Furthermore, our microscopic theory with the generalized Elliot formula suggests that an additional strain on the top layer stemming from the fabrication process could be a solution to explain such an unusual phenomenon.

Results and discussion

Stacking sequence under various circumstances

We investigate the stacking sequence of bilayers at various conditions: heterogeneous TMDs (WSe2/WS2 and WS2/MoSe2), different substrates (SiO2 and hBN), and homogeneous TMDs (WSe2/WSe2 and WS2/WS2) (Fig. 1). In each case, one layer (bottom layer) has been exfoliated mechanically on the substrate. Another layer (top layer) has been exfoliated on top of the PMMA-coated substrate, which is then dry transferred to the aforementioned layer (see methods and Supplementary Fig. 1). Individual monolayers, as well as the vertical heterostructures were confirmed by Raman spectra (Supplementary Fig. 2) and the height profile in atomic force microscopy (AFM) (Supplementary Fig. 3). The interface of the stacked bilayer has not been touched by contaminants, such as PMMA, during dry transfer to ensure a clean interface (Supplementary Note 3).

Fig. 1: Dark-exciton modulation in TMDs heterostructures.
figure 1

a Schematic along with optical images of WS2/WSe2 heterostructure on SiO2/Si substrate. PL spectra of individual monolayer (dotted lines) with heterostructure (solid line); A red color peak at heterostructure indicates a dark exciton peak (SQ) near the A-exciton peak of WS2, which is deconvoluted for detailed analysis (left). b Similar for the inverted stacking sequence of WSe2/WS2. Dark exciton peak (SeQ) emerges near A-exciton peak of the top WSe2, in stark contrast with that in a. c, d The effect of different substrates with h-BN. Dark exciton peaks (SQ and SeQ) are still persistent with corresponding stacking sequences. e, f The effect of different materials in combination with WS2-MoSe2. Dark exciton peaks still appear near the top layer. g, h Emergence of dark exciton peaks at homogeneous bilayers of each WS2 and WSe2.

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Figure 1a, b represents a schematic for vertical TMD heterostructures of top-WS2/bottom-WSe2 and top-WSe2/bottom-WS2 on SiO2/Si substrate together with optical images (inset). We find clear PL peaks appearing around 2.0 eV for the WS2 monolayer (green dotted lines) and around 1.64 eV for the WSe2 monolayer (blue dotted lines), corresponding to the A-excitons of the layers. An additional feature (marked as SQ) emerges near 1.89 eV below the A-exciton peak of the top layer (WS2) at WS2/WSe2 heterostructure (Fig. 1a). Meanwhile, in the reverse stacking sequence of the WSe2/WS2 heterostructure, another feature (marked as SeQ) emerges around 1.56 eV below the A-exciton of the top WSe2 layer, whereas the previous feature (SQ) near the WS2 layer notably disappears (Fig. 1b and Supplementary Fig. 8). The presence of such additional peaks close to the top layer is well distinguished from trions or bi-exciton peaks, as shown in the deconvoluted curves (Fig. 1a left, 1b, right and Supplementary Fig. 9)8,17. This unusual stacking sequence has not been reported to date. We confirmed the reproducibility of such additional peaks in multiple heterostructures (a total of 35) for both stacking sequences (Supplementary Fig. 10). We note that all heterostructures were randomly stacked and the stacking angle between the layers was further confirmed by second harmonic generation (SHG) measurements (Supplementary note 7). Therefore, the observed features are not specific to a stacking angle. This suggests that our additional peaks originate neither from interlayer excitons8,10 nor moiré excitons17,18, as they require precise angle alignment between the layers. Furthermore, the energies of these additional peaks (1.89 and 1.56 eV) are distinct from reported interlayer excitons (1.35–1.42 eV) for the WS2-WSe2 heterostructure8,17. Moreover, we observed weak interlayer exciton peak at an energy of 1.35 eV in MoSe2/WSe2 heterostructure at room temperature (Supplementary Fig. 12). This is well matched with previously reported results10,12,19, which is rarely observed as it is contingent upon a specific stacking angle.

We now consider the substrate effect involving possible strains, charge traps, and dielectric screening by fabricating both heterostructures on the top of h-BN layer (Fig. 1c, d). Both features (SQ and SeQ) are still visible for the corresponding stacking sequences, although the peak positions were slightly upshifted (Supplementary Fig. 13). The SQ peak is also visible with other substrates including quartz and HfO2 (Supplementary Fig. 14). We further confirm the existence of these peaks by encapsulating both sides of heterostructures by h-BN as well as SiO2 to exclude possible environmental effects (Supplementary Fig. 15). These results confirm that the additional features are independent of the substrate, which could be related to the strong interfacial coupling of the hetero-bilayer. We also investigate the stacking sequence with other TMDs combinations (Fig. 1e, f). The SQ peak still appears around 1.85 eV in the WS2/MoSe2 heterostructure (Fig. 1e). Meanwhile, a similar additional peak (SeQ) emerges in the MoSe2/WS2 heterostructure (Fig. 1f), but the energy is upshifted, different from the downshifted energy in W-based materials. We further investigated several hetero-bilayers with other types of TMDs, for example, the MoS2-WSe2 or MoS2-WS2 heterostructure (Supplementary Note 10). All these bilayer heterostructures clearly revealed the emergence of additional features near the top layer.

To investigate if the additional peak appears only in heterostructures, we also study the PL in homogeneous bilayers by exfoliating the intact bilayer. The homogeneous bilayer WS2 still clearly shows the SQ peak, which is well distinguished from the monolayer WS2 (Fig. 1g). Similarly, the SeQ peak is also observed in bilayer WSe2 (Fig. 1h). This implies that the additional peak does not necessarily originate from the nature of the hetero-bilayer but is inherent from the homogeneous bilayer, congruent with previous reports20,21,22. Furthermore, the exfoliated bilayer interface remains intact and hence the presence of such an unusual peak reassures the clean interface in our fabricated hetero-bilayers (Supplementary Fig. 4). We also observed strong and distinct SQ peak on the spatially uniform and bubble-free flat heterostructure surface (Supplementary Fig. 17), which differs from previously reported localized exciton due to the presence of bubbles23.

Additionally, monolayer (or any odd number of layers) TMD has broken inversion symmetry, whereas bi-layer (or any even number of layers) preserved inversion symmetry24,25. We have also checked how the broken inversion symmetry affects the additional features (SQ or SeQ). We first measured the layer-dependent (even and odd layer) PL at homogeneous WS2 as well as WSe2 (Supplementary Fig. 18). The additional feature was still present, except for further downshifting as the number of layers increased, as observed previously26. We further checked the heterogeneous heterostructure with the different number of layers, which follows the similar stacking sequence as observed in bilayer heterostructure (see details in Supplementary note 13). We further confirmed similar features SQ or SeQ depending on the corresponding stacking sequence at low temperatures (77 K) (Supplementary Fig. 20). These additional peaks were still observed after annealing the sample at 250 °C for 12 h in helium environment (Supplementary Fig. 21). Additionally, we detected similar peaks (SQ or SeQ) on the doped materials such as Re-doped WS2 and Nb-doped WSe2 samples (Supplementary Fig. 22). All of these observations reassure these additional peaks to be intrinsic, but not related to defect states.

Q-band downshifts via band renormalization

The emergence of these unusual peaks in PL can be explained in terms of the Q-band downshift in the top layer via band renormalization in the heterostructure region, which has been demonstrated previously for WS2-WSe2 hetero-bilayer8 and bilayer of MoSe227. In contrast, such peaks (SQ/SeQ) was absent in absorption measurements due to the indirect nature of SQ or SeQ peak (see details in Supplementary Note 15). The band structure of individual monolayer WS2 or WSe2 has K- and Q-bands, as illustrated in Fig. 2a, b resulting in bright K-K and indirect dark K-Q excitons6. We now propose that dark excitons become active and visible due to the downshifted Q-band at the heterobilayer region. For example, in WS2/WSe2 heterostructure (Fig. 2c), the Q-band is downshifted compared to K-band in the top WS2 layer in the heterostructure region. The momentum indirect phonon-assisted emission from intralayer K-Q exciton becomes stronger by accumulated electron population at the Q-band transferred from the K-band in the same WS2 layer as well as additional electron population transferred from WSe2 to WS2 layer due to type-II band alignment in the heterostructure (Supplementary Fig. 24a, b)8,28,29. Additionally, the reduction of relaxation time in heterostructure, measured by time-resolved photolumenesence, can be attributed to fast charge transfer facilitated by both direct K-K band and further indirect K-Q band transition (Supplementary Fig. 24c). This gives rise to strong emission of the SQ peak. Similarly, for the opposite stacking, K-Q excitons become active in the top WSe2 layer by band renormalization. Furthermore, the K-Q population is less significant in the bottom WS2 layer (Fig. 2d), consequently yielding a relatively low SeQ intensity. The microscopic origin for the stacking sequence dependence of the additional peaks will be discussed further below and is ascribed to strain appearing in the upper layer during the stacking procedure.

Fig. 2: Band renormalization at heterostructure.
figure 2

a, b Schematic band diagram of individual monolayer WS2 and WSe2 along with the corresponding PL spectrum, where bright exciton appears in K-band, unlike dark exciton in Q-band. c, d Schematic band alignment at WS2/WSe2 (WSe2/WS2) heterostructure, where Q-band is downshifted at the top layer, converting dark exciton into bright exciton. e Schematic and optical image of the device for STS measurements. f STS profiles at individual monolayer WS2 (green dotted line), WSe2 (blue dotted line), and heterostructure (solid line). Mid-gap state SQ (red) emerges closely to CB, due to the downshifting of the Q-band at the heterostructure region.

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Electronic band structure

To confirm the Q-band downshift, we examine the position of Q-band in the top-WS2 layer in WS2/WSe2 heterostructure via scanning tunneling spectroscopy (STS). The schematic of WS2/WSe2 heterostructure on SiO2/Si substrate (top) and the optical image of the device (bottom) are shown in Fig. 2e. The dI/dV clearly reveals the additional peak (red) near the conduction band edge in the heterostructure (Fig. 2f), resembling PL peaks in Fig. 1. This is well contrasted with the simple band profiles from individual WS2 or WSe2 layer (Fermi level is shifted to the middle of the band edges and see Supplementary Note 18).

Power-dependent PL measurements

To elucidate the underlying mechanism of the seemingly inconsistent stacking sequence, we measure the laser (532 nm) power-dependent PL of the stacking sequence of WSe2/WS2 heterostructure on the SiO2/Si substrate (Fig. 3a). As the power rises, the SeQ peak intensity (IQ) in the top WSe2 layer decreases compared to the A-exciton peak (IK) (Fig. 3a, right). This reduction of the SeQ intensity is ascribed to the charge screening of saturated electron density in the Q-band at high laser power. Variances of neutral exciton peak (X0) and emerging trion peak (XT) at high power region are not appreciable (Fig. 3b). This tendency holds true for SeQ peak. At higher power above ~130 µW, the trion state of SeQ (SeQT) emerges. We observe two screening regions in the power-law dependence with \(I={P}^{\alpha }\) where I is intensity, p is power and α is an exponent. For the neutral exciton, the exponent α1 = 0.88, steady in the low power region, and α2 = 0.65, saturated in high power region due to conversion of the neutral exciton to trion30. This is again confirmed by the emergence of the trion peaks with a single exponent α1 = 0.62 in the high power region. The SeQ peak reveals the screening behavior with similar two exponents like neutral exciton in WSe2, except a much lower exponent (α2) in high power region due to the high carrier screening at the Q-band compared to the K-band (X0). This similar power exponent reassures that the nature of the SeQ peak originates from the intra-band exciton (K-Q).

Fig. 3: Laser power-dependent PL measurements.
figure 3

a Laser power-dependent PL spectra in WSe2/WS2 heterostructure on SiO2/Si substrate. Inset; the corresponding contour plot. The black dotted box represents the SeQ peak region. Variation of the intensity ratio of Q to K-band (IQ/IK) of the top-WSe2 layer with laser power (right). b, c Deconvoluted PL peak energies and the corresponding peak intensities with laser power. d Laser power-dependent PL spectra in the opposite stacking sequence of WS2/WSe2 heterostructure on the h-BN substrate. Inset: contour plot of the spectra and the black dotted box represents the SQ peak region. Variation of IQ/IK of the top-WS2 layer with laser power on SiO2 and the h-BN substrate (right). e, f Deconvoluted PL peak energies and the corresponding peak intensities of the top-WS2 layer with laser power on the h-BN substrate.

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We next study the power-dependence in the WS2/WSe2 heterostructure on the hBN substrate (Fig. 3d) as well as on the SiO2 substrate (Supplementary Fig. 27). The SQ peak appears in the top WS2 layer, which is still persistent with increasing power. The intensity ratio of IQ/IK decreases, yielding a similar tendency to the WSe2/WS2 heterostructure, independent of the substrate of hBN or SiO2 related to trap-charges (Fig. 3d, right)30. The XT peak in the WS2 layer emerges from the beginning of low power region due to enhanced carrier density on the h-BN substrate (Fig. 3e). The presence of both X0 and SQ peaks at low power again indicates that the SQ peak originates from the intra-band nature (Fig. 3e, f). In addition, the trion state of the SQ peak (SQT) emerges in the low power region of about 60 µW again due to enhanced carrier density, and furthermore, the energy is downshifted due to strong charge screening. The power dependence of exciton peaks is similar to those of the corresponding peaks in the top-WSe2 layer except for the elevated power exponent, which is ascribed again to the enhanced carrier density (Fig. 3f)30.

Inhomogeneous distribution of dark excitons

We investigate homogeneity of the K-Q excitons over the sample via PL energy and intensity mapping, as shown in Fig. 4a, b (schematic and optical image in Fig. 4a top and inset). The light-bluish milky region with an energy range of 1.88–1.90 eV is referred to as K-Q excitons (Fig. 4a), which is inhomogeneously distributed over the entire heterostructure region. For clarity, we plotted the normalized SQ peak distribution over the heterostructure region (details in Supplementary Note 20), where the SQ population is dominant at the heterostructure region compared to the intrinsic inlayer WS2 exciton intensity (Fig. 4b). Three PL spectra have been extracted from the representative spots (A, B, C). Most of the light-bluish milky region, for example, marked as region A, represents the dominant Q-band peak with downshifted energy as well as high intensity compared to the K-band (Fig. 4c, top). In region B, the Q-peak intensity is reduced and comparable to the K intensity as well as the PL energy is slightly upshifted (Fig. 4c, middle). Moreover, the Q-peak intensity is quenched further with more energy upshift in region C (Fig. 4c, bottom). Some other spots including low-energy (e.g., red circle) regions are attributed to the artifacts, such as air bubbles and contaminants. Such inhomogeneous energy and intensity of the Q-band can be explained as a metric of coupling strength between layers, \(\sigma \,=\frac{{I}_{Q}-{I}_{K}}{{I}_{Q}+{I}_{K}}\,\). σ is positive for strong coupling and negative for weak coupling and σ ≈ 0 for medium strength (Fig. 4d).

Fig. 4: Theoretical modeling and application of Q-exciton as an optical power switch.
figure 4

a, b PL energy and intensity mapping of WS2/WSe2 heterostructure at the box region shown in inset optical image of a. Inhomogeneous SQ distribution over the heterostructure region. The SQ intensity is dominant at the heterostructure region as indicated IQ > IK (arrow) in b. c Extracted PL spectra, corresponding to A, B, and C positions: positive (IQ—IK) in A, which primarily covers the heterostructure region, IQ—IK ≈ 0 in B, and negative (IQ—IK) in C. d Schematic demonstration of the Q-band renormalization depending on the strong, medium, and weak coupling at the interface. e, f Experimental (top) and theoretical (down) photoluminescence for oppositely-stacked WSe2-WS2: A compressive strain of respectively 0.3 and 0.15% in the top layer leads to peaks corresponding to SQ and SeQ in the experimental data. g Power-dependent PL spectra at high IQ intensities with strong coupling region. h In strong coupling regime, σ changes the polarity from positive to negative with increasing the laser power and acts as an optical power switch (red dots). This crossing point of K and Q- exciton intensity with laser power is clearly visible in the inset contour plot. In medium coupling regime, σ does not alter the polarity, although the power law is similar to that of strong coupling regime (green dots).

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Microscopic model for the origin of unusual peaks

The variation of the coupling strength from position to position indicates possible inhomogeneous strain in the bilayered heterostructure. E2g peak shift from Raman spectra could be monitored as a metric for strain. For example, the E2g peak was slightly shifted at each layer in the heterostructure compared to the individual layer due to minute local strain but varied at different positions (Supplementary Fig. 29), making it difficult to identify strain at each layer of the heterostructure. However, in practice, the bottom layer was directly exfoliated on the target substrate, which was strongly anchored on the substrate. The top layer was transferred later on to the bottom layer via dry transfer, in which the top layer was anchored and then strained via vdW interaction (see Methods and Supplementary Fig. 30). Yet, the interlayer coupling varies with positions, resulting in different strains (Fig. 4a–c). We introduce a generalized Elliot formula including indirect phonon-assisted recombination of momentum-dark excitons in our theoretical model for calculating the PL emission of strained TMD monolayers31. By applying a small compressive strain on the top layer (top-WS2 layer: 0.3% and top-WSe2 layer: 0.15% strain in heterostructures), we can successfully reproduce the experimental data (Fig. 4e, f) for W-based materials. While the theoretical results shown in Fig. 4e, f are obtained for 77 K for better visualization, a strain-induced increase of the energy separation between K-Q and K-K excitons (Supplementary Fig. 31)31 could lead to visible phonon-sidebands also at higher temperatures, in accordance to experimental observations of SeQ and SQ. In contrast, Mo-based material showed K-Q dark exciton higher in energy compared to K-K bright exciton even in the presence of compressive strain (Details in Supplementary Fig. 31). This leads to a negligible phonon-assisted PL. Therefore, the presence of a strained top layer, leads to a new peak in WS2/MoSe2, corresponding to lower energy K-Q states, while no new phonon-assisted peaks are observed in MoSe2/WS2 due to higher energy K-Q excitons, resulting in negligible phonon-assisted PL (Supplementary Fig. 32).

To explore this phenomenon further, we measured the laser path-dependent PL for the same sample (WS2/WSe2 heterostructure) by fabricating our heterostructure on top of the transparent substrate (quartz) and illuminating the laser from the top (bottom) heterostructure (Supplementary Fig. 33). The presence of the SQ peak from the top WS2 layer (which was not directly attached to the substrate) in both cases, again confirmed that this phenomenon is independent of the optical path.

Optical power switch

Finally, we investigate the variation of the PL intensity with laser power in the strong coupling region (A), which was demonstrated in Fig. 4g, where IQ is reduced, and IK is elevated with increasing laser power. Interestingly, the polarity of σ in the strong coupling region is clearly inverted with laser power (red dot, Fig. 4h). For example, σ is positive in low power regime, whereas σ is negative in high power regime. This distinct polarity flipping can be utilized as an optical power switch for monitoring solar panels. However, such a distinct polarity cannot be realized in an intermediate or weak coupling regime, as shown in the bottom panel of Fig. 4h, although the power law is similar to that in strong coupling.

Discussion

In summary, we have observed additional PL peaks in TMD heterostructures that strongly rely on the stacking sequence of TMD monolayers. We ascribe these peaks to momentum-dark K-Q excitons, that follow a similar power law as typical K-K excitons. Numerous scenarios have been explored to explain such an unexpected stacking sequence dependence, including different substrates, different TMD materials, homogeneous bilayers, different number of layers, doping concentration, symmetric encapsulation of hBN or SiO2, optical pathway, and low temperature. However, all these scenarios are not feasible to explain such unusual stacking sequence-dependent properties. On the other hand, our microscopic many-particle theory suggests layer-dependent strain as one possible solution, where assuming the top layer becomes more strained than the bottom layer due to actual stacking conditions. In reality, the strain landscape might be even more complicated as bubbles are known to appear in the heterostructure region. Thus, further studies are needed to fully understand the observed stacking sequence dependence. However, experimentally observed dark excitons of TMD heterostructures with inevitable use of substrate can be engineered for further exciton dynamics and can be useful as an optical power switch. In addition, intralayer indirect excitons in heterostructures may hold significant potential for future valleytronic applications.

Methods

Fabrication of vdW heterostructures

Various TMDs including, WS2, WSe2, MoS2, and MoSe2 were mechanically exfoliated for heterostructure stacking. SiO2, quartz, HfO2 films, and h-BN layers were used as the substrate. The presence of monolayer TMDs was confirmed by PL, Raman, and AFM. The aligned dry transfer technique was adopted for stacking of the selected flake on another designated flake. A water-soluble layer, polyvinyl alcohol (PVA), followed by polymethyl methacrylate (PMMA) was spin-coated on the substrate with two steps of 5 s at 500 rpm and subsequently 60 s at 3000 rpm. The coated substrate was then baked over the glass transition temperature. Monolayer TMDs were exfoliated on PVA/PMMA-coated substrate. The whole substrate was submerged in warm water such that PVA was dissolved and consequently, TMD/PMMA film was scooped by a customized holder with a 2-3 mm size hole (see Supplementary Fig. 1). The holder was then annealed for 12 min at 120 °C. This holder was loaded on the customized dry-transfer machine. Second, the bottom TMD was exfoliated directly on substrate, which was further aligned with the top TMD/PMMA. The temperature was maintained at 110 °C during stacking to remove residuals including contaminants and bubbles. Finally, the top PMMA was removed by rinsing in acetone-IPA-ethanol solutions sequentially.

Atomic force microscopy (AFM)

AFM was performed in a Hitachi AFM5000II probe station using NSG30_SS tips in a tapping mode. Gwyddion software was used for extracting the flake height by the thickness difference between substrate and desired flake.

Photoluminescence (PL) and Raman spectroscopy

PL and Raman were carried out in a commercial Ntegra Spectra II confocal system under room temperature with 532 nm laser excitation. The laser beam was focused onto the sample by using an X100 objective lens (with a numerical aperture of 0.7).

Device fabrication and STS measurements

For STS measurement we fabricated our heterostructure on an insulating layer (300 nm SiO2) to avoid the band renormalization in TMDs due to the metal substrate. The contact electrode was constructed for grounding our sample. STS measurements were performed in an ultra-high vacuum (UHV) chamber with a base pressure of around 2.0 ×10−11 Torr by using first-generation low-temperature Scienta Omicron (Germany) scanning tunneling microscopy (LT-STM) at room temperature. Electrochemically etched tungsten (W) tips were used for the measurement. Surface oxides of the tips had been removed by electron bombardment inside the UHV chamber. For taking STS data we used the conventional lock-in technique with a voltage modulation of 16 mVrms at a frequency of 817 Hz.

Mechanical cleaning

The fabrication process often introduces residues, which reduces carrier mobility. Especially post-lithography cleaning is therefore required. By using contact mode AFM, the tip pressure allows pushing the contamination at the edges of the scan area, so that mechanical cleaning can be possible (Supplementary Fig. 25). This significantly improves the electronic properties and is much more beneficial for high vacuum measurement, as suggested previously32.

Electronic bandgap determination

Band onset is determined by dI/dV spectrum. The bandgap of the material, the energy difference by conduction band minimum (CBM) to valance band maximum (VBM) in dI/dV spectrum, was extracted by taking a logarithm of dI/dV spectrum33.

Theoretical modeling

Although the hybridization of dark states has been shown to have a significant impact on their energetic position and associated photoluminescence34,35, the observed stacking-sequence-dependent peaks SQ and SeQ require to go beyond stacking-symmetric interlayer coupling effects. For this purpose, we consider a stacking-induced asymmetric strain and consider how this would change the monolayer PL34,35. We microscopically model the monolayer PL including indirect recombination of momentum-dark excitons via the recently introduced Elliott formula31 \(I(E)=\frac{2{|M|}^{2}[{I}_{d}+{I}_{ind}(E)]}{{({E}_{0,K-{K}^{-E}})}^{2}+{(\gamma+{\varGamma }_{0,K-K})}^{2}}\) with a state-independent inhomogeneous broadening Γinh reflecting the experimental linewidth. Here \({I}_{d}=\gamma {N}_{0,K-K}\) provides the direct radiative recombination PL with rate γ while \({I}_{ind}(E,\, t)={\sum }_{\alpha,\beta,\pm }{c}_{\alpha ;\beta }^{\pm }{N}_{\alpha }(t)\frac{{\varGamma }_{\alpha }+{\varGamma }_{inh}}{{({E}_{\alpha }\pm {{\in }}_{\beta }-E)}^{2}+{({\varGamma }_{\alpha }+{\varGamma }_{inh})}^{2}}\) provides the indirect recombination of excitonic states α = Q,v with energy Eα (having center-of-mass momentum Q in valley v). The indirect emission is assisted by absorption/emission of phononic modes β with energy εβ (via coefficients c depending on exciton-phonon interaction31). Finally, Γα and Nα provide respectively the excitonic scattering rate and thermalized occupation, both depending crucially on the exciton band alignment. The latter is evaluated by solving the Wannier equation after introducing a generalized Keldysh potential for the Coulomb interaction, where we include the dielectric constants ε for the environment (ε = 3.8 and 1 for respective SiO2 and air) and the pristine/strained electronic dispersion obtained from DFT studies36. In this way we microscopically predict in W-based materials a strain-induced increase of the energy separation between bright K-K and momentum-dark K-Q excitons, with a consequent increase in the corresponding occupations. This in turn results in an increased indirect emission from monolayer K-Q states in comparison to the direct one, as shown in Fig. 4e, f. In particular, the strain in the top layer induces new peaks in good agreement with the experimental observation. The phonon-assisted emission has a lower efficiency than direct recombination, which is, however, compensated by a higher occupation of the K-Q excitons compared to the bright excitons. The higher relative occupation of K-Q excitons is induced by the energy alignment in tungsten-based materials and is further triggered by compressive strain. Furthermore, at decreased temperatures (77 K), we find an increased relative occupation of K-Q excitons and hence more intense phonon-sidebands31. In contrast, in Mo-based materials, K-Q excitons are higher in energy than the bright-excitons even in the presence of compressive strain, Supplementary Fig. 31. This leads to a negligible phonon-assisted PL. Nevertheless, the strain-induced reduction of the dark-bright exciton separation (Supplementary Fig. 32) could potentially lead to a resonance similar to SeQ in MoSe2/WS2, if additional activation mechanisms are present, e.g., via the interplay of strain and defects37. The microscopic evaluation of these mechanisms goes beyond the scope of this work.