Main

Van der Waals heterostructures incorporating transition metal dichalcogenide (TMD) bilayers open up new avenues for exploring strong correlations using transport and optical spectroscopy. In contrast to similar structures in III–V semiconductors, these heterostructures exhibit possibilities for exotic material combinations, the creation of moiré superlattices exhibiting narrow electronic bands8,9,10,11 and strong binding of spatially separated interlayer excitons12. Recently, transport experiments in twisted bilayer graphene demonstrated strongly correlated electron physics in a single system1,2,3,4,5, ranging from superconductivity to a Mott insulator state as the filling factor ν is varied. In fact, this system realizes a two-dimensional (2D) Fermi–Hubbard model on a triangular lattice with a tunable electron density.

In parallel, optical spectroscopy in van der Waals heterostructures have revealed the prevalence of many-body hybrid light–matter states, termed exciton-polarons6,7, in the excitation spectra of electron- or hole-doped monolayers. Advances in material quality and device fabrication has led to the observation of moiré physics of non-interacting excitons in TMD heterobilayers13,14,15,16. The potential of this new system for many-body physics was recently revealed in a demonstration of a long-lived interlayer exciton condensate17. Here we describe experiments in a heterostructure incorporating a molybdenum diselenide/hexagonal boron nitride/molybdenum diselenide (MoSe2/hBN/MoSe2) homobilayer that in several ways combine the principal developments in these two fields to demonstrate interaction-induced incompressible states of electrons. We provide an unequivocal demonstration of the hybridization of inter- and intralayer excitons by coherent hole tunnelling11,15,18 between the two MoSe2 layers: the avoided crossings in the optical reflection not only show the formation of dipolar excitons with strong optical coupling but also reveal the existence of moiré bands of interlayer excitons. We then demonstrate that intralayer exciton-polaron resonances provide a sensitive tool to investigate correlated electronic states in the bulk. Equipped with this spectroscopic tool, we observe strong layer pseudospin paramagnetism19,20 and an incompressible Mott-like state of electrons at half-filling of each layer.

Device structure and basic characterization

Figure 1a shows a schematic of the device structure. By using a double-gate structure, we control the electric field Ez and the chemical potential μ of the device independently (see Methods for details). Figure 1b (top) is a schematic image of a dipolar exciton formed by coherent coupling of an interlayer exciton (IX) and an intralayer exciton (X) via hole tunnelling. Figure 1b (bottom) shows a sketch of electrons in a moiré lattice, probed by intralayer excitons.

Fig. 1: Device structure and basic characterization.
figure 1

a, Schematic of the device structure. Vtg (Vbg) is the applied voltage to the top (bottom) gate. b, Schematic of coupled inter- and intralayer exciton (top) and electrons in a moiré lattice probed by the intralayer exciton (bottom). Purple planes correspond to MoSe2 layers and dashed lines indicate a moiré lattice. The pink (light blue) circles with a + (−) sign indicate holes (electrons) forming excitons, and the electron–hole pair enclosed by the red (yellow) ellipse indicates intralayer (interlayer) exciton. The green double arrow in the top panel indicates tunnel coupling (t) of holes through a monolayer hBN barrier. The light green circles in the bottom panel indicate electrons filling the moiré lattice. c, Spatial map of integrated photoluminescence from 1.59 eV to 1.65 eV. The blue and yellow dashed lines indicate the boundary of the monolayer MoSe2 and MoSe2/hBN/MoSe2 regions, respectively. The inset shows representative PL spectra of monolayer MoSe2 and MoSe2/hBN/MoSe2 measured at the positions indicated with the blue and yellow stars in the main figure, respectively. 1L, monolayer; 2L, bilayer.

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Figure 1c shows a spatial map of total photoluminescence (PL) from the device. Here both the top and bottom gate voltages are at zero volts. We observe PL from regions with monolayer MoSe2, but not from bilayer MoSe2, where two MoSe2 flakes are in direct contact (the region around the point indicated by the white arrow in Fig. 1c). In contrast, the MoSe2/hBN/MoSe2 region shows bright PL. This indicates that the heterostructure becomes a direct-bandgap system owing to the reduction of the interlayer hybridization of the valence bands at the Γ point21, due to the presence of monolayer hBN. Typical PL spectra of the monolayer MoSe2 and the MoSe2/hBN/MoSe2 region are shown in the inset of Fig. 1c: there are pronounced intralayer exciton luminescence peaks in both regions. We observe two distinct exciton peaks in the MoSe2/hBN/MoSe2 region. This strain-induced energy difference between the PL from the top and bottom layers varies across the sample (Supplementary Information section 2).

Coherent interlayer hole tunnelling and dipolar excitons

We first analyse the electric field (Ez) dependence of the elementary optical excitations of the MoSe2/hBN/MoSe2 region in the absence of itinerant electrons or holes. To this end, we scan the top and bottom gate voltages together to change Ez while keeping the system in the charge-neutral regime. The obtained PL spectrum is depicted in Fig. 2a: using the top (Vtg) and bottom (Vbg) gate voltage dependence, we determine that the PL spectra around 1.632 eV and 1.640 eV stem from an intralayer exciton in the top and bottom layer (Xtop and Xbot), respectively. For high values of |Ez| depicted in the top and bottom parts of the colour-coded PL spectrum, we observe PL lines with a strong Ez dependence: we identify these PL lines as originating from interlayer excitons with a large dipole moment leading to a sizeable Stark shift.

Fig. 2: Electric field dependence of PL and differential reflectance at charge neutrality.
figure 2

a, b, Gate dependence of PL (a) and differential reflectance (b) of MoSe2/hBN/MoSe2. Top and bottom gate voltages are scanned together to tune the electric field at a constant chemical potential (scanned along the dashed line L4 shown in Fig. 3c). The intensity of the PL in a is shown on a log scale. c, d, Magnified plots of b. The corresponding region of c and d is indicated by the blue and green dashed rectangles in b, respectively. e, f, Schematic of the energy bands and the exciton energy alignment under electric fields where the excitons are in resonance. CB and VB denote the conduction and valence bands, respectively, and EB denotes the exciton binding energy. Green double arrows represent hole tunnelling. Panels e and f are the cases for opposite direction of electric field, and correspond to c and d, respectively.

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The spectra for a positive (negative) Vtg regime correspond to the interlayer exciton IX↑ (IX↓), which has a hole in the bottom (top) layer and an electron in the top (bottom) layer. The associated dipole moment of the interlayer exciton changes its polarity for Vtg ≈ 0. By extrapolating the IX↑ and IX↓ PL lines and finding their crossing point, we estimate the energy difference between the inter- and the intralayer exciton resonances at Ez = 0, which allows us to determine their binding energy difference to be 0.1 ± 0.01 eV (Supplementary Information section 6).

Figure 2b shows the differential reflectance (ΔR/R0) spectrum obtained for the same range of gate voltage scan as that of Fig. 2a. Here ΔR/R0 ≡ (R − R0)/R0, with R and R0 denoting the reflectance signal from the MoSe2/hBN/MoSe2 region and background reflectance, respectively. In accordance with the PL data (Fig. 2a), we find Xtop and Xbot resonances around 1.632 eV and 1.640 eV, respectively. Moreover, for Vtg ≳ 7.5 V (Vtg ≱ −7.5 V), we observe IX↑ (IX↓) hybridizing exclusively with Xtop (Xbot). Figure 2c, d shows magnified plots of the regions highlighted with blue and green dashed lines in Fig. 2b, confirming avoided crossing of an intralayer exciton with multiple interlayer excitons. We first note that the observation of a sizeable reflection signal from IX↑ away from the avoided crossing suggests that it is possible to resonantly excite long-lived interlayer excitons in these structures. The hybridization of IX↑ lines with Xtop, together with the lack of an avoided crossing with Xbot in Fig. 2c, unequivocally shows that avoided crossings originate exclusively from the coherent hole tunnelling schematically shown in Fig. 2e. Our observation, proving that the hole tunnel coupling is much larger than that of the electron, is consistent with the band alignment expected from first-principles band-structure calculations22. This conclusion is also confirmed by the data depicted in Fig. 2d, where avoided crossing originates from the coherent-hole-tunnelling-induced hybridization of IX↓ and Xbot schematically shown in Fig. 2f.

One of the most remarkable features of the spectra depicted in Fig. 2c, d is the existence of multiple avoided crossings associated with three interlayer exciton resonances separated in energy by about 5 meV. This interlayer exciton fine structure demonstrates the existence of a moiré superlattice8,10,11,13,14,15,16, originating from a small twist angle between the two MoSe2 layers (Methods). The presence of an hBN tunnel barrier strongly suppresses the strength of the associated moiré potential, rendering it sizeable for only the interlayer excitons8.

Charge configuration detection by exciton-polaron spectroscopy

The presence of itinerant charges drastically alters the optical excitation spectrum23. Recent theoretical and experimental work established that the modified spectrum originates from dynamical screening of excitons by electrons or holes6,7, leading to the formation of a lower-energy attractive polaron (AP) branch. Concurrently, the exciton resonance evolves into a repulsive polaron (RP) (Supplementary Information section 1). The strong sensitivity of the RP resonance energy to changes in electron density renders it an ideal spectroscopic tool for sensing the electron density n in the same layer24,25. The strain-induced resonance energy difference between Xtop and Xbot, ensuring different energies for the corresponding RPtop and RPbot, together with the much weaker sensitivity of RPtop (RPbot) to the electron density nb (nt) in the bottom (top) layer, allows us to determine the charging configuration of the two layers simultaneously. As we are predominantly interested in the low-carrier-density regime where the quasiparticle (bare exciton) weight of the RP resonance is close to unity, we will refer to it as the exciton resonance.

Figure 3a, b shows the gate voltage dependence of ΔR/R0 at E = 1.632 eV and E = 1.640 eV, which correspond to the top (Xtop) and bottom (Xbot) intralayer exciton resonance energy in the charge-neutral regime, respectively. The insets to these figures show a line cut through the dispersive neutral exciton reflection spectrum, indicating the exciton energies at which we monitor ΔR/R0. As a small increase of electron density from about 0 to 1 × 1011 cm−2 results in a change of ΔR/R0 from about −1 to ≳0, the blue regions in Fig. 3a, b correspond to the charge-neutral regime of each layer. The red and white regions in turn, correspond to the electron- or hole-doped regime of each layer. This all-optical determination of the charge map of the bilayer provides an invaluable tool for monitoring the bulk properties of 2D materials.

Fig. 3: Gate dependence of differential reflectance spectrum of MoSe2/hBN/MoSe2.
figure 3

a, b, Two gates dependence maps of differential reflectance around the top (a) and bottom (b) intralayer exciton resonances (E = 1.632 eV and E = 1.640 eV, respectively). The insets of a and b show the differential reflectance spectrum at (Vtg, Vbg) = (0 V, 0 V) (indicated with the white stars in the maps). The magenta and cyan dots in the insets indicate the points where E = 1.632 eV and E = 1.640 eV, respectively. c, Charge configuration diagram obtained by derivative of the differential reflectance spectrum with respect to energy (sum of the derivatives at E = 1.632 eV and E = 1.640 eV). The charge configuration for each layer is indicated by p, i and n, which correspond to hole doped (p), neutral (i) and electron doped (n), and shown in the order of (top, bottom). d–f, Gate dependence of differential reflectance along the dashed lines L1 (d), L2 (e) and L3 (f) shown in a–c. Magenta and cyan dashed lines indicate the top (E = 1.632 eV) and bottom (E = 1.640 eV) exciton resonance energies, respectively. \({{\rm{AP}}}_{{\rm{L}}}^{{\rm{C}}}\) and \({{\rm{RP}}}_{{\rm{L}}}^{{\rm{C}}}\) stand for intralayer attractive and repulsive polarons, where L is ‘top’ or ‘bot’ stands for top or bottom layer, and C is + or − stands for hole or electron as Fermi sea carriers. Charge configuration is written in green together with the green dashed lines indicating the charge configuration transition point.

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To enhance the sensitivity of the charge map and to visualize the charge configuration of both layers at the same time, we evaluate and overlay the derivative of ΔR/R0 (d(ΔR/R0)/dE) obtained for both layers (see also Supplementary Information section 7). The resulting charge map, depicted in Fig. 3c, is closely reminiscent of the charging diagram used to characterize gate-defined quantum dots26. The blue regions in Fig. 3c correspond to gate voltages where the charge configuration changes, allowing us to clearly separate the regions (t, b) where the top or bottom layer is neutral (t = i or b = i), electron doped (t = n or b = n) or hole doped (t = p or b = p).

We show the typical gate voltage dependence of ΔR/R0 in Fig. 3d, e obtained while scanning the two gates along the lines L1 and L2, indicated in Fig. 3a, b, respectively. In both plots, we confirm the emergence of the AP resonance and the associated blueshift of the exciton or RP energy around the charge configuration transition points, confirming the assignment obtained from d(ΔR/R0)/dE in Fig. 3c.

Figure 3f shows the gate voltage scan along L3, indicated in Fig. 3c, where we fixed Vbg at 4 V and scanned Vtg. As we increase the chemical potential by sweeping Vtg from negative to positive, we find that initially the bottom layer gets electron doped, because the (single particle) conduction band energy of the bottom layer is lower than that of the top layer for the chosen Vbg. For Vtg ≃ 4 V, electrons are introduced into the top layer as well. In the absence of interactions, we would expect the electron density in the bottom layer to become independent of a further increase of Vtg, due to screening of Vtg by the free electrons in the top layer. We observe different behaviour in Fig. 3f: increasing Vtg results in a decrease and eventually total depletion of electrons from the bottom layer. The underlying physics is understood by considering the intralayer exchange interaction, which is maximal if all electrons occupy a single layer. If the ensuing reduction in total repulsive Coulomb energy exceeds the kinetic energy cost of having all electrons in a single layer, layer polarization is favoured, leading to the observed depletion of electrons from the bottom layer. This phenomenon, termed negative compressibility, was previously observed in transport experiments in bilayer semiconductor systems27.

Interaction-induced incompressible states

The results we present in the ‘Coherent interlayer hole tunnelling and dipolar excitons’ section establish the existence of a moiré superlattice for interlayer excitons. The absence of coherent electron tunnelling, however, indicates that the corresponding electronic moiré subbands in the top and bottom layers do not hybridize. Therefore, our homobilayer realizes a rather unique system exhibiting flat bands with layer and valley-spin degrees of freedom. Moreover, negative compressibility (‘Charge configuration detection by exciton-polaron spectroscopy’ section) indicates that electron–electron interactions are prominent even at relatively high electron densities (n ≈ 1 × 1012 cm−2) where several moiré bands in one layer are occupied. In this section, we explore electron correlation effects by focusing on the low-n regime of the charging map (Fig. 3c) where the (i, i), (i, n), (n, i) and (n, n) regions coalesce. The high sensitivity of the exciton/RP resonance energy, as well as the AP oscillator strength, to changes in electron density once again forms the backbone of our investigation.

Figure 4a shows the gate voltage dependence of ΔR/R0 at energy of 1.6320 eV. The choice of the energy at which we monitor ΔR/R0 maximizes the sensitivity to the shift of Xtop and is indicated by the magenta point in the inset of Fig. 4a. We now choose the horizontal and vertical voltage axes to be VE = 0.5Vtg − 0.5Vbg and Vμ = (7/15)Vtg + (8/15)Vbg. With this choice, vertical (Vμ axis) and horizontal (VE axis) cuts through the reflectance map leave Ez and μ unchanged, respectively. Remarkably, Fig. 4a shows a periodic modulation of the reflectance as a function of Vμ, particularly in the low-n regime. Moreover, the modulation of the Xtop and Xbot reflectance are correlated and symmetric with respect to the VE = −1 V axis (Supplementary Fig. 4), indicating that for this value of VE, the conduction band edge energy of the two layers is aligned.

Fig. 4: Gate dependence of intralayer exciton resonance in the low-electron-density regime.
figure 4

a, Gate dependence map of differential reflectance around the top intralayer exciton resonance (E = 1.6320 eV). The inset of a shows the differential reflectance spectrum at (VE, Vμ) = (−1 V, 0.175 V) (indicated with the white star in the map). b, Gate dependence map of the top intralayer exciton resonance energy differentiated by Vμ.

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To gain further insight, we first determine the repulsive polaron resonance energy (\({E}_{{{\rm{X}}}_{{\rm{top}}}}\)) by fitting the reflectance spectrum with a dispersive Lorentzian line shape (Supplementary Information section 3) and then plot the derivative of \({E}_{{{\rm{X}}}_{{\rm{top}}}}\) with respect to Vμ in Fig. 4b. The resulting map shows a remarkable checkerboard pattern that is complementary for the top and bottom layers (Supplementary Fig. 4). As the blueshift of Xtop (Xbot) resonance while increasing Vμ corresponds to filling of electrons in the top (bottom) layer, the complementary checkerboard patterns in Fig. 4b and Supplementary Fig. 4f indicate a layer-by-layer filling of electrons28.

The observed periodicity in Fig. 4 indicates the existence of moiré subbands for electrons. In anticipation of the subsequent discussion, we define a layer filling factor νL (where L is ‘top’ or ‘bot’, indicating top or bottom layer, respectively) such that νL = 1/2 corresponds to 1 electron per moiré unit cell of a single layer, and a total filling factor ν as ν = νtop + νbot. From a capacitive model of our device, we determine that ν = 1/2 coincides with electron density of n = 2 × 1011 cm−2. At this low electron density, the rs parameter, which describes the ratio of interaction energy to kinetic energy, is estimated to be rs ≳ 14. The density periodicity corresponds to a moiré superlattice lattice constant of λmoiré = 24 nm by assuming a triangular superlattice (see Methods for the estimation of rs and λmoiré). We indicate the values of ν corresponding to ν = 1/2, 1, 3/2, 2 with blue dashed lines in Fig. 4b. We confirmed that the same periodicity also appears in the AP photoluminescence intensity (Supplementary Fig. 4g, h).

Figure 5a, b shows the VE dependence of ΔR/R0 for fixed Vμ for ν = 1/2 and ν = 1, respectively. In Fig. 5a, we find an abrupt shift of exciton energy together with complete oscillator strength transfer between APtop and APbot, demonstrating that the electrons are completely transferred from one layer to the other. Figure 5c, e shows the extracted Xbot and Xtop energies around ν = 1/2. Remarkably, the abrupt jump in excitonic resonance is pronounced at ν = 1/2, and smeared out for both lower (ν = 0.35) and higher filling factors (ν = 0.65). These measurements show that abrupt transfer of practically all of the roughly 1,500 electrons within the region we monitor optically is linked to the emergence of an interaction-induced incompressible state in the lowest moiré subband at ν = 1/2 filling. As the filling factor is increased or decreased towards ν = 1/2, the electronic system shows an ever-stronger layer pseudospin paramagnetism, due to the enhanced role of interactions, but otherwise exhibits a continuous interlayer transfer of electrons as a function of Ez that would be expected from a compressible state. Close to ν = 1/2, there is a phase transition to an incompressible state that can be accommodated in either the top or the bottom layer (Fig. 5g). Remarkably, interlayer charge transfer takes place upon changing VE by only 1.9 mV: the corresponding change in the single-particle energy detuning between the top and bottom layers (26 μeV) is much smaller than kBT (360 μeV; where kB is the Boltzmann constant and T is the temperature) (Supplementary Information section 5).

Fig. 5: Electric field dependence of differential reflectance spectrum in the low-electron-density regime.
figure 5

a, b, Electric field (VE) dependence of differential reflectance spectrum for fixed Vμ at ν = 1/2 and ν = 1. The charge configuration of the top and bottom layers is indicated by (νtop, νbot) in green. c, d, VE dependence of Xbot resonance energy around total filling of ν = 1/2 (c) and ν = 1 (d). e, f, VE dependence of Xtop resonance energy around total filling of ν = 1/2 (e) and ν = 1 (f). In c–f, the cyan curves are without offset and other curves are displaced by 0.5 meV (purple) and 1.0 meV (magenta). g, h, Schematic of the charge configuration with density of states at a filling of ν = 1/2 (g) and ν = 1 (h).

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Figure 5b shows that for ν = 1, ΔR/R0 is characterized by three plateau-like regions. We attribute the abrupt jumps in the excitonic resonance energy to the transition from (νtop, νbot) = (0, 1), through (1/2, 1/2), to (1, 0) configurations (Fig. 5h). This explanation is confirmed by the corresponding changes in the oscillator strength of the AP resonances of the top and bottom layers. In the (1, 0) and (0, 1) configurations, we measure a reflectance signal either from APtop or APbot, consistent with full-layer polarization. In the (1/2, 1/2) configuration, we find the oscillator strength of APtop and APbot to be identical and equal to half the value obtained under (1, 0) for APtop. The extracted excitonic resonance energy for Xbot and Xtop around ν = 1 is shown in Fig. 5d, f, respectively. The plateau structure of the (1/2, 1/2) state with abrupt changes of electron density difference between the layers at VE = −1.3 V and VE = −0.5 V is clearly visible at ν = 1, but is smeared out for both lower (ν = 0.85) and higher fillings (ν = 1.15). The emergence of the stabilized (1/2, 1/2) plateau at ν = 1 strongly suggests that there is mutual stabilization of the incompressible electronic state due to the interlayer electron–electron interactions. The energy gap of the incompressible (1/2, 1/2) state is estimated to be 5.5 meV (Supplementary Information section 6). The reflectance data for higher fillings (ν = 3/2 and ν = 2) are shown in the Supplementary Information (Supplementary Fig. 5): in stark contrast to the (1/2, 1/2) configuration at ν = 1, a plateau at the (1, 1) electron configuration is missing at ν = 2 filling, indicating that the state with the corresponding integer fillings is not sufficiently stabilized by the interlayer interactions.

Finally, we also perform the measurement under a perpendicular magnetic field Bz = 7 T (Supplementary Information). In Supplementary Fig. 6, we find that the plateau structure observed for ν = 2 under full valley polarization of electrons is identical to that observed for Bz = 0 T, although the total number of electronic states per moiré subband is halved due to giant valley-spin susceptibility of electrons in MoSe2 (ref. 24). This observation shows that the incompressibility is determined by filling of each moiré site by a single electron, irrespective of its (valley) degeneracy. The observation supports that our identification of n = 2 × 1011 cm−2, yielding half-filling of a single-layer moiré subband. We also find that the reflectance spectrum does not show any valley polarization at Bz = 7 T for the ν ≱ 1/2 state. However, the resilience against valley polarization is not sufficient to claim antiferromagnetic order.

Discussion

The experiments we describe in the ‘Interaction-induced incompressible states’ section demonstrate the existence of Mott-like incompressible electronic states for half-filling of the lowest moiré subband. Unlike previous reports13,14,15,16, our experiments are carried out for a long moiré superlattice lattice constant of λmoiré = 24 nm and an rs parameter of rs ≳ 14. The weakness of the moiré potential stemming from the hBN layer separating the two MoSe2 layers in turn ensures that the on-site interaction strength is larger than the depth of the moiré potential. In this sense, the homobilayer system realizes a rather unique regime that goes beyond the standard Fermi–Hubbard model.

In addition to establishing twisted TMD homobilayers as a promising system for investigating Mott–Wigner physics10,29,30,31 originating from strong electronic correlations, our experiments open up new avenues for exploring interactions between dipolar excitons and electrons confined to flat bands. In particular, the structure we analysed could be used to realize and study Bose–Fermi mixtures consisting of degenerate electrons strongly interacting with an exciton condensate generated by resonant laser excitation. The phase diagram of such a mixture is not fully understood32, but is expected to provide a rich playground for many-body physics33.

Methods

Device fabrication

The device structure is shown in Extended Data Fig. 1. All MoSe2, graphene and hBN flakes were obtained by mechanical exfoliation of bulk crystals. The flakes were assembled together using the dry transfer technique34 in an argon-filled glove box. The crystal axis of top and bottom MoSe2 layers were aligned to be close to 0° using a tear-and-stack technique35. We expect to have the effect of moiré potential from a small twist angle of MoSe2 layers, but not between MoSe2 and hBN layers due to the large lattice constant mismatch (\({a}_{{{\rm{M}}{\rm{o}}{\rm{S}}{\rm{e}}}_{2}}=3.32\,\AA \) (ref. 22) and ahBN = 2.51 Å (ref. 36)). The metal electrodes to graphene layers are formed by Ti/Au (5 nm/145 nm). The contact to the bottom graphene gate is formed by Cr/Au (3 nm/147 nm) using the 1D contact technique34 by etching the hBN layer with reactive ion etching using CHF3/O2 as mixture gas.

Optical spectroscopy

The PL measurements were performed using a HeNe laser (633 nm) as an excitation source. The reflectance measurements were performed using a single-mode fibre-coupled broadband light-emitting diode with a centre wavelength of 760 nm and a bandwidth of 20 nm. In both PL and reflectance measurements, we used a long working distance apochromatic objective lens with a numerical aperture of 0.65 (attocube LT-APO/LWD/VISIR/0.65). All optical spectroscopy measurements were performed at cryogenic temperature (T ≈ 4 K). All data in the main text were obtained at zero magnetic field, Bz = 0 T.

Capacitance parameters

The thickness of the top and bottom hBN layers was estimated to be 63.3 nm and 52.7 nm, respectively, from the white light reflectance spectrum. We used 3.7 as the dielectric constant for hBN37. We expect the uncertainty to be about 10% for both hBN thickness and the dielectric constant, which gives around 14% error in the calculation of carrier density.

Estimation of r s parameter and moiré periodicity from electron density

The rs parameter (Wigner–Seitz radius divided by the effective Bohr radius) that describes the ratio between kinetic energy to Coulomb energy is given by \({r}_{{\rm{s}}}=1/{a}_{{\rm{B}}}^{\ast }\sqrt{{\rm{\pi }}n}\) (ref. 19). Here \({a}_{{\rm{B}}}^{\ast }=4{\rm{\pi }}\varepsilon {\varepsilon }_{0}{\hslash }^{2}/{m}_{{\rm{e}}}^{\ast }{e}^{2}=0.91\,{\rm{n}}{\rm{m}}\) is the effective Bohr radius in an encapsulated MoSe2 monolayer, \({m}_{{\rm{e}}}^{\ast }\) is the effective electron mass, \(\varepsilon =({\varepsilon }_{\parallel ,{{\rm{MoSe}}}_{2}}+{\varepsilon }_{\parallel ,{\rm{hBN}}})/2\) is the lattice dielectric constant19, ε0 is the vacuum permittivity, ħ is the reduced Planck constant and e is the elementary charge. For the calculation we used \({m}_{{\rm{e}}}^{* }=0.7{m}_{{\rm{e}}}\) (ref. 38), where me is the bare electron mass and the in-plane dielectric constants \({\varepsilon }_{\parallel ,{{\rm{MoSe}}}_{2}}=17.1\) and \({\varepsilon }_{\parallel ,{\rm{hBN}}}=6.93\) (ref. 37). For an electron density of n = 2 × 1011 cm−2 (ν = 1/2), we obtain rs ≈ 14, which is reduced to rs ≈ 7 at density of n = 8 × 1011 cm−2 (ν = 2). We emphasize that the above values for the rs parameter are underestimated. At these low densities, the interelectron separation, \({r}_{0}=1/\sqrt{n{\rm{\pi }}} > 10\,{\rm{n}}{\rm{m}}\), is much larger than the MoSe2 layer thickness \({d}_{{{\rm{MoSe}}}_{2}}\approx 0.7\,{\rm{nm}}\). In this limit39, the screening of the interactions is dominated by the hBN dielectric; therefore, a better approximation is to take the dielectric constant to be ε = ε∥,hBN, which results in rs ≈ 24 for an electron density of n = 2 × 1011 cm−2 (ν = 1/2) and rs ≈ 12 for n = 8 × 1011 cm−2 (ν = 2).

The moiré superlattice lattice constant λmoiré is estimated from the following relation by assuming a triangular moiré superlattice: \({A}_{{\rm{moir}}\mathop{{\rm{e}}}\limits^{ \acute{} }}=(\sqrt{3}/2){\lambda }_{{\rm{moir}}\mathop{{\rm{e}}}\limits^{ \acute{} }}^{2}=1/{n}_{{\rm{moir}}\mathop{{\rm{e}}}\limits^{ \acute{} }}\), where Amoiré is the moiré superlattice unit cell area and nmoiré is the electron density that corresponds to one electron occupation per moiré superlattice unit cell. We used nmoiré = 2 × 1011 cm−2, which is obtained from the periodicity of the differential reflectance modulation as we discussed in the main text, which gives λmoiré ≈ 24 nm. The twist angle θ between MoSe2 layers is then estimated to be θ = 0.8° using the following relation: \({\lambda }_{{\rm{moir}}\mathop{{\rm{e}}}\limits^{ \acute{} }}={a}_{{{\rm{MoSe}}}_{2}}/\theta \), where \({a}_{{{\rm{MoSe}}}_{2}}=3.32\,{\rm{\AA }}\) is the lattice constant of MoSe2 (ref. 22).

Verification of moiré periodicity estimation using interlayer exciton fine-structure analysis

Here we estimate the moiré periodicity from the fine-structure energy scale of the interlayer exciton. In the limit when the moiré potential is weak compared with the kinetic energy scale, umklapp scattering by the moiré potential is the dominant origin of the fine structure of the interlayer exciton. In this limit, the energy difference of the two lowest energy interlayer excitons is approximately described by the kinetic energy of the interlayer exciton at the momentum \({q}_{{\rm{m}}{\rm{o}}{\rm{i}}{\rm{r}}\mathop{{\rm{e}}}\limits^{ \acute{} }}=4{\rm{\pi }}/(\sqrt{3}{\lambda }_{{\rm{m}}{\rm{o}}{\rm{i}}{\rm{r}}\mathop{{\rm{e}}}\limits^{ \acute{} }})\), where λmoiré is moiré periodicity9,11. Using the energy splitting of the first- and second-lowest energy interlayer exciton ΔE1,2, λmoiré is obtained using

$$\Delta {E}_{1,2}\approx \frac{{\hbar }^{2}{q}_{{\rm{moir}}\mathop{{\rm{e}}}\limits^{ \acute{} }}^{2}}{2M}$$
(1)
$${\lambda }_{{\rm{moir}}{\rm{ \acute{{\rm{e}}} }}}\approx \frac{2h}{\sqrt{6M\Delta {E}_{1,2}}}$$
(2)

where the total mass of exciton is \(M={m}_{{\rm{e}}}^{* }+{m}_{{\rm{h}}}^{* }=1.3{m}_{e}\), \({m}_{{\rm{e}}}^{* }\) is the effective electron mass, \({m}_{{\rm{h}}}^{* }\) is the effective hole mass (\({m}_{{\rm{e}}}^{* }=0.7{m}_{{\rm{e}}}\) from ref. 38 and \({m}_{{\rm{h}}}^{* }=0.6{m}_{{\rm{e}}}\) from ref. 21) and h is the Planck constant. From Fig. 2c in the main text, the energy splitting is ΔE1,2 ≈ 4.4 meV, which corresponds to λmoiré ≈ 19 nm. At a different spot, from Supplementary Fig. 3d, the energy splitting is ΔE1,2 ≈ 3.7 meV, which corresponds to λmoiré ≈ 20 nm. Both values are qualitatively similar to what we obtained from the density periodicity of the reflectance signal (λmoiré ≈ 24 nm from Fig. 4b and λmoiré ≈ 25 nm from Supplementary Fig. 10). We note that the third-lowest energy interlayer exciton line observed in reflectance is not captured well by this simple model. In reality, the energy spacing of the interlayer excitons depends on the magnitude of moiré potential, and the analysis we present here could only yield a rather rough estimate of the moiré periodicity.

Effect of strain on moiré periodicity

A moiré superlattice emerges from a twist angle, lattice constant difference or a combination of the two. In the main text, we attributed our superlattice to be originating from this twist angle. Here we discuss how much strain-induced lattice constant modification changes the moiré periodicity. Moiré periodicity for twist angle of θ and lattice constant difference ratio of two layers δ (ref. 40), which corresponds to strain difference of two layers, is expressed as:

$${\lambda }_{{\rm{m}}{\rm{o}}{\rm{i}}{\rm{r}}\mathop{{\rm{e}}}\limits^{ \acute{} }}=\frac{{a}_{{{\rm{M}}{\rm{o}}{\rm{S}}{\rm{e}}}_{2}}(1+\delta )}{\sqrt{2(1+\delta )(1-\,\cos \theta )+{\delta }^{2}}}$$
(3)

Extended Data Fig. 2a plots the relation between the biaxial strain difference (δ) and twist angle (θ) for fixed λmoiré using equation (3) (\({a}_{{{\rm{MoSe}}}_{2}}=3.32\,{\rm{\AA }}\)). In the main text, we have shown that there is an intralayer exciton energy difference between the top and bottom layers, which is about 8 meV. By assuming this energy difference to arise from the strain difference between the two layers, we extract the amount of strain difference to be on the order of about 0.1–0.25%. We base our rough estimate on values reported in literature: uniaxial strain MoS2: 70 meV %−1 (ref. 41), 45 meV %−1 (ref. 42), 48 meV %−1 (ref. 43); biaxial strain MoSe2: 33 meV %−1 (ref. 44). For λmoiré ≈ 24 nm, the moiré periodicity within this strain range is dominated by twist angle and hardly modified by strain (only 1.4% reduction of λmoiré for 0.25% strain assuming fixed twist angle of 0.8°, as we show in Extended Data Fig. 2b). However, to introduce a moiré periodicity of about 24 nm with only strain difference, more than 1.3% of strain difference is required, which is unlikely from the magnitude of the energy splitting of the intralayer excitons. The spectrum at another spot, which we show in Supplementary Information section 2, exhibits 5 meV energy splitting of the top- and bottom-layer intralayer excitons. From this energy difference, the amount of strain difference is estimated to be in the order of about 0.07–0.15%, which results in only 0.5% of λmoiré reduction, assuming fixed twist angle of 0.8°.

First-principles calculation of the tunnel coupling and the moiré potential

We perform density functional theory (DFT) calculations using Quantum ESPRESSO45. We use projector-augmented-wave pseudopotentials with a generalized gradient approximation (Perdew–Burke–Ernzerhof functional) from PSlibrary 1.0.0 (ref. 46). To reduce computation cost, we neglect spin–orbit interaction and used non-relativistic pseudopotentials. Computations are performed with 48 cores on a high-performance computing cluster.

We take lattice structure parameters for MoSe2 from ref. 47, which gives a lattice constant \({a}_{{{\rm{MoSe}}}_{2}}=3.32\,{\rm{\AA }}\). The lattice constant of hBN given in ref. 36 is ahBN = 2.504 Å. We uniformly stretched the hBN lattice by −0.56% and set the lattice constant of hBN as \({a}_{{\rm{hBN}}}^{{\rm{stretch}}}=2.49\,{\rm{\AA }}\) to form a 3 × 3 MoSe2 and 4 × 4 hBN commensurate supercell. We take the interlayer distance of MoSe2 and hBN from ref. 48, and set the thickness of vacuum layer between MoSe2 and hBN as dSe−hBN = 3.36 Å. We use a plane-wave cutoff energy of 60 Rydberg and a charge density cutoff energy of 480 Rydberg. The Brillouin zone is sampled with a 9 × 9 × 1 k-point grid. We use a slab geometry with a 30-Å-thick vacuum layer.

Extended Data Fig. 3a–c shows the actual R-stacked (0° twist angle) MoSe2/hBN/MoSe2 supercell for different lattice displacement configurations that we use for the calculation. α = h, X, M denote hexagon centre, chalcogen and metal site of TMDs, and \({{\rm{R}}}_{{\rm{h}}}^{\alpha }\) denotes the specific lattice displacement for R-stacking where the α site of top MoSe2 is aligned with the h site of the bottom MoSe2. Extended Data Fig. 3d–f shows the calculated band structure shown with the mini-Brillouin zone. We take the vacuum level as the energy reference (0 eV). Owing to zone folding, the K and K′ points of MoSe2 come to the γ point of the mini-Brillouin zone (the K point of hBN comes to the κ point of the mini-Brillouin zone). Therefore, the lowest conduction bands and the highest valence bands at the γ point are the ones from MoSe2. As the interlayer hybridization effect is substantially smaller than the directly contacting TMD heterostructure8,9,10, these band structures look almost the same except for a slightly visible energy splitting of \({{\rm{R}}}_{{\rm{h}}}^{{\rm{h}}}\) displacement. Though the energy modulation is substantially smaller, the calculation shows about 0.5 meV modulation for the lowest conduction band and 5 meV modulation for the highest valence band dependent on the lattice displacement (Extended Data Table 1). In addition, the calculation shows 11-meV tunnel splitting of the valence band edge at \({{\rm{R}}}_{{\rm{h}}}^{{\rm{h}}}\) displacement, which is qualitatively in good agreement with experimentally observed avoided crossings of intralayer and interlayer excitons mediated by hole tunnelling. We emphasize that these results are rather qualitative, and in reality, it depends on many factors such as interlayer distance, outer dielectric environment and so on.