Long-range spatial extension of exciton states in van der Waals heterostructure

Domain: chemistry
Published: 2026-03-05
Editorial outcome: major_revision

Abstract
Narrow lines in photoluminescence (PL) spectra of excitons are characteristic of low-dimensional semiconductors. These lines correspond to the emission of exciton states in local minima of a potential energy landscape formed by fluctuations of the local exciton environment in the heterostructure. The spatial extension of such states was in the nanometer range. In this work, we present studies of narrow lines in PL spectra of spatially indirect excitons (IXs) in a MoSe2 WSe2 van der Waals heterostructure. The narrow lines vanish with increasing IX density. The disappearance of narrow lines correlates with the onset of IX transport, indicating that the narrow lines correspond to localized exciton states. The narrow lines extend over distances reaching several micrometers and over areas reaching ca. ten percent of the sample area. This macroscopic spatial extension of the exciton states, corresponding to the narrow lines, indicates a deviation of the exciton energy landscape from random potential and shows that the excitons are confined in moire´ potential with a weak disorder.

Introduction
Narrow lines with linewidths ≲ 1 meV in PL spectra of excitons are ubiquitous in low-dimensional semiconductors. The narrow lines found in a GaAs quantum dot correspond to the emission of exciton states in the dot [1]. The narrow lines observed in GaAs quantum wells originate from the emission of exciton states in local minima of random potential formed by fluctuations of the local exciton environment in the heterostructure, e.g. quantum well width and materials fluctuations [2–5].

Recent studies revealed narrow lines in PL spectra of excitons in van der Waals (vdW) heterostructures composed of single atomic layers of transition-metal dichalcogenides (TMD). The narrow lines observed in a TMD electrostatically defined trap correspond to the emission of exciton states in the trap [6]. The narrow lines were also observed for excitons in local minima of random potential in monolayer TMD [7–11] and for excitons confined by strain in the regions of the heterostructure flake edges [12], heterostructure wrinkles [13], or nanopillars [14–16].

Local minima in exciton potential landscape can be also formed in moire´ superlattices in TMD heterostructures [17–21]. The moire´ potentials can be affected by atomic reconstruction [22–24]. Narrow lines were observed for excitons in bilayer TMD heterostructures with moire´ potentials [25–33].

Similar to other low-dimensional semiconductors, such as GaAs heterostructures outlined above, narrow lines are ubiquitous in vdW heterostructures. In addition to TMD heterostructures with the exciton confinement caused by electrostatic traps [6], random potentials [7–11], or strain [12–16] and TMD bilayer heterostructures with moire´ potentials [25–33] outlined above, narrow lines were also observed in TMD bilayer heterostructures where moire´ potentials are suppressed by hBN spacers [34] and in TMD trilayer heterostructures [35].

In contrast to random potentials, which are characteristic of semiconductor heterostructures with 2D layers formed by several monolayers (like GaAs heterostructures) or single monolayers (like TMD heterostructures), moire´ potentials are periodic in the heterostructure plane. Their lateral period is typically in the range of ca. 10 nm, exceeding the exciton Bohr radius ca. 1 nm and providing a confining potential for excitons [17–21]. Narrow lines are observed both in heterostructures with moire´ potentials and in heterostructures without moire´ potentials, as outlined above, and both these types of heterostructure have disorder potentials. Since both disorder and moire´ potentials can produce the narrow-line emission, the roles of the disorder and moire´ potentials in the origin of the narrow lines remain unclear as outlined, in particular, in recent studies of TMD heterostructures [34].

The major difference between moire´ potentials and disorder potentials in the origin of the narrow lines is a spatial ordering for the former. The lateral extension of the narrow lines is given by the lateral extension of the corresponding exciton states. The extension of localized exciton states in a random potential is typically on the order of nanometers, and even extension of delocalized exciton states, given by the mean free path, is typically in the nanometer range for excitons in semiconductor heterostructures as overviewed in Ref. [36]. In contrast, exciton states in a periodic lattice potential, such as a moire´ potential, can extend over long distances limited by imperfections of the periodic potential in the heterostructure.

A strong disorder potential both limits the spatial extension of localized exciton states and reduces the diffusivity of delocalized excitons. Therefore, heterostructures with longer extension of localized exciton states may provide more efficient exciton transport with higher diffusivity and mean free path. The long lifetimes of IXs give an opportunity to realize long-range efficient exciton transport and IX transport in TMD heterostructures is intensively studied [36–58]. Recent studies show the long-range IX transport [57], the long-range IX mediated spin transport [58], and the high IX diffusivity and mean free path [36]. In this work we verify, in particular, if the heterostructure presenting this efficient exciton transport [36, 57, 58] is characterized by a long spatial extension of localized exciton states.

In earlier studies of narrow lines, outlined above, the spatial extension of the narrow lines was limited by the spatial resolution of the optical experiments, ca. 1 µm. This short extension of exciton states associated with the narrow lines is characteristic of disordered potentials as shown, in particular, in studies of narrow lines in GaAs heterostructures [2–5] where moire´ potentials are absent and narrow lines originate from the emission of exciton states in local minima of random potential in the heterostructure. In this work, we studied narrow lines in PL spectra of IXs in a MoSe2 WSe2 vdW heterostructure. We observed that the narrow lines extend over distances reaching several microns. This macroscopic spatial extension of the exciton states, corresponding to the narrow lines, indicates a deviation of the excitonic energy landscape from random potential. An ordering in the local environment of excitons, such as a moire´ potential disordered weakly, is consistent with the observed long-range spatial extension of the exciton states. The long-range extension of exciton states facilitates efficient exciton transport in the heterostructure.

Results
In the studied MoSe2 WSe2 vdW heterostructure, the adjacent MoSe2 monolayer and WSe2 monolayer form the separated electron and hole layers and IXs are formed by electrons and holes confined in these separated layers [59]. Twisting between the MoSe2 and WSe2 monolayers with the twist angle δθ ∼ 1.1◦ produces a moire´ potential with the moire´ superlattice period b ∼ a / δθ ∼ 17 nm (a is the lattice constant) [17–21]. The heterostructure details and the optical measurements are outlined in Supplementary Information (SI).

Energies of narrow-line exciton states
Figure 1 shows narrow lines in PL spectra of IXs in the heterostructure. With increasing density, the energies of the narrow lines stay fixed (Fig. 1), more data on the density dependence of the narrow lines is presented in Fig. S2 in SI. This indicates that each narrow line corresponds to an exciton state with a low sensitivity to the average exciton density. An exciton in a moire´ cell with certain occupations of neighbor cells is consistent with such a state [25–33]. For such exciton states, adding an exciton to a neighbor cell leads to an increase by the inter-cell interaction energy exceeding the linewidth of the narrow line so, at low densities, cells with statistically distinct occupations of the neighbor cells can produce narrow PL lines with the lack of continuous energy shift with density [25–33].

In contrast, statistical averaging over different exciton states gives the broad PL line with the energy monotonically increasing with average exciton density n (Fig. 1 and Fig. S2 in SI). This average energy shift can be approximated by the mean field ’capacitor’ formula δEz = 4πe2d n / ε [60] (d ∼ 0.6 nm is the separation between the electron and hole layers and ε ∼ 7.4 is the dielectric constant for the heterostructure [61]) and, in particular, can be used for estimating n.

Disappearance of narrow lines and onset of exciton transport
The narrow lines vanish with increasing density and, at high densities, a broad PL line dominates the spectrum (Fig. 1a,b). In Fig. 1c, the relative intensity of the narrow lines in the PL spectra is compared with IX transport in the heterostructure. The former is presented by the ratio of the sum of spectrally integrated intensities of the narrow lines to the spectrally integrated intensity of the broad line in the PL spectrum, and the latter is presented by the 1/e decay distance of IX transport d1/e measured in Ref. [57]. The opportunity to achieve with varying density both IX localization and long-range IX transport, studied in Ref. [57], enables such comparison. This comparison shows that the disappearance of narrow lines with increasing density in the range of the pump power Pex ∼ 1–50 µW correlates with the onset of IX transport (Fig. 1c). The anticorrelation with transport indicates that the narrow lines correspond to localized excitons. However, this anticorrelation does not establish the nature of localization that may be caused by a disorder potential or by an ordered moire´ potential in the heterostructure.

Figure 1 shows that the narrow lines vanish with the onset of IX transport, however, they do not re-appear at the higher densities where IX re-entrant localization, outlined in Ref. [57], is observed. This is consistent with the narrow line association with the exciton localization in local minima of a potential energy landscape formed by variations of the local exciton environment in the heterostructure. The re-entrant localization at the higher densities due to insulating phase, such as the Mott insulator and the Bose glass [62], is of a different origin. In particular, for the higher densities when most of the moire´ cells are occupied, the particle transport from cell to cell leads to double occupancy that creates a gap for particle-hole excitations, consequently making the state insulating [62]. Figure 1 shows that narrow lines are not characteristic of this high-density insulating phase.

g factor of narrow-line exciton states
For all narrow lines, the measured excitonic g factor is g ∼ −1.55 ± 0.07 as shown in Fig. S6 and outlined in SI. Excitonic g factor is determined by the local atomic registry and the measured g factor corresponds to Hh site in the moire´ potential of the MoSe2 / WSe2 heterostructure with, in turn, Hh stacking [17–20, 25, 63]. For TMD heterostructures with moire´ potentials, a coincidence of g factor for all narrow lines was found in Ref. [25]. The g factor specific for a certain local atomic registry (Hh in our case) shows that the narrow lines correspond to excitons in the specific site (Hh in our case) of the moire´ potential.

The same local atomic registry Hh may extend over a considerable part of the sample in (reconstructed) moire´ potentials that makes the measured g factor essentially insensitive to the location of the exciton state in the sample [63]. In particular, for the exciton Bohr radius much smaller than the moire´ site, excitons can be localized by random potential fluctuations within the moire´ site as outlined in Ref. [34]. Therefore, the same and atomic-registry-specific g factor of narrow lines is insufficient to establish the nature of localization of the corresponding exciton states that may be caused by a strong disorder or by an ordered moire´ potential in the heterostructure.

Spatial extension of narrow-line exciton states
Figure 2 shows x − Energy maps of the exciton PL. In these maps, the narrow lines are revealed by the spectrally narrow enhancements of the PL intensity. Figure 2 shows that narrow lines and, in turn, the corresponding exciton states, can extend over long distances reaching several micrometers.

Figure 2 shows the extension of the narrow-line exciton states in the x-direction. The measured x − Energy maps at different y locations allow building the x − y maps for the exciton states corresponding to the narrow lines. Examples of the x − y maps for the exciton states are presented in Fig. 3. (Figures S7 and S10 in SI show x − Energy maps for all measured y, covering essentially the entire heterostructure area, and x − y maps for many of the narrow-line exciton states seen in the x − Energy maps.) Figure 3 shows that the narrow lines extend over distances reaching several micrometers and over areas reaching ca. ten percent of the measured sample area.

The observed macroscopic spatial extension of exciton states, corresponding to the narrow lines, indicates a deviation of the exciton energy landscape from random potential.

A strong disorder potential does not produce macroscopically extended localized exciton states. In particular, no such extension was observed in any semiconductor system, including GaAs and vdW heterostructures outlined in the introduction, where narrow lines originate from the emission of exciton states localized in local minima of random potential formed by fluctuations of the local exciton environment in the heterostructure, e.g. stress, dielectric, electrostatic, and materials fluctuations.

In turn, the observed macroscopic spatial extension of localized exciton states, corresponding to the narrow lines, indicates ordering in the local environment of excitons: The exciton state at a certain energy, corresponding to the narrow line, extends over macroscopic length and area (Figs. 2 and 3). A moire´ potential is consistent with such ordering and long-range spatial extension of localized exciton states. We note that different narrow lines and their corresponding exciton states are extended over different regions of the heterostructure (e.g. compare the regions for different narrow lines in Figs. 3a and 3b). This indicates that the local environment for excitons fluctuates over the heterostructure, however, the fluctuations are small enough to allow the long-range extension of the individual exciton states. Therefore, the long-range extension of the narrow lines shows that the excitons are confined in moire´ potential with a weak disorder.

Moire´ potentials with a weak disorder can host long-range ballistic exciton transport due to exciton superfluidity in periodic potentials [62]. A strong disorder destroys superfluidity [62]. Therefore, the weakness of disorder in the moire´ potential, revealed by the long-range extension of the exciton states (Figs. 2 and 3), suggests an opportunity to observe the long-range ballistic transport of excitons in this weakly disordered moire´ potential. The studies of exciton transport in the same heterostructure show the long-range IX transport [57], the long-range IX mediated spin transport [58], and the high IX diffusivity and mean free path [36].

The extension of exciton states over distances reaching several micrometers raises a question of distinguishing such states from delocalized excitons states and a question if such extended states can be called localized states. In this work, we qualitatively discuss an exciton state confined in a region, even of a large area, as a localized state and exciton states, which can travel over different localization regions, as delocalized states. The long-range extension of localized exciton states and their small energy difference facilitates exciton transport over different localization regions in the heterostructure.

In summary, we studied narrow lines in PL spectra of IXs in a MoSe2 WSe2 heterostructure. We found that the disappearance of narrow lines correlates with the onset of IX transport, indicating that the narrow lines correspond to localized exciton states. We found that the narrow lines extend over distances reaching several micrometers and over areas reaching ca. ten percent of the sample area. This macroscopic spatial extension of exciton states, corresponding to the narrow lines, indicates a deviation of the exciton energy landscape from random potential and shows that the excitons are confined in moire´ potential with a weak disorder. The long-range extension of exciton states facilitates efficient exciton transport.

Methods
Van der Waals heterostructure. The MoSe2 WSe2 heterostructure was assembled using the dry-transfer peel technique [64]. The heterostructure details are presented in SI.

Optical measurements. Excitons were generated by a continuous-wave Ti:sapphire laser with the excitation energy Eex = 1.689 eV resonant to DX in WSe2 heterostructure layer (a cw semiconductor laser with similar Eex was used for the data in Figs. S3b-d, S4, S5, and S12 in SI). PL spectra were measured using a spectrometer with a resolution of 0.2 meV and a liquid-nitrogen-cooled CCD. A spectrometer slit along the x direction was used to select a δy = 1.3 µm wide portion of the sample emission. An emission that is outside the selected δy = 1.3 µm area did not enter the spectrometer and, therefore, was separated. The spectrometer showed on the CCD detector the x − Energy map where the x axis is along the spectrometer slit and the orthogonal axis on the CCD detector shows the spectrum for each x position. For the x − Energy maps in Fig. 2 and in Figs. S3, S4, S7-S9 in SI, we used a defocused laser excitation covering the entire heterostructure that allowed obtaining the x − Energy maps without moving and precise positioning of the laser beam. The x − Energy images were measured with the step 1.3 µm and the signal integration within 1.3 µm in the y direction given by the slit (the slit positions in the measurements of the x − Energy maps are shown in Fig. S7h in SI). For the x − y maps, the step in the slit position and the signal integration within the slit 1.3 µm gave the spatial resolution in the y direction. NA = 0.64 of the lens gave the spatial resolution 0.7 µm in the x direction.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by Zhou et al. entitled “Long-range spatial extension of exciton states in van der Waals heterostructure” studies narrow lines in the PL emission of Interlayer Excitons in a hBN encapsulated MoSe2/WSe2 heterostructure. The work demonstrates narrow line PL emission extending over several microns. The authors attribute the long-range narrow line formation to an ordered varying potential landscape allowing for consistent trapping of IXs at potential minima. The data is good quality, the paper is well written, and the long-range mapping of narrow line energies is intriguing. However, given the large body of work that already exist on moiré trapped excitons in twisted TMD heterostructures, I have some concerns about the papers methodology and impact that should be addressed before recommendation for publication in Nature Communications.\n\nI am unclear about the authors methodology in this experiment. The authors state “The x−Energy images were measured with the step 1.3 µm and the signal integration within 1.3 µm in the y direction given by a slit (the slit positions in the measurements of the x−Energy maps are shown in Fig. S4h).” I interpret this to mean that the authors used a slit to image only a 1.3 µm wide portion of their sample emission along y. They then must have stepped their laser in 1.3 µm steps in x and measured the PL emission on their CCD. If this is correct, then it is unclear to me how the authors have adequately separated any extraneous emission that is outside the 1.3 µm area they claim to measure in x. If this is incorrect, then I believe the authors should revise their language and include a figure demonstrating their optical set up.\n\nOne concern I have is in the way the authors characterize the disappearance of their narrow lines with power. Figure 1 c only shows the integrated narrow line emission divided by the integrated broader gaussian peak; however, it would make sense that the narrow lines scale slower with power than the broader emission. The authors should also show the total integrated emission from the narrow lines and gaussian subtracted, position dependent narrow line emission for higher powers like in Figure 2.\n\nI have a concern about the characterization of the narrow lines as moiré trapped exciton emission. The authors vary their excitation power to control their exciton density and point to the disappearance of the narrow line emission at high power along with the onset of IX transport in reference [40] as evidence of the moiré becoming over filled. However, reference [34] demonstrates narrow line emission in a heterostructure where the moiré is suppressed. Because of this, the validity of the main conclusion of this paper is dependent on the interpretation that large spatial extent narrow line emission can only be caused by a moiré. This argument could be strengthened by the introduction of a control device. If the authors are correct in their interpretation, the narrow lines present in an hBN separated heterostructure would not remain over a few microns and it would be easier to attribute these results to a moiré rather than another ordered or coherent state.\n\nI also have a concern about the reproducibility of these results. The few micron extent of the narrow line emission is a unique and interesting result. However, given the phenological nature of the results presented in this paper, similar results should be shown over multiple samples, and a lack of such results should be shown on a control as discussed in the previous comment to sufficiently convince the reader of this papers conclusion.\n\nA minor comment I have pertains to figure and reference presentation. The figures have inconsistent letter placement, for example in Figure 1, the letters a and b appear in the top right of the plots and c appears in the top left. In reference [42] superfluid is misspelled. The authors should double check for errors.", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "In this manuscript, the authors probed the spatial extension of narrow PL emission lines of interlayer excitons (IX) in MoSe2- WSe2 heterobilayer at cryogenic temperatures. By showing the correlation between the disappearance of narrow PL lines and the onset of IX transport, they showed that narrow lines originate from localized exciton states. Additionally, the observed spatial extension of the exciton states over distances of several micrometers was attributed to the localization of excitons in moiré potentials with weak disorder, rather than the random potential.\n\nWhile the demonstration of the spatial extension of narrow emission lines over several micrometers is interesting and important for obtaining periodic quantum emitters from these heterobilayers, the results/evidences and discussion provided by the authors to support their claim are not sufficient enough for the publication of the manuscript in its current version in Nature Communications.\n\nIn the following, I give some additional comments and questions:\n1. Temperature-dependent PL measurements need to be performed along the extended region of the narrow emission lines to see whether they show the same behavior or not at different locations. (Major)\n2. How do the lifetimes of localized interlayer excitons change spatially in the extended region of the narrow lines? In the case of moiré potentials with weak disorder, one can expect to obtain a uniform lifetime along the extended region. Did the authors measure the lifetimes along the extended region of the narrow lines? The position-dependent lifetime mapping would also be valuable in further strengthening their claim of having moiré potentials with weak disorder.(Major)\n3. The presence of uniformity of the moiré pattern should be verified through one of the imaging techniques, such as TEM, piezoresponse force microscopy, etc. (Major)", "reviewer_4": "This paper appears to be a sequel to a series of publications by the same group of authors [1. Nature Communications (2024) 15:9454; 2. Nature Photonics (2024) 18, 823]. The subject matter studied is of high interest to the scientific community. I would support the publication of this manuscript if the authors can provide additional information to clarify the following issues:\n1. In the introduction, the authors should cite their previously published papers on the same subject and explain what new information the current study provides. This will give readers a clearer perspective on the continued development of this research direction.\n2. The authors state that \"the disappearance of narrow lines correlates with the onset of IX transport.\" Please specify the pump power at which this correlation occurs.\n3. The caption of Fig. 1 mentions that “a higher-energy IX line appearing at ∼1.41 eV at high IX densities was attributed to the appearance of moiré cells with double occupancy in Ref. [41].” Please explicitly mark the ∼1.41 eV line in Fig. 1, as it is not easily identifiable to readers.\n4. If the narrow lines observed several microns away from the excitation spot correspond to the same exciton states, then their energies should be nearly identical, and the PL line shapes at different positions should correlate with the center-of-mass wavefunction of the excitons in the periodic moiré potential. The authors should provide a figure (as supplementary information) showing detailed line shapes of all narrow lines within the relevant spectral range at various x positions. This will allow readers to examine whether the lines indeed originate from the same exciton states. Such analysis will also be valuable for future theoretical work aiming to model PL line shapes and exciton diffusion characteristics in this system.\n5. To better understand the influence of exciton diffusion from a tightly focused excitation spot, the authors should compare PL maps obtained using a 2-μm illumination spot with PL maps obtained under uniform illumination (i.e., an excitation spot significantly larger than 10 μm)."}, "author_response": "Response to Remarks of Reviewer 1 (Reviewer’s comments are marked by italic font, changes in the\nmanuscript are marked by blue font, references correspond to the revised manuscript).\nThe manuscript by Zhou et al. entitled “Long-range spatial extension of exciton states in van der Waals\nheterostructure” studies narrow lines in the PL emission of Interlayer Excitons in a hBN encapsulated\nMoSe2/WSe2 heterostructure. The work demonstrates narrow line PL emission extending over several\nmicrons. The authors attribute the long-range narrow line formation to an ordered varying potential\nlandscape allowing for consistent trapping of IXs at potential minima. The data is good quality, the paper\nis well written, and the long-range mapping of narrow line energies is intriguing. However, given the\nlarge body of work that already exist on moiré trapped excitons in twisted TMD heterostructures, I have\nsome concerns about the papers methodology and impact that should be addressed before\nrecommendation for publication in Nature Communications.\nWe thank Reviewer 1 for a high estimate of our work and useful remarks. We address remarks of\nReviewer 1 in the revised manuscript as outlined below.\nI am unclear about the authors methodology in this experiment. The authors state “The x−Energy images\nwere measured with the step 1.3 μm and the signal integration within 1.3 μm in the y direction given by a\nslit (the slit positions in the measurements of the x−Energy maps are shown in Fig. S4h).” I interpret this\nto mean that the authors used a slit to image only a 1.3 μm wide portion of their sample emission along y.\nThey then must have stepped their laser in 1.3 μm steps in x and measured the PL emission on their CCD.\nIf this is correct, then it is unclear to me how the authors have adequately separated any extraneous\nemission that is outside the 1.3 μm area they claim to measure in x. If this is incorrect, then I believe the\nauthors should revise their language and include a figure demonstrating their optical set up.\nYes, this is correct (with a clarification that for x-Energy maps in Fig. S4 that is Fig. S7 in the revised\nmanuscript, we used defocused laser excitation covering the entire sample so stepping the laser beam was\nnot needed). The 1.3 m wide portion of the sample emission in the y direction was selected by the\nspectrometer slit and any extraneous emission that is outside the 1.3 μm area did not enter the\nspectrometer and, therefore, was separated. As to the x direction, the emission from different x positions\non the sample appear at different x locations on the CCD detector. To address this remark of Reviewer 1,\nwe include an additional clarification of the optical measurements in SI: “A spectrometer slit along the x\ndirection was used to select a y = 1.3 m wide portion of the sample emission. An emission that is\noutside the selected y = 1.3 μm area did not enter the spectrometer and, therefore, was separated. The\nspectrometer showed on the CCD detector the x-Energy map where the x axis is along the spectrometer\nslit and the orthogonal axis on the CCD detector shows the spectrum for each x position. For the x-Energy\nmaps in Fig. 2 in the main text and in Figs. S3, S4, S7-S9, we used a defocused laser excitation covering\nthe entire heterostructure that allowed obtaining the x-Energy maps without moving and precise\npositioning of the laser beam.”\nOne concern I have is in the way the authors characterize the disappearance of their narrow lines with\npower. Figure 1 c only shows the integrated narrow line emission divided by the integrated broader\ngaussian peak; however, it would make sense that the narrow lines scale slower with power than the\nbroader emission. The authors should also show the total integrated emission from the narrow lines and\ngaussian subtracted, position dependent narrow line emission for higher powers like in Figure 2.\nIndeed, the narrow lines scale slower with power than the broader emission. To address this remark of\nReviewer 1, we add in SI new Fig. S3 showing the total integrated emission from the narrow lines (in\nFig. S3a) and gaussian subtracted, position dependent narrow line emission for higher powers like in\nFigure 2 (in Figs. S3b-d, we performed these measurement to address this remark of Reviewer 1). We\nalso add the corresponding description of this new figure: “Figure 1c in the main text shows the relative\nintensity of the narrow lines presented by the ratio of the sum of spectrally integrated intensities of the\nnarrow lines to the spectrally integrated intensity of the broad line in the PL spectrum. Figure S3a shows\nthe sum of spectrally integrated intensities of the narrow lines and shows the spectrally integrated\nintensity of the broad line in the PL spectrum. The intensities of narrow lines grow slower with power\nthan the intensity of the broad line (Fig. S3a), consistent with the data in Fig. 1c in the main text. Figures\nS3b-d show x-Energy maps for higher excitation powers. With increasing density, the energies of the\nnarrow lines stay fixed (Fig. 3b-d), consistent with the lack of change of energies of narrow lines with\ndensity discussed in the main text.”\nI have a concern about the characterization of the narrow lines as moiré trapped exciton emission. The\nauthors vary their excitation power to control their exciton density and point to the disappearance of\nthe narrow line emission at high power along with the onset of IX transport in reference [40] as evidence of\nthe moiré becoming over filled. However, reference [34] demonstrates narrow line emission in a\nheterostructure where the moiré is suppressed. Because of this, the validity of the main conclusion of this\npaper is dependent on the interpretation that large spatial extent narrow line emission can only be caused\nby a moiré. This argument could be strengthened by the introduction of a control device. If the authors\nare correct in their interpretation, the narrow lines present in an hBN separated heterostructure would\nnot remain over a few microns and it would be easier to attribute these results to a moiré rather than\nanother ordered or coherent state.\nOur interpretation is that large spatial extent of narrow line emission is caused by a relatively small\ndisorder. In contrast, heterostructures with a large disorder show a much shorter spatial extent of narrow\nline emission as outlined in the manuscript. In contrast to Ref. [34] where MoSe and WeSe monolayers\n2 2\nare separated by hBN that suppresses the moiré potential, in our heterostructure, MoSe and WeSe\n2 2\nmonolayers are adjacent and twisting between the MoSe and WSe monolayers with the twist angle  ~\n2 2\n1.1o produces a moiré potential with the moiré superlattice period b ~ a/ ~ 17 nm. This is confirmed by\nthe measured g factor g ~ 15.5, which corresponds to H h site of the moiré potential as outlined in our\nh\nmanuscript: The g factor specific for a certain local atomic registry (H h in our case) shows that the\nh\nnarrow lines correspond to excitons in the specific site (H h in our case) of the moiré potential. Narrow\nh\nlines can appear for excitons localized in local minima of moiré potential or for excitons localized by\nrandom potential fluctuations within the moiré site as outlined in Ref. [34]. However, for a random\npotential the spatial extent of narrow line emission is short. The large spatial extent of narrow line\nemission indicates a deviation of the exciton energy landscape from random potential and shows that the\nexcitons are confined in moiré potential with a weak disorder, as outlined in our manuscript.\nGaAs heterostructures, where the moire potential is clearly absent, can serve as a control device, and,\nindeed, in GaAs heterostructures the narrow lines do not remain over a few microns that agrees with the\nremark of Reviewer 1. In response to this remark, we add a clarification in the manuscript: “In earlier\nstudies of narrow lines, outlined above, the spatial extension of the narrow lines was limited by the spatial\nresolution of the optical experiments, ca. 1 m. This short extension of exciton states associated with the\nnarrow lines is characteristic of disordered potentials as shown, in particular, in studies of narrow lines in\nGaAs heterostructures [2-5] where moiré potentials are absent and narrow lines originate from the\nemission of exciton states in local minima of random potential in the heterostructure.”\nI also have a concern about the reproducibility of these results. The few micron extent of the narrow line\nemission is a unique and interesting result. However, given the phenological nature of the results\npresented in this paper, similar results should be shown over multiple samples, and a lack of such results\nshould be shown on a control as discussed in the previous comment to sufficiently convince the reader of\nthis papers conclusion.\nWe checked that the long-range extension of the narrow PL lines is reproducible in multiple repeated\nmeasurements. We agree that it would be good to show similar results over multiple samples. However,\nso far the long-range extension of the narrow PL lines was realized in one sample in this work. This work\ndemonstrates the existence of the long-range extension of exciton states in TMD heterostructures.\nStudying this phenomenon in other samples is the subject for future works. Furthermore, in response to a\nremark of Reviewer 4, we added in the introduction citations of our previous works [1. Nature\nCommunications (2024) 15:9454; 2. Nature Photonics (2024) 18, 823] and outlined the relation between\nlong-range extension of localized exciton states and efficient exciton transport. The efficient exciton\ntransport was also realized in one sample, the same sample where the long-range extension of the narrow\nPL lines was observed. In response to this remark of Reviewer 1, we add a clarification in SI: “So far, the\nlong-range extension of the narrow PL lines and the corresponding exciton states was realized in one\nsample in this work. Other studies of GaAs and TMD heterostructures show short extension of the narrow\nlines, below the spatial resolution of the optical measurements, as outlined in the main text. A shorter\nextension of exciton states likely originates from stronger disorder. This work demonstrates the existence\nof the long-range extension of exciton states in TMD heterostructures. Studying this phenomenon in other\nsamples is the subject for future works. As outlined in the introduction in the main text, the long-range\nextension of the narrow PL lines is observed in the heterostructure presenting the long-range IX transport\n[2], the long-range IX mediated spin transport [3], and the high IX diffusivity and mean free path [4]. So\nfar, this efficient exciton transport [2-4] was realized in one sample. Studying this efficient exciton\ntransport in other samples and, in particular, verifying its relation to the long-range extension of the\nnarrow PL lines is also the subject for future works.”\nA minor comment I have pertains to figure and reference presentation. The figures have inconsistent\nletter placement, for example in Figure 1, the letters a and b appear in the top right of the plots and c\nappears in the top left. In reference [42] superfluid is misspelled. The authors should double check for\nerrors.\nWe thank Reviewer 1 for noting this. We move all letters to the top right of the plots, double check for\nerrors, and correct typos in the manuscript.\nResponse to Remarks of Reviewer 2.\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the\nNature Communications initiative to facilitate training in peer review and to provide appropriate\nrecognition for Early Career Researchers who co-review manuscripts.\nWe thank Reviewer 2 for the participation in reviewing our manuscript.\nResponse to Remarks of Reviewer 3 (Reviewer’s comments are marked by italic font, changes in the\nmanuscript are marked by blue font, references correspond to the revised manuscript).\nIn this manuscript, the authors probed the spatial extension of narrow PL emission lines of interlayer\nexcitons (IX) in MoSe2-WSe2 heterobilayer at cryogenic temperatures. By showing the correlation\nbetween the disappearance of narrow PL lines and the onset of IX transport, they showed that narrow\nlines originate from localized exciton states. Additionally, the observed spatial extension of the exciton\nstates over distances of several micrometers was attributed to the localization of excitons in moiré\npotentials with weak disorder, rather than the random potential.\nWhile the demonstration of the spatial extension of narrow emission lines over several micrometers is\ninteresting and important for obtaining periodic quantum emitters from these heterobilayers, the\nresults/evidences and discussion provided by the authors to support their claim are not sufficient enough\nfor the publicatin of the manuscript in its current version in Nature Communications.\nWe thank Reviewer 3 for noting the importance of the demonstration of the spatial extension of narrow\nemission lines over several micrometers presented in our work and for useful remarks. We address\nremarks of Reviewer 3 in the revised manuscript as outlined below.\nIn the following, I give some additional comments and questions:\n1. Temperature-dependent PL measurements need to be performed along the extended region of the\nnarrow emission lines to see whether they show the same behavior or not at different locations. (Major)\nTo address this remark of Reviewer 3, we performed Temperature-dependent PL measurements. These\nmeasurements show that the narrow emission lines show the same behavior at different locations. To\noutline this, we add in SI new Fig. S4 and its description: “x-Energy maps measured at different\ntemperatures are shown in Fig. S4. These measurements show that the long-range extension of the narrow\nlines is also observed at higher temperatures, the energies of the narrow lines practically do not change\nwith temperature, and the intensities of the narrow lines reduce with increasing temperature, similarly at\ndifferent locations.”\n2. How do the lifetimes of localized interlayer excitons change spatially in the extended region of the\nnarrow lines? In the case of moiré potentials with weak disorder, one can expect to obtain a uniform\nlifetime along the extended region. Did the authors measure the lifetimes along the extended region of the\nnarrow lines? The position-dependent lifetime mapping would also be valuable in further strengthening\ntheir claim of having moiré potentials with weak disorder.(Major)\nTo address this remark of Reviewer 3, we performed lifetime measurements with the available\ninstrumentation. Unfortunately, in these measurements, the signal is insufficient to measure the position-\ndependent lifetime for a single selected narrow line. Therefore, we measured the position-dependent\nlifetime for spectrally integrated PL. The spectral integration combines in the measurements all narrow\nlines in the range E < 1.46 eV of the PL spectrum. Lifetimes of different narrow lines may be different.\nThe x-Energy maps in the manuscript show the emergence of different narrow lines at different positions\nthat may contribute to variations of the spectrally integrated lifetime along x. Our measurements show\nthat these variations are rather small, roughly within the accuracy of the measurements. To outline the\nlifetime measurements, we add in SI new Fig. S5 and its description: “We measured the PL lifetime using\nexcitation by a pulsed semiconductor laser and detecting the signal by APD. In these measurements, the\nsignal is insufficient to measure the position-dependent lifetime for a single selected narrow line.\nTherefore, we measured the position-dependent lifetime for spectrally integrated PL (Fig. S5). The\nspectral integration combines in the measurements all narrow lines in the range E < 1.46 eV of the PL\nspectrum. Lifetimes of different narrow lines may be different. The x-Energy maps in this work show the\nemergence of different narrow lines at different positions that may contribute to variations of the\nspectrally integrated lifetime along x. The measured lifetime varies roughly within the accuracy of the\nmeasurements along the heterostructure (Fig. S5). The measurements of the position-dependent lifetime\nfor a single selected narrow line is the subject for future work.”\n3. The presence of uniformity of the moiré pattern should be verified through one of the imaging\ntechniques, such as TEM, piezoresponse force microscopy, etc. (Major)\nWe do not perform TEM, piezoresponse force microscopy, etc. measurements, in particular, because the\nstudied sample is a unique sample, which shows the long-range IX transport [57], the long-range IX\nmediated spin transport [58], the high IX diffusivity and mean free path [36], as we outline in the revised\nintroduction in response to a remark of Reviewer 4, and the long-range spatial extension of the narrow\nlines considered in this manuscript. We hesitate risking this unique sample by performing the above\nmeasurements, which may compromise the sample quality. Therefore, we limit the measurements by the\noptical measurements outlined in this manuscript. To address this remark of Reviewer 3, we add a\nclarification in SI: “This work demonstrates the existence of the long-range extension of exciton states in\nTMD heterostructures. Studying this phenomenon in other samples and by other experimental techniques\nis the subject for future work.”\nResponse to Remarks of Reviewer 4 (Reviewer’s comments are marked by italic font, changes in the\nmanuscript are marked by blue font, references correspond to the revised manuscript).\nThis paper appears to be a sequel to a series of publications by the same group of authors [1. Nature\nCommunications (2024) 15:9454; 2. Nature Photonics (2024) 18, 823]. The subject matter studied is of\nhigh interest to the scientific community. I would support the publication of this manuscript if the authors\ncan provide additional information to clarify the following issues:\nWe thank Reviewer 4 for a high estimate of our work and useful remarks. We address remarks of\nReviewer 4 in the revised manuscript as outlined below.\n1. In the introduction, the authors should cite their previously published papers on the same subject and\nexplain what new information the current study provides. This will give readers a clearer perspective on\nthe continued development of this research direction.\nTo address this remark of Reviewer 4, in the introduction, we add citations of our previous works [1.\nNature Communications (2024) 15:9454; 2. Nature Photonics (2024) 18, 823] and outline the relation\nbetween long-range extension of localized exciton states and efficient exciton transport that explain what\nnew information the current study provides: “A strong disorder potential both limits the spatial extension\nof localized exciton states and reduces the diffusivity of delocalized excitons. Therefore, heterostructures\nwith longer extension of localized exciton states may provide more efficient exciton transport with higher\ndiffusivity and mean free path. The long lifetimes of IXs give an opportunity to realize long-range\nefficient exciton transport and IX transport in TMD heterostructures is intensively studied [36-58]. Recent\nstudies show the long-range IX transport [57], the long-range IX mediated spin transport [58], and the\nhigh IX diffusivity and mean free path [36]. In this work we verify, in particular, if the heterostructure\npresenting this efficient exciton transport [36,57,58] is characterized by a long spatial extension of\nlocalized exciton states.” “The long-range extension of exciton states facilitates efficient exciton transport\nin the heterostructure.” Refs. [57,58] in this addition are [1. Nature Communications (2024) 15:9454; 2.\nNature Photonics (2024) 18, 823] noted by Reviewer 4. We also add citations of these works [57,58] and\nrelated clarification on p. 4: “the weakness of disorder in the moiré potential, revealed by the long-range\nextension of the exciton states (Figs. 2 and 3), suggests an opportunity to observe the long-range ballistic\ntransport of excitons in this weakly disordered moiré potential. The studies of exciton transport in the\nsame heterostructure show the long-range IX transport [57], the long-range IX mediated spin transport\n[58], and the high IX diffusivity and mean free path [36].”\n2. The authors state that \"the disappearance of narrow lines correlates with the onset of IX transport.\"\nPlease specify the pump power at which this correlation occurs.\nWe add a specification of the pump power at which this correlation occurs: “the disappearance of narrow\nlines with increasing density in the range of the pump power P ~ 1 – 50 W correlates with the onset of\nex\nIX transport (Fig. 1c)”.\n3. The caption of Fig. 1 mentions that “a higher-energy IX line appearing at ∼1.41 eV at high IX\ndensities was attributed to the appearance of moiré cells with double occupancy in Ref. [41].” Please\nexplicitly mark the ∼1.41 eV line in Fig. 1, as it is not easily identifiable to readers.\nAs shown in Ref. [58] (former Ref. [41]), the high-energy line is a broad line centered at ∼ 1.41 eV and\nobserved as a shoulder for the high-density spectra considered in this manuscript. Since the high-energy\nline is not well seen in Fig. 1 and is not discussed in the text, we remove this sentence from the Fig. 1\ncaption. We thank Reviewer 4 for noting this sentence, which is not needed for this manuscript.\n4. If the narrow lines observed several microns away from the excitation spot correspond to the same\nexciton states, then their energies should be nearly identical, and the PL line shapes at different positions\nshould correlate with the center-of-mass wavefunction of the excitons in the periodic moiré potential. The\nauthors should provide a figure (as supplementary information) showing detailed line shapes of all\nnarrow lines within the relevant spectral range at various x positions. This will allow readers to examine\nwhether the lines indeed originate from the same exciton states. Such analysis will also be valuable for\nfuture theoretical work aiming to model PL line shapes and exciton diffusion characteristics in this\nsystem.\nTo address this remark, we add in SI new Fig. S9 showing line shapes of narrow lines within the relevant\nspectral range at various x positions. We also add a description outlining the nearly identical narrow line\nenergies at various x positions and outlining a possible use of these data for future theoretical work as\nnoted by Reviewer 4: “Figures S9a and S9b show the x-Energy map and the corresponding spectra of\nnarrow lines at various x positions. The lineshapes of the narrow lines are fit by the narrow Gaussians in\nFig. S9c. Figure S9d shows the detailed spectra at various x positions for the narrow line at 1.3768 eV,\nwhich extends over long distances. The detailed lineshapes show a nearly identical energy of the narrow\nline at various x positions. The measured lineshapes may be useful for future theoretical work aiming to\nmodel PL line shapes, spectral diffusion, and exciton diffusion characteristics in this system.”\n5. To better understand the influence of exciton diffusion from a tightly focused excitation spot, the\nauthors should compare PL maps obtained using a 2-μm illumination spot with PL maps obtained under\nuniform illumination (i.e., an excitation spot significantly larger than 10 μm).\nTo address this remark of Reviewer, we add in SI new Fig. S12 comparing the x-Energy PL maps\nobtained using a tightly focused excitation spot and using defocused illumination of the entire sample.\nSimilar to the measurements with the defocused laser excitation, the spectra obtained with the focused\nlaser excitation show narrow lines and allow reconstructing x-Energy map of narrow lines, however the\naccuracy is lower. We outline this in the added description: “For the x-Energy maps in Fig. 2 in the main\ntext and in Figs. S3, S4, S7-S9, we used a defocused laser excitation covering the entire heterostructure.\nThe defocused laser excitation allows obtaining x-Energy maps of narrow lines without moving the laser\nexcitation spot (Fig. S12a). For the data in Fig. S12b, we used a laser beam focused to a spot ~ 2 m in\ndiameter and moved it in the x direction to obtain spectra for various x positions. Similar to the spectra\nobtained with the defocused laser excitation, the spectra obtained with the focused laser excitation show\nnarrow lines, which are revealed by the spectrally narrow enhancements of the PL intensity (Fig. S12).\nThe spectra obtained with the focused laser excitation at various x positions allow reconstructing an x-\nEnergy map (Fig. S12b), however, the accuracy of this map is lower than the accuracy of the x-Energy\nmap measured with the defocused laser excitation where moving and precise positioning of the laser\nbeam is not needed (Fig. S12a).”\nx-E energy PL maps obtained with the defocused excitation are more accurate and their density\ndependence (shown in new Fig. S3b-d) may also give relevant information regarding the influence of\nexciton diffusion. As shown in Fig. 1c and supported by Fig. S3, narrow lines vanish with increasing\nexciton diffusion and this is probably the strongest effect of exciton diffusion on narrow lines.\nUnderstanding the effect of exciton diffusion on narrow lines is the subject for future theoretical work."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors responded to all of our comments. The paper is now suitable for publicaiton.", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "COMMENTS TO THE AUTHORS\nI thank the authors for their efforts in addressing my questions and comments. They have performed temperature dependent- PL and lifetime measurements to address my first and second questions. I have one remaining minor comment related to my first comment. However, my third comment, as detailed below has not yet been adequately addressed and still needs to be resolved before I recommend the publication of current manuscript in Nature Communications.\n\nMy additional comments and questions:\n1. For my first comment, authors performed the temperature-dependent PL measurements in the temperature range of 1.7 K to 10 K. They should also discuss their results in terms of the linewidths of the narrow emitter lines at each position to see whether they are correlated. (Minor)\n2. Complementary to the fourth comment of Reviewer 1, to answer also my third comment, authors need to fabricate another sample with (thin ~2nm) or without top hBN layer, and perform one of the characterization techniques I suggested to show the uniformity of the moiré pattern. In that way, they can also show the reproducibility of their results in another sample. It might also help to measure the lifetime of individual narrow PL lines at different positions as suggested before.", "reviewer_4": "The authors have answered all my questions. I am satisfied with the revised manuscript and supplemental material. I would support the publication of the revised version in Nature Communications."}, "author_response": "A point-by-point response to the Reviewers' comments (Reviewer’s comments are marked by italic\nfont, changes in the manuscript are marked by blue font).\nReviewer #1 (Remarks to the Author):\nThe authors responded to all of our comments. The paper is now suitable for publicaiton.\nWe thank Reviewer 1 for reviewing our manuscript and for useful comments.\nReviewer #2 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the\nNature Communications initiative to facilitate training in peer review and to provide appropriate\nrecognition for Early Career Researchers who co-review manuscripts.\nWe thank Reviewer 2 for the participation in reviewing our manuscript.\nReviewer #3 (Remarks to the Author):\nI thank the authors for their efforts in addressing my questions and comments. They have performed\ntemperature dependent-PL and lifetime measurements to address my first and second questions. I have\none remaining minor comment related to my first comment. However, my third comment, as detailed\nbelow has not yet been adequately addressed and still needs to be resolved before I recommend the\npublication of current manuscript in Nature Communications.\nWe thank Reviewer 3 for reviewing our manuscript and for useful comments. Our response to additional\ncomments and questions of Reviewer 3 is presented below.\nMy additional comments and questions:\n1. For my first comment, authors performed the temperature-dependent PL measurements in the\ntemperature range of 1.7 K to 10 K. They should also discuss their results in terms of the linewidths of the\nnarrow emitter lines at each position to see whether they are correlated. (Minor)\nIn SI section Lineshape, we’ve noted a nearly identical narrow line energy at various x positions. In\nresponse to this comment of Reviewer 3, we add a note on a nearly identical narrow line linewidth at\nvarious x positions: “The detailed lineshapes show nearly identical energy and linewidth of the narrow\nline at various x positions.”\n2. Complementary to the fourth comment of Reviewer 1, to answer also my third comment, authors need\nto fabricate another sample with (thin ~2nm) or without top hBN layer, and perform one of the\ncharacterization techniques I suggested to show the uniformity of the moiré pattern. In that way, they can\nalso show the reproducibility of their results in another sample.\nIt is not possible for us to make an additional sample with (thin ~2nm) or without top hBN layer, and …\nshow the uniformity of the moiré pattern. We would like to note that another sample with (thin ~2nm) or\nwithout top hBN layer would not have the same low disorder characteristics, in particular, because the\nhBN layer in our sample protects the MoSe /WSe layers of the heterostructure from disorder, thus\n2 2\nallowing the uniformity, and removing or substantially thinning the hBN layer would not allow keeping\nsimilar low disorder required for both the long-range extension of the narrow PL lines and the efficient IX\ntransport in transport studies in Refs. [36,57,58] in the heterostructure as outlined in our paper. The\ncurrent characterisation we've already provided within the paper supports our claims: we measured the\nspatial extension of narrow lines in PL spectra that shows the long-range spatial extension.\nIt might also help to measure the lifetime of individual narrow PL lines at different positions as suggested\nbefore.\nThe signal is insufficient for these measurements. We noted this in SI: “In these measurements, the signal\nis insufficient to measure the position-dependent lifetime for a single selected narrow line.”\nReviewer #4 (Remarks to the Author):\nThe authors have answered all my questions. I am satisfied with the revised manuscript and supplemental\nmaterial. I would support the publication of the revised version in Nature Communications.\nWe thank Reviewer 4 for reviewing our manuscript and for useful comments."}]

Decision letter