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Naomi Tabudlong Paylaga, Chang-Ti Chou, Chia-Chun Lin, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Raman Sankar, Yang-hao Chan, Shao-Yu Chen, Wei-Hua Wang

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[Monolayer indium selenide: an indirect bandgap material exhibits efficient brightening of dark excitons](https://mdr.nims.go.jp/datasets/f5328bc3-a232-49c5-bd3d-a6123b0a076e)

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Monolayer indium selenide: an indirect bandgap material exhibits efficient brightening of dark excitonsnpj | 2D materials and applications ArticlePublished in partnership with FCT NOVA with the support of E-MRShttps://doi.org/10.1038/s41699-024-00450-3Monolayer indium selenide: an indirectbandgap material exhibits efficientbrightening of dark excitonsCheck for updatesNaomi Tabudlong Paylaga1,2,3, Chang-Ti Chou 3, Chia-Chun Lin1,2,3, Takashi Taniguchi 4,Kenji Watanabe 5, Raman Sankar 6, Yang-hao Chan 3,7 , Shao-Yu Chen 8,9 &Wei-Hua Wang 3Atomically thin indium selenide (InSe) exhibits a sombrero-like valence band, leading to distinctiveexcitonic behaviors. It is known that the indirect band gap of atomically thin InSe leads to a weakemission from the lowest-energy excitonic state (A peak). However, the A peak emission ofmonolayer(ML) InSe was observed to be either absent or very weak, rendering the nature of its excitonic stateslargely unknown. Intriguingly, we demonstrate that ML InSe exhibits pronounced PL emissionbecause of the efficient brightening of the momentum-indirect dark excitons. The mechanism isattributed to acoustic phonon-assisted radiative recombination facilitated by strong exciton-acousticphonon coupling and extended wavefunction in momentum space. Systematic analysis of layer-,power-, and temperature-dependent PL demonstrates that a carrier localization model can accountfor the asymmetric line shape of the lowest-energy excitonic emission for atomically thin InSe. Ourwork reveals that atomically thin InSe is a promising platform for manipulating the tightly bound darkexcitons in two-dimensional semiconductor-based optoelectronic devices.Tightly bound excitons in two-dimensional (2D) semiconductors ascribedto the reduced dielectric screening of Coulomb interactions have attractedintensive attention from the fundamental exciton science to emergingphotonic and optoelectronic applications1–5. Bright excitons exhibit greatoscillator strengths, but their ultrashort radiative lifetimes on the sub-picosecond timescale hinder potential optoelectronic and catalytic appli-cations that require a long exciton lifetime. In contrast, dark excitons, suchas spatial- and momentum-indirect excitons, have a higher populationdensity and a longer lifetime of approximate nanoseconds; nevertheless,these excitons are optically inaccessible due to their weak coupling to light.Therefore, exploring mechanisms that can effectively brighten the darkexciton is crucial for extending the functionality of indirect gap materials6,7.Layered III–VI post-transition metal chalcogenides are semi-conductors with distinctive optical properties, including a strong layer-dependence of the optical band gap8–11, broad spectral response12–14, indirectband structures10,15,16, and strong photoresponsivity17,18. The band edge ofatomically thin indium selenide (InSe) is considered indirect, owing to aweak inversion of the hole dispersion of the maximum-energy valenceband10,15,16,19,20. Figure. 1a shows the electronic band structure of monolayer(ML) InSe calculated by employingGWapproximation to thequasi-particleenergy. The topmost valence band (v1) has a sombrero-like band edge10,15,16,in contrast to the lower valence bands with parabolic dispersions and bandmaxima at the Γ point. Next, we focus on two bright excitons and amomentum-indirect dark exciton: A and B excitons are bright excitons,consisting of electrons at the lowest conduction band (c1) and holes at the v1and second-to-topmost valence band (v2), respectively. The sombrero-likevalence band leads to a most energetic-favorable and highly degeneratedmomentum-indirect dark excitons at the van-Hove singularity around the Γpoint (marked as dark excitons in Fig. 1a), leading to intriguing opticalproperties in ML InSe19. In particular, experimental demonstrations of Aexciton luminescence of ML InSe are still under debate, as recent worksreported either undetectable8,9,15 or very weak signals10,21. Therefore,1Molecular Science Technology Program, Taiwan International Graduate Program, Academia Sinica, Taipei 10617, Taiwan. 2National Central University, Zhongli,Taoyuan 320317, Taiwan. 3Institute of Atomic andMolecular Sciences, AcademiaSinica, Taipei 10617, Taiwan. 4ResearchCenter forMaterials Nanoarchitectonics,National Institute forMaterials Science, 1-1Namiki, Tsukuba305-0044, Japan. 5ResearchCenter for Electronic andOpticalMaterials, National Institute forMaterialsScience, 1-1 Namiki, Tsukuba 305-0044, Japan. 6Institute of Physics, Academia Sinica, Taipei 115201, Taiwan. 7Physics Division, National Center of TheoreticalSciences, Taipei 10617, Taiwan. 8Center of Atomic Initiative for New Materials, National Taiwan University, Taipei 10617, Taiwan. 9Center for Condensed MatterSciences, National Taiwan University, Taipei 10617, Taiwan. e-mail: yanghao@gate.sinica.edu.tw; shaoyuchen@ntu.edu.tw; wwang@sinica.edu.twnpj 2D Materials and Applications |            (2024) 8:12 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41699-024-00450-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41699-024-00450-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41699-024-00450-3&domain=pdfhttp://orcid.org/0000-0002-7189-6284http://orcid.org/0000-0002-7189-6284http://orcid.org/0000-0002-7189-6284http://orcid.org/0000-0002-7189-6284http://orcid.org/0000-0002-7189-6284http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-4702-2517http://orcid.org/0000-0003-4702-2517http://orcid.org/0000-0003-4702-2517http://orcid.org/0000-0003-4702-2517http://orcid.org/0000-0003-4702-2517http://orcid.org/0000-0002-9113-5319http://orcid.org/0000-0002-9113-5319http://orcid.org/0000-0002-9113-5319http://orcid.org/0000-0002-9113-5319http://orcid.org/0000-0002-9113-5319http://orcid.org/0000-0003-3423-9768http://orcid.org/0000-0003-3423-9768http://orcid.org/0000-0003-3423-9768http://orcid.org/0000-0003-3423-9768http://orcid.org/0000-0003-3423-9768http://orcid.org/0000-0003-1737-6648http://orcid.org/0000-0003-1737-6648http://orcid.org/0000-0003-1737-6648http://orcid.org/0000-0003-1737-6648http://orcid.org/0000-0003-1737-6648mailto:yanghao@gate.sinica.edu.twmailto:shaoyuchen@ntu.edu.twmailto:wwang@sinica.edu.twinvestigating the luminescent properties of ML InSe and understanding theexcitonic behavior of the indirect excitons is essential.In this work, we investigate the excitonic properties of atomicallythin InSe by employing photoluminescence (PL) spectroscopy. Weobserve robust PL emission originating from amomentum-indirect darkexciton in a high-quality hexagonal boron nitride (h-BN)-encapsulatedML InSe at T ¼ 36 K, despite that an indirect band structure typicallyleads to weak luminescence. Additionally, we detect an apparentasymmetric PL line shape in the lowest-energy emission for atomicallythin InSe at cryogenic temperatures, suggesting that the long-lived darkexcitons undergo carrier localization. By analyzing the PL emission witha carrier localization model, the energy of the A peak of ML InSe can beextracted as 2.58 eV, consistent with the excitation energy of the Aexciton in our calculation.Results and discussionThe γ-phase InSe crystals are grownby theBridgmanmethod22.We employh-BN flakes to encapsulate the atomically thin InSe to ensure high samplequality. The InSe and h-BN flakes are mechanically exfoliated and stackedusing the dry transfer method23 in a nitrogen environment. An inertenvironment for exfoliation and stacking is essential for maintaining thecrystalline quality, especially for atomically thin InSe24,25. Figure. 1b shows aschematic of an h-BN-encapsulated InSe and an optical micrograph of atypical h-BN-encapsulated InSe sample in upper and lower panels,respectively. The details of sample preparation and atomic forcemicroscopytopography of the InSe sample are provided in Supplementary Note 1. Thecrystalline quality of h-BN-encapsulated ML and bilayer (BL) InSe ischaracterized by Raman spectroscopy, as shown in Fig. 1c. Notably, weobserve four intrinsic Raman peaks from both ML and BL InSe at 116.1cm�1, 178.3 cm�1, 199.4 cm�1, and 227.1 cm�1, corresponding to A01 1ð Þ,E00 2ð Þ,A002 1ð Þ, andA01 2ð Þ vibrationalmodes26,27, respectively, confirming theirhigh crystalline quality.To study the PL emission of ML InSe, we employ a cavity-stabilizeddiode laser at 3.05 eV as the excitation light source (SupplementaryNote 1).Figure 1d shows the spectra of the lowest energy PL emission of h-BN-encapsulated InSe from ML to 18 L at T ¼ 36 K. The PL energies corre-sponding to the lowest-energy emissions, denoted as A peaks, are in goodagreement with previous reports for bulk and few-layer InSe9,28. The layerdependence of the A peaks significantly blueshift with decreasing thicknessdue to the quantum confinement effect8–10,21,29. Intriguingly, we observe thepronounced A peak of ML InSe. Moreover, in addition to A peaks, weobserve distinct PL emissions from B excitons at higher energy regimes,denoted hereafter as B peaks, down to BL InSe (Supplementary Note 2),highlighting the ultrawide spectral response from 1.3 eV to 2.9 eV of InSefromML tobulk. Figure 1e summarizes thePLenergiesof theAandBpeaksfor atomically thin and bulk InSe, as well as calculated excitation energies ofA excitons for ML, BL, and bulk InSe. The calculation detail of theabsorption spectra is described inSupplementaryNote 3.The energies of theA peaks for ML, BL, and bulk InSe coincide well with the calculated tran-sition energies, strongly substantiating the assignment of the A peak ofML InSe.Figure 1d further reveals the layer dependence of the PL intensities:MLand BL InSe manifest pronounced PL of A peak, and the PL intensitygenerally decreaseswith the increasingnumberof layers.Due to the in-planemirror symmetry of InSe, the A transition from in-plane photoexcitationassociated with the s to the pz orbitals is prohibited. However, spin-orbitcoupling (SOC) can mix the px; py and pz orbitals, enabling A exciton tocouple to in-plane polarized light30. This SOC strength increases withdecreasing thickness, rendering a maximum effect in ML InSe30. It is notedthat in our measurement setup with the objective lens in the out-of-planeFig. 1 | Band structures and photoluminescence spectra of atomically thin InSe.a Electronic band structure ofML InSe with the transition of bright A and B excitonsand momentum-indirect dark exciton denoted. b Upper panel: Schematic of h-BNencapsulated ML InSe. Lower panel: Optical micrograph of a typical h-BN encap-sulatedML InSe sample. The area of ML InSe is outlined by a white dashed line. Thescale bar is 10 µm. c Raman spectra of the ML and BL InSe. d PL spectra of the Apeaks of InSe from ML to 18 L. e Extracted PL peak energies of the A and B peaksalong with the theoretically calculated excitation energies of the A excitons of ML,BL, and bulk InSe.https://doi.org/10.1038/s41699-024-00450-3 Articlenpj 2D Materials and Applications |            (2024) 8:12 2direction, the in-plane polarized light is effectively collected from thesamples, facilitating the detection of A peak emission from ML InSe.The PL of indirect transition governed by phonon-assisted processes istypically weak. The pronounced luminescence from A peak of ML InSesuggests the strong exciton-phonon coupling that efficiently brightens thedark exciton. Figure 2a, b plot the T-dependent PL spectra for ML and BLInSe, respectively. The observed PL peak continuously evolves fromT ¼ 36K up to higher T, indicating that same type of exciton is involved. Figure 2c,d show the T dependence of the integral intensity of A peak of ML and BLInSe, respectively, revealing amonotonic increasewith decreasingT. SuchTdependence has been reported for the indirect PL emission in the studies ofother 2D semiconductors31,32. At T ¼ 36 K, absorbing LA phonons in MLInSe can efficiently upconvert the dark excitons to the bright A exciton statebecause of matching momentum and energy. However, phonons can alsolead to non-radiative recombination, competing with the radiative pro-cesses. Our results suggest that the non-radiative recombination becomesdominant at higherT range, resulting in suppression of thePL intensitywithincreasing T.In Figure. 2e, we compare the normalized PL spectra of the A peak ofML InSewith the excitationpowers of 1W/cm2 and300W/cm2.ThePL lineshape is almost identical over an excitation power range of more than twoorders of magnitude. In addition, we observe no additional emission at thelow energy side, which can be associated with other mechanisms, such asbiexcitonic recombination, secondary radiative recombination, and defect-bound excitonic states33,34. ML InSe exhibits a small blueshift, which can beattributed to state filling of the photoexcited carriers35–37.We plot the powerdependence of the PL intensities of theApeaks ofML andBL InSe in Fig. 2f.The PL intensities of bothMLandBL InSe can bewell-described by a powerlaw of the exponent of 0.96 and 1.02.We thus can rule out that the observedPL emission of ML and BL InSe is due to defect-bound excitons or biexci-tons because they exhibit a sublinear33 or superlinear power law34,38,respectively.Next, we discuss the spectral line shape of PL emission and its indi-cation of the excitonic properties of the atomically thin InSe. Figure 3a plotsthenormalizedPL spectra from theApeaksofML to18 L InSe to emphasizethe evolution of line shapes. A peak ofML to 10 L InSe exhibits asymmetricline shapes with low-energy tails and steep high-energy shoulders. In con-trast, the A peak of 18 L and B peaks (detailed in SupplementaryNote 2) aresymmetric. To analyze these asymmetric line shapes of the A peaks, weemploy a carrier localization model39, as depicted in Fig. 3b, c. This modelaccounts for a spatial fluctuating potential, leading to a randomdistributionof surface potential energy. This potential fluctuation in the conduction andvalence bands causes spatial variations in exciton energy. We consider twocompeting processes of exciton dynamics: (1) carrier localization, whereFig. 2 | Excitonic properties of atomically thin InSe. a, b The normalized PLspectra of ML and BL InSe at different temperatures. The PL spectra are shiftedvertically for clarity. c, d The temperature dependence of PL intensity of A peak ofML and BL InSe. eNormalized PL spectra of ML InSe taken at 1W/cm2 and 300W/cm2. f Excitation power dependence of PL intensities of the A peaks of ML and BLInSe. The black line shows the linear power law fittings, yielding exponents αML andαBL to be very close to unity for ML and BL InSe.https://doi.org/10.1038/s41699-024-00450-3 Articlenpj 2D Materials and Applications |            (2024) 8:12 3photoexcited electrons and holes migrate to local energy minima (char-acteristic time ~ τC) and (2) exciton radiative recombination (radiativerecombination time ~ τR). Here, all other decay paths of photoexcitedelectrons and holes are assumed to be negligible.Quantitatively, in the carrier localization model, the PL line shape canbe described by the following equation39:IðEÞ ¼ y0 þ I0ðEÞexp�τR2τCerfcEexc � Effiffiffi2pσE� �� �� �ð1ÞwhereEexc is the excitonpeak energy, σE is the average potentialfluctuationsenergy, and I0 Eð Þ is the exciton population function.We approximate I0 Eð Þby aVoigt function40,41, which consists of the contributions of homogeneous(wL) and inhomogeneous broadening (wG) with Lorentzian and Gaussianforms, respectively. The Voigt function can be written asI0ðEÞ ¼ A2ln2π3=2wLw2GZ 1�1e�t2ffiffiffiffiffiffiffiffiffiffiffiln2 wLwGq� �2þffiffiffiffiffiffiffiffiffi4ln2p x�xcwG� t� �2 dt ð2Þwhere xc is the peak center. There are 5 free parameters in the Voigt-basedcarrier localizationmodel, including Eexc, σE ,wL,wG, and τR=τC . As shownin Fig. 3d, the PL spectra of the A peaks for ML and BL InSe are well-fitted,validating the carrier localization model to account for the asymmetric PLline shape.Figure 3e shows the fitting results of PL spectra of ML InSe along withthe corresponding Voigt and error functions in log scale. Without carrierlocalization (τC≫τR, the excitonic PL emission exhibits typically symmetricline shape. However, for atomically thin InSe, the excitonic ground state ismomentum-indirect, leading to amuch longer radiative lifetime than brightexcitons with a direct band gap. For ML InSe, the dark excitons are char-acterized by a long radiative recombination lifetime; τR is estimated to be onthe order of 10 ns (SupplementaryNote 4). Also, previous time-resolved PLmeasurements had shown a long radiative lifetime of 8 ns for few-layerInSe28, suggesting that τR may significantly greater than τC . Under suchconditions, the carrier localization effect takes place: Higher energy excitonsmigrate to lower energy sites before direct recombination, leading to sup-pressed exciton recombination at the high-energy side, as illustrated inFig. 3b, c. The energy redistribution is observable through the PL line shape,providing valuable insights into the excitonic behavior in atomically thinFig. 3 | PL line shape analysis of atomically thin InSe and carrierlocalizationmodel. aNormalized PL spectra of theApeaks of InSe fromML to 18 L.b Schematic of the carrier localization model when the carrier localization time issmaller than the recombination time. c PL spectra of the A peaks of ML and BL InSeare fitted by the carrier localization model. d Asymmetric PL line shape of ML InSeand various functions employed for fitting. e A schematic diagram depicts thepresence of potentialfluctuations-induced energy redistribution and the asymmetricline shape.https://doi.org/10.1038/s41699-024-00450-3 Articlenpj 2D Materials and Applications |            (2024) 8:12 4InSe layers. Significantly, our fitting results of the asymmetric PL emissionsupport the long τR of the momentum-indirect dark excitons in atomicallythin InSe.We note that the carrier localizationmodel is not exclusive to excitonsbut is equally applicable to other quasiparticles, for instance trions andbiexcitons, in 2D semiconductors that exhibit a similar interplay of differentrecombination processes. This is particularly relevant in atomically thin 2Dmaterials, where the potential fluctuations in the atomic layers are greatlysusceptible to extrinsic effect due to a weak dielectric screening and a sub-stantial surface-to-volume ratio. Nonetheless, it is noted that the carrierlocalization effect becomes negligible when τR is substantially shorter thanτC . For excitons with short τR, such as bright excitons, and 2D materialswith uniform potential energy and thus long τC , the carrier localizationmodel become no longer applicable.To assess the role of phonons participating in the luminescence, weextract the PL peak energy forML and BL InSe as a function of temperature(T) using carrier localizationmodel, as shown in Fig. 4a.While the PL peakpositions of both ML and BL InSe show a monotonically decreasing trend,ML InSe manifests a stronger T dependence compared with BL InSe. Thesample temperature is calibrated to quantitatively access the phonon effect(SupplementaryNote 5). TheT-dependent PLpeak energy can be describedby O’Donnell-Chen model42–44:EG Tð Þ ¼ EG 0ð Þ � SEph cothEph2kBT� �� 1  ð3ÞwhereEG 0ð Þ represents theoptical bandgapatT ¼ 0 K,S is adimensionlessconstant associated with the strength of exciton-phonon coupling, and Ephis the average phonon energy. The ML and BL InSe data are well describedby this equation, yielding S= 14 and S = 3 and Eph = 23.5meV and23.1meV for ML and BL InSe, respectively. Figure 4b plots the S values forML, BL, 3 L, and 18 L InSe; The fitting parameters are summarized inSupplementary Note 6. Our results show a distinct enhancement of S as thethickness of InSe decreases, with a significantly large S for ML InSe. Thelarge exciton-phonon coupling strength of ML InSe can be attributed tostrong scattering of hole inML InSe by longitudinal acoustic (LA) phononsat the sombrero-like valence band maximum, in contrast to much weakerLA scattering for BL and bulk InSe45. Moreover, electrons in ML InSe issubjected to large scattering from the longitudinal optical (LO) phonons,compared with the one in bulk InSe45, further enhancing the exciton-phonon coupling. In addition, strong electron-phonon coupling inML InSehas also been attributed to a large deformation potential resulting fromheavy holes46 or a combination of strong piezoelectric properties and a lackof inversion symmetry47.The phonon-assisted brightening of the indirect dark excitons is fur-ther examined by considering the exciton band structure in ML InSe. Wecalculate the exciton dispersion of ML InSe, and the two lowest-energybands associated with bright A excitons are shown in Fig. 4c. The first bandexhibits an energy minimum at Qmin 0.2 Å−1, originating from the weaklyinverted, topmost valence band, as shown in Fig. 1a. The weakly invertedvalence band is attributed to the vanishing momentum matrix elementbetween the topmost valence band and the lowest conduction band fromFig. 4 | Strong exciton-phonon coupling and brightening of the indirect excitonsofML InSe. aTemperature dependence of the PL peak energies of theA peaks ofMLand BL InSe. The solid curves are fitted by O’Donnell-Chen model as described inEq. (3). b Extracted exciton-phonon coupling strengths of ML, BL, 3 L, and 18 LInSe, showing an enhancement trend with a decreasing layer number and strongexciton-phonon coupling forML InSe. c Exciton dispersion ofML InSe showing thetwo low excitonic bands associated with the A exciton atQ ¼ 0. d Envelope functionof the A exciton of ML InSe in momentum space.https://doi.org/10.1038/s41699-024-00450-3 Articlenpj 2D Materials and Applications |            (2024) 8:12 5k � p theory48. The dark exciton at the lowest energy, denoted as the kmaxΓexciton, consists of the hole located at the kmax and the electron at the Γvalley. From the exciton dispersion, the corresponding activation energy(εact) is defined as the energy difference between the optically active state atQ ¼ 0 and the Qmin exciton state. ML InSe exhibits a very small εact of8meV, which is easily susceptible to excitation, leading to highly probablephonon-assisted emission. For a tightly bound exciton in real space with atiny Bohr radius, its exciton distribution greatly extends in momentumspace49, which can enhance exciton-phonon coupling. To be quantitative,we calculate theA exciton envelop function ofML InSe, as shown in Fig. 4d.The exciton envelope function ranges well beyond k 0.15 Å-1 from the Γpoint, signifying a large overlap with the lowest-energy bands of bright Aexcitons at Q ¼ 0, facilitating efficient phonon-assisted radiative recombi-nation. It is noted that the aforementioned strong exciton-phonon couplingofML InSe favors ahot phononeffect50–52.Moreover, fromthe calculatedLAphonon of ML InSe, a dispersion of approximately 7meV with k 0.3 Å-1along theΓ-Kdirection45,52,53 canprovide theneeded energy andmomentumto activate themomentum-indirect kmaxΓ exciton through acoustic phononabsorption.Finally, we discuss the carrier localization effects in InSe with differentnumber of layers. Figure 5a compares the τR=τC of the A peak fromML to4 L InSe. All the τR=τC for the A peaks are larger than unity, substantiatingthe dominance of carrier localization in the relaxation path τC < τR (Sup-plementary Note 7). Moreover, the τR=τC increases with decreasingthickness up to τR=τC 14 for ML InSe. The inset of Fig. 5a shows theextracted average potential fluctuations energy (σE) of ML to 4 L InSe,revealing that σE increases with decreasing layer number. Because ofreduced screening in thinner 2D materials, the larger σE may be associatedwith the shorter τC of thinner InSe, which can account for the layer numberdependence of the τR=τC . We note that the σE ofML InSe is approximately100meV, which is comparable to the dielectric disorder resulting fromvariation in thedielectric screeningof h-BNand scatteredpolymer residue54.The dominance of carrier localization effects on A peak is due to thelong lifetime. Figure 5b shows the PL spectra of theA andBpeaks for the 3 LInSe sample.Asdiscussed above, the asymmetric line shapeof theApeaksof3 L InSe suggests that carrier localization dominates the recombinationpath. We found that the PL of the B peaks of 3 L InSe exhibits much lessasymmetry,which can also bewell-fittedwith the carrier localizationmodel.Figure 5c compares the layer dependence of τR=τC for the A and B peaks of2–4 L InSe, signifying that the τR=τC of the A peaks is larger than that of theB peaks. It is noted that τC for the A and B peaks of the same InSe sampleshould be similar because the localization time is determined by the mag-nitude of the potentialfluctuation and is insensitive to the band structure. Inother words, τR becomes the determining factor to govern the value ofτR=τC in theA andBpeaks. Because theApeak PL emission arises from themomentum-indirect exciton that exhibits a much longer lifetime, it can beinferred that A peak exhibits a larger τR=τC as shown in Fig. 5a, leading todifferent extents of asymmetry for the A and B peaks as shown in Fig. 5b.In Fig. 5c, we further compare the observed τR=τC values with those ofother material systems to which the carrier localization model isapplied39,55–57. Generally, when τC < τR is valid and carrier localization is thedominant relaxation path, the carriermigration is relatively fast, signifying ahigh diffusion coefficient. For InSe, the relatively large diffusion coefficient(~10 cm2/s) and high carrier mobility58 thus favor the carrier localizationeffect. We found that the A peaks of ML and BL InSe exhibit much higherτR=τC than quantum wells composed of compound semiconductors39,55–57,suggesting the distinct properties of the potential fluctuation and carrierrecombination associated with the tightly bound indirect excitons of 2Dsemiconductors.In addition to the carrier localization model, we note that the virtualphonon relaxation process32 with multiple phonon scattering cannot becompletely excluded to explain the asymmetric PL line shape. However, thecarrier localization model with the Voigt function for exciton population isfound to comprehensively account for the extended layer-, temperature-,and power-dependent PL spectra of atomically thin InSe, hence justifies ourchoice of the fitting model.In conclusion, the PL of high-quality, h-BN-encapsulated InSe sam-ples, ranging from ML to bulk, is systematically measured and analyzed.Notably, we observe strong PL emission from the A peak of ML InSe,although its band structure suggests a dark exciton. The brightening of themomentum-indirect exciton of ML InSe can be attributed to the phonon-assisted process facilitated by the strong exciton-phonon coupling and theFig. 5 | Comparison of the A and B peaks of atomically thin InSe and the carrierlocalization model. a Extracted τR=τC from ML to 4 L InSe at T = 36 K. Inset: theextracted average potential fluctuations energies of atomically thin InSe fromML to4 L. b PL spectra of the A and B peaks of 3 L InSe and the fitting curves by the carrierlocalization model. c Comparison of the τR=τC extracted from the A and B peaks ofatomically thin InSe from BL to 4 L and other quantum wells composed of com-pound semiconductors.https://doi.org/10.1038/s41699-024-00450-3 Articlenpj 2D Materials and Applications |            (2024) 8:12 6extendedwave function inmomentum space. Furthermore, atomically thinInSe exhibits an apparent asymmetric PL for A peaks, which can be wellaccounted for by the carrier localization model with spatial potential fluc-tuation.Wedemonstrate the distinct excitonic behaviors, including a strongexciton-phononcoupling, a long exciton lifetime, anda large binding energycoexisting in the ML III-VI metal chalcogenides. In the context of theemerging interest in the excitonic physics of 2D semiconductors, our resultspave the way for developing dark-exciton-mediated energy harvesting,quantum catalysts, and optoelectronics.MethodsSample fabricationBottom h-BN flakes were mechanically exfoliated on 300 nm SiO2/n-Sisubstrates. The polymer residue from the tape was removed by annealing ina furnace at 500 °C in Ar/O2 mixed gas. The InSe flakes were mechanicallyexfoliated on a PDMS gel film in a glovebox in a nitrogen environment andthen transferred to the bottom h-BN through the dry transfer method. Tocomplete the structure, h-BN flakes on PDMS were transferred onto theInSe/h-BN stacking with careful alignment to ensure encapsulation.Optical spectroscopy measurementsThe samples weremounted in a Janis (ST-500) cryostatmaintained at a highvacuum (5× 10�6 Torr). The photoexcitationwas performedwith a custom-built cavity-stabilized diode laser with a wavelength of 407 nm. We employfused-silica optics to reduce the residual luminescence fromglassy optics.Weemployed a 50x objective lens (NA: 0.5) to focus the laser to a spot size ofapproximately ~1 μmon the sample and to collect the PL signal under back-scattering geometry. The collected signal was directed to a Horiba iHR550 spectrometer and then dispersedwith reflective grating (150 lines/mmor600 lines/mm for PL; 1800 lines/mm for Raman spectroscopy) and detectedby the liquid-nitrogen cooled charge-coupled device of Horiba Symphony IIdetection system. An additional 537 nm longpass filter was added to thecollectionpathduring themeasurementof thebulkAtransitionPL topreventsignal mixing from the B transition overtone. The Raman spectroscopy isperformed with a diode-pumped solid-state laser with an energy of 2.33 eV.Data availabilityAll data supporting the findings of this study are included in the paper andits Supplementary Information files. 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Y.-h.C. acknowledges support byNational Center for High-Performance Computing in Taiwan. S.-Y.C.acknowledges support from the Center of Atomic Initiative for NewMaterials, National Taiwan University (Grant No. 111 L9008, 112 L9008,and 113 L9008) and from the Featured Areas Research Center Programwithin the framework of the Higher Education Sprout Project by the Min-istry of Education of Taiwan.Author contributionsN.T.P., S.-Y.C., and W.-H.W. conceived and designed this project. N.T.P.fabricated the deviceswith assistance fromC.-C.L. S.-Y.C. constructed thePL and Raman spectroscopy measurement setup. N.T.P. and S.-Y.C.performed the optical measurements. N.T.P., S.-Y.C., and W.-H.W.performed data analysis. K.W. and T.T. provided the h-BN crystals. R.S.provided InSe crystals. C.-T.C., J.R., and Y.-h.C. performed theoreticalmodeling and calculations. The manuscript was written through the con-tributions of all authors. All authors have approved the final version of themanuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41699-024-00450-3.Correspondence and requests for materials should be addressed toYang-hao Chan, Shao-Yu Chen or Wei-Hua Wang.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.https://doi.org/10.1038/s41699-024-00450-3 Articlenpj 2D Materials and Applications |            (2024) 8:12 8https://doi.org/10.1038/s41699-024-00450-3http://www.nature.com/reprintsOpen 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) 2024https://doi.org/10.1038/s41699-024-00450-3 Articlenpj 2D Materials and Applications |            (2024) 8:12 9http://creativecommons.org/licenses/by/4.0/ Monolayer indium selenide: an indirect bandgap material exhibits efficient brightening of dark excitons Results and discussion Methods Sample fabrication Optical spectroscopy measurements Data availability References Acknowledgements Author contributions Competing interests Additional information