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Saroj B. Chand, John M. Woods, Jiamin Quan, Enrique Mejia, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Andrea Alù, Gabriele Grosso

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[Interaction-driven transport of dark excitons in 2D semiconductors with phonon-mediated optical readout](https://mdr.nims.go.jp/datasets/d1b99de6-4d18-46c5-91c5-df1a86964eb4)

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Interaction-driven transport of dark excitons in 2D semiconductors with phonon-mediated optical readoutArticle https://doi.org/10.1038/s41467-023-39339-yInteraction-driven transport of dark excitonsin 2D semiconductors with phonon-mediated optical readoutSaroj B. Chand1, John M. Woods1, Jiamin Quan1, Enrique Mejia1,Takashi Taniguchi 2, Kenji Watanabe 3, Andrea Alù 1,4,5 &Gabriele Grosso 1,5The growingfield of quantum information technology requires propagation ofinformation over long distances with efficient readout mechanisms. Excitonicquantum fluids have emerged as a powerful platform for this task due to theirstraightforward electro-optical conversion. In two-dimensional transitionmetal dichalcogenides, the coupling between spin and valley provides excitingopportunities for harnessing, manipulating, and storing bits of information.However, the large inhomogeneity of single layers cannot be overcome by theproperties of bright excitons, hindering spin-valley transport. Nonetheless, therich band structure supports dark excitonic states with strong binding energyand longer lifetime, ideally suited for long-range transport. Here we show thatdark excitons can diffuse over several micrometers and prove that thisrepulsion-driven propagation is robust across non-uniform samples. The long-range propagation of dark states with an optical readout mediated by chiralphonons provides a new concept of excitonic devices for applications in bothclassical and quantum information technology.Since the discovery of excitonic resonances inMoS21,2, transitionmetaldichalcogenides (TMDs) have emerged as a promising platform forapplications in classic and quantum technology. The atomic-sizethickness of TMDs imbues excitons with a unique combination ofproperties that make them attractive in many scientific fields, includ-ing optoelectronics, sensing, quantum information, and spintronics3.The large binding energy makes excitons in TMDs robust at roomtemperature, and the combination of their short lifetime and in-planedipole makes them extremely bright. The broken inversion symmetryand large spin–orbit coupling gives rise to a unique spin-valley cou-pling that provides an extra degree of freedom for manipulating andstoring information4. These aspects have been mostly explored inbright excitons, which are bound states formed by an electron and ahole in electron bands with the same spin orientation at the K (or K’)valley. On one hand, bright excitons are optically active, with a largeoscillation strength and high quantum yield. However, their shortlifetime, weak interaction, and short transport range are severe lim-itations for the potential of TMDs in spintronics and quantumtechnology5. On the other hand, the band structure of TMDs gives riseto a variety of excitonic complexes6, and it allows for the formation ofseveral lower-energy dark excitons that cannot directly recombineoptically due to the non-conservation of spin or momentum. Spin-forbidden excitons are made of carriers in electron bands with oppo-site spinorientations in the samevalley (K andK’), and contrary to theirbright counterpart, they possess an out-of-plane transition dipole andtwo orders of magnitude longer lifetime7–11. Despite their dark nature,spin-forbidden excitons can inefficiently recombine optically due tothe small spin crossover of the conductionbandor through theRashbaReceived: 28 November 2022Accepted: 8 June 2023Check for updates1Photonics Initiative, Advanced Science Research Center, City University of New York, New York, NY 10031, USA. 2International Center for MaterialsNanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 3Research Center for Functional Materials, NationalInstitute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 4Department of Electrical Engineering, City College of the City University of New York,New York, NY 10031, USA. 5Physics Program, Graduate Center, City University of New York, New York, NY 10016, USA. e-mail: ggrosso@gc.cuny.eduNature Communications |         (2023) 14:3712 11234567890():,;1234567890():,;http://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/0000-0002-4297-5274http://orcid.org/0000-0002-4297-5274http://orcid.org/0000-0002-4297-5274http://orcid.org/0000-0002-4297-5274http://orcid.org/0000-0002-4297-5274http://orcid.org/0000-0002-2577-1755http://orcid.org/0000-0002-2577-1755http://orcid.org/0000-0002-2577-1755http://orcid.org/0000-0002-2577-1755http://orcid.org/0000-0002-2577-1755http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39339-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39339-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39339-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39339-y&domain=pdfmailto:ggrosso@gc.cuny.edueffect12. Moreover, the spin-flip process required for dark excitonrecombination can be controlled by an external magnetic field13,14.Although the dynamics and transport of interlayer excitons in van derWaals heterostructures15–17 and intervalley excitons18 have beenrecently investigated, the dynamics and the transport properties ofspin-forbidden dark excitons have not been fully explored yet19, due tothe challenges associated with their detection using far-field spectro-scopy techniques11,20,21. However, efficient phonon interaction pro-cesses can open recombination paths for dark excitons with out-of-plane emission that allow for efficient optical readoutwithout the needfor auxiliary external fields.In this work, we study exciton dynamics by employing high-resolution, spatially-resolved photoluminescence (PL) spectroscopytechniques that enable the observation of their transport up to severalmicrometers. We demonstrate the long-range transport of dark exci-tons by resolving spectrally their emission away from the excitationposition. We show that dark exciton transport is driven by mutualrepulsive interaction, and it depends on the exciton density at theexcitation region (see Fig. 1a). The information of the quantum state ofdark excitons is then read out via the emission of lower-energy exci-tons resulting from exciton-phonon scattering. Our conclusions arecorroborated by twomain observations: (i) the size of the dark excitoncloud increases as a function of density; and (ii) dark excitons areshown to diffuse in different energy landscapes, including flat, down-hill, and uphill. This last observation can be explained by the strongrepulsive interaction energy that overcomes the energy differencebetween the excitation and detection point. The larger interactionenergy of dark excitons compared to the bright counterpart isexplained by their higher density. The cartoon in Fig. 1a illustrates themain mechanisms and findings of our experiments. We note that thisrobust transport is peculiar to dark excitons as, differently frominterlayer excitons in heterostructures, the stronger binding energycombined with high interaction energy can compensate for largelyinhomogeneous energy landscapes. Moreover, this effect is clearlyobservable in naturally n-doped samples, indicating that it can be usedin simple device geometries without the need for external gating orcomplex heterostructures in which the exciton properties stronglydepend on the twist angle.ResultsPL spectroscopy of encapsulated WS2 monolayerFigure 1b (bottom panel) shows the measured emission spectrum atT = 7K of theWS2monolayer encapsulated within two thin hBN layers.In order to study the spin-valley coherence properties and identify theseveral excitonic complexes in the sample22,23, we excite it with a right-handed circularly polarized (σþ) laser quasi-resonant with the B exci-ton and detect the emission of right-handed circularly polarized (σþ)or left-handed circularly polarized (σ�) light. The top panel shows thechirality, defined as the degree of polarization and calculated asρ= I +�I�I + + I�, where I + and I� is the emission intensity of σþ and σ�polarized light, respectively. The sample shows several excitonic peakswhose energy and polarization degrees are in good agreement withprevious studies in n-doped WSe2 and WS222–24. Peaks labeled with X0,XX0, X�T , X�S , and XX� are the bright A neutral exciton, neutralFig. 1 | Bright anddark excitoncomplexes inmonolayerWS2. aUpon irradiation,exciton density follows a Gaussian distribution imprinted by the laser. In the high-density region, the energy of dark excitons increases due to the repulsive inter-action. The extra interaction energy drifts dark excitons allowing them to propa-gate over large areas of the sample and optically recombine far away from theexcitation spot. The rainbow ramp represents the energy landscape used in ourexperiments in which the energy increases linearly due to strain. The energy ofbright excitons does not change significantly due to the smaller density and, as aresult, they do not drift and their emission is limited to the region of the laserexcitation. b Emission spectra of WS2 at T = 7K excited with σ+ circular polarizationand collected with σ+ (blue line) and σ− (red line). The top panel shows the chiralityof the emission ρ= I + �I�I + + I�, where I + and I� are the emission intensity of σ+ and σ−polarized light, respectively. The peaks of bright and dark exciton complexes arehighlighted by vertical dashed lines. The band compositions of the relevant darkexciton species are illustrated in the cartoon in (c–f).Article https://doi.org/10.1038/s41467-023-39339-yNature Communications |         (2023) 14:3712 2biexciton, triplet trion, singlet trion, and negatively charged biexciton.These exciton complexes are the results of the Coulomb interactionbetween X0 and carriers in the K or K’ valley. The energy detuning withrespect to the bright exciton of all exciton complexesmeasured in thiswork is in excellent agreement with previous works and is reported inSupplementary Tables S1 and S3. We identify two peaks in the low-energy shoulders of the bright trions that have been previouslyattributed to intervalley momentum-forbidden (I0) and intravalleyspin-forbidden (D0) excitons23,25. The energy difference between I0 andD0 is the result of short-range Coulomb exchange interactions in WS2,measured to be ∼6.5meV in agreement with theoretical predictions26.The presence of dark excitons is more evident in the chirality plot (toppanel of Fig. 1b), where clear features emerge at the low energy side ofX�T and X�S . We note that the chirality of D0 should be suppressed dueto its out-of-plane optical transition23. The nonvanishing chirality of D0that emerges in ourmeasurements is due to thenearbypresenceof thestrong emission of X�S and of a strongly-polarized unknown peak at∼ 3meVbelow. Amorecareful analysis returns amuch lower degreeofpolarization for D0 of 7% (see Section 3 of the SI). The peaks labeledwith TI and D− have been attributed to the intervalley momentum-forbidden and intravalley spin-forbidden dark trions22–24. The peak T1was recently ascribed to the optical recombination of D− mediated byelectron-electron intervalley scattering22,27. Several peaks are observedin the lower energy side of the spectrum and are attributed to thereplicas of D− and D0 due to the interaction with chiral phonons. Weidentify D�i as replicas of D− generated by the interaction with valleyphonons i=K1,K2,K3,Γ523,25,28,29. Theoretical calculations of the energyof these phonon modes, the rationale for the assignment of thesepeaks, and the discussion on additional peaks observed in theexperimental spectrum are reported in the Supplementary Informa-tion. The band composition of the dark exciton complexes relevant forthe rest of the discussion is illustrated in Fig. 1c–f. Note that thedynamics of D�i and T1 can be ascribed to the one of D− and D0, but,differently from D− and D0, their emission is out-of-plane23 and theyappearmore clearly in the emission spectrum of Fig. 1b. In the text, werefer to D0 and D− as dark excitons and dark trions if not otherwisespecified.Diffusion of exciton complexes in WS2First, we study howdifferent exciton populations diffuse bymeasuringthe size of their cloud, obtained by spectrally filtering the corre-sponding peaks as discussed in the Supplementary Information.Figure 2a shows the map of the PL emission from the WS2 monolayerused for this experiment taken by scanning the sample with galvan-ometer mirrors in confocal mode (see Fig. S1). During the samplefabrication process, we introduce tensile strain gradients in a largeregion of the monolayer to create an energy landscape for excitons.The details of the fabrication process and the characterization of strainare discussed later in the text and in the SI. The position in the sampleand the direction of the energy gradient are indicated in Fig. 2a.Figure 2b–d is the spatially resolved emission of X0, T1 andD�P =D�K3 +D�Γ5, respectively, taken with an EMCCD. Figure 2b showsthat the extension of the bright exciton cloud vanishes completelyaround 4μmaway from the excitation location. The diffusion of brightexcitons is driven by their non-uniform concentration at the excitationspot and it is constrained by the short lifetime, in the order of 1–5 ps,and weak interaction30,31. Differently from the bright exciton, theclouds of dark species extend over a larger area, and the emission isstill detected more than 8μm away from the excitation location,double the distance of its bright counterpart. A comparison of theintensity profiles is illustrated in Fig. 2e. Twomain factors contribute tothe longer diffusion of dark excitons: the long lifetime allows them totravel further distances, and the larger interaction energy generates adrift potential that pushes dark excitons away from the excitationlocation. These features are well suited for the requirements of longtransport in quantum technologies.Density-dependent propagation of dark excitons in WS2The strong interaction of dark excitons is further demonstrated bypower-dependent measurements. The evolution of the emissionspectrum for increasing pump power, reported in Fig. S7 in the SI,shows a smaller redshift due to band renormalization for dark excitonspecies when compared to the bright counterparts32–34. This suggeststhat dark excitons are affected by stronger repulsive interactions. Theeffect of this interaction ismanifestedmoreclearly by the expansionofFig. 2 | Spatial emission of bright and dark excitons. a Photoluminescence mapof the WS2 monolayer encapsulated within thin layers of hBN. White dashed linesindicate regions of the sample characterized by a linear strain gradient that createsincreasing exciton energy in the direction of the arrow. Yellow lines indicate asample region with no significant strain gradient. The green circle highlights theposition of the laser for the diffusionmeasurement in (b–d). Images of the cloud ofbbright excitons, cdark excitonsT1, andddark trionphonon replicaD�P =D�K3 +D�Γ5shows the diverse propagation dynamics. The edges of the WS2 flake are high-lighted by white lines. The images in c–e are acquired with a CW laser with power600μW. e Spatial intensity profiles along the positive horizontal direction for thelaser, bright, and dark excitons.Article https://doi.org/10.1038/s41467-023-39339-yNature Communications |         (2023) 14:3712 3the dark exciton cloud. Figure 3a–c illustrates how the diffusion ofD�P increases when the pump power is increased by two orders ofmagnitude. The horizontal line profiles of the intensity plotted inFig. 3d confirm the expansion of the dark exciton cloudwith increasingexciton density corroborating the repulsive nature of exciton–excitoninteractions. Exciton diffusion is evaluated bymeasuring the full-widthhalf maximum (FWHM) of the spatial emission. Differently from X0 inwhich the FWHM shows a constant value of 2.4μm, the FWHM of darkexcitons increases from 2.8 to 3.9μm when the pump power isincreased by two orders of magnitude. Diffusion measurements of X0and T1, the filtering, and the fitting procedures are shown in Supple-mentary Figs. S8–S11. In addition, we estimate the characteristic dif-fusion length of dark excitons by fitting the line profiles away from thecenter of laser excitation with the two-dimensional diffusion modelwith a point source according to nðrÞ∼ e�r=LDffiffiffiffiffiffiffir=LDp 16,19,35. To investigate theeffect of exciton–exciton interaction, we monitor the diffusion lengthat different excitation powers. Figure 3f shows that LD increases from1.4 to 2.4μm in the unstrained (+x) directions and from 1 to 1.6μm inthe strained (−x) region. The asymmetry in the diffusion length in thestrained uphill and unstrained directions is attributed to the differentenergy landscapes. Figure S12 in the SI shows the characterization ofstrain along the horizontal direction x. After a small interface region, inthe negative x direction, strain creates a smooth and linear uphillenergy gradient that limits diffusion. In both directions, the increase ofthe characteristic diffusion length LD with increasing excitation powerindicates an enhanced diffusion at high exciton densities. Similarly tointerlayer excitons16,17, exciton interactions promote longer diffusion.We note how thediffusion length of dark excitons is comparable to theone of interlayer excitons despite the shorter lifetime (τ), suggesting adifferent interaction strength. As discussed below, we attribute theincrease in diffusion length of dark excitons to the large interactionenergy. The diffusion constant (D= LD2=τ) of dark excitons in theunstrained region ranges from 40 to 240 cm2/s, in agreement withprevious reports19. Differently from dark excitons, the diffusion lengthofX0 is not density-dependent and stays constant at around 1 μmfor allpump powers. A similar value for the diffusion length of bright exci-tons has been already reported19 and attributed to the combinedcontribution of a rapid and slower decay rate36. However, we note that,while our imaging method is capable of detecting very weak signals,point sources in the sample can appear diffuse, affecting the spatialresolution. A comparisonwith confocal imaging is discussed in Section5 and Figure S11 of Supplementary Information.Exciton diffusion in different energy landscapesThe differential between the diffusibility of dark excitons and otherexcitonic complexes allows us to analyze how exciton diffusioninteracts with a varying potential energy landscape. We note that,despite the large inhomogeneity of the sample (Fig. 2a and Fig. S12),longdiffusionoccurs in thedarkexciton species, indicating robustnessagainst energy barriers due to strain and impurities. To further provethat dark excitons are a resilient way to transport energy and infor-mation across samples, we study exciton diffusion in a linear energygradient. Tensile strain is carefully produced in the encapsulated WS2monolayer during the transfer process (more details in the SI) suchthat a linear energygradient is created in a regionextendingover6μm.Tensile strain reduces the energy band-gap (EBG) at the K valley andtransforms the energy landscape for excitons whose energy readsEX ðxÞ= EBGðxÞ � EB. Here we assume that the exciton binding energy(EB) does not depend on the spatial coordinate x, because strain onlyweakly affects it in the strain range investigated in this work37. UniaxialFig. 3 | Density-dependent propagation of dark excitons in WS2. a–c Power-dependent diffusion of the dark trion phonon replicas (D�P =D�K3 +D�Γ5). d Intensityprofiles at different excitation powers show as the cloud of D�P expands due toincreasing interaction. The colored areas indicate the regions used to extract thediffusion length along the strained (−x) and unstrained (+x) directions.e Comparison of the full-width half maximum (FWHM) of the exciton cloud of X0,D�P , and T1 as a function of the pumppower. fDiffusion lengthof the of X0 andD�P asa function of the pump power. The error bars in e and f indicate the error of thefitting.Article https://doi.org/10.1038/s41467-023-39339-yNature Communications |         (2023) 14:3712 4strain variations in TMDs have been shown to modify the emissionenergy of the neutral excitons X0 with a uniaxial strain gauge factor of−45meV/%38,39. Although this value has been measured at room tem-perature in previous experiments, it is a good approximation to esti-mate the strain magnitude in our samples. Figure 4a shows theposition-dependent PL spectra measured by exciting and collectingthe emission spectra in the same position along one direction. Notethat, for the sake of visibility, in all color maps of Fig. 4 the spectra(plotted as columnsof themap) arenormalized to theirmaximum, andthe energy of bright (dark) exciton complexes is highlighted by red(other colors) circular markers. The variation of intensity of the peaksas a functionof spacecanbe appreciated in the raw spectra reported inSupplementary Fig. S13. Figure 4a shows that as the position moves inthe direction of increasing applied strain, a clear linear energy shift upto 20meV is visible for all excitons, confirming the presence of a one-dimensional linear strain gradient in the sample from 0 to 0.4%. Thisstrain distribution allowsus to study excitondiffusion in both downhilland uphill energy landscapes. We perform energy-resolved diffusionexperiments by decoupling detection and excitation so that excitonsare excited in one location and their emission spectrum is measuredfar away from it (see Supplemental Fig. S1). The schematic of theexperiment is illustrated in the top panel of Fig. 4b, c. When excitonsdiffuse away from the excitation spot, they relax toward their mini-mum energy state before recombining radiatively28 and their emissionenergy depends on the local conditions, including the variation of EBGdue to strain (Fig. 1a). This process is illustrated in Fig. 5c.First, we study downhill diffusion by creating the exciton gas in apoint of the sample with zero strain corresponding to high potentialenergy and by measuring the emission spectra at different positionsalong the strain gradient. Differently from other exciton species, theenergy of the negatively charged biexciton (XX�) and the bright trions(X�T and X�S ) does not change over the entire strain landscape. Due totheir low binding energy (∼ 20–40meV), very short lifetime (∼ 30ps),and large mass24,40, the diffusion of these species is limited. However,at the excitation location, they have the strongest emission (seeFig. S7), which can be picked by our large NA objective lens even at adistance of a few micrometers. This conclusion is supported by theconstant emission energy, showing that the recombinationoccursonlyin the same location of the excitation. However, we observe significantdiffusion of bright excitons up to 2.5μm, in agreement with the dif-fusion measurements of Fig. 2c. Despite its short lifetime, X0 has largebinding energy, and the tensile strain generates a funneling effectsimilar to what was observed in previous works41. Very good correla-tion emerges between the energy landscape created by the straingradient and the emission energyofdark excitoncomplexes, includingI0, D0, D−, T1, D�P . The downhill transport of dark excitons is due tothe combination of repulsion interaction and the funneling forceFf un = � ∇EðxÞ generated by the strain gradient.Finally, we study uphill diffusion by exciting the sample in alocation with high strain, corresponding to low potential energy. Fig-ure 4c shows that the emission energy of all bright excitons detectedaway from the excitation spot is constant, despite the increase ofpotential energy.This indicates that they are notpropagating, and theyare observed in the spectrum only because the large NA objective lenspicks up their strong out-of-plane emission at the excitation spot. Thisresult also confirms the weak interaction energy among bright excitonspecies in TMDs10. Surprisingly, dark exciton complexes show aremarkable blueshift as a function of the distance, indicating that theyFig. 4 | Exciton diffusion in different energy landscapes. a The energy landscapegenerated by strain is characterized by measuring the emission spectrum of themonolayer WS2 in different positions. Excitation and detection are done at thesame location as described in the top panel. The color map in the bottom panelshows the spectral emission normalized to themaximumatdifferent positions. Theenergy corresponding to the exciton complexes is highlighted by circular markersof different colors: red (X0, XX0, X�T , X�S , XX�), green (I0), yellow (D0), blue (T1), androse (D�K3 and D�Γ5). All exciton species show an energy shift up to 20meV corre-sponding to a linear tensile strain gradient from 0 to 0.4%. b Downhill diffusion ofexcitons created at the top of the potential gradient (top panel) and measured atincreasing distances from the generation location. The diffusion of the dark exci-tons appears clearly from the spectral response as a function of the distance fromthe excitation (bottom panel). c Uphill diffusion of excitons created at the bottomof the potential gradient (top panel) andmeasured at increasing distances from thegeneration location. Differently from the downhill case, only dark excitons pro-pagate far away from the excitation spot. After propagation due to repulsiveinteractions, dark excitons relax towards the bottom of their dispersion andrecombine radiatively via different paths, generating I0, T1, and D�P .Article https://doi.org/10.1038/s41467-023-39339-yNature Communications |         (2023) 14:3712 5overcome the potential gradient. Dark excitons can diffuse away fromthe excitation spot, relax to the energy landscape—which representsthe lowest energy state available to them at that particular position—and then recombine radiatively. This observation together with theincrease of the diffusion length as a function of pump power indicatesthe buildup of interaction energy in the dark exciton population. Thiseffect is particularly evident for I0,D0, T1,D�Γ5, andD�K3. This experimentsets a lower limit for the blueshift induced by repulsive interaction to20meV, limited by the energy landscape size. The interaction-drivenpropagation of dark excitons is confirmed in similar diffusion experi-ments performed at lower excitation powers (Fig. S16) and differentenergy landscapes, including shorter energy ramps (Fig. S17) andalmost flat energy landscapes (Fig. S18).Dark exciton interaction and relaxation pathwaysWe estimate the strength of the exciton-exciton interaction with thetheoretical model based on Coulomb interactions between excitonsdescribed in refs. 42,43. While the direct contribution to the interac-tion is generally weak and vanishes for small values of center of massmomentum, the exchange contribution is always repulsive andfinite43,44, with an approximate strength of Uex ∼ao2Eb, where Eb is thebinding energy and ao is the Bohr radius. Therefore, themain sourceofinteraction is provided by the exchange term which is spin and valleydependent4. Bright and dark excitons have comparable bindingenergy45,46 and radius26, and their exchange interaction strength Uex isexpected to be similar. Since the overall interaction energy is given byΔE =n0Uex , the longer propagation and the blueshift of dark excitonsobserved experimentally is attributed to a higher density. To estimatethe order of magnitude of the exciton density, we solve a set of cou-pled rate equations for the population dynamics of bright, intervalleyand dark exciton states as illustrated in Fig. 5a. Themodel accounts forthe radiative lifetime9, nonradiative phonon-mediated interbandtransitions47, and exciton–exciton annihilation processes48. Uponexcitation, the bright exciton X0 can relax towards the I0 or D0 state.The X0→I0 transition is induced by the interaction with chiral phononsK3 which promote the spin-preserved intervalley scattering of elec-trons in the conduction bands from K to K’ (see Fig. 1f). The X0→D0transition is induced by the interaction with chiral phonons Γ5 whichpromote the spin-flipping intravalley scattering of electrons in theconduction bands. In n-doped samples, the transition I0→D0 is medi-ated by scattering with free electrons. Figure 5b shows that in a largerange of generation rates (G), the density of dark excitons is almostthree orders of magnitude larger than the one of the bright and theintervalley excitons, indicating that ΔE is expected to be greatest fordark excitons, explaining the experimental observation. We note thatthe effect of exciton-exciton annihilation is only significant for valuesof generation ratesmuch larger than the ones used in the experiments.We use a power density of ∼ 5 � kW cm�2 that, for an absorption effi-ciency of ∼ 5%49, corresponds to a generation rate of G∼ 2 �109cm�2 ps�1 returning a density ofD0 in the order of 8 � 1011 cm�2. Fora blueshift ofΔE =20meV as in the caseof Fig. 4b, c, the correspondinginteraction strength is Uex ∼ 2:5 � 10�11meV cm2, which is one order ofmagnitude larger compared to the one measured in bright excitons50.This deviation can originate from the different properties betweendark and bright excitons, or from the limitations of the model used toestimate the dark exciton density discussed in the SI. However, thecalculated density is compatible with the ones extracted from theexperimental data when the interaction strength Uex is estimated byusing the values of the bright exciton radius and binding energyreported in the literature (more details in the SI)45,51,52. The estimatedrange of experimental values for the dark exciton density is1 � 1012 � 3 � 1012 cm�2. The agreement between the calculated andmeasured values of the dark exciton density suggests that the longpropagation of dark excitons is promoted by strong exciton–excitoninteractions due to their high density and long lifetime.We note that bright single and triplet trions can relax toward thedark trion state via an electron scattering process with a characteristictime similar to the one of the neutral excitons. Therefore, the densityofD− is expected tobe similar to theoneofD047. However, due to its thesmaller binding energy, the interaction energy experienced by D−should be smaller.It is worth noting that despite the similar lifetime of I0 and D0, ourmodel returns a density for I0 which is orders of magnitude smallerthan the one of D0. While D0 is the lowest exciton state of the system,the dynamics of I0 is governed by the fast I0→D0 scattering channelthat quickly depopulates I0 in favor of D0. Interestingly, in the experi-ment of Fig. 4c, I0 recombines at higher energy with respect to theexcitation location despite its low density and weak interaction. Eventhough it is energetically unfavorable, the transition D0→ I0 can occurwhen the kinetic energy of dark excitons is larger than ∼ 20meV47,compatible to the one observed in our experiments. This observationsuggests that I0 generates from the relaxation of the transported D0.Fig. 5 | Dark exciton interaction and relaxation paths. a Scheme of the energylevels and possible transitions among different exciton species. The transition DIshown with dash lines indicates the scattering D0→ I0 that occurs only for highvaluesof exciton kinetic energy.bCalculatedexcitondensity at the steady state as afunction of the generation rate for the excitonic states illustrated in (a). c Thestrong interaction of D0 at the excitation position provides initial momentum topropagate uphill. The transported hot dark excitons can then thermalize towardthe bottom of their energy dispersion via scattering with low energy Γ acousticphonons. From this state, they can radiatively recombine in the form of a darkexciton D0, dark trions D−, phonon replica D�p , T1, or I0.Article https://doi.org/10.1038/s41467-023-39339-yNature Communications |         (2023) 14:3712 6Further diffusion measurements over varying energy landscapesand with different excitation powers are reported in SupplementaryFig. S16–S18.DiscussionOur polarization measurements (Fig. 1b), consistent with previousworks, indicate that dark exciton replicas posses spin-valleycoherence7,23,25. Although spin-valley polarization cannot be mea-sured directly from D0 and D− due to their out-of-plane optical transi-tion, it can be observed in their phonon replicas that reveal the valleyinformation of the original dark states. Therefore, due to their robustdiffusion over several micrometers, dark excitons are capable oftransporting spin information across large areas and inhomogeneouslandscapes. This information can then be read out in the far-field viaout-of-plane emission generated by phonon or electron interaction.The transport and relaxation pathways for the dark exciton D0 areschematically illustrated in Fig. 5c. The system is initially excited tocreate dark excitons as shown in Fig. 5a. The high density of darkexcitons favors the strong exciton–exciton repulsive interaction thatincreases the overall energy of the dark exciton population at theexcitationposition. This repulsive interaction at the excitationpositionprovides initial momentum to excitons which diffuse away eventowards locations of the sample with higher energy landscape. Thetransported hot dark excitons can then thermalize toward the bottomof their energy dispersion via scattering with low energy Γ acousticphonons28. From this state they can radiatively recombine in the formof dark excitons D0, dark trions D−, phonon replica, T1 or I0.In summary, we have reported the observation of a robust drift-diffusion process in dark excitons in WS2 that extends over severalmicrometers in an inhomogeneous energy landscape. We provideevidence that the diffusion is driven by repulsion, stemming from thehigh exciton density. Due to the introduction of an advanced fabri-cation process and imaging setup, we are able to study diffusion ofdark excitons in a linear potential energy gradient. With theseexperiments, we show that dark excitons are promising transportmeans for spin-valley information for applications in quantum infor-mation technology.MethodsSample preparation2D materials are mechanically exfoliated using a standard scotch tapemethod on a Polydimethylsiloxane (PMDS) stamp (X4 WF Film fromGel-Pak) and then transferred onto a Si/SiO2 substrate. We use hBNflakes of thickness around ∼ 20nm as our bottom and top protectivelayers. Monolayers ofWS2 are initially exfoliated from the bulk crystals(from HQ Graphene) and then transferred onto the hBN substrate.During this process, strain is imprinted on the monolayer by using alarge contact angle between theWS2 and the hBN layer. Finally, the 2Dmaterial stack is encapsulated with the top hBN layer and the resultingheterostructure is baked at 200° for 5min in air to remove nano-bubbles created at theWS2/hBN interfaces during the transfer process.More details in the SI.PL and hyperspectral measurementPL and spectroscopy measurements are carried out in a home-builtconfocal microscope setup coupled to a closed-cycle cryostat. PLexperiments are performed by exciting the samples non-resonantlywith a continuous-wave green laser (532 nm). The laser spot size on thesample is 2μm. PLmaps are taken with a galvanometermirror scannerin a 4f configuration. The diffraction-limited spatial resolution isapproximately 350 nm. An objective lens with NA=0.9 and free-space-coupled avalanchephotodiodes areused for high-efficiency collection.In detection, the excitation laser isfiltered outwith a 550-nm long-passfilter. Spectra are obtained by directing the signal to a spectrometerwith gratings of 600G/mm. Images of the exciton clouds are takenwith an EMCCD camera in the low noise mode. The complete schemeof the optical setup is in Fig. S1.Data availabilityThe datasets generated during and/or analyzed during the currentstudy are available from the corresponding authors on request.Code availabilityThe codes used to generate the data are available from the corre-sponding authors upon request.References1. Splendiani, A. et al. Emerging photoluminescence in monolayerMoS2. Nano Lett. 10, 1271–1275 (2010).2. Mak, K. F., Lee, C., Hone, J., Shan, J. & Heinz, T. F. Atomically thinMoS2: a new direct-gap semiconductor. Phys. Rev. Lett. 105,136805 (2010).3. 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Commun. 7, 1–8 (2016).AcknowledgementsG.G. acknowledges support from theNational Science Foundation (NSF)(grant No. DMR-2044281), support from the physics department ofthe Graduate Center of CUNY and the Advanced Science ResearchCenter through the start-up grant, and support from the ResearchFoundation through PSC-CUNY award 64510-00 52. A.A. and J.Q. weresupported through the Simons Foundation and the Air Force Office ofScientific Research. K.W. and T.T. acknowledge support from the Ele-mental Strategy Initiative conducted by the MEXT, Japan (grant No.JPMXP0112101001) and JSPSKAKENHI (grant No. 19H05790, 20H00354,and 21H05233).Author contributionsS.B.C. and G.G. conceived the idea/concept and defined the experi-mental and theoretical work. S.B.C. prepared samples and performedexperimental measurements with the assistance of J.M.W. and J.Q. E.M.implemented the control software for data collection. S.B.C. performedDFT simulations and theoretical calculations. G.G. and S.B.C. analyzedthe data. T.T. and K.W. grew hBN samples. G.G. supervised the project.All authors discussed the results and commented on the paper.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-39339-y.Correspondence and requests for materials should be addressed toGabriele Grosso.Peer review information Nature Communications thanks the anon-ymous reviewer(s) for their contribution to thepeer reviewof thiswork. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023Article https://doi.org/10.1038/s41467-023-39339-yNature Communications |         (2023) 14:3712 9http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Interaction-driven transport of dark excitons in 2D semiconductors with phonon-mediated optical readout Results PL spectroscopy of encapsulated WS2 monolayer Diffusion of exciton complexes in WS2 Density-dependent propagation of dark excitons in WS2 Exciton diffusion in different energy landscapes Dark exciton interaction and relaxation pathways Discussion Methods Sample preparation PL and hyperspectral measurement Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information