Introduction

Atomically thin heterostructures created by stacking van der Waals materials mark a new frontier in condensed matter physics1,2,3. When two monolayer crystals of the same lattice symmetries overlay on each other, a moiré superlattice may form due to a small mismatch in their lattice constants or angular alignment4,5. The latter—the twist angle between the two layers—provides a powerful tuning knob of the electronic properties of the heterostructure. Seminal results have been obtained in twisted bilayer graphene, where superconducting and correlated insulating states are created by fine control of the twist angle6,7,8,9. In semiconductors, such as transition metal dichalcogenides (TMDC) heterobilayers, the moiré lattice has a period on the length scale of an exciton, thereby providing a unique opportunity to create coupled exciton lattices hitherto unavailable in any other systems. A wide variety of phenomena, tunable with the twist angle, may become possible, ranging from single quantum-dot arrays and topological bands to strongly correlated states10,11,12,13,14.

To search for the effects of moiré lattices on excitons, split-exciton states have been reported in TMDC bilayers with very small twist angles, demonstrating localization of exciton states likely in moiré supercells15,16,17,18. However, increasing the twist angle has led to the suppression of measurable features of moiré excitons. In WS2/MoSe2 heterobilayers, it was suggested that the resonant interlayer hybridization amplifies the moiré superlattice effects on the electronic structure19; yet only a single resonance was resolved as the twist angle deviates significantly from 0° or 60°18. Existence of moiré superlattice for exciton in large-twist-angle bilayers and nontrivial effects of the twist-angle on excitons remain largely unexplored in experiments.

In this work, we show moiré excitons in heterobilayer of a wide range of twist angles and demonstrate tuning of their properties by the moiré lattice or the twist angle. Utilizing the inter- and intralayer hybrid excitons in WS2/MoSe2 bilayers, we reveal the formation of moiré reciprocal lattices with Brillouin zones of different sizes at different twist angles. We furthermore show how the moiré reciprocal lattices drastically change the properties of the moiré excitons, such as their resonance energies, oscillator strengths, and inter-/intralayer mixing. The twist-angle dependence of the moiré exciton states is well-explained by an analytical theory model based on band folding in the moiré lattice, which also consistently explain the dependence on the spin–orbit splitting of the conduction band, valley selection rules, atomic stacking orders, and the lattice symmetries. Comparing the experimental results with the model, we obtain the effective mass of the interlayer excitons, the interlayer electron-tunneling strength.

Results

The devices used in this work are WS2/MoSe2 heterobilayers with different twist angles θ, capped by few-layer hexagonal boron nitride (hBN). Details of sample fabrication and calibration of θ have been described elsewhere20,21 and provided in the “Method” section. Figure 1a shows the optical microscope image of a heterobilayer, where the sharp edges of two monolayers are aligned. The twist angle is θ = 59.8° ± 0.3°, determined optically using polarization-dependent second-harmonic-generation measurements20,21 (see Supplementary Figs. 1 and 2 and Supplementary Notes 3 and  4 for details).

Fig. 1: Hybrid excitons in rotationally aligned WS2/MoSe2 bilayers.
figure 1

a An optical microscope image of a hexagonal boron nitride (hBN)-capped heterobilayer. Red and blue solid lines outline the MoSe2 and WS2 monolayers, respectively. The inset is an atomistic model of the WS2/MoSe2 heterobilayer. Red and blue spheres represent MoSe2 and WS2 monolayers, respectively. The black solid diamond represents the moiré unite cell. b Top panels illustrate a unit cell of R-stacking (left) and H-stacking (right) WS2/MoSe2 bilayers. Middle panels depict the corresponding band alignment of WS2 and MoSe2, where X labels the intralayer transition and IX labels the nearly resonant interlayer transition that shares the same hole state. Solid and dashed lines correspond to states of opposite spins. Bottom panels illustrate the alignment between an intralayer MoSe2 A exciton state (red) and the interlayer exciton state (blue) that it hybridizes with. c Reflectance contrast (RC) spectra for, from bottom to top, a monolayer WS2 (blue), monolayer MoSe2 (red), R-stacking bilayer (orange), and H-stacking bilayer (purple). The dots are the data, and solid lines are fits. The spectra are displaced vertically for easier reading. d Summary of the fitted ratio of the oscillator strength between LHX and UHX, and the corresponding detuning δ.

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Identification and analysis of inter- and intralayer hybrid excitons

We first characterize exciton hybridization in closely aligned heterobilayers, with a twist angle θ ~ 0° or 60°. In such bilayers, the Brillouin zones of the two layers closely overlap in momentum space to form nearly direct bandgaps for both the inter- and intralayer transitions (top panels of Fig. 1b). At the same time, the hole band offset is large, but the conduction-band offset is small between WS2 and MoSe2 (middle panels of Fig. 1b). Therefore interlayer electron tunneling is expected between states of the same spin and valley, which leads to hybridization between the corresponding intra- and interlayer exciton transitions that share the same hole state (bottom panels of Fig. 1b).

Making use of the large difference in oscillator strength between spatially direct and indirect excitons, we identify the formation of hybrid states via the reflectance contrast (RC) spectra at 4 K: \({\rm{RC}}=\frac{{{\rm{R}}}_{{\rm{sample}}}\,-\,{{\rm{R}}}_{{\rm{sub}}}}{{{\rm{R}}}_{{\rm{sub}}}}\), where Rsample and Rsub are reflection spectrum taken from sample and substrate respectively (see Supplementary Fig. 3 and Note 5). The interlayer exciton has an oscillator strength two to three orders of magnitude weaker than that of the intralayer exciton, due to separation of the electron and hole wavefunction22,23,24, so it is typically too weak to be measurable in absorption or RC spectroscopy where the noise level is typically 1% or higher (Supplementary Fig. 4 and Note 6). However, when interlayer excitons hybridize with intralayer ones via electron or hole tunneling, the hybrid states acquire an oscillator strength through the intralayer exciton component. Therefore, we can identify the hybrid excitons via their spectral weight in the absorption spectra of the heterobilayer.

As shown in Fig. 1c, the MoSe2 monolayer region of the device (as marked on Fig. 1a) shows a strong intralayer MoSe2 A exciton resonance near 1.65 eV, while the WS2 monolayer has no exciton resonances nearby. In the bilayer, stacking of the WS2 layer is expected to lead to a red shift of MoSe2 A exciton resonance18 while also introducing an interlayer exciton transition, between an electron in WS2 and a hole in MoSe2. The interlayer exciton has a negligible oscillator strength and should not be observable in RC. However, two clearly resolved resonances appear in our bilayer, both with significant spectral weight (top two spectra in Fig. 1c). The same two resonances are also measured in photoluminescence (see Supplementary Fig. 5 and Note 7). We therefore identify them as the inter- and intralayer hybrid states, the lower (LHX) and upper hybrid excitons (UHX). Both LHX and UHX inherit an oscillator strength from their intralayer component18, with the ratio fLHX/fUHX controlled by their intralayer exciton fractions, which in turn is controlled by the energy detuning δ = EIX − EX between the uncoupled interlayer (EIX) and intralayer (EX) resonances. Therefore fLHX/fUHX greater or less than one corresponds to positive or negative detuning δ. There are multiple pairs of intra- and interlayer excitons that can hybridize. We focus on the transition region of MoSe2 A exciton first and label these states as MoA excitons, of which the hole is always in the highest MoSe2 valence band. Other pairs will be analyzed later.

As clearly seen in Fig. 1c, in the R-stacking bilayers (θ = 2. 1°), fLHX/fUHX >1, suggesting the uncoupled interlayer state lies above the intralayer one, or δR > 0. In contrast, in the H-stacking bilayer (θ = 59.8°), fLHX/fUHX <1, suggesting δH < 0. These results are consistent with the spin selection rules of the excitonic transitions illustrated in Fig. 1b25. Assuming the interlayer exciton-binding energy is about the same in R- and H-stacking bilayers26, and the difference δR − δH is comparable to the spin–orbit splitting of WS2 conduction band (Fig. 1b).

To analyze the results quantitatively, we first obtain the energies, ELHX and EUHX, and oscillator strengths of the hybrid states by fitting the RC spectra using the transfer matrix method, where the hybrid excitons are modeled as Lorentz oscillators (see “Methods”, Supplementary Fig. 3 and Note 5)25,27. The fitted spectrum agrees well with the data, as shown in Fig. 1c. Describing the hybrid modes with the coupled oscillator model, we have \({E}_{{\rm{LHX}}}-{E}_{{\rm{UHX}}}=-\sqrt{4{J}^{2}+{\delta }^{2}}\), and \(\frac{{f}_{{\rm{LHX}}}}{{f}_{{\rm{UHX}}}}=\frac{\sqrt{{\delta }^{2}\,+\,4{J}^{2}}\,+\,\delta }{\sqrt{{\delta }^{2}\,+\,4{J}^{2}}\,-\,\delta }\) (see Supplementary Note 1 for details). Thereby using the fitted ELHX,UHX and fLHX,UHX, we can obtain δ and J. As summarized in Fig. 1d, we obtain J ~ 20 meV for both R- and H-stacking and δR − δH = 25.9 ± 0.5 meV, consistent with the spin–orbit splitting of WS228, confirming the hybrid states are formed by spin-conserved interlayer electron tunneling.

Twist-angle dependence of moiré-lattice-induced hybrid excitons

To study tuning of the hybrid excitons by the moiré lattice, we perform the same measurements and analysis as discussed above on 30 samples with different twist angles, and obtain how the exciton energies, oscillator strengths and interlayer tunneling vary with the changing moiré lattice. We define θ0 < 30° as the angular deviation from aligned bilayers of R- or H-stacking. θ0 = ∣θ∣ for R stacking and θ0 = ∣60° − θ∣ for H-stacking.

As shown in Fig. 2a, the MoA hybrid exciton doublets are clearly resolved for θ0 up to 6°, which would correspond to a tuning of the moiré lattice constant by nearly threefold29. The spectral weights of the doublets evolve continuously with the twist angle, reflecting the continuous increase of fLHX/fUHX and δ with θ0 (middle panel of Fig. 2c). At the same time, the interlayer coupling J decreases continuously (bottom panel of Fig. 2c). These observations show clearly moiré lattice induce hybridization and tuning of intralayer and interlayer excitons, as we explain below.

Fig. 2: Twist-angle dependence of the hybrid excitons.
figure 2

a RC spectra of bilayers of different twist angles θi, for i = 1–6. The corresponding θi and extracted detuning δi = EIX,i − EX,i are labeled by each spectrum. The spectra are displaced vertically for easier reading. b Schematics of MoA intralayer (red) and interlayer (blue) exciton bands at the different twist angles θi. The interlayer exciton band is displaced in the momentum space with increasing θi. The moiré superlattice leads to band folding and formation of a new interlayer exciton state at the Γ point q = 0 (blue oval), with the same angular momentum as the intralayer exciton state (red oval). c Ratio of the oscillator strengths of LHXMoA and UHXMoA, detuning, and inter- and intralayer exciton coupling strength as a function of the twist angle θ, obtained from the RC spectra. The gray solid lines in the middle panel are quadratic fits based on Eq. (1). The blue solid lines in the bottom panel are the theoretical values based on Eq. (2).

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We illustrate in Fig. 2b the MoA exciton bands at different twist angles, corresponding to the six samples shown in Fig. 2a. The intralayer MoSe2A exciton transition (red band) remains direct, with the band minimum at zero center-of-mass momentum qX ~ 0, irrespective of the twist angle. It is close in energy with the interlayer exciton formed by a hole from the same MoSe2 valence band but an electron from a WS2 conduction band. This interlayer exciton band has the band minimum also at zero center-of-mass momentum: qIX ~ 0, when θ ~ 0° (θ1 in Fig. 2a, b) or 60° (θ6 in Fig. 2a, b), neglecting the small lattice constant mismatch.

As the two lattices rotate relative to each other by θ (θ2 to θ5 in Fig. 2a, b), the Brillouin zones of MoSe2 and WS2 also rotate by θ. The interlayer exciton band minimum shifts away from the intralayer exciton band minimum by momentum KW − KM for R stacking, where KM and KW are, respectively, the Brillouin zone corners for MoSe2 and WS2 layers. Due to this momentum mismatch, hybridization between intralayer MoSe2A excitons and the interlayer state at the band minimum is not allowed.

However, interlayer electron tunneling in the moiré lattice can lead to the formation of new moiré miniband states to hybridize with the optically bright intralayer excitons. As illustrated in Figs. 2b and  3c, three interlayer excitons \({\left|{{\boldsymbol{q}}}_{i}\right\rangle }_{\text{IX}}\) overlap with the optically bright intralayer exciton, where the center-of- mass momentum qi, measured relative to the band minimum of interlayer exciton, correspond to q1 = KM − KW for R stacking, with q2,3 connected to q1 by 2π/3 and 4π/3 rotations, respectively, via moiré reciprocal lattice vectors. These three interlayer states are offset from their band minimum by the kinetic energy \({\hslash }^{2}{q}_{i}^{2}/(2{M}_{\text{IX}})\), for i = 1, 2, 3, and MIX the total mass of interlayer exciton. These three states can couple due to the moiré lattice and therefore, superpose to form moiré miniband states, of which one interlayer exciton state shares the same angular momentum as the intralayer MoSe2A exciton at qX ~ 0, giving rise to the hybrid doublet we observe in angularly misaligned bilayers (see Supplementary Note 2).

Fig. 3: Hybridization in commensurate moiré lattices compared with aligned bilayers.
figure 3

a RC spectra of bilayers with θ = 2.1°, 20.1°, 40.3°, and 59.8°. The LHX has a higher (lower) spectral weight than UHX in bilayers with θ = 2.1° and 40.3° (θ = 59.8° and 20.1°). Dots are the data, solid lines are fits, and dashed lines are the fitted individual hybrid exciton resonances. b Summary of the fitted parameters for the RC spectra in a, showing similarities between bilayers with θ = 2.1° and 40.3° and between bilayers with θ = 59.8° and 20.1°. c–e Schematics of the Brillouin zones of twisted bilayers. The red (blue) hexagons depict the Brillouin zones of MoSe2 (WS2) monolayers. In c, the twist angle is 6∘. The green arrows indicate vectors q1, q2, and q3, which represent the momentum shift between the Brillouin zone corners of the two monolayers. In d, θ = 21.8°, a commensurate moiré lattice is formed, with the corresponding moiré Brillouin zone depicted by the black hexagons. The yellow arrow represents the moiré reciprocal lattice base vector that connects KM and \({{\boldsymbol{K}}}_{{\rm{W}}}^{\prime}\). In e, θ = 38.2°, which is another commensurate angle dual to 21.8°, and KM and KW become equivalent states in the moiré Brillouin zone.

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When θ deviates more from 0° or 60°, the interlayer exciton formed in the moiré lattice continuously blueshifts because of the increasing kinetic energy, which explains the measured continuous blueshift of the LHX and UHX resonances, and the continuous increase of the spectral weight of LHX compared to UHX.

Theoretical analysis of moiré-lattice-induced hybrid excitons

To analyze our results more quantitatively, we develop an analytical microscopic theory based on the above understanding (see Supplementary Note 2 for details). Comparing it with the measured twist-angle dependence of the hybrid states, we obtain the key band parameters of the bilayer, including the interlayer exciton effective mass and interlayer coupling strength.

We first compare the measured detuning δ with θ0 and the interlayer exciton kinetic energy. As discussed above, δ is given by:

$$\delta ({\theta }_{0})={\delta }_{0}+\frac{{\hslash }^{2}{{\boldsymbol{q}}}_{1}^{2}}{2{M}_{\text{IX}}},$$
(1)

where δ0 is the detuning at θ = 0° or 60° for bilayers close to R- and H-stacking, respectively. q1 is equal to 4π/(3aM), and aM is the moiré period approximated by \({a}_{0}/\sqrt{{\theta }_{0}^{2}+{\epsilon }^{2}}\), for a0 the monolayer lattice constant and ϵ the lattice constant mismatch \(| {a}_{0}-{a}_{0}^{\prime}| /{a}_{0}\) between the two layers. Equation (1) shows that δ increases quadratically with θ0. As θ0 increases from 0° to 6°, aM changes by nearly threefold, and δ − δ0 changes by sevenfold29. Fitting the measured δ vs. θ0 with Eq. (1), we find the interlayer exciton total mass MIX to be (6.9 ± 3.2)m0 and (1.41 ± 0.28)m0 for R- and H-stacking heterobilayers, respectively, for m0 the electron free mass. These values are greater than the sum of the monolayer electron and hole effective masses. One possible reason is that the electron and hole effective masses in the bilayer may have been modified due to effects such as lattice relaxation30.

We note that strain may lead to deviation in the twist angle and detuning. To minimize the effect of strain, we use devices of high structural integrity as verified by imaging and SHG (see Supplementary Note 4 and Supplementary Fig. 2 for details). The clear trends in the twist-angle dependence shown in Fig. 2 suggest that the effect of strain is relatively small and mainly leads to additional fluctuations in the measurement results above the measurement uncertainties (indicated by the error bars).

From our microscopic theory, we can also estimate the conduction-band interlayer tunneling parameter w from the coupling strength J through the relation

$$J=\frac{\sqrt{3}w}{{\mathcal{A}}}\sum _{{\boldsymbol{k}}}{\phi }_{{\boldsymbol{k}}+\frac{{m}_{{\rm{h,IX}}}}{{M}_{{\rm{IX}}}}{{\boldsymbol{q}}}_{1}}^{* }{\psi }_{{\boldsymbol{k}}},$$
(2)

where ϕk and ψk are, respectively, the relative-motion wavefunction for interlayer and intralayer excitons with the normalization \((1/{\mathcal{A}}){\sum }_{{\boldsymbol{k}}}| {\psi }_{{\boldsymbol{k}}}{| }^{2}=1\) and \((1/{\mathcal{A}}){\sum }_{{\boldsymbol{k}}}| {\phi }_{{\boldsymbol{k}}}{| }^{2}=1\). Here, \({\mathcal{A}}\) is the system area, and mh,IX is the hole mass for the interlayer exciton. Because of the momentum shift (mh,IX/MIX)q1 in the integral of Eq. (2), J decreases with increasing θ0, which agrees with the experimentally observed angle dependence of J (Fig. 3c). At small θ0, J can be approximated by \(\sqrt{3}w\). Using our experimentally measured value of J at θ0 ~ 0, we estimate the interlayer tunneling w to be 11.5–14.0 meV for both R- and H-stacking bilayers.

When the twist angle θ0 is greater than 6°, the hybrid exciton doublets become hard to be resolved, likely because there is a large blue detuning and the UHX has a vanishing oscillator strength (see Supplementary Fig. 6 and Note 8).

Moiré excitons in commensurate moiré lattices at twist angles near 21.8° and 38.2°

Remarkably, pronounced and well-resolved doublets reappear in heterobilayers with θ = 20.1° ± 0.3 and 40.3° ± 0.3, as shown in Fig. 3. In the bilayer with 20.1° twist angle, the LHX has a smaller spectral weight than UHX has, corresponding to a negative detuning (δ = − 5.6 meV), which is similar to H-stacking bilayers formed at θ ~ 60°. In contrast, in the bilayer with 40.3° twist angle, the LHX has a larger spectral weight than UHX has, corresponding to a positive detuning (δ = 11.8 meV), which is similar to R-stacking bilayers formed at θ ~0°. In both devices, the coupling strength J ~8 meV is weaker than but of the same order of magnitude as aligned bilayers with θ close to 0° or 60°.

The revival of hybrid excitons in these two bilayers can be understood as a direct result of interlayer tunneling induced by a moiré lattice that is nearly commensurate with the monolayer lattices. The two twist angles are close to the two special commensurate angles 21.8° and 38.2°, respectively22, where the corresponding periodic moiré reciprocal lattices have the largest reciprocal lattice constant, \(1/\sqrt{7}\) of the monolayer reciprocal lattice constant. Corners of the Brillouin zones of the two monolayers become connected by primitive moiré reciprocal lattice vectors, as illustrated in Fig. 3d and e. The MoSe2and WS2band minima overlap again in the moiré reciprocal lattice, allowing strong nearly resonant tunneling between the intra- and interlayer states. Specifically, when θ ≈ 21.8°, K-valley of MoSe2 and \({{\rm{K}}}^{\prime}\)-valley of WS2 are connected by moiré reciprocal lattice vectors and are folded into equivalent momentum in the moiré Brillouin zone (Fig. 3d). The corresponding hybridized excitons have the same valley configuration as those in bilayers with θ ~ 60°, which is consistent with the observed negative detuning. When θ ≈ 38.2°, K-valley of MoSe2 and K-valley of WS2 are folded into equivalent momentum in the moiré Brillouin zone (Fig. 3e), and the corresponding hybridized excitons have the same valley configuration as those in bilayers with θ ~ 0°, consistent with the observed positive detuning. Moreover, since interlayer tunneling only needs one Umklapp scattering by a moiré reciprocal lattice vector, the tunneling strength remains of the same order of magnitude as in angularly aligned bilayers. Therefore, the strong revival of the hybrid excitons and their similarities with the angularly aligned bilayers show again the critical role of moiré lattice in interlayer tunneling.

Moiré excitons formed with different intralayer excitons

In the above discussion, we have focused on hybrid states formed with the MoSe2A excitons, which feature large spectral weight, relatively narrow linewidths, and well-resolved doublets at small detunings. Hybrid states can also form with higher-energy bands, including the A excitons of WS2 (WA), and B excitons of WS2 and MoSe2. The B excitons have broader linewidths than the A excitons; as a result, the doublets are not well resolved. The A excitons of WS2 have a broader linewidth than A excitons of MoSe2 and generally a larger detuning. We observe well-resolved WA doublets only in bilayers with θ ~ 0°, corresponding to hybrid excitons formed by a hole in the WS2 valence band and an electron tunneling between the MoSe2 and WS2 conduction bands (Fig. 4).

Fig. 4: Comparison of MoA and WA hybrid states.
figure 4

a Band diagram of R-stacking MoSe2/WS2 bilayers. The conduction and valence bands are represented by broad continuous bands. Exciton states are represented by the horizontal solid lines. Arrows represent spin-conserved exciton transitions, which are lowered in energy from the band-to-band transition by binding energy. EBX and EBIX denote the binding energies for the intra- and interlayer transitions, respectively. b RC spectra of both MoA and WA hybrid excitons from bilayers with θ ~ 0°. Dots are the data, and solid lines are fits. The corresponding θ and detuning δM and δW obtained from fitting are labeled by each spectrum.

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It is interesting to compare the detuning for MoA and WA states for θ ~ 0°, which we label as δMoA and δWA, respectively. As shown in the schematic electronic band diagram in Fig. 4a, neglecting exciton-binding energies, the detuning of the interlayer transition from the intralayer one is the same magnitude but opposite signs between the MoA and WA states. The sum of the two detuning should be zero. However, this is different from our observation that both the LHX states have larger spectral weight for both MoA and WA states. This can be understood as due to the weaker binding energy of interlayer excitons compared to intralayer ones, resulting from electron–hole separation. The difference in intra- and interlayer exciton-binding energies, \(\Delta {E}_{{\rm{B}}}^{{\rm{R}}}={E}_{{\rm{BX}}}-{E}_{{\rm{BIX}}}\), adds to both δMoA and δWA. Assuming \(\Delta {E}_{{\rm{B}}}^{{\rm{R}}}\) is approximately the same for the MoA and WA state, the sum of δMoA and δWA becomes twice of ΔEB, or, \(\Delta {E}_{{\rm{B}}}^{{\rm{R}}}=1/2({\delta }_{{\rm{MoA}}}+{\delta }_{{\rm{WA}}})\). From our measurements of MoA and WA states in bilayers with θ < 1°, we estimate \(\Delta {E}_{{\rm{B}}}^{{\rm{R}}}\) of 10–16 meV (Fig. 4b). In comparison, values between 17 meV and 100 meV have been reported based on first principle calculations31,32,33,34 or measurements on homo-bilayers35,36,37. The relatively small values of 10–16 meV we measured may possibly be due to difference in the materials or a smaller interlayer exciton Bohr radius as a result of strong interlayer tunneling.

Discussion

In summary, we demonstrate hybrid states formed between momentum-direct, moiré-induced interlayer, and intralayer excitons in twisted WS2/MoSe2 bilayers, opening the door to studies of excitonic phenomena in twist bilayers.

Deviation of the twist angle from 0° or 60° not only does not suppress moiré excitons but provides a sensitive tuning knob of the moiré excitons’ properties. Persistence of the moiré excitons, or, the moiré lattice, is clearly manifested in the interlayer tunneling strength, which remains within the same order of magnitude over the measured range of twist angles. It is possible because momentum conservation between the twisted layers is restored by the moiré lattice, or Umklapp scattering by the moiré reciprocal lattice vector. Remarkably, while large detuning between the interlayer and intralayer states suppressed hybridization over a range of angles, pronounced hybrid moiré excitons due to strong interlayer tunneling reappears near twist angles of 21.8° and 38.2°. At these angles, moiré-lattices are formed commensurate with the monolayer lattices, bringing angularly shifted valleys of the two monolayers into equivalent momentum in the same moiré Brillouin zone, thereby enabling strong interlayer tunneling. The resulting hybrid exciton states resemble the features in heterobilayers with θ = 60° and 0°, respectively. These results are direct manifestations of the discrete translational symmetry of the underlying moiré superlattice, which enables transitions that otherwise would not conserve momentum.

Since the hybrid excitons are formed in moiré reciprocal lattices, their properties dependent sensitively on the moiré period, or the twist angle. Utilizing the twist angle degree of freedom, we demonstrate tuning of the moiré exciton properties and furthermore obtain fundamental parameters of the bilayer system that are difficult to measure otherwise, including conduction-band splitting of WS2 induced by spin–orbital coupling, the effective mass of the interlayer excitons in R- and H-stacking bilayers, interlayer electron-tunneling strength, and the difference of binding energies between intra- and interlayer excitons at the presence of interlayer tunneling.

These hybrid excitons inherit large oscillator strengths from the intralayer component that may allow strong exciton–photon coupling while, at the same time, inherit static dipole moment from the spatially indirect interlayer component that leads to long-range interactions. In a moiré lattice, both the oscillator strength and dipole interactions depend sensitively on, and can be tuned by the twist angle. The twisted WS2/MoSe2 bilayers may provide tunable, nonlinear, exciton and polariton lattice systems for exotic states of matter, such as topological excitons and exciton crystals, with novel applications in nanophotonics and quantum information science10,11,12,13,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53.

Methods

Sample fabrication

Monolayer MoSe2, WS2, and few-layer hBN flakes are obtained by mechanical exfoliation from bulk crystals. A polyethylene terephthalate (PET) stamp was used to pick up the top hBN, WS2 monolayer, MoSe2 monolayer, and the bottom hBN under a microscope. After picking up all the layers, PET stamp was then stamped onto the sapphire substrate, and the PET was dissolved in dichloromethane for six hours at room temperature.

Optical measurements

For low-temperature measurements, the sample is kept in a 4 K cryostat (Montana Instrument). The excitation and collection are carried out with a home-built confocal microscope with an objective lens with a numerical aperture (NA) of 0.42. For reflection contrast measurement, white light from a tungsten halogen lamp is focused on the sample with a beam size of 10 μm in diameter. The spatial resolution is improved to be 2 μm by using pinhole combined with confocal lenses. The signal is detected using a Princeton Instruments spectrometer with a cooled charge-coupled camera.