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Johannes Figueiredo, Marten Richter, Mirco Troue, Jonas Kiemle, Hendrik Lambers, Torsten Stiehm, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Ursula Wurstbauer, Andreas Knorr, Alexander W. Holleitner

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[Laterally extended states of interlayer excitons in reconstructed MoSe2/WSe2 heterostructures](https://mdr.nims.go.jp/datasets/4d451784-28b2-4511-bd26-328c3f6b2109)

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Laterally extended states of interlayer excitons in reconstructed MoSe2/WSe2 heterostructuresnpj | quantummaterials ArticlePublished in partnership with Nanjing Universityhttps://doi.org/10.1038/s41535-025-00820-0Laterally extended states of interlayerexcitons in reconstructed MoSe2/WSe2heterostructuresCheck for updatesJohannes Figueiredo1,2, Marten Richter3 , Mirco Troue1,2, Jonas Kiemle1,2, Hendrik Lambers4,Torsten Stiehm4, Takashi Taniguchi5, Kenji Watanabe6, Ursula Wurstbauer4, Andreas Knorr3 &Alexander W. Holleitner1,2Heterostructures made from 2D transition-metal dichalcogenides are known as ideal platforms toexplore excitonic phenomena ranging from correlatedmoiré excitons to degenerate interlayer excitonensembles. So far, it is assumed that the atomic reconstruction appearing in some of theheterostructures gives rise to a dominating localization of the exciton states.We demonstrate that thecenter-of-mass wavefunction of the excitonic states in reconstructed MoSe2/WSe2 heterostructurescan extend well beyond the moiré periodicity of the investigated heterostructures. The results arebased on real-space calculations yielding a lateral potential map for interlayer excitons within thestrain-relaxed heterostructureswith weak randomdisorder, as expected for realistic samples, and thecorresponding real-space center-of-mass excitonic wavefunctions. We combine the theoreticalresults with cryogenic photoluminescence experiments, which support the computed level structureand relaxation characteristics of the interlayer excitons.Monolayers of group-VI transition-metal dichalcogenides (TMDs) haverisen to prominence in solid-state research for over a decade due to theirunique light-matter interactions1–3. The optical response of TMDs and theirvan derWaals heterostructures is dominated by Coulomb-bound electron-hole pairs, known as excitons, that can be tuned by dielectric engineering,doping and heterostructuring and are subjected to rich interaction aswell asspin- and multivalley physics4–9. A prominent example are interlayer exci-tons (IXs), which particularly form across the interface of a type-II bandalignment such as in MoSe2/WSe2 heterostructures8–13. The correspondingspatial separation of the electron and hole in adjacent layers gives rise to anincreased out-of-plane electric dipole and a reduction of their wavefunc-tion’s spatial overlap. The latter significantly enhances the radiative lifetimesof the IXs to the order of tens to hundreds of nanoseconds8,9. Moreover,lateral moiré superlattices emerge in lattice-mismatched and/or twistedTMD hetero-bilayers and -trilayers14–17. At small twist angles and/or smalllattice mismatches, reconstructions can lead to the formation of lateraldomains with a rather constant atomic registry and a correspondingpotential landscape for electrons and holes, which typically results instrongly localized excitons16–20.While the long lifetimes and theout-of-planeelectric dipoleposition IXs aspromising candidates for studyingmany-bodyphenomena across a large range of density regimes21, many findings areassumed to be limited by localization effects7,19,22–26.The present work aims to explain the impact of reconstruction on theexcitonic wavefunction in real space. A particular question is whetherspatially extended IX states can evolve in reconstructed heterostructures atcertain twist angles. Generally, the amplitude of the periodic potentiallandscape changes as a function of the twist angle. Recent studies suggestmuch lower energy amplitudes in the potential landscape for IXs in H-typeMoSe2/WSe2 heterostructures (twist angle close to 60°) than in R-type ones(twist angle close to 0°)27. We combine theoretical calculations of excitonicstates in reconstructed MoSe2/WSe2 heterostructures close to 60° withexperimental photoluminescence measurements on correspondinglydesigned samples. The calculations yield a two-dimensional potentiallandscape for the IXs within the relaxed heterostructures and a real-spacerepresentation of the IX center-of-mass wavefunctions of electrons andholes, as well as theoretical linear absorption and photoluminescence1Walter Schottky Institute andPhysics Department, Technical University ofMunich, AmCoulombwall 4a, Garching, 85748, Germany. 2MunichCenter for QuantumScience and Technology (MCQST), Schellingstr. 4, Munich, 80799, Germany. 3Institut für Physik und Astronomie, Nichtlineare Optik und Quantenelektronik,Technische Universität Berlin, Hardenbergstr. 36, EW 7-1, Berlin, 10623, Germany. 4Institute of Physics and Center for Soft Nanoscience (SoN), University ofMünster, Wilhelm-Klemm-Str. 10, Münster, 48149, Germany. 5International Center for Materials Nanoarchitectonics, National Institute for Materials Science,Tsukuba, 305-0044, Japan. 6Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, 305-0044, Japan.e-mail: marten.richter@tu-berlin.de; andreas.knorr@tu-berlin.de; holleitner@wsi.tum.denpj Quantum Materials |           (2025) 10:96 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41535-025-00820-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41535-025-00820-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41535-025-00820-0&domain=pdfmailto:marten.richter@tu-berlin.demailto:andreas.knorr@tu-berlin.demailto:holleitner@wsi.tum.dewww.nature.com/npjquantmatsspectra of the IXs in the energy and time domains. The theory is formulatedin real space, covering several moiré unit cells to simulate a spatially inho-mogeneous exciton distribution. Most importantly, the lateral extension ofthe wavefunction of the first IX eigenstates suggests delocalized excitonsover more than a hundred nanometers, particularly for twist angles largerthan 1.3° with respect to 60°. Therefore, the delocalized IXs extend wellbeyond the moiré periodicity and are impacted by potential fluctuationswithin the two-dimensional plane.Moreover, the energetically lowest stateslocated in different moiré unit cells form a single luminescence peak, whichparticularly dominates the calculated photoluminescence in the quasi-equilibrium regime. At smaller twist angles with respect to 60°, the calcu-lated depth of the exciton potential increases, leading to a distribution ofclearly localized exciton states, as it is consistent with earlier work28,29.Moreover, depending on the twist angle, the type of localized states coversquantum dot-like states to networks of quantum wire-like states, again inagreementwith literature7.A relatively small twist angle variation canutterlychange the nature of the exciton states, forming the described resonances inphotoluminescence for small twist angles close to 60°. The calculations areconsistent with the presented photoluminescence experiments performedon several samples. We present low-power and time-resolved photo-luminescence spectra of IXs at cryogenic temperatures, suggesting that theinvestigated exciton ensembles are in a dilute regime. The temporal evo-lution of the photoluminescence after excitation shows higher IX lumi-nescence peaks with a distinct spectrum, which is consistent with thetheoretical results. The overall observed Lorentzian-type lineshape supportsthe interpretation of a spatially extended IX ground state within the plane ofthe heterostructure for twist angles close to 60°.ResultsExperimental photoluminescenceWe experimentally investigate three different samples with similar pho-toluminescence properties, all of which consist of an hBN-encapsulatedMoSe2/WSe2 H-type heterostructure at a twist angle close to 60°, asdetermined independently by magneto-photoluminescence measure-ments (cf. Figs. S1 and S2). For the rest of the manuscript, all twist anglesmentioned are given as a relative deviation from the ideal 60° of a typicalH-type heterostructure. Figure 1a depicts a scheme of the investigatedheterostructure highlighting an IX, where the hole and the electron residein the different TMDmonolayers.We utilize a pump laser at an energy ofElaser = 1.94 eV to excite the charge carriers in both monolayers, thusforming IXs. Figure 1b shows the resulting photoluminescence spectra forexcitation powers from 722 nWdown to 778 fW, where the luminescencesignal is close to the overall noise floor. The single luminescence peak atEIX = 1.398 eV is interpreted as the recombination of IXs, and it keeps asharp Lorentzian-type lineshape with a FWHM ≈ 4 meV throughout theinvestigated range of excitation powers. Figure S2 shows the power seriesof photoluminescence spectra on the two other samples. For all threesamples, the photon count of the low-power spectra suggests a small IXdensity in the dilute, single-exciton regime. Moreover, the lack of ablueshift of the emission energy with increasing pump power supportsthis interpretation of having IXs within the dilute regime, where mutualinteractions between excitons can be neglected. Given the long photo-luminescence lifetime of several tens of ns, we interpret the underlying IXstates as the excitonic ground states at the chosen positions of the threesamples, respectively30,31.We further observe that the single luminescencepeaks with a sharp Lorentzian-type lineshape as in Fig. 1b exhibit a spatialprofile that exceeds the point-spread function of the utilized opticalsystem by hundreds of nanometers even in the dilute, single-excitonregime (cf. Fig. S3).Calculation of exciton states in strained heterostructuresIn the next step, we complement the experimental results with theoreticalcalculations of H-type heterostacks of MoSe2 and WSe2 monolayers. Theunderlying theoretical model including the used parameters is described inref. 32, which contains calculations of linear spectra for R-typeheterostructures. The model starts with a continuum mechanics theory todescribe the lattice relaxation leading to reconstruction (based onrefs. 20,32,33). The calculated displacement and strain fields form apotential for electrons and holes, which allows the calculation of interlayerstates. Instead of the frequently used quasi-momentum space, the excitonstates are calculated in real space to capture the effects of imperfections anddisorder that disrupt translational symmetry within the plane of theheterostructures32,34,35. The interlayer states are used as the basis for aquantum dynamic model to describe the IXs, including exciton-phononscattering. The quantum dynamic calculation incorporates rates using aBorn-Markov approximation after polaron transformation to account formulti-phonon processes32. The calculation of optical spectra uses para-meters from refs. 33,36–43. The utilized approach enables us to study thestructure of both localized and spatially extended exciton states with respectto the center-of-mass motion of electrons and holes.The calculation starts with the displacement field of the MoSe2 andWSe2 monolayers that form the top and bottom layers of the reconstructedlattice. The corresponding displacement field ub/t(r) of the bottom and toplayers is obtainedbyminimizing the intra- and interlayer lattice energy20,32,33,usually for an 8 x 4 supercell of the moiré structure32. We do not aim forperfectminimization, but leave some residual error to simulate the disorderpresent in the experimental samples. In turn, disorder does not come fromFig. 1 | Interlayer excitons in H-type MoSe2/WSe2 heterostructures. a Schematicside-view of a MoSe2/WSe2 heterostructure encapsulated in hBN, featuring aninterlayer exciton (IX). bPower series of experimental photoluminescence spectra ata bath temperature of Texpbath ¼ 1:65 K. The IXs emit light at EIX = 1.398 eV with asingular Lorentzian-type shape with a full-width-at-half-maximum (FWHM) γ1.The lineshape is retained down to the lowest investigated excitation powers.c Simulated total potential Vtot landscape within the x–y plane of a strain relaxed,reconstructed bilayer at a twist angle of 1.3° wrt. 60°. The dashed line highlights acertain crystal direction, as discussed in the text. Scale bar marks 10 nm. dAveragedtheoretical photoluminescence spectrum with an apparent FWHM of γtheo at a timedelay of 10 ns after the excitation and a theoretical bath temperature ofT theobath ¼ 1:65 K. Note that a zero detuning describes the lowest 1s IX energy withoutany influence from strain potentials.https://doi.org/10.1038/s41535-025-00820-0 Articlenpj Quantum Materials |           (2025) 10:96 2www.nature.com/npjquantmatsimpurities or other sources in our model, but from slight imperfections ofthe reconstruction.The relative interlayer displacement leads to a potential V intere=h ðrÞ forelectrons and holes37. We modified the formula from37,38 such thatV intere=h ðre=hÞ ¼ VexcP3n¼1 cosðφnðre=hÞ þ φexc;sÞ, and that φn(r) is alsomodifiedby thedisplacementfieldu(t/b)(r) of the top andbottom layer (please,see ref. 32 for the definitions and values of Vexcφn(re/h) and φexc,s). Addi-tionally, the intralayer strain created by lattice reconstruction leads toanother contribution to the potential V intrae=h ðrÞ36 acting on electrons andholes, it has the form V intrae=h ðre=hÞ ¼ ΔEe=hgap∂ux∂x þ ∂uy∂y� �, where changesin displacementfieldudescribe stress leading to a band gap change (cf.32,36 fordetails). The formulas from refs. 36,37 were modified in ref. 32 to includedisplacement due to reconstruction. The modification shifts the currentposition in the interlayer potential by the strain fields. More details of themodification canbe found in ref. 32. The total potentialV totðrÞ ¼ V intraðrÞ þ12V interðrÞ is just the sum of the interlayer and intralayer contribution (theinterlayer contribution is a band gap shift38, which we distribute evenly toelectrons and holes as an initial assumption32). Thus,Vtot(r) describes the fullstrain potential.For an exemplary twist angle deviation of 1.3°, the calculated potentiallandscape Vtot for IXs exhibits a periodic pattern [Fig. 1c]. The potentiallandscape matches experimental findings of a kagome-like lattice rearran-gement dominated by hexagonal areas16,17. Moreover, the calculations pre-dict the potential to feature shallow minima when compared to R-typeheterostructures of the same materials, e.g. of around − 12meV for 1.3°.Our theory also provides photoluminescence spectra (see below for thespecific calculation of the spectra), which again consider theexciton–phonon interactions in the calculation. Exemplarily, Fig. 1d showsthe calculatedphotoluminescence (PLtheo) for the alreadydiscussed1.3° caseat a time delay of 10 ns after initial injection of interlayer excitons and atheoretical temperatureofT theobath ¼ 1:65K.The timedelay is long enough forthe simulation to mimic a quasi-equilibrium near steady-state photo-luminescence before a complete radiative recombination.Generally, the exciton wavefunctions in real spacemust be calculated asa functionof the spatial coordinates tounderstand the lateral characteristics ofthe excitonic states within the reconstructed heterostructures, includingdisorder.We start with a factorization of the excitonwavefunctionΨ(re, rh) =ψ(R)ϕ(r) into the relative ϕ(r) and the center-of-mass (COM) ψ(R)part. We obtain the 1s relative wavefunction ϕ1s(r) after solving the Schrö-dinger equation EϕðrÞ ¼ � ℏ22mrΔr þ VehðrÞh iϕðrÞ using finite differenceswith reduced mass mr and a Rytova-Keldysh potential Veh(r) (cf.32 for theused formandparameters).We include only one species of interlayer excitonand disregard its spin state.We calculate the potentialVCOM(R) acting on theCOM wavefunction by convoluting the electron and hole potentials Ve(r) =Vh(r) = Vtot(r) with ϕ1s(r) (cf.32,44): VCOMðRÞ ¼Rd2rjϕ1sðrÞj2Ve R� mhM r� �þ Vh Rþ meM r� �� �with total mass M and electron and holemassesme andmh.We numerically compute the lowest 2500 eigenstates andeigenstates of the COM Schrödinger equation � ℏ22M ΔR þ VCOMðRÞh iψðRÞ ¼ EψðRÞ for the COM wavefunction using finite differences as thebasis for the COM exciton states. If the 2500 eigenstates do not cover thespectroscopically relevant states, we choose a smaller supercell (4 x 2 insteadof 8 x 4). For the 1.3° example, Fig. 2a shows both the interlayer potential andthe absolute value of the wavefunction of the first IX eigenstate along thewhite, dashed line of the reconstructed bilayer shown in Fig. 1c. Figure 2b–dgives the absolute values of the wavefunctions Ψ for the first three COM IXeigenstates as a function of the spatial coordinates x and y. Eachwavefunctionvaries in space because of the impact of the small spatial imperfections ofVtotwithin the plane of the heterostructure [cf. arrow in Fig. 2a and details in Fig.S4]. For comparison, the red dashed line in Fig. 2b highlights the directionthat resembles the x-axis of Fig. 2a.In real experimental samples, there are often small twist angle devia-tions across the lateral extension of the sample. Therefore, we compare ourresults at 1.3° to those with a relative twist angle of 1.1° and 0.9° (cf. Fig. S5),whichmaymanifest in other parts of a sufficiently large sample. Figure 3a–cdepicts the absolute value of the real space COMwavefunctionΨ for each ofthe twist angles 1.3°, 1.1°, and 0.9°. Our results suggest that the qualitativenature of the lowest energy state changes from the 1.3° configurationtowards slightly smaller twist angles. Particularly for 0.9°, the wavefunctionseems to be localized within one moiré unit cell. To quantitatively capturethis observation, Fig. 3d–f depicts the lateral size (calculated asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihðR� hRiÞ2ip) of the COM wavefunction for the 2500 lowest IX eigen-states for each of the three twist angles, as it is plotted as a function of theexciton energy excluding polaron shifts. Note that themaximal possible sizein Fig. 3d–f is limited to the computational domain, so reaching the max-imum effectively means a full delocalization of the corresponding state.The calculatedwavefunction extensions and an inspection of theCOMwavefunction in real space show that the lowest energy eigenstate at 1.3° isalreadydelocalized [Fig. 3a and reddot inFig. 3d] and is thusnot confined toone moiré unit cell (dashed line with expected moiré periodicitya1:3�m ¼ 14:3 nm)45. This means that for 1.3° the influence of the moirépotential is so small that delocalized states are close to the case without amoiré potential. Such delocalization is not the case for smaller twist angles,as exemplarily shown in Fig. 3e, f, where the first states are small and ratherlocalized inside a moiré unit cell at the respective angles (a1:1�m ¼ 16:9 nmand a0:9�m ¼ 20:5 nm). We note that comparing the wavefunction’s size tothe calculated moiré periodicity as an absolute number is a simplification,ignoring details about the shape of the wavefunction within the two-dimensional plane. For instance, someof the 2500 states for the smaller twistangles, e.g. 0.9°, exhibit rather one-dimensional quantum wire-like statesalong ridges of the potential landscape (cf. Fig. S6). Nonetheless, for moststates and in general, the size given in Fig. 3d–f is a good indication of thewavefunction’s (de)localization.Calculation of optical spectraNext, we discuss the calculated linear absorption spectrum of the computedstates. In general, the temporal Fourier transform of the dipole-dipolecorrelation function trðσ�α ðtÞσþα ð0ÞρÞ (with σþα ¼ ∣αi g ∣) yields theFig. 2 | Real space representation of the center-of-mass wavefunction of the firstexcitonic states. a Top curve: absolute value of the center-of-mass (COM) wave-function Ψ for the first COM IX eigenstate along the crystallographic direction of areconstructed bilayer at 1.3° twist angle as shown in Fig. 1c. Bottom curve: Corre-sponding total IX potential along the same direction. The arrow indicates one ofmany small variations of Vtot caused by the introduced imperfections within thesimulated sample. b–dAbsolute value of the COMwavefunction of the first three IXeigenstates in real space. The dashed red line in (b) corresponds to the x-axis in (a).https://doi.org/10.1038/s41535-025-00820-0 Articlenpj Quantum Materials |           (2025) 10:96 3www.nature.com/npjquantmatsabsorption lineshape Lα(ω− Eα) for an exciton state α centered around theexciton to ground state energy Eα. After applying the polaron transforma-tion from the framework of ref. 32, the lineshape function becomestrðσ�α ðtÞBα�ðtÞBαþð0Þσþα ð0ÞρÞ, where the operators Bα± describe the nuclearreorganization initiated after exciton-photon interaction. A linear absorp-tion spectrum αpp0ðωÞ for the incoming and detected polarization p and p0can be calculated viaαpp0 ðωÞ ¼XαDαg � epD�αg � e�p0Lαðω� EαÞ; ð1Þwith the coupling element Dαg = ∫drD(r)ψα(r), where D(r) uses a mod-ification of definition from ref. 37 to include displacement due to strain (cf.32for details). Initially, before the excitation, there are no excitons present inlinear absorption, and all bright exciton states can be accessed in theexperiment. Figure 3g–i shows the linear absorption spectra for 1.3°, 1.1°,and 0.9° vs. the detuning energy, while a zero energy defines the lowestinterlayer 1s exciton energy without the influence of the strain potentials.Note that all spectra in the following are averaged over calculations forslightly different strain fieldsmade from the distribution of strainfieldswithdifferent residual disorder. Furthermore, we include only one type of exci-ton: i.e. interlayer 1s excitons, in particular, so that signatures of singlet/triplet excitons46 are not included in the simulation.The gray areas in Fig. 3g–i highlight the exciton states with the lowestenergy for each twist angle with a finite absorption strength within thedisordered ensemble. Comparing Fig. 3g–i, it becomes obvious that thepotential wells exhibit deeper minima for smaller twist angles. Moreover,our calculations imply an increasing spectral distinction between peakswithincreasing twist angles. This is visible in the linear absorption spectrum andthe energetic position of all 2500 states in the size plots. The increasingdistinction is especially visible in the shape of the lowest peak, whichtransforms fromthe asymmetric shape at 0.9° to thepureLorentzian format1.3°. The trend reflects the transition from a distribution of clearly localizedstates with anUrbach tail visible in the spectrum at (0.9°)47,48, to amixture ofdelocalized and localized states (1.1°), and finally to only delocalized states(1.3° and above).It is important to note that not all states within the linear absorptionspectra are expected to be visible in photoluminescence, which explains thestriking difference between the absorption spectrum for 1.3° as depicted inFig. 3g and the corresponding photoluminescence spectrum introduced inFig. 1d. The reason for the difference is discussed in the following. Thephotoluminescence spectra are calculated via the correlation functiontrðσþα ðtÞσ�α ð0ÞρÞ (or after polaron transformationtrðBαþð0Þσþα ð0ÞBα�ðtÞσ�α ðtÞρÞ),whereα indices the exciton state,which yieldsthe photoluminescence lineshape function L*(Eα − ω). The lineshapefunction for photoluminescence (mainly caused by electron-phononinteraction) is mirrored at the exciton energy Eα compared to the line-shape function in linear absorption. Note that the center of the lineshapefunction does not coincide with its maximum but with the zero phonontransition due to an imbalance of phonon emission and absorption pro-cesses. While for linear absorption all optically active states contribute, theemission in photoluminescence is determined by the exciton density dis-tribution ραα, which approaches a quasi-equilibrium Boltzmann distribu-tion for a long time. Overall, the PL spectrum takes the form:PLpðωÞ ¼XαDαg � epD�αg � e�pL�αðEα � ωÞραα; ð2Þassuming that it is measured on time scales long enough for the coherenceinduced by the exciting laser to have already decayed, and that theapproximate steady-state limit can be applied to describe the emissionprocess.Fig. 3 | Spatial extension of the excitonic states andabsorption spectra for a varying twist angle.a–c Absolute value of the COMwavefunction of thefirst IX eigenstate in real space for a relative twistangle of (a) α = 1. 3°, (b) 1.1°, and (c) 0.9°. d In-planesize of the wavefunction with some adjustments forthe periodic computation domain. We calculate theexpectation value for all 2500 exciton eigenstates vs.their energy for 1.3° with each dot in the graphrepresenting one state. The lowest-energy state, as in(a), is highlighted in red. The dashed horizontal lineindicates the size of the moiré cell for the specifictwist angle. e, f Corresponding representations for1.1° and 0.9°. g Calculated linear absorption spec-trum for the computed states at 1.3°. Gray areahighlights the spectral range of the first states, asindicated by the gray area in (b). h, i Similar repre-sentations for 1.1° and 0.9°. Note that a zerodetuning describes the lowest 1s IX energy withoutany influence from strain potentials. The shift tolower energies is consistent with a trend to excitonlocalization for smaller relative twist angles.https://doi.org/10.1038/s41535-025-00820-0 Articlenpj Quantum Materials |           (2025) 10:96 4www.nature.com/npjquantmatsTime-resolved photoluminescenceFigure 4a shows calculated time-resolved photoluminescence spectraT theobath ¼ 100mK for 1.3°. For the luminescence simulation, an initialGaussian exciton density distribution of excitons at higher energies isassumed tomimic the scattering from intra-layer excitons. At short times inthe 10 ps regime, the spectrum is dominated by emission lines at higherenergies corresponding to a higher density population of IXs [cf. Figs.S7 and S8]. These lineshave a direct correspondence to the ones in the linearabsorption spectra [cf. Fig. 3g]. With the same argument, the amplitudes ofthe emission peaks vary as a function of time delay. Within a time-scale oftens to hundreds of ps, the energetically lowest state in PL increases inrelative amplitude, i.e., the exciton states from the ground state peakdominate the photoluminescence at the longest times [cf. Fig. 4a]. Thiscorresponds to the emission characteristic for an initially lower density ofthe IX population realized in experiment by a low excitation power andverified by a single Lorentzian-type emission line as in Fig. 1b (see also [cf.Figs. S7 and S8]). Figures 4b, c show equivalent theoretical photo-luminescence spectra for 1.1° and0.9° (cf. alsoFig. S7).Theoverall dynamicsis similar for the other angles, even though the nature of the involved statesdiffers. Similar to the results for absorption, themaindifference occurs in thelineshape variation and a varying spectral position of the levels.In Fig. 4d, we complement the theoretical work with experimentaltime-resolved photoluminescence on sample 3 at Texpbath < 100mK. Apartfrom the lowest energetic emission already discussed, the first spectrum at 1ns delay features higher-lying excitonic states, which disappear for longerdelays. We fit the spectrum by three Lorentzian curves with an FWHM of14, 17, and 21meV, respectively. Since the utilizedCCDhas a comparativelylong gate time (2 ns), the first spectrum at a nominal time delay of 1 nscaptures all events predicted by our calculations in the ps range. In turn, weinterpret the two higher-lying luminescence peaks to arise from an overlayof multiple excitonic states at an energy higher than the ground state, asdepicted inFig. 4a–c for short delay times.At later delay timesof 2–10ns, thelowest energetic emission dominates the photoluminescence and agreesqualitatively with the calculated spectra at 1.3°. The lines are much broaderthan in the measurement shown in Fig. 1b. The broadening is very likelycaused by a larger area contributing to the signal in this particular experi-mental setup (as discussed below). Consequently, the experimental signalshould be described by a weighted superposition of theoretical results ofdifferent twist angles, i.e. the results complement the interpretation of theexperimental data originating from areas with a reconstructed hetero-structure close to 1.3° on our samples, but likely with contributions reachingdown to 0.9° and lower.DiscussionOur calculations predict a comparatively flat potential for IXs at small twistangles relative to the 60° in H-type MoSe2/WSe2 heterostructures [e.g. Fig.1c] and, consequently, the emergence of delocalized IX ground states with aspatial extension of several tens of nanometers (cf. Fig. 2). We note thatpredictions of the calculated spatial extension are merely limited by thecomputational domain and thus, computational power. Possible defects,sample edges, and cracks truncate the extension of the wavefunctions, butcan still be considered in the simulations32. For a particular relative twistangle of 1.3°, we observe that a spatially extended ground state dominatesthe calculated luminescence for long times after excitation (>100 ps) [cf.Figs. 4a, 1d]. The corresponding absorption and luminescence exhibit aLorentzian-type lineshape [cf. Figs. 3g, 1d]with a prevailing FWHMof γtheoon the order of one meV [Fig. 1d], fundamentally limited by the electron-phonon interaction32. The spectral width and lineshape are sensitive mea-sures of the IXs’ interaction with their environment, and thereby, asdemonstrated in Fig. 3, also for the localization and extension of thewavefunction in a spatially slightly varying potential landscape. As locali-zation is related to the potential depths and disorder, it will particularlyinfluence the luminescence energy and linewidth of localized states ratherthan the ones of delocalized states. In other words, the spatially integratedphoton absorption and emission via localized IX states reflects the potentialvariation of the various moiré cells, where the IXs are generated orrecombine [cf. Figs. 3h, i and 4b, c]. Note that the simulated potentialvariations based on the chosen disorder parameters due to incompletereconstruction are on the order of 100s of μeV [e.g. arrow in Fig. 2a and Fig.S4] and that, in theory, a zero detuning describes the lowest 1s IX energywithout any influence from strain potentials. In turn, Fig. 3g suggests thatthe Lorentzian-type lineshape evolves as soon as the impact of the potentialvariations is negligible, while localized states give rise to broadened non-Lorentzian lineshapes [Figs. 3h, i]. When all previous arguments are com-bined, it follows that, starting at 1.3° the underlying wavefunction of thelowest energy state is delocalized in the simulations, and a deviation from aLorentzian-type emission may reflect the impact of the potential variationsin the simulated area of a heterostructure.Experimentally, we observe luminescence peaks on the investigatedsamples with a Lorentzian-type lineshape, which prevail down to excitationpowers for which the signal is only limited by the apparent noise level of theutilized optical circuitry [Fig. 1b]. Comparing γ1 = 3.8meV of sample 1 toγtheo [Fig. 1d], the experimental value reflects the imperfections of theinvestigated sample within the optical focus, but still, the observedLorentzian-type lineshape suggests that the underlying wavefunctions arelaterally extended (for γ2, γ3 of samples 2 and 3, see Fig. S1). Sincewe do notobserve any blueshift of the emission energy of the experimental lumines-cence as in Fig. 1b, the investigated exciton ensembles can be considered tobe in thedilute limit. In turn, theunderlying IX states canbe considered tobethe lowest energy states that contribute to the luminescence in a quasi-stationary limit. Still, the time-resolved photoluminescence spectra at shorttime scales comprise light from states at higher energy. For instance, higher-lying states dominate the luminescence at short time scales [Fig. 4d and Fig.S8], which seems to be consistent with all of the shown, calculated spectra atFig. 4 | Time-resolved photoluminescence spectra of interlayer excitons.a Calculated photoluminescence spectra for a heterostructure with a relative twistangle of 1.3° at time delays ranging from 10 ps to 10 ns with respect to the excitationpulse. The temperature isT theobath ¼ 100mK. b, cCorresponding theoretical results for1.1° and 0.9°. d Experimental time-resolved photoluminescence spectra of IXs atTexpbath < 100mK. Excitation with a laser power of 150 nW and an energy of 1.937 eV.All spectra are normalized.https://doi.org/10.1038/s41535-025-00820-0 Articlenpj Quantum Materials |           (2025) 10:96 5www.nature.com/npjquantmatsshort time scales for α = 1.3°, 1.1°, and 0.9° [Fig. 4a–c], where exciton-phonon scattering eventually leads to relaxation towards the lowest IXstate. On purpose, we present calculations for more than just one relativetwist angle α tomake clear that an experiment with an optical focus of oneto two micrometers collects light from presumably many specific dis-order-, strain- andmoiré-potential variations, as well as corresponding IXstates (cf. Figs. S7. For instance, the rather large value of γ03 ¼ 11:5meVfor sample 3 as in Fig. 4d turns out to decrease to γ3 = 7.5meV when theheterostructure is measured in the setup as sample 1 in Fig. 1b (cf. Fig. S1)and the lineshapes becomemore Lorentzian. Furthermore, if we comparethe experimental lineshapes at different positions of a sample (e.g. Figs.S9 and S10) to lineshapes of the calculated photoluminescence spectra fordifferent angles (cf. Fig. 4a–c and Fig. S7 at 10 ns), we tentatively concludethat the lineshape can be a qualitative indicator for assigning the twistangles under investigation.Onabroaderperspective,wenote that thementionedelectron-phononinteraction is the fundamental limit of the observed experimental FWHMsas is consistent with previous measurements on the temporal coherence ofIX ensembles with a Lorentzian-type luminescence profile30,31. The laterpublications report temporal coherence times of 100s of femtoseconds,which translates to an FWHM on the order of a few meV as is consistentwith γtheo. In other words, the measured Lorentzian-type lineshape can beunderstood to be dominated by a homogeneous broadening: namely, thatfor delocalized states, the physical interpretation allows for a momentumpicture to account for the center-of-mass motion. In this picture, theemission results from low-energy, i.e., near-zero-momentum COM-states.Every scattering mechanism resulting in the occupation of non-zero COMmomentum states is connected to dephasing. Last but not least, the deducedlaterally extendedwavefunctions of the IXground states show the possibilityof degenerate exciton ensembles in real samples of twisted TMD hetero-structures with atomic reconstruction and small imperfections. We note,however, that the current work emphasizes a single-exciton delocalization,since it does not consider any exciton-exciton interactions as it would beneeded to describe any transition to possible condensation andmany-bodyphenomena of excitonic boson or fermion ensembles49. Exciton gas prop-erties, including condensation effects, typically require a momentum-resolved description to discuss and compare their properties as closely aspossible with respect to ideal, textbook-like quantum gases. Therefore, it isimportant to show that delocalized states can occur due to a dominatingkinetic energy of the interlayer excitons in comparison to the moiré loca-lization potential or the occurring disorder. We argue that delocalization atlow densities is a prerequisite for observing exciton gas properties andmany-body effects in analogy to ideal quantum gases. In order to experi-mentally disentangle the discussed single-exciton delocalization from pos-sible many-body phenomena, a combination of experiments on the verysame heterostructure seems to be necessary. This includes the demonstra-tion of an extended spatial coherence of the luminescence31, a reduction ofthe (Lorentzian-type) linewidth30, and a (possible) increase of photonemission at zeromomentumasdetected in a back-focal plane geometry50, allfor an increasing number of excitons and a lowering of the bathtemperature.In conclusion, our work combines theoretical calculations of excitonicstates in reconstructedMoSe2/WSe2 bilayers close to 60° with experimentalphotoluminescence measurements in equivalent samples. The calculationsyield a potential for IXs within the strain-relaxed bilayers and randomdisorder, real-space excitonic wavefunctions, linear absorption spectra, andtime-resolved photoluminescence spectra, as is consistent with earlierwork20,33. Importantly, the theoretical results predict the reconstructedbilayer to exhibit a relatively flat potential landscape at 1.3° twist angle andabove, with an energetically lowest state featuring a single Lorentzian-typephotoluminescence peak. This peak also dominates for later times in theregime of quasi-equilibrium. Both predictions are consistent with ourexperiments, which include low-power as well as time- and spatio-resolvedphotoluminescence spectra at cryogenic temperatures.The calculated sizeofthe wavefunction of the first COM IX eigenstates within the two-dimensional plane at 1.3° twist angle suggests a delocalization of the exci-ton well beyond the moiré periodicity. Such a spatial extension is inagreement with the existence of degenerate interlayer exciton ensembles insimilar MoSe2/WSe2 bilayers, but the current work does not consider anyexciton-exciton interactions needed to describe any transition to possiblecondensation and many-body phenomena of excitonic boson or fermionensembles49, as well as lateral transport dynamics of IXs based on exciton-exciton and exciton-phonon interactions51–54.MethodsExperimental detailsOur heterostructures are stacked with the PDMS method on SiO2/Si sub-strates and encapsulatedwith hBN.We showphotoluminescence data fromthree different samples 1, 2 and 3 (Fig. 1b, Fig. S1). All samples feature anH-type heterostructure close to 60° twist (Fig. S2). The widths of the lowestinterlayer exciton emission peak are γ1 = 3.8 meV, γ2 = 3.3meV and γ3 =7.5meV. All power series show the lack of an energetic blueshift, suggestingnegligible interparticle interactions within the investigated range of laserpower. We note that the dilute limit mentioned in the main manuscriptrefers to this regime of negligible interactions between interlayer excitons.Moreover, spatial images of the excitonic photoluminescence are detectedby utilizing a CCD camera, and the profiles are compared to the reflectedlaser profile (cf. Fig. S3). For the time-resolvedphotoluminescence spectra asin Fig. 4d of themainmanuscript, we utilize a pulsed laser and a gatedCCD.The detection CCD features a gate time of 2 ns.We use a pulsed laser diode(PicoQuant PDL 800-D) at 1.937 eV with a pulse duration of 70 ps. Therepetition rate and the average power are set to 125 kHz and approximately150 nW on the sample, respectively. The repetition frequency generates atime delay of 8 μs between the laser pulses that far exceeds the interlayerexciton lifetime.Simulation detailsThe reconstructed strainmaps are calculated byminimizing the strain field-dependent energy with a minimization algorithm using the numericallibrary TAO/PETSC55, starting from a Gaussian random strain field dis-tribution (though the dependence on the initial distribution is minimal).The minimization algorithm halts if a typical gradient accuracy for theenergy minimization of Eq. (1) of ref. 32 of 0.1 meV/nm is reached or if therelative gradient is below 0.0001 (for details see TAO/PETSCdocumentation56,57, for TaoSetTolerances and the parameters gatol andgrtol). 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ANL-21/39 -Revision 3.19, Argonne National Laboratory (2023).AcknowledgementsThe authors gratefully acknowledge the German Science Foundation (DFG)for financial support via Grants HO 3324/16-1, No. 290642686, 443274199,and 556436549 (WU 637 4-2, 7-1, 8-1), KN 427/11-2 and KN 427/15-1, No.556436549, and the clusters of excellence MCQST (EXS-2111) ande-conversion (EXS-2089), and the priority program 2244 (2DMP) viaHO3324/13-2.K.W.andT.T. acknowledgesupport from theJSPSKAKENHI(GrantNumbers21H05233and23H02052), theCREST(JPMJCR24A5), JSTand World Premier International Research Center Initiative (WPI),MEXT, Japan.Author contributionsJ.F., M.T., H.L., and T.S. performed the experiments. MR performed thetheoretical calculations. A.W.H., U.W., M.R., and A.K. perceived the project.J.F., M.T., and J.K. fabricated the samples. T.T. and K.W. supplied the high-quality hBN. All authors contributed to the writing, read, and approved themanuscript.FundingOpen Access funding enabled and organized by Projekt DEAL.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41535-025-00820-0.Correspondence and requests for materials should be addressed toMarten Richter, Andreas Knorr or Alexander W. Holleitner.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in anymedium or format, as longas you give appropriate credit to the original author(s) and the source,provide a link to the Creative Commons licence, and indicate if changeswere made. The images or other third party material in this article areincluded in the article’s Creative Commons licence, unless indicatedotherwise in a credit line to the material. If material is not included in thearticle’sCreativeCommons licence and your intended use is not permittedby statutory regulation or exceeds the permitted use, you will need toobtain permission directly from the copyright holder. To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2025https://doi.org/10.1038/s41535-025-00820-0 Articlenpj Quantum Materials |           (2025) 10:96 8https://petsc.org/https://petsc.org/https://doi.org/10.1038/s41535-025-00820-0http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/www.nature.com/npjquantmats Laterally extended states of interlayer excitons in reconstructed MoSe2/WSe2 heterostructures Results Experimental photoluminescence Calculation of exciton states in strained heterostructures Calculation of optical spectra Time-resolved photoluminescence Discussion Methods Experimental details Simulation details Data availability Code availability References Acknowledgements Author contributions Funding Competing interests Additional information