# Fileset

[Sokolowski_2023_2D_Mater._10_034003.pdf](https://mdr.nims.go.jp/filesets/d0cba716-61db-47e7-8091-ed3a5fcb5a7f/download)

## Creator

Nikodem Sokolowski, Swaroop Palai, Mateusz Dyksik, Katarzyna Posmyk, Michał Baranowski, Alessandro Surrente, Duncan Maude, Felix Carrascoso, Onur Cakiroglu, Estrella Sanchez, Alina Schubert, Carmen Munuera, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Joakim Hagel, Samuel Brem, Andres Castellanos-Gomez, Ermin Malic, Paulina Plochocka

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Twist-angle dependent dehybridization of momentum-indirect excitons in MoSe<sub>2</sub>/MoS<sub>2</sub> heterostructures](https://mdr.nims.go.jp/datasets/c74627ef-8c35-4e24-9d58-99d8c25f29a4)

## Fulltext

Twist-angle dependent dehybridization of momentum-indirect excitons in MoSe2/MoS2 heterostructures2D MaterialsPAPER • OPEN ACCESSTwist-angle dependent dehybridization ofmomentum-indirect excitons in MoSe2/MoS2heterostructuresTo cite this article: Nikodem Sokolowski et al 2023 2D Mater. 10 034003 View the article online for updates and enhancements.You may also likeAnalyses of the5p65d–(5p66p+5p65f+5p55d2) transitionsof eight times ionized iridium to eleventimes ionized mercury spectra (Ir IX–HgXII)S S Churilov and Y N Joshi-IX International Conference Days ofApplied MathematicsO Valbuena, E Gelvez-Almeida and E D V-Niño-IX Ophiuchi: A High-Velocity Star Near aMolecular CloudG. H. Herbig-This content was downloaded from IP address 144.213.253.16 on 21/10/2023 at 03:57https://doi.org/10.1088/2053-1583/acdbdbhttps://iopscience.iop.org/article/10.1088/0031-8949/57/5/003https://iopscience.iop.org/article/10.1088/0031-8949/57/5/003https://iopscience.iop.org/article/10.1088/0031-8949/57/5/003https://iopscience.iop.org/article/10.1088/0031-8949/57/5/003https://iopscience.iop.org/article/10.1088/0031-8949/57/5/003https://iopscience.iop.org/article/10.1088/0031-8949/57/5/003https://iopscience.iop.org/article/10.1088/0031-8949/57/5/003https://iopscience.iop.org/article/10.1088/0031-8949/57/5/003https://iopscience.iop.org/article/10.1088/0031-8949/57/5/003https://iopscience.iop.org/article/10.1088/0031-8949/57/5/003https://iopscience.iop.org/article/10.1088/1742-6596/2515/1/011001https://iopscience.iop.org/article/10.1088/1742-6596/2515/1/011001https://iopscience.iop.org/article/10.1086/431759https://iopscience.iop.org/article/10.1086/4317592D Mater. 10 (2023) 034003 https://doi.org/10.1088/2053-1583/acdbdbOPEN ACCESSRECEIVED30 March 2023REVISED15 May 2023ACCEPTED FOR PUBLICATION6 June 2023PUBLISHED16 June 2023Original Content fromthis work may be usedunder the terms of theCreative CommonsAttribution 4.0 licence.Any further distributionof this work mustmaintain attribution tothe author(s) and the titleof the work, journalcitation and DOI.PAPERTwist-angle dependent dehybridization of momentum-indirectexcitons in MoSe2/MoS2 heterostructuresNikodem Sokolowski1, Swaroop Palai1, Mateusz Dyksik2, Katarzyna Posmyk1,2, Michał Baranowski2,Alessandro Surrente2, Duncan Maude1, Felix Carrascoso3, Onur Cakiroglu3, Estrella Sanchez3,Alina Schubert3, Carmen Munuera3, Takashi Taniguchi4, Kenji Watanabe5, Joakim Hagel6,Samuel Brem7, Andres Castellanos-Gomez3, Ermin Malic6,7 and Paulina Plochocka1,2,∗1 Laboratoire National des Champs Magnétiques Intenses, UPR 3228, CNRS-UGA-UPS-INSA, Grenoble and Toulouse, France2 Department of Experimental Physics, Faculty of Fundamental Problems of Technology, Wroclaw University of Science and Technology,Wroclaw, Poland3 Materials Science Factory, Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC), Madrid E-28049, Spain4 International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Ibaraki 305-004, Japan5 Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-004, Japan6 Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden7 Department of Physics, Philipps University of Marburg, 35037 Marburg, Germany∗ Author to whom any correspondence should be addressed.E-mail: paulina.plochocka@lncmi.cnrs.frKeywords:moiré, interlayer exciton, heterostructure, transition metal dichalcogenide, photoluminescence, twist angle, MoSe2/MoS2Supplementary material for this article is available onlineAbstractThe moiré superlattice has emerged as a powerful way to tune excitonic properties in two-dimensional van der Waals structures. However, the current understanding of the influence of thetwist angle for interlayer excitons (IXs) in heterostructures is mainly limited to momentum-direct K–K transitions. In this work, we use a judicious combination of spectroscopy andmany-particle theory to investigate the influence of the twist angle on momentum-indirect IXs of aMoSe2/MoS2 heterostructure. Here, the energetically lowest state is a dark and strongly hybridizedΓK exciton. We show that increasing the twist angle from an aligned structure (0◦ or 60◦) gives riseto a large blue shift of the IX, which is a manifestation of the strong dehybridization of this state.Moreover, for small twist angle heterostructures, our photoluminescence measurements revealcontributions from two IX states, which our modelling attributes to transitions from differentmoiré minibands. Our finding contributes to a better fundamental understanding of the influenceof the moiré pattern on the hybridization of momentum-dark IX states, which may be importantfor applications in moiré-tronics including novel quantum technologies.1. IntroductionTwo-dimensional (2D) van der Waals crystals, withtheir inherent weak interlayer bonding, have enableda new paradigm of heterostructure engineering [1, 2].For 2D materials, lattice-matching constraints areno longer obstacles (in contrast to the case of epi-taxial heterostructures), while the twist angle betweenthe layers provides a convenient handle to tunetheir electronic properties, facilitating access to exoticphysics phenomena [3–13]. A prominent exampleis provided by the transition metal dichalcogenides(TMDs) homo- and heterostructures. These repres-ent a unique system in which spin, valley, excitonic,and many-body physics are heavily intertwined andinvestigated [3, 9–21]. Initially, these heterostructureswere described as simple type II quantum wells withelectrons and holes located in the adjacent TMD lay-ers. This band alignment leads to the formation of theinterlayer exciton (IX) [14,19, 22–26]. However,moredetailed investigations of this excitonic complex havequickly shown that the simple type II quantum wellpicture is not sufficient to capture all of the intriguingphysics of TMD heterostructures.Currently, it is generally accepted that the opto-electronic properties of TMD stacks are determinedby the interplay of two effects: (i) the formationof a moiré pattern [7, 8, 16–18, 20, 27–30], and© 2023 The Author(s). Published by IOP Publishing Ltdhttps://doi.org/10.1088/2053-1583/acdbdbhttps://crossmark.crossref.org/dialog/?doi=10.1088/2053-1583/acdbdb&domain=pdf&date_stamp=2023-6-16https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://orcid.org/0000-0003-4945-8795https://orcid.org/0000-0002-5974-0850https://orcid.org/0000-0003-4078-4965https://orcid.org/0000-0003-0202-4340https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-3858-4174https://orcid.org/0000-0002-3384-3405https://orcid.org/0000-0002-4019-6138mailto:paulina.plochocka@lncmi.cnrs.frhttps://doi.org/10.1088/2053-1583/acdbdb2D Mater. 10 (2023) 034003 N Sokolowski et alFigure 1. Schematic picture of electron and holes transferbetween different valleys in a MoSe2/MoS2 heterostructure.The two intralayer excitons and the hybridized interlayerexciton are indicated by red/black and green ovals,respectively.(ii) the interlayer hybridization of the states [31–36].A moiré pattern is created due to the mismatchof the TMD lattice constants and/or to the twistbetween two adjacent layers. It results in a slowly vary-ing periodic potential, which can be treated as anin-plane superlattice of quantum dots [16, 20, 28],which affects locally the optical selection rules [17,18, 37]. On the other hand, the hybridization arisesfrom the overlap of the atomic wave functions ofthe two adjacent layers. Therefore, it mostly affectsstates derived from the chalcogen atomic orbitals,such as the states around the Γ point in the Brillouinzone. This interlayer coupling is well recognized forthe TMD homobilayers (and thicker forms) as it isresponsible for their indirect band gap character [38–40]. Band structure modelling shows that a similarsituation occurs in MoSe2/MoS2 [41] and MoS2/WS2[34] heterostructures, where the hybridization loc-ates the valence band maximum at the Γ point, whilethe minimum of the conduction band remains at theK point (spatially, electrons are located in MoS2), asschematically shown in figure 1. Therefore, the lowestenergy excited state is a hybrid exciton, indirect bothin real and k-space [34]. This feature distinguishesthese heterostructures from the most often invest-igated MoSe2/WSe2 heterostructures, which are welldescribed as type II quantum wells, with the valenceand conduction bands extrema located in adjacentlayers but still at the same K points of the Brillouinzone [34, 42]. Thus, in terms of the band struc-ture, we can expect that MoSe2/MoS2 andMoS2/WS2heterostructures should be more related to TMDhomobilayers [33, 43–46], rather than to the directbandgap MoSe2/WSe2 stack, which leads to very dif-ferent optical properties.As the twist between the layers modifies theirspatial separation [47], it is natural to expect thatthe energy of the hybridized states strongly dependson the twist angle. Simultaneously, states closeto K points should be only weakly affected byinterlayer hopping (coupling), since the correspond-ing orbital functions are mostly localized on thetransition metal atoms situated in the central layerof the chalcogenide-metal-chalcogenide sandwich.These expectations are supported by band struc-ture calculations [33, 34] and corroborated by exper-imental studies of homobilayers and MoSe2/WSe2heterostructure [28, 42–46]. The characteristic fea-ture for the hybridized states (K–Γ orK–Λ) is a strongblue shift of the IX energy with increasing twist angle(when moving away from high symmetry alignment– 0◦ or 60◦) [43–46], which is absent (very weak)for direct K–K transitions [28, 42]. While so far theeffect of hybridization has been investigated mainlyin homobilayers [33, 43–46, 48], here we report onthe distinct fingerprint of the hybridized nature ofthe ground excitonic transition in MoSe2/MoS2 het-erostructures, which distinguishes it from the muchmore intensively investigated MoSe2/WSe2 stack. Weshow that the photoluminescence (PL) spectrum oftheMoSe2/MoS2 heterostructure reveals a strong blueshift of the ground excitonic transition, with increas-ing twist angle. This can be understood as a pro-cess of layer decoupling and dehybridization of thebands, which is fully confirmed by our theoreticalmodelling. We conclude that the band structure andoptical spectrum of the investigated stacks resemblethose observed for twisted homobilayers. Finally, wedemonstrate that the optical spectrum of the hybrid-ized exciton is affected by the moiré pattern. Forhighly aligned samples, we observe a double peakstructure of the hybridized IX. Our measured excit-ation power and temperature dependence, togetherwith band structure calculations, suggests that thedouble peak originates from the state filling of themoiré bands.2. Results and discussionAtomically thin flakes were obtained by mechanicalexfoliation and stacked using a deterministic transfermethod [49]. We fabricated six MoS2/MoSe2 hetero-structures with different interlayer twist angle, encap-sulatedwith hexagonal boron nitride (hBN) on a SiO2substrate (for fabrication details see section 4.1).Figure 2(a) shows an optical microscope image ofone of the heterostructures (Sample A, characterizedby a twist angle of 57.2◦. Details of Samples B–Fare in SI. 1–3). The contour of the MoSe2 and MoS2monolayers are indicated by black and red curves,and the overlap region of the TMD layers is indicatedby a green dashed line with chequered pattern. Wedetermine the interlayer twist angle between mono-layers by comparing the second harmonic generation(SHG) intensity as well as polarization-resolved SHGof the monolayers and the heterostructure, which arepresented in figure S1 [27, 50].22D Mater. 10 (2023) 034003 N Sokolowski et alFigure 2. (a) Microscope image of the MoS2/MoSe2 heterostructure encapsulated in hBN. (b) Photoluminescence and reflectivitycontrast spectra. The relevant intralayer and interlayer exciton resonances are indicated.Figure 2(b) shows representative optical spectrafrom the heterostructure area of sample A. In thereflectivity contrast spectrum (RC), both A and Bexcitons of MoS2 and MoSe2 are visible. Similarly,in the PL spectrum, we observe emission related toboth monolayers. The peak at ∼ 1.9 eV correspondsto MoS2 and probably stems from a mixture of trionand shallow defect-induced state emission as it isred-shifted by ∼ 35 meV from the A exciton reson-ance that is visible in the RC spectra [38, 51, 52]. Inthe PL spectrum of MoSe2 located between 1.5 and1.66 eV, we observe a peak corresponding to the freeXA exciton at 1.64 eV and a dominating trion peak∼30 meV below [51, 53], followed probably by somedefect state emission.At lower energies, we observe a pronounced peakat around 1.3–1.4 eV, which is only visible in the het-erostructure region. We attribute this emission to thehybridized IX [25, 34, 41], with the electron locatedat the K point of the MoS2 layer, and the hole loc-ated at the hybridized valence band at the Γ pointof the Brillouin zone [41], as schematically depictedin figure 1. The indirect character of the IX trans-ition, together with the spatial separation of elec-trons and holes, makes the oscillator strength of theIX transition too weak to be observed in the RCspectra. However, this momentum-dark state can beobserved in the PL spectrum due to its high occu-pation as the energetically lowest state. Its emissionis indirect and is driven by phonons, resulting inphonon sidebands [34]. Intriguingly, the IX PL spec-trum exhibit a double-peak structure, which is fur-ther discussed below. First, we discuss the hybridizednature of the IX exciton in the investigated hetero-structure focusing on the dominant low-energy IX PLpeak.Figure 3(a) illustrates the evolution of the PLspectrum for different twist angles. The energy ofthe PL maximum blue shifts when the twist angledeviates from the 0◦ or 60◦ stacking, which corres-ponds to R-type and H-type stacking, respectively.Fitting the PL peaks with a double Gaussian (dashedlines), we can extract the energy of the ground (dom-inating) IX1 transition as a function of the twistangle. The shift of the IX1 transition is summar-ized as open circles in figure 3(b). The blue shiftcan be as high as 100 meV moving from ∼0◦ to6− 7◦. This strong dependence of the IX emissionenergy on the twist angle is in stark contrast with thebehavior observed in MoSe2/WSe2 heterostructures[28, 42, 47], where the IX recombination stems froma momentum direct K–K transition. This can beexplained by the different character of the IX in theMoSe2/MoS2 heterostructure. The strong blue shiftof the IX transition in the MoSe2/MoS2 heterostruc-ture resembles closely the behavior of IX excitons intwisted homobilayers [33, 43–46], which points to itshybridized nature related to the states close to theΓ point.To obtain a deeper insight into the mechan-ism driving the evolution of the IX states with thetwist angle, we employ the exciton density-matrixformalism, using input fromdensity functional the-ory (DFT) calculations [8, 33, 34] (see also SI. 6). TheHamiltonian operator of excitons in twisted rigid lat-tices consists of three different contributions,H∝ E+V(Θ, r)+T(Θ, r). (1)Here, E is the exciton dispersion for the decoupledmonolayers, V(Θ, r) is the spatially periodic elec-trostatic potential and T(Θ, r) is the hybridizationHamiltonian taking into account the overlappingelectronic wave functions giving rise to hybrid excitonstates [34]. Similarly to V(Θ, r), T(Θ, r) is also spa-tially periodic as the interlayer distance varies withinthe moiré supercell. Therefore, both components32D Mater. 10 (2023) 034003 N Sokolowski et alFigure 3. (a) Photoluminescence spectra of samples withdifferent twist angles. (b) Calculated energy shift of moiréexcitons in R-type stacking taking into account onlyconfinement length (black) and total blue shift includingthe dehybridization effect (green), which agrees with theshift of the dominating IX1 transition (open circles).determine the twist-angle dependent moiré poten-tial. With the increasing twist angle, the period ofthe moiré superlattice decreases, which induces delo-calization of the IX over many moiré supercells.This results in a spectral blue shift of the excitontransitions [8, 33] (see also figure 4). However, theconfinement length increase (the wave function delo-calization) gives rise to only a moderate blue shiftof the IX transition, as shown by the black curvein figure 3(b). The most significant contribution tothe blue shift arises from the increase of the aver-age interlayer distance throughout the supercell whenincreasing the twist angle. This consequently leads toa significant dehybridization of the moiré excitonsand a major blue shift (see also SI. 6). The sumof both, moiré period and dehybridization contri-butions in R-type stacking, is represented by thegreen line in figure 3(b), which nicely reproduces theobserved blue shift. This result strongly supports theFigure 4. Calculated exciton band structure for the K−Γexciton at (a) 1◦ and (b) 5◦ twist angle obtained by solvingthe eigenvalue problem of equation (4). For the lowest twistangle (1◦), we observe multiple minibands with the flatlowest-lying band indicating at least one trapped state. Withincreasing twist angle (5◦), the exciton blue shifts, and themultiple band structure vanishes. In addition, the groundstate is no longer trapped and exhibits a nearly parabolicdispersion. The inset presents the scheme of the Brillouinzone (BZ) of isolated monolayers and the moiré Brillouinzone (MBZ). The Γ points are chosen to coincide..hybrid nature of the IX transition in theMoSe2/MoS2heterostructure.Our model also allows us to explain the double-peak structure of the PL spectrum for small twistangles. The exciton band structure calculationspresented in figure 4 show a change in the band dis-persion and in the number of bands with the twistangle. For small twist angles (1◦), our calculationspredict multiple exciton bands separated by severalto tens meV. Flat bands indicate that the exciton isspatially localizedwithin themoiré trapping potential[8], while dispersive bands correspond to the spatiallydelocalized IX.We attribute the double peak structureof IX PL spectrumobserved for small twist angles het-erostructure to the recombination of excitons relatedto the different bands. For larger twist angles, the IXexciton band blue shifts and the multiband structurevanishes. This yields a single parabolic delocalizedband, which results in a single PL peak.To support our finding, we have performedmeas-urements as a function of the excitation power.Figure 5(a) shows the normalized PL spectra (fromsample B with a 0.4◦) for different excitation powers.For the lowest excitation powers, the PL is composedof only one peak. With increasing power, an addi-tional peak on the high energy side emerges. Theintensity increases with excitation power for bothpeaks, as demonstrated in figure 5(b). To quantifythis effect, we fit the power dependence of IX1 andIX2 with a power law. The lower energy peak exhib-its a sublinear power dependence with an exponentαIX1 = 0.71, characteristic of a trapping potential(with a low density of states), which experiencesa gradual saturation (state filling) with increasingexcitation power [14, 37, 54, 55]. The increase of the42D Mater. 10 (2023) 034003 N Sokolowski et alFigure 5. (a) Photoluminescence spectra and (b) IX1 andIX2 integrated intensities as a function of the excitationpower. The lines are fitted using a power function where αis the exponent.high energy peak (IX2) with the excitation power isslightly superlinear (αIX2 = 1.17), which may pointto the delocalized exciton character of this trans-ition or to a much higher density of states. Therefore,the power-dependent measurements corroborate theassignment of the double-peak structure to a moiréinduced multiband structure of the IX exciton forheterostructures with a small twist angle. In addition,both IX transitions exhibit a blue shift with increas-ing excitation power (figure S3), which can be attrib-uted to the repulsive dipolar interactions betweenIXs caused by their permanent out-of-plane dipolemoments [56, 57]. To further support our claim,in figure S4 we present the temperature-dependentPL measurements performed on the same sample.With increasing temperature we observe that theIX2-related emission quenches faster as comparedto IX1. This observation is consistent with the power-dependent measurements and support the strongerlocalization of the IX1 transition as comparedwith IX2.3. ConclusionCombining experiment and theory, we have shownthat the ground exciton state in the MoSe2/MoS2heterostructure is a momentum dark and stronglyhybridized interlayer ΓK exciton state. Its proper-ties are determined by the combined effect of themoiré potential and the hybridization of MoS2 andMoSe2 valence bands around the Γ point. We observea strong blue shift of the K–Γ transition with increas-ing twist angle. This can be explained by the dehybrid-ization of the exciton when the twist angle movesaway from 0◦ or 60◦. This behavior resembles twistedhomobilayers and distinguishes MoSe2/MoS2 fromthe most intensively investigated MoSe2/WSe2 het-erostructure.We also show that themultiple peaks weobserve in the PL spectrum of MoSe2/MoS2 hetero-structures with a small twist results from the moirépattern-driven exciton miniband formation.4. Methods4.1. Samples fabricationsTMD monolayers and hBN flakes were obtained bythe mechanical exfoliation technique. The TMDs andpart of hBN used in the fabrication are commer-cially available. Synthetic MoSe2 grown by chem-ical vapor transport has been purchased from HQgraphene. Natural MoS2 from Molly Hill mine,Québec, Canada, hBN for samples E and F is providedfrom Japan. For the remaining samples, hBN waspurchased from HQ Graphene. For all the exfoli-ations, we used Nitto tape (Nitto Denko corp. SPV224). Monolayer thickness of MoS2 and MoSe2 wasconfirmed by transmittance and reflection measure-ments before their transfer [58]. The heterostructureswere stacked by dry pick-up method [49, 59, 60] anddeposited on SiO2 substrates.4.2. Spectroscopy measurementsTo perform spectroscopy measurements, the sampleswere mounted on the cold finger of a helium flowcryostat. All of the measurements were performed ata temperature of T= 5K unless otherwise specified.The excitation laser was focused and the PL was col-lected by a 50×microscope objective (Mitutoyo Inc.)having a numerical aperture of 0.55. The resultingspot size had a diameter of approximately ≃ 1µm.For PL measurements, the excitation was providedby a continuous-wave frequency-doubled solid-statelaser emitting at 532 nm. A fs-pulsed Ti:Sapphirelaser with an average power of 15 mW was usedfor SHG measurements. Additionally, for polariza-tion resolved SHG, was polarized bymeans of a Glan–Thompson polarizer and an achromatic half-waveplate. The polarization state of the second harmonicsignal was controlled by making use of the same half-wave plate and was analyzed by a linear polarizer.The reflectance, PL and SHG signals were spectrallyresolved by a 30 cm long monochromator equippedwith a 150 grooves mm−1 grating and detected by aliquid nitrogen cooled CCD camera.52D Mater. 10 (2023) 034003 N Sokolowski et al4.3. TheoryIn order to obtain access to the moiré exciton energylandscape we consider a Hamiltonian formulated insecond quantization. For this purpose, we start ina decoupled monolayer basis and take into accountthe moiré potential as periodic modifications to thedecoupled exciton energies [8, 33, 61]. Importantly,in the rigid lattice case we have two components of themoiré potential [48, 61]: the electrostatic alignmentshift [8] and interlayer hybridization [33, 34, 61]. Thedecoupled exciton energies are obtained by solvingtheWannier equation [22], which gives us the bindingenergies for the intra/IX states. This allows us to writethe Hamiltonian in exciton basis as [8, 30, 33, 34, 61]H0 =∑LQξEξLQXξ†L,QXξL,Q+∑LQξ,gVξL(g)Xξ†L,Q+gXξL,Q+∑LL ′,QξgTξLL ′(g)Xξ†L,Q+gXξL ′,Q+ h.c (2)with L= (le, lh) as a compound layer index, Q asthe center-of-mass momentum, ξ = (ξe, ξh) as theexciton valley index and g= G2 −G1 as the recip-rocal lattice vectors of the rigid superlattice (given bythe difference of the reciprocal lattice vectors of thetwo different layers). Additionally, X(†) are annihila-tion (creation) operators for the non-hybrid excitons.Here, EξLQ is the non-hybridized exciton dispersionthat is calculated from the Wannier equation [22].Furthermore, VξL(g) is the periodic electrostatic shiftof the moiré excitons, determined by the local atomicalignment [8]. In this work, the predominant com-ponent of the moiré potential stems from the excitonhybridization that is described by the tunneling termin the Hamiltonian reading [33, 34]TξLL ′(g) =[δlh,l ′h (1− δle,l ′e )tcξelel ′e(g)FξLL ′(βLL ′g)− δle,l ′e (1− δlh,l ′h )tvξhlhl ′h(g)F∗ξLL ′(−αLL ′g)].(3)Here, FξLL ′(q) =∑kΨξ∗L (k)ΨξL ′(k+ q) are theexciton form factors. Furthermore, we have intro-duced αij(βij) =mc(v)i( j)/(mci +mvj ) with the massesextracted from [62]. The Kronecka deltas ensuresingle carrier tunneling processes. Furthermore,tλξλlλl ′λ(g) are the Fourier coefficients of the real-spacetunneling potential, where λ= (c,v) is the bandindex. The Fourier coefficients take into account thetwist-angle dependence of the tunneling strength (cf.figure S5).We transform equation (2) to a zone-foldedhybrid moiré exciton basis [8, 33, 61], Y†ξηQ =∑gL Cξη∗Lg (Q)Xξ†L,Q+g, where Q is now restricted to thefirst mini-Brillouin zone. Here, η is the new excitonband index, Cξη∗Lg (Q) are the mixing coefficientsdetermining the relative mixing between sub-bandsand intra/IXs. Moreover, Y†ξηQ is the zone-foldedmoiré exciton creation operator. Consequently, weobtain the following eigenvalue problemEξLQ(g)CξηLg (Q)+∑g ′VξL(g′ − g)CξηLg ′(Q)+∑L ′g ′TξLL ′(g ′ − g)CξηL ′g ′(Q) = EξηQCξηLg (Q). (4)Solving equation (4) numerically gives a microscopicaccess to the final hybrid moiré exciton energies EξηQ.Data availability statementAll data that support the findings of this study areincluded within the article (and any supplementaryfiles).AcknowledgmentsThis work received funding from the EuropeanUnion’s Horizon 2020 research and innova-tion program under Grant Agreements 956813(2Exciting) and 755655 (ERC-St G 2017 project 2D-TOPSENSE). M B acknowledges National ScienceCentre Poland within the SONATA BIS program(Grant No. 2020/38/E/ST3/00194) and OPUS LAP(2021/43/I/ST3/01357). Funding was also receivedfrom the Ministry of Science and Innovation(Spain) through the Project PID2020-115566RB-I00 and the EU FLAG-ERA project ‘To2Dox’ underthe program PCI2019-111893-2. This study hasbeen partially supported through the EUR GrantNanoX no ANR-17-EURE-0009 in the frameworkof the ‘Programme des Investissements d’Avenir’.M D acknowledges the support from the PolishNational Agency for Academic Exchange (Grant No.BPN/BKK/2021/1/00002/U/00001). E M acknow-ledges support from the European Unions Horizon2020 research and innovation programme underGrant Agreement No. 881603 (Graphene Flagship)as well as Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) via SFB 1083 (ProjectB9) and DFG Project 504846924. K W and T Tacknowledge support from JSPS KAKENH I (GrantNos. 19H05790, 20H00354 and 21H05233).ORCID iDsMateusz Dyksik https://orcid.org/0000-0003-4945-8795Michał Baranowski https://orcid.org/0000-0002-5974-0850Alessandro Surrente https://orcid.org/0000-0003-4078-49656https://orcid.org/0000-0003-4945-8795https://orcid.org/0000-0003-4945-8795https://orcid.org/0000-0003-4945-8795https://orcid.org/0000-0002-5974-0850https://orcid.org/0000-0002-5974-0850https://orcid.org/0000-0002-5974-0850https://orcid.org/0000-0003-4078-4965https://orcid.org/0000-0003-4078-4965https://orcid.org/0000-0003-4078-49652D Mater. 10 (2023) 034003 N Sokolowski et alEstrella Sanchez https://orcid.org/0000-0003-0202-4340Kenji Watanabe https://orcid.org/0000-0003-3701-8119Joakim Hagel https://orcid.org/0000-0002-3858-4174Andres Castellanos-Gomezhttps://orcid.org/0000-0002-3384-3405Paulina Plochocka https://orcid.org/0000-0002-4019-6138References[1] Geim A K and Grigorieva I V 2013 Van der Waalsheterostructures Nature 499 419–25[2] Novoselov K, Mishchenko A, Carvalho A and Castro Neto A2016 2D materials and Van der Waals heterostructuresScience 353 aac9439[3] Mueller T and Malic E 2018 Exciton physics and deviceapplication of two-dimensional transition metaldichalcogenide semiconductors npj 2D Mater. Appl. 2 29[4] Jorio A 2022 Twistronics and the small-angle magic Nat.Mater. 21 844–5[5] Dean C R et al 2013 Hofstadter’s butterfly and the fractalquantum hall effect in moiré superlattices Nature497 598–602[6] Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras Eand Jarillo-Herrero P 2018 Unconventionalsuperconductivity in magic-angle graphene superlatticesNature 556 43–50[7] Alexeev E M et al 2019 Resonantly hybridized excitons inmoiré superlattices in Van der Waals heterostructures Nature567 81–86[8] Brem S, Linderälv C, Erhart P and Malic E 2020 Tunablephases of moiré excitons in Van der Waals heterostructuresNano Lett. 20 8534–40[9] Tang Y et al 2020 Simulation of Hubbard model physics inWSe2/WS2 moiré superlattices Nature 579 353–8[10] Regan E C et al 2020 Mott and generalized Wigner crystalstates in WSe2/WSe2 moiré superlattices Nature 579 359–63[11] Xu Y, Liu S, Rhodes D A, Watanabe K, Taniguchi T, Hone J,Elser V, Mak K F and Shan J 2020 Correlated insulating statesat fractional fillings of moiré superlattices Nature 587 214–8[12] Huang X et al 2021 Correlated insulating states at fractionalfillings of the WSe2/WSe2 moiré lattice Nat. Phys. 17 715–9[13] Perea-Causin R, Erkensten D, Fitzgerald J M, Thompson J J,Rosati R, Brem S and Malic E 2022 Exciton optics, dynamicsand transport in atomically thin semiconductors APL Mater.10 100701[14] Rivera P et al 2015 Observation of long-lived interlayerexcitons in monolayer MoSe2–WSe2 heterostructures Nat.Commun. 6 6242[15] Rivera P, Yu H, Seyler K L, Wilson N P, Yao W and Xu X 2018Interlayer valley excitons in heterobilayers of transition metaldichalcogenides Nat. Nanotechnol. 13 1004–15[16] Yu H, Liu G-B, Tang J, Xu X and Yao W 2017 Moiré excitons:from programmable quantum emitter arrays tospin-orbit–coupled artificial lattices Sci. Adv. 3 e1701696[17] Wu F, Lovorn T and MacDonald A 2018 Theory of opticalabsorption by interlayer excitons in transition metaldichalcogenide heterobilayers Phys. Rev. B 97 035306[18] Wu F, Lovorn T and MacDonald A H 2017 Topologicalexciton bands in moiré heterojunctions Phys. Rev. Lett.118 147401[19] Rivera P, Seyler K L, Yu H, Schaibley J R, Yan J, Mandrus D G,Yao W and Xu X 2016 Valley-polarized exciton dynamics in a2D semiconductor heterostructure Science 351 688–91[20] Seyler K L, Rivera P, Yu H, Wilson N P, Ray E L,Mandrus D G, Yan J, Yao W and Xu X 2019 Signatures ofmoiré-trapped valley excitons in MoSe2/WSe2 heterobilayersNature 567 66–70[21] Smoleński T et al 2021 Signatures of Wigner crystal ofelectrons in a monolayer semiconductor Nature 595 53–57[22] Ovesen S, Brem S, Linderälv C, Kuisma M, Korn T, Erhart P,Selig M and Malic E 2019 Interlayer exciton dynamics in Vander Waals heterostructures Commun. Phys. 2 23[23] Kang J, Tongay S, Zhou J, Li J and Wu J 2013 Band offsetsand heterostructures of two-dimensional semiconductorsAppl. Phys. Lett. 102 012111[24] Miller B, Steinhoff A, Pano B, Klein J, Jahnke F, Holleitner Aand Wurstbauer U 2017 Long-lived direct and indirectinterlayer excitons in Van der Waals heterostructures NanoLett. 17 5229–37[25] Baranowski M et al 2017 Probing the interlayer excitonphysics in a MoS2/MoSe2/MoS2 Van der Waalsheterostructure Nano Lett. 17 6360–5[26] Merkl P et al 2019 Ultrafast transition between excitonphases in Van der Waals heterostructures Nat. Mater.18 691–6[27] Zhang N, Surrente A, Baranowski M, Maude D K, Gant P,Castellanos-Gomez A and Plochocka P 2018 Moiréintralayer excitons in a MoSe2/MoS2 heterostructure NanoLett. 18 7651–7[28] Tran K et al 2019 Evidence for moiré excitons in Van derWaals heterostructures Nature 567 71–75[29] Jin C et al 2019 Observation of moiré excitons inWSe2/WSe2 heterostructure superlattices Nature567 76–80[30] Schmitt D et al 2022 Formation of moiré interlayer excitonsin space and time Nature 608 499–503[31] Wilson N R et al 2017 Determination of band offsets,hybridization and exciton binding in 2D semiconductorheterostructures Sci. Adv. 3 e1601832[32] Fang H et al 2014 Strong interlayer coupling in Van derWaals heterostructures built from single-layer chalcogenidesProc. Natl Acad. Sci. 111 6198–202[33] Brem S, Lin K-Q, Gillen R, Bauer J M, Maultzsch J,Lupton J M and Malic E 2020 Hybridized intervalley moiréexcitons and flat bands in twisted WSe2 bilayers Nanoscale12 11088–94[34] Hagel J, Brem S, Linderälv C, Erhart P and Malic E 2021Exciton landscape in Van der Waals heterostructures Phys.Rev. Res. 3 043217[35] Merkl P et al 2020 Twist-tailoring coulomb correlations inVan der Waals homobilayers Nat. Commun. 11 2167[36] Tagarelli F et al 2023 Electrical control of hybrid excitontransport in a Van der Waals heterostructure Nat. Photon.1–7[37] Tan Q, Rasmita A, Zhang Z, Novoselov K and Gao W-B 2022Signature of cascade transitions between interlayer excitonsin a moiré superlattice Phys. Rev. Lett. 129 247401[38] Splendiani A, Sun L, Zhang Y, Li T, Kim J, Chim C-Y, Galli Gand Wang F 2010 Emerging photoluminescence inmonolayer MoS2 Nano Lett. 10 1271–5[39] Kuc A, Zibouche N and Heine T 2011 Influence of quantumconfinement on the electronic structure of the transitionmetal sulfide T S2 Phys. Rev. B 83 245213[40] Raja A et al 2018 Enhancement of exciton–phonon scatteringfrom monolayer to bilayer WSe2 Nano Lett. 18 6135–43[41] Su X, Ju W, Zhang R, Guo C, Zheng J, Yong Y and Li X 2016Bandgap engineering of MoS2/MX2 (MX2 =WS2, MoSe2and WSe2) heterobilayers subjected to biaxial strain andnormal compressive strain RSC Adv. 6 18319–25[42] Choi J et al 2021 Twist angle-dependent interlayer excitonlifetimes in Van der Waals heterostructures Phys. Rev. Lett.126 047401[43] Villafañe V, Kremser M, Hübner R, Petríc M M,Wilson N P, Stier A V, Müller K, Florian M, Steinhoff A andFinley J J 2023 Twist-dependent intra-and interlayerexcitons in moiré MoSe2 homobilayers Phys. Rev. Lett.130 0269017https://orcid.org/0000-0003-0202-4340https://orcid.org/0000-0003-0202-4340https://orcid.org/0000-0003-0202-4340https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-3858-4174https://orcid.org/0000-0002-3858-4174https://orcid.org/0000-0002-3858-4174https://orcid.org/0000-0002-3384-3405https://orcid.org/0000-0002-3384-3405https://orcid.org/0000-0002-4019-6138https://orcid.org/0000-0002-4019-6138https://orcid.org/0000-0002-4019-6138https://doi.org/10.1038/nature12385https://doi.org/10.1038/nature12385https://doi.org/10.1126/science.aac9439https://doi.org/10.1126/science.aac9439https://doi.org/10.1038/s41699-018-0074-2https://doi.org/10.1038/s41699-018-0074-2https://doi.org/10.1038/s41563-022-01290-6https://doi.org/10.1038/s41563-022-01290-6https://doi.org/10.1038/nature12186https://doi.org/10.1038/nature12186https://doi.org/10.1038/nature26160https://doi.org/10.1038/nature26160https://doi.org/10.1038/s41586-019-0986-9https://doi.org/10.1038/s41586-019-0986-9https://doi.org/10.1021/acs.nanolett.0c03019https://doi.org/10.1021/acs.nanolett.0c03019https://doi.org/10.1038/s41586-020-2085-3https://doi.org/10.1038/s41586-020-2085-3https://doi.org/10.1038/s41586-020-2092-4https://doi.org/10.1038/s41586-020-2092-4https://doi.org/10.1038/s41586-020-2868-6https://doi.org/10.1038/s41586-020-2868-6https://doi.org/10.1038/s41567-021-01171-whttps://doi.org/10.1038/s41567-021-01171-whttps://doi.org/10.1063/5.0107665https://doi.org/10.1063/5.0107665https://doi.org/10.1038/ncomms7242https://doi.org/10.1038/ncomms7242https://doi.org/10.1038/s41565-018-0193-0https://doi.org/10.1038/s41565-018-0193-0https://doi.org/10.1126/sciadv.1701696https://doi.org/10.1126/sciadv.1701696https://doi.org/10.1103/PhysRevB.97.035306https://doi.org/10.1103/PhysRevB.97.035306https://doi.org/10.1103/PhysRevLett.118.147401https://doi.org/10.1103/PhysRevLett.118.147401https://doi.org/10.1126/science.aac7820https://doi.org/10.1126/science.aac7820https://doi.org/10.1038/s41586-019-0957-1https://doi.org/10.1038/s41586-019-0957-1https://doi.org/10.1038/s41586-021-03590-4https://doi.org/10.1038/s41586-021-03590-4https://doi.org/10.1038/s42005-019-0122-zhttps://doi.org/10.1038/s42005-019-0122-zhttps://doi.org/10.1063/1.4774090https://doi.org/10.1063/1.4774090https://doi.org/10.1021/acs.nanolett.7b01304https://doi.org/10.1021/acs.nanolett.7b01304https://doi.org/10.1021/acs.nanolett.7b03184https://doi.org/10.1021/acs.nanolett.7b03184https://doi.org/10.1038/s41563-019-0337-0https://doi.org/10.1038/s41563-019-0337-0https://doi.org/10.1021/acs.nanolett.8b03266https://doi.org/10.1021/acs.nanolett.8b03266https://doi.org/10.1038/s41586-019-0975-zhttps://doi.org/10.1038/s41586-019-0975-zhttps://doi.org/10.1038/s41586-019-0976-yhttps://doi.org/10.1038/s41586-019-0976-yhttps://doi.org/10.1038/s41586-022-04977-7https://doi.org/10.1038/s41586-022-04977-7https://doi.org/10.1126/sciadv.1601832https://doi.org/10.1126/sciadv.1601832https://doi.org/10.1073/pnas.1405435111https://doi.org/10.1073/pnas.1405435111https://doi.org/10.1039/D0NR02160Ahttps://doi.org/10.1039/D0NR02160Ahttps://doi.org/10.1103/PhysRevResearch.3.043217https://doi.org/10.1103/PhysRevResearch.3.043217https://doi.org/10.1038/s41467-020-16069-zhttps://doi.org/10.1038/s41467-020-16069-zhttps://doi.org/10.1038/s41566-023-01198-whttps://doi.org/10.1103/PhysRevLett.129.247401https://doi.org/10.1103/PhysRevLett.129.247401https://doi.org/10.1021/nl903868whttps://doi.org/10.1021/nl903868whttps://doi.org/10.1103/PhysRevB.83.245213https://doi.org/10.1103/PhysRevB.83.245213https://doi.org/10.1021/acs.nanolett.8b01793https://doi.org/10.1021/acs.nanolett.8b01793https://doi.org/10.1039/C5RA27871Fhttps://doi.org/10.1039/C5RA27871Fhttps://doi.org/10.1103/PhysRevLett.126.047401https://doi.org/10.1103/PhysRevLett.126.047401https://doi.org/10.1103/PhysRevLett.130.026901https://doi.org/10.1103/PhysRevLett.130.0269012D Mater. 10 (2023) 034003 N Sokolowski et al[44] Yan W, Meng L, Meng Z, Weng Y, Kang L and Li X-A 2019Probing angle-dependent interlayer coupling in twistedbilayer WS2 J. Phys. Chem. C 123 30684–8[45] Van Der Zande A M et al 2014 Tailoring the electronicstructure in bilayer molybdenum disulfide via interlayertwist Nano Lett. 14 3869–75[46] Liu K, Zhang L, Cao T, Jin C, Qiu D, Zhou Q, Zettl A, Yang P,Louie S G and Wang F 2014 Evolution of interlayer couplingin twisted molybdenum disulfide bilayers Nat. Commun.5 4966[47] Nayak P K et al 2017 Probing evolution oftwist-angle-dependent interlayer excitons in MoSe2/WSe2Van der Waals heterostructures ACS Nano 11 4041–50[48] Linderälv C, Hagel J, Brem S, Malic E and Erhart P 2022 Themoiré potential in twisted transition metal dichalcogenidebilayers (arXiv:2205.15616)[49] Castellanos-Gomez A, Buscema M, Molenaar R, Singh V,Janssen L, Van Der Zant H S and Steele G A 2014Deterministic transfer of two-dimensional materials byall-dry viscoelastic stamping 2D Mater. 1 011002[50] Hsu W-T, Zhao Z-A, Li L-J, Chen C-H, Chiu M-H,Chang P-S, Chou Y-C and Chang W-H 2014 Secondharmonic generation from artificially stacked transitionmetal dichalcogenide twisted bilayers ACS Nano8 2951–8[51] Cadiz F et al 2017 Excitonic linewidth approaching thehomogeneous limit in MoS2-based Van der Waalsheterostructures Phys. Rev. X 7 021026[52] Mak K F, He K, Shan J and Heinz T F 2012 Control of valleypolarization in monolayer MoS2 by optical helicity Nat.Nanotechnol. 7 494–8[53] Ross J S et al 2013 Electrical control of neutral and chargedexcitons in a monolayer semiconductor Nat. Commun.4 1474[54] Kremser M, Brotons-Gisbert M, Knörzer J, Gückelhorn J,Meyer M, Barbone M, Stier A V, Gerardot B D, Müller K andFinley J J 2020 Discrete interactions between a few interlayerexcitons trapped at a MoSe2–WSe2 heterointerface npj 2DMater. Appl. 4 8[55] Li W, Lu X, Wu J and Srivastava A 2021 Optical control ofthe valley Zeeman effect through many-exciton interactionsNat. Nanotechnol. 16 148–52[56] Nagler P et al 2017 Interlayer exciton dynamics in adichalcogenide monolayer heterostructure 2D Mater.4 025112[57] Brotons-Gisbert M, Baek H, Campbell A, Watanabe K,Taniguchi T and Gerardot B D 2021 Moiré-trappedinterlayer trions in a charge-tunable WSe2/MoSe2heterobilayer Phys. Rev. X 11 031033[58] Frisenda R et al 2017 Micro-reflectance and transmittancespectroscopy: a versatile and powerful tool to characterize2D materials J. Phys. D: Appl. Phys. 50 074002[59] Rebollo I, Rodrigues-Machado F, Wright W, Melin G andChampagne A 2021 Thin-suspended 2D materials: facile,versatile and deterministic transfer assembly 2D Mater.8 035028[60] Haley K L, Cloninger J A, Cerminara K, Sterbentz R M,Taniguchi T, Watanabe K and Island J O 2021 Heatedassembly and transfer of Van der Waals heterostructureswith common nail polish Nanomanufacturing1 49–56[61] Hagel J, Brem S and Malic E 2022 Electrical tuningof moiré excitons in MoSe2 bilayers 2D Mater.10 014013[62] Kormányos A, Burkard G, Gmitra M, Fabian J, Zólyomi V,Drummond N D and Fal’ko V 2015 k·p theory fortwo-dimensional transition metal dichalcogenidesemiconductors 2D Mater. 2 0220018https://doi.org/10.1021/acs.jpcc.9b08602https://doi.org/10.1021/acs.jpcc.9b08602https://doi.org/10.1021/nl501077mhttps://doi.org/10.1021/nl501077mhttps://doi.org/10.1038/ncomms5966https://doi.org/10.1038/ncomms5966https://doi.org/10.1021/acsnano.7b00640https://doi.org/10.1021/acsnano.7b00640https://arxiv.org/abs/2205.15616https://doi.org/10.1088/2053-1583/1/1/011002https://doi.org/10.1088/2053-1583/1/1/011002https://doi.org/10.1021/nn500228rhttps://doi.org/10.1021/nn500228rhttps://doi.org/10.1103/PhysRevX.7.021026https://doi.org/10.1103/PhysRevX.7.021026https://doi.org/10.1038/nnano.2012.96https://doi.org/10.1038/nnano.2012.96https://doi.org/10.1038/ncomms2498https://doi.org/10.1038/ncomms2498https://doi.org/10.1038/s41699-020-0141-3https://doi.org/10.1038/s41699-020-0141-3https://doi.org/10.1038/s41565-020-00804-0https://doi.org/10.1038/s41565-020-00804-0https://doi.org/10.1088/2053-1583/aa7352https://doi.org/10.1088/2053-1583/aa7352https://doi.org/10.1103/PhysRevX.11.031033https://doi.org/10.1103/PhysRevX.11.031033https://doi.org/10.1088/1361-6463/aa5256https://doi.org/10.1088/1361-6463/aa5256https://doi.org/10.1088/2053-1583/abf98chttps://doi.org/10.1088/2053-1583/abf98chttps://doi.org/10.3390/nanomanufacturing1010005https://doi.org/10.3390/nanomanufacturing1010005https://doi.org/10.1088/2053-1583/aca916https://doi.org/10.1088/2053-1583/aca916https://doi.org/10.1088/2053-1583/2/2/022001https://doi.org/10.1088/2053-1583/2/2/022001 Twist-angle dependent dehybridization of momentum-indirect excitons in MoSe2/MoS2 heterostructures 1. Introduction 2. Results and discussion 3. Conclusion 4. Methods 4.1. Samples fabrications 4.2. Spectroscopy measurements 4.3. Theory References