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Garima Gupta, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Kausik Majumdar

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[Observation of ~100% valley-coherent excitons in monolayer MoS2 through giant enhancement of valley coherence time](https://mdr.nims.go.jp/datasets/846e2b37-88bd-4370-9f63-19d1be8fb77a)

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Observation of ~100% valley-coherent excitons in monolayer MoS2 through giant enhancement of valley coherence timeGupta et al. Light: Science & Applications          (2023) 12:173 Official journal of the CIOMP 2047-7538https://doi.org/10.1038/s41377-023-01220-4 www.nature.com/lsaART ICLE Open Ac ce s sObservation of ~100% valley-coherent excitons inmonolayer MoS2 through giant enhancement ofvalley coherence timeGarima Gupta1, Kenji Watanabe 2, Takashi Taniguchi 3 and Kausik Majumdar 1✉AbstractIn monolayer transition metal dichalcogenide semiconductors, valley coherence degrades rapidly due to a combination offast scattering and inter-valley exchange interaction. This leads to a sub-picosecond valley coherence time, making coherentmanipulation of exciton a highly challenging task. Using monolayer MoS2 sandwiched between top and bottom graphene,here we demonstrate fully valley-coherent excitons by observing ~100% degree of linear polarization in steady statephotoluminescence. This is achieved in this unique design through a combined effect of (a) suppression in exchangeinteraction due to enhanced dielectric screening, (b) reduction in exciton lifetime due to a fast inter-layer transfer tographene, and (c) operating in the motional narrowing regime. We disentangle the role of the key parameters affectingvalley coherence by using a combination of calculation (solutions of Bethe-Salpeter and Maialle-Silva-Sham equations) and acareful choice of design of experiments using four different stacks with systematic variation of screening and exciton lifetime.To the best of our knowledge, this is the first report in which the excitons are found to be valley coherent in the entirelifetime in monolayer semiconductors, allowing optical readout of valley coherence possible.IntroductionThe bound state of an electron and a hole, an exciton, is asuperposition of the conduction and valence band states inthe K and K0 valleys in monolayer transition metal dichal-cogenides (TMDs)1–3. K and K0 valley excitons are selectivelygenerated by circularly polarized light excitation of oppositehelicities4–7. On linearly polarized excitation, a hybridK� K0exciton is generated in a state of valley coherence8,9. How-ever, valley coherence degrades rapidly due to a combinedeffect of fast scattering and inter-valley exchange10–12. Thereported values of valley coherence time lie in the range of98� 520 fs12–17, much shorter than the exciton radiativelifetime of � 1 ps18–20. This makes optical read out of strongexciton valley coherence a highly challenging task. To be ableto use these coherent excitons as a qubit for quantuminformation processing, a longer valley coherence time isdesirable to perform any manipulation on it. Any technique21that enhances this valley coherence time significantly is thusof high scientific importance.Here we demonstrate a 100% degree of linear polariza-tion (DOLP) in photoluminescence (PL) peak of A1sexciton in a monolayer of MoS2 encapsulated with few-layer-graphene (FLG) at the top and bottom. Such acomplete retention of the generated valley coherence insteady-state PL implies the achievement of a large valleycoherence time, the measured value of which is only lim-ited by the lifetime of the exciton. This suggests that thevalley coherence time has been significantly enhanced ascompared to the reported values to date12–16.ResultsDepending on the linear polarization direction of theexcitation light, the excitons are generated at specific center-of-mass momentum ðQÞ values [where Q ¼ keþkh, keðkhÞ© The Author(s) 2023OpenAccessThis article is licensedunder aCreativeCommonsAttribution 4.0 International License,whichpermits use, sharing, adaptation, distribution and reproductionin any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate ifchangesweremade. The images or other third partymaterial in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to thematerial. Ifmaterial is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtainpermission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.Correspondence: Kausik Majumdar (kausikm@iisc.ac.in)1Department of Electrical Communication Engineering, Indian Institute ofScience, Bangalore, India2Research Center for Functional Materials, National Institute for MaterialsScience, Tsukuba, JapanFull list of author information is available at the end of the article1234567890():,;1234567890():,;1234567890():,;1234567890():,;www.nature.com/lsahttp://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/0000-0002-1467-3105http://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/0000-0002-6544-7829http://orcid.org/0000-0002-6544-7829http://orcid.org/0000-0002-6544-7829http://orcid.org/0000-0002-6544-7829http://orcid.org/0000-0002-6544-7829http://creativecommons.org/licenses/by/4.0/mailto:kausikm@iisc.ac.indenoting the electron (hole) crystal momentum] in theexciton band at time t ¼ 0 (Fig. 1). During its lifetime, theexciton undergoes scattering and exchange interaction,which, coupled together, degrades the valley coherence. Thepolarization state can be represented by the pseudospinvector S in the Bloch sphere. At t ¼ 0; the direction of S isparallel to the exchange-induced magnetic field (denoted bythe precession frequency Ω). Considering x� polarizedexcitation, the system is generated in a pure state repre-sented by Sx ¼ 1, Sy ¼ 0; Sz ¼ 0. When excitons scatter todifferentQ values, the pseudospin precesses on experiencinga finite torque around ΩðQÞ due to which it becomes amixed state represented by a density matrix operator ρ. TheQ-space and the Bloch sphere representations of thismechanism are shown in Fig. 1a, b. On decoupling thedensity matrix in terms of the number trace and tracelessmatrix S:σ (where σ denotes the Pauli matrices), the overalldynamics of the valley pseudospin as described by theMaialle-Silva-Sham (MSS) mechanism (see SupplementaryNote 1 for proof) is given by10:dSðQÞdt¼ Ω Qð Þ ´ S Qð Þ þXQ0WQQ0 S Q0ð Þ � S Qð Þ½ � � 1τS Qð Þ þ Gð1ÞHere WQQ0 is the rate of any generic momentum scatteringmechanism, e.g., exciton-impurity and exciton-phononscattering. τ is the net exciton lifetime given by 1=τ ¼1=τr þ 1=τnr þ 1=τfilter , where τr; τnr; and τfilter are theradiative, non-radiative, and filtering timescale. Filtering is anon-radiative process where the excitons are scattered out ofthe light cone, e.g., scattering to lower energy states,interlayer transfer to graphene, etc., in which case, the lightcollection is limited to t � τfilter . G represents the excitongeneration rate. On recombination, the DOLP of this mixedstate is given by Sxh i, averaged overQ values within the lightcone (see Supplementary Note 2 for proof).The possible ways to improve the valley coherencetime are: (a) by minimizing scattering ðWQQ0 Þ inside thelight cone such that S does not accumulate randomphase by precessing around Ω Qð Þ, or by enhancingWQQ0 such that the whole operation is pushed towardsthe motional narrowing regime (simulation results inFig. 1c); (b) by screening the electron–hole interactionwhich results in reduced exchange interaction, and inturn a suppressed Ω (middle panel of Fig. 1d). However,a side-effect of the enhanced screening is an incrementin the exciton lifetime due to a reduction in the bindingenergy. This can be overcome by (c) reducing τ byintroducing a fast-filtering mechanism21–23 (bottompanel of Fig. 1d).To understand the interplay among these factorssystematically, we prepare four different stacks ofmonolayer MoS2 combined with hBN and FLG, whichare: (1) hBN-MoS2-hBN (HMH), (2) FLG-hBN-MoS2-hBN-FLG (GHMHG), (3) MoS2-FLG-hBN (MGH), and(4) FLG-MoS2-FLG (GMG) (see Methods). We obtainan exciton DOLP of 44:5 ð± 10Þ% in the HMH stack,37 ð± 9Þ% in the GHMHG stack, 77 ð± 5Þ% in the MGHEnergyExciton band dispersionaDOLPLow scatteringregimet = τΩt = 0t = 0 t = �filtert = �′r (> �r)t = �rt = 0t = 0SW QQ′MotionalnarrowingregimeScreeningScreening + filteringScattering ratecCScreening(↓ exchange + ↑ lifetime)+QQScreening +Filtering(↓ exchange +↓ lifetime)Energyde- h+e- h+e- h+bS (t = 0) yS (t = τ)x (|H〉)–z (|�z–〉)– x (|V 〉)z (|�z+〉)ΩSFig. 1 Mechanism of exciton valley decoherence and the factors affecting it. a Top view of a ring inside the light cone of the exciton bandshowing the exciton decoherence dynamics due to scattering WQQ′ within the light cone (purple dashed arrows) and subsequent precessionbecause of inter-valley exchange induced pseudo-magnetic field (black solid arrows). b The direction of S at generation ðt ¼ 0Þ and recombinationðt ¼ τÞ are shown in the Bloch sphere. c The calculated DOLP as a function of scattering rate with motional narrowing regime shown in shade. d Thedecoherence of S (green arrows) with time due to scattering (dashed purple arrow) and precession around Ω (black arrows) for an exciton for threedifferent scenarios. The effect of screening versus screening + filtering on valley decoherence is compared in the three rowsGupta et al. Light: Science & Applications          (2023) 12:173 Page 2 of 7stack, and 96 ð± 6Þ% in the GMG stack on 633 nm nearresonant laser excitation at 5 K. Linear polarization-resolved representative PL spectra in Fig. 2a–d and thebar diagram (with error bars) in Fig. 2e compare theDOLP numbers in all the four stacks (more spectra inFigs. S3-S6). Interestingly, there are several spots wherewe observe � 100% DOLP in the GMG stack (Fig. 2dand Fig. S6). In Fig. S7a, we show similar results of~100% DOLP obtained from few-layer grapheneencapsulated monolayer WS2 (GWG) stack. We alsoperform polarization dependent time-resolved photo-luminescence (TRPL) measurement and obtain a peakDOLP of 97.6% in the GWG stack (see Fig. S7b).We would like to highlight some additional observa-tions on the GMG stack before the main analysisbegins: (1) As a result of FLG encapsulation, the PLspectra predominantly consist of the clean A1s excitonpeak22. The spectra of FLG encapsulated monolayerMoS2, MoSe2, and WS2 on 532 nm excitation are shownin Fig. 2f, clearly indicating suppression of spuriouspeaks from defect-bound excitons and other excitoniccomplexes. We also observe a clear A2s peak located at44 (32) meV higher than the A1s peak in MoS2 (MoSe2and WS2) due to enhanced screening24. (2) We also geta very high degree of circular polarization (DOCP) of81:6 ð± 2Þ% in the GMG stack, much larger comparedto the 20:5 ð± 9Þ% DOCP in the HMH stack (Fig. 2g, hand more spectra in Fig. S8-S9). The in-plane nature ofΩ explains this observation that DOCP is smaller thanDOLP for 2D excitons, which is consistent with pre-vious reports10,25 (see Supplementary Note 3). Thisindicates that starting with a linear polarization (that is,on the equator of the Bloch sphere) is the mostfavourable scenario to maintain valley coherence com-pared with any other (elliptical) polarization. (3) Theinitial and the final state in the Raman scattering pro-cess coincides with the A2s and the A1s exciton level,respectively, on 633 nm excitation at 5 K in the GMGstack. This dual resonance enhances the intensity of theRaman peaks significantly (represented by the dashedlines in Fig. 2d, h) and enables the observation of otherless commonly observed modes distinctly (Fig. S10).The fact that the excitation laser is resonant with the 2sstate in the GMG stack and we observe almost fullycoherent 1s excitonic emission from the stack, it islikely that the generated 2s excitons relax throughpolarization preserving processes, such as, dipole-coupled radiative transition (2s→ 2p→ 1s).In order to establish the different degrees of screeningin the stacks, we plot the PL spectra for the HMH,GHMHG, and the GMG stack obtained from 532 nmexcitation in Fig. 3a–c. The A2s � A1s energy separation1.6 1.8 2.0 2.2A2sA1sA2sA1sA1sA1sA–1sA1sA–1sA1sA2s 1.88 1.90 1.92 1.940.00.10.20.30.91.01.85 1.90 1.95010.20.40.60.81.0011.88 1.90 1.92 1.940.00.20.40.60.81.0    1.80 1.85 1.90 1.950101H/HH/VEnergy (eV)37.7%MGHGHMHG~100%HMHEnergy (eV)Energy (eV)40%Normalized PL intensityNormalized PL intensityEnergy (eV)DOLPNormalized PL intensitya bc deg hHMHGHMHGMGH GMG81.2%11.6 %σ+/σ+σ+/σ–GMGMoSe2WS2MoS2f∗79.6%∗A1sFig. 2 Excitonic emission from different stacks along with DOLP and DOCP. a–d PL spectra with near-resonant 633 nm linearly polarizedexcitation in co-(H/H) and cross- (H/V) polarized detection configuration in the a hBN-MoS2-hBN (HMH) stack, b FLG-hBN-MoS2-hBN-FLG (GHMHG)stack, c MoS2-FLG-hBN (MGH) stack, and the d FLG-MoS2-FLG (GMG) stack at T = 5 K. A1s and A�1s represent the exciton and the trion peak,respectively. e Bar graph comparing the experimental DOLP values along with the error bars in the four stacks. f PL spectra obtained from FLG-TMD-FLG stack (using 532 nm excitation) for monolayer MoS2, MoSe2, and WS2 showing the prominent A1s and A2s peaks. The A2s � A1s separation isaround 44 ð32ÞmeV in MoS2 (MoSe2, WS2). The peaks marked as * and ⋅ are the 2D and the G Raman peaks of the FLG. g, h Representative PL spectrataken with circularly polarized excitation in co- ðσ þ =σþÞ and cross- ðσ þ =σ�Þ polarized detection configuration. The corresponding DOCP valueof the A1s exciton in (g) the HMH and (h) the GMG stack at T = 5K is shown in the inset. The peaks indicated by the dashed lines in (d), (h) representthe prominent Raman peaks in the GMG stack due to dual resonanceGupta et al. Light: Science & Applications          (2023) 12:173 Page 3 of 7obtained is 144.5 meV in the HMH stack, which reducesto 60 and 45 meV in the GHMHG and GMG stack,respectively. To get an estimate of the A1s excitonbinding energy change (Fig. 3d), we obtain the con-tinuum of the exciton energy spectrum by numericallysolving the Bethe-Salpeter equation26 using a two-bandHamiltonian. In the calculation, the parameters arefitted such that the experimentally obtained A2s � A1senergy separation matches with the calculated one(Supplementary Note 4 and Fig. S2). The calculated A1sexciton binding energy is 379 meV in the HMH system,which reduces to 162.5 and 122 meV in the GHMHGand the GMG stacks respectively due to grapheneinduced screening.One immediate consequence of such a screening isthe reduction in the inter-valley exchange interaction25.The exchange interaction is composed of two compo-nents—the short-range part and the long-range part.The short-range component is zero at Q= 0 due to thethree-fold rotational symmetry condition and is negli-gible at higher Q values. The long-range part is givenby27:JLRQ / �Pk ψ kð Þ�� ��2E2gV Qð Þ Qj j2 ð2ÞHerePkψðkÞj j2A ¼ ψðreh ¼ 0Þj j2 is the electron–holewavefunction overlap at zero relative separationreh ¼ 0ð Þ, k is the reciprocal space wave-vector, Egdenotes the bandgap of MoS2, and V ðQÞ is theelectron–hole coulomb interaction. The dielectric screen-ing modulates the following factors: (a) ψðreh ¼ 0Þj j2 - dueto a reduction in the 2D exciton binding energy24,28; (b)Eg - due to bandgap renormalization effect in monolayerTMDs29,30; and (c) V ðQÞ - due to suppressed electron-hole interaction31. In Fig. 3e, we show the variation in JLRQwith Q within the light cone, and hence the screeninginduced suppression of the long-range exchange in oursamples (see Supplementary Note 4).Another consequence of screening is the enhancementof the exciton radiative lifetime due to a reducedelectron–hole wavefunction overlap. Here, a longer1.91s 2s2.02.12.4A1sA1sA1sA2sA2sA2sB1sA–1sA–1s1.90 1.95 2.05 2.10 2.150.20.40.60.81.0144.2 meV1.90 1.95 2.0043.3 meV1.85 1.90 1.95 2.0063.5 meVNormalized PL intensityPeak position (eV) B.E.(HMH) = 379 meVB.E.(GHMHG) = 162.5 meVB.E.(GMG) = 122 meV106 10710010–1Normalized ⎮J QLR⎮⎮Q0⎮⎮Q0⎮⎮Q⎮(m–1)Energy (eV)a b cd eHMH GHMHG GMGHMHGHMHGGMGHMHGHMHGGMGExciton eigen energy statesFig. 3 Evidence of graphene induced screening of electron-hole exchange interaction. a–c Top panel: Schematic representation of thescreening effect in the different stacks. Bottom panel: PL spectra taken with 532 nm excitation highlighting the different degree of dielectricscreening in our samples. The A2s � A1s separation for the HMH stack (144.5 meV), GHMHG stack (63.5 meV), and the GMG stack (43.3 meV) isindicated by the dashed arrows. * in (c) is the 2D Raman peak of FLG. d Eigen energies (open symbols) and the corresponding binding energies ofthe A1s exciton obtained from the Bethe-Salpeter equation. The solid symbols denote experimental data. The dashed lines are the correspondingcalculated continuum levels. e Calculated value of the normalized long-range exchange potential variation inside the light cone ð Qj j< Q0j jÞ. Inset:light emitting region ð Qj j< Q0j jÞ of the exciton band highlighted by the light coneGupta et al. Light: Science & Applications          (2023) 12:173 Page 4 of 7lifetime is undesirable as it leads to a larger valley deco-herence (Fig. 1a). To estimate the exciton lifetime and itsrole in the valley decoherence, we carry out TRPL mea-surements on our samples (see Methods and Fig. S11).The TRPL values obtained in our stacks are as follows:<5 ps in the HMH stack, 6–8 ps in the GHMHG stack,and <5 ps in the GMG stack. The uncertainty in thelifetime in the HMH and GMG stacks arises as it issmaller than the 10% limit of our Instrument ResponseFunction (IRF) width32. Nonetheless, several reportssupporting these numbers are already available in litera-ture18,19,22,33. Moreover, the qualitative trend in theexciton lifetime testifies the anticipated trend in Fig. 1d,showing a clear enhancement in the exciton lifetime inthe GHMHG stack, as compared to the other two stacks.Obtaining an average DOLP of only 44.5 (±10)% impliesan ultra-short valley coherence time in the HMH stack, inagreement with previous reports12,15. On the other hand,in the GHMHG stack, an increased exciton lifetime is anevidence of screening induced enhancement of excitonlifetime as a result of introducing top and bottom FLG.Due to the opposing roles of reduced exchange andincreased lifetime, we do not observe any improvement inthe exciton DOLP in this sample compared with theHMH sample.This side-effect of screening driven enhanced excitonlifetime is eliminated in the GMG stack through filtering,where light collection from the long-lived excitons isprohibited due to a fast transfer of excitons to graphene.The extracted timescale corresponds to the graphene-transfer-limited exciton lifetime in this system. Thistimescale is similar to that in the HMH stack, and inagreement with previous report22,33. Therefore, the sig-nificant exciton DOLP difference between the HMH andthe GMG stacks is attributed to screening modifiedexchange interaction without any confounding effect dueto a change in the exciton lifetime.To obtain a quantitative understanding, we solve thesteady-state form of the MSS equation:G ¼ 1τS Qð Þ �Ω Qð Þ ´ S Qð Þ �XQwQ2 sin2 α2S Q0ð Þ � S Qð Þ½ �ð3ÞWe obtain the DOLP ð Sxh iÞ for the A1s exciton in theHMH and the GMG stack (Supplementary Note 4). ForWQQ0 , the exciton-impurity scattering rate expression isused (Supplementary Note 4.2). τ is the exciton lifetime, αis the angle between the initial (Q) and the final Q0ð Þ stateand w is an overall scaling factor. We neglect the con-tribution of exciton-phonon scattering in decoherence at5 K. Plotte d in Fig. 4 is the variation in Sxh i as a functionof the w for the two stacks. In both the cases, the V shapedvariation is understood as follows: For small w, anincrease in the scattering degrades the valley coherencedue to enhanced exciton precession around Ω (Fig. 1a).However, this effect is non-monotonic, as on significantlyenhancing the scattering rate, the DOLP starts increasingafter reaching a minimum. This phenomenon is referredto as motional narrowing10,34, and arises due to a can-cellation of the accumulated randomness in the phaseinformation of S. Mathematically, the system is in themotional narrowing regime when the exciton scatteringfrequency becomes larger than the precession frequency,leading to a longer pseudospin coherence time.We take the extracted homogeneous (Lorentzian com-ponent of the Voigt fitting) linewidth (Γhom) of the co-polarized exciton PL peak as the experimental analogue ofthe scattering rate. The experimental DOLP as a functionof the Lorentzian linewidth, superimposed on the simu-lation results, is shown in Fig. 4. Both in GMG and HMHstacks, the extracted value of Γhom is much larger than theexciton lifetime limited linewidth, as obtained from TRPL.This suggests that the impurity scattering rate is similar inboth the samples, and it dominates over other linewidthbroadening mechanisms. This is also evident from theupward trend of the experimental DOLP with Γhom, whichis in excellent agreement with the rising side of Sxh iversus w in the simulation. This suggests that the wholeoperation lies in the motional narrowing regime in boththe samples.HMHGMGLorentzian linewidth (meV)DOLP100 102 104 106 108 10100.40.60.81.010–2 10–1 100 101 102Motional narrowingFig. 4 Comparison of the experimental data with the steady-statesolution of the MSS equation. Simulation results comparing theexciton DOLP Sxh ið Þ as a function of the scaling factor wð Þ of thescattering rate (bottom axis) for the HMH stack (solid black trace) andthe GMG stack (dashed black trace). The downward trend in the left-hand side (unshaded region) is the low-scattering regime, and theupward trend in the right-hand side (shaded region) is the motionalnarrowing regime. Overlapped on the simulation results is theexperimentally obtained DOLP variation with the Lorentzian linewidthΓhomð Þ of the corresponding co-polarized PL spectrum (top axis) forthe HMH stack (solid spheres) and the GMG stack (open circles). Theupward trend of the experimental data suggests that both the stacksare operating in the motional narrowing regimeGupta et al. Light: Science & Applications          (2023) 12:173 Page 5 of 7DiscussionWe conclude that the combined effect of enhancedscreening, reduced lifetime due to interlayer transfer, andmotional narrowing helps us to achieve ~100% excitonDOLP in our FLG-capped MoS2 sample. This is a directevidence of valley coherence being maintained during theentire exciton lifetime by cutting down the decoherencechannels before the spontaneous emission. Achieving~100% steady-state DOLP also indicates that the truevalley coherence timescale is beyond few picoseconds inthe GMG stack, which is in excellent agreement withsimulation result obtained by solving time-dependentMSS equation (see Fig. S12).The combination of such ~100% polarization, coupledwith background-free, narrow linewidth emission, makesthe GMG stack a promising substrate for spectral diffu-sion-free, indistinguishable single photon source. As theinitialized coherence in the exciton is shown to be stayingprotected for a longer time, the results have intriguingprospects on performing experiments involving coherentmanipulation of exciton and building quantum systemoperating at these timescales.MethodsSample preparationAll the stacks in this paper are prepared first bymechanically exfoliating the layered material on a Poly-dimethylsiloxane (PDMS) film, and then its subsequenttransfer in a controlled manner underneath a microscopeon a Si substrate covered with 285 nm thick thermallygrown SiO2. The thickness of the few-layer graphene ischosen to be 2–3 nm in all the stacks. The thickness ofhBN layers is in the range of 20–30 nm apart from theGHMHG stack. To ensure strong screening in theGHMHG stack, the hBN thickness is kept at ~5 nm. Afterthe preparation of the entire stack, the samples areannealed at 200 °C for 5 h (pressure ~10-6 torr) to ensurebetter adhesion between successive layers.Sample characterizationAll the measurements are taken in a closed-cycle opticalcryostat (Montana Instruments) at 5 K using a ×50 long-working-distance objective having a numerical aperture of0.5. To measure the exciton DOLP¼ ðIH=H � IH=VÞ= IH=H þ IH=V� �� �, we place the analyzerin the parallel IH=H� �and perpendicular IH=V� �directionin the collection path relative to the excitation polariza-tion direction. For the DOCP measurements, a quarter-wave plate is inserted just before the objective lens, andaligned at 45° with respect to the incoming linearlypolarized light. The time-resolved photoluminescencemeasurement is carried out using a 531 nm laser con-trolled by the PDL-800 D driver (laser pulse width is48 ps). We use a single photon counting detector fromMicro Photon Devices and the time correlated measure-ments are taken using the PicoHarp 300 TCSPC system(PicoQuant). We use a combination of two bandpass fil-ters to get the time resolved counts of the A1s exciton at5 K: 650 (FWHM - 55 nm) and 635 nm bandpass filter(FWHM - 10 nm) for the HMH and the GHMHG stack,and 650 (FWHM - 55 nm) and 660 nm bandpass filter(FWHM - 10 nm) for the GMG stack. The instrumentresponse function (IRF) has an FWHM of 52 ps, andshows a decay of 23 ps. The deconvolution of the TRPLdata with the IRF is carried out using the QuCoa software(PicoQuant).AcknowledgementsThis work was supported in part by a Core Research Grant from the Scienceand Engineering Research Board (SERB) under Department of Science andTechnology (DST), a grant from Indian Space Research Organization (ISRO), agrant from MHRD under STARS, and a grant from MHRD, MeitY and DST NanoMission through NNetRA. K.W. and T.T. acknowledge support from the JSPSKAKENHI (Grant Numbers 19H05790 and 20H00354).Author details1Department of Electrical Communication Engineering, Indian Institute ofScience, Bangalore, India. 2Research Center for Functional Materials, NationalInstitute for Materials Science, Tsukuba, Japan. 3International Center forMaterials Nanoarchitectonics, National Institute for Materials Science, Tsukuba,JapanAuthor contributionsG.G. and K.M. designed the experiment. G.G. prepared the samples. K.W. andT.T. provided the hBN crystals. G.G. and K.M. performed the measurements andanalyzed the data. G.G. and K.M. co-wrote the manuscript.Conflict of interestThe authors declare no competing interests.Supplementary information The online version contains supplementarymaterial available at https://doi.org/10.1038/s41377-023-01220-4.Received: 11 February 2023 Revised: 16 June 2023 Accepted: 27 June 2023References1. Chernikov, A. et al. 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Light: Science & Applications          (2023) 12:173 Page 7 of 7 Observation of &#x0007E;100% valley-coherent excitons in monolayer MoS2 through giant enhancement of valley coherence time Introduction Results Discussion Methods Sample preparation Sample characterization Acknowledgements