The role of momentum-dark excitons in the elementary optical response of bilayer WSe2

Jessica Lindlau1, Malte Selig2,3, Andre Neumann1, Leo Colombier1, JonghwanKim4, Gunnar Berghauser2, Feng Wang4, Ermin Malic2, and Alexander Hogele1

1Fakultat fur Physik, Munich Quantum Center, and Center for NanoScience (CeNS),Ludwig-Maximilians-Universitat Munchen, Geschwister-Scholl-Platz 1, D-80539 Munchen, Germany

2Chalmers University of Technology, Department of Physics, SE-412 96 Gothenburg, Sweden3Institut fur Theoretische Physik, Nichtlineare Optik und Quantenelektronik,

Technische Universitat Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany and4Department of Physics, University of California at Berkely, Berkely, 94720 California, U.S.A.

(Dated: October 26, 2017)

Monolayer (ML) transition metal dichalcogenides (TMDs) undergo substantial changes in the single-particleband structure and excitonic optical response upon the addition of just one layer. As opposed to the single-layerlimit, the bandgap of bilayer (BL) TMD semiconductors is indirect which results in reduced photoluminescencewith richly structured spectra that have eluded a detailed understanding to date. Here, we provide a closed in-terpretation of the elementary optical responses of BL WSe2 as a representative material for the wider class ofTMD semiconductors. By combining theoretical calculations with comprehensive spectroscopy experiments,we identify the crucial role of momentum-indirect excitons for the understanding of basic absorption and emis-sion spectra ubiquitously exhibited by various TMD BLs. Our results shed light on the origin of quantum dot(QD) formation in monolayer and bilayer crystals and will facilitate further advances directed at opto-electronicapplications of layered TMD semiconductors in van der Waals heterostructures and devices.

Semiconductor TMDs exhibit remarkable properties in theML limit, including a direct bandgap at the K and K′ points ofthe hexagonal Brillouin-zone (BZ) [1, 2] with unique spin andvalley physics [3] of value for novel opto-valleytronic appli-cations [4–8]. In addition to bright excitons [9–11] composedof electrons and holes in K (or K′) valleys with collinear out-of-plane spin projections, tungsten-based MLs feature lowest-lying spin-dark excitons [12–15] as combinations of K (orK′) electrons and holes of opposite spin [16–21]. The realmof both momentum-direct excitons is expanded by the no-tion of momentum-indirect excitons involving electrons andholes in different valleys [22–28]. In ML WSe2, for example,momentum-indirect excitons can be constructed from elec-trons and holes in opposite K and K′ valleys [23, 26, 28],involve holes in the Γ valley [27], or electrons in Q (or Q′)pockets that reside roughly halfway between the Γ and K (orK′) points of the first BZ [26].

In bilayer TMDs, the single-particle band gap is indirectbecause of a downshift of the conduction band (CB) energyat Q well below K and an upshift of the valence band (VB)edge at the Γ point upon the addition of a second layer [1, 29–31]. In the specific case of BL WSe2 crystals, the lowestCB minimum is located at Q, while the VB maximum at Kexceeds the one at Γ only by 40 ± 30 meV according toangle-resolved photoemission spectroscopy [32]. The asso-ciated photoluminescence (PL) spectra are thus dominated bymomentum-indirect transitions interconnecting electrons andholes in dissimilar valleys [30, 33–39]. The BL emission isconsistently less efficient, with PL from short-lived direct ex-citons [38] red-shifted by a few tens of meV from the MLpeak emission, and a second peak with larger red-shift andlonger lifetimes [38] attributed to momentum-indirect exci-tons composed of electrons in the K or Q valleys and holes inthe K or Γ valleys [35–39]. A detailed understanding of bothpeaks, however, has remained elusive [40] despite the signif-

icance of BL TMDs as hosts of novel single-photon sources[41, 42], finite valley polarization [38] or potential utilizationof the spin-layer locking effect in charged BLs [43].

Here we present a comprehensive study, carried out bothin experiment and theory, of exciton manifolds in BL WSe2.Using cryogenic spectroscopy of BL regions subjected tostrain at unintentional disorder, we identify brightening ofmomentum-indirect excitons that in many cases is accompa-nied by the formation of quantum dots (QDs) with intenseemission of non-classical light. Complementary experimentsreveal the energy level hierarchy of all excitons involved indetermining the fundamental optical response of BL WSe2.These findings, in good quantitative agreement with theoret-ical calculations, not only explain the intricate details of theBL PL spectra and the origin of the QD PL, they can be alsogeneralized to other representatives of TMD materials to fa-cilitate a detailed understanding of opto-electronic propertiesof BL and multi-layer semiconductors.

Cryogenic optical studies (see Supplementary Informationfor details) of ML and BL WSe2 were carried out on a flakeshown Fig. 1a that was obtained by standard exfoliation ontoa Si/SiO2 substrate. Extended ML and BL regions (markedwith arrows) were identified by their respective contrast in theoptical micrograph of Fig. 1a. The dashed square indicatesthe region of the cryogenic hyperspectral raster-scan PL maprecorded with a home-built confocal microscope. The falsecolor map in Fig. 1b shows PL peak maxima in the spectralrange of 1.43 − 1.59 eV, highlighting extended homogeneousregions of bright and dim ML and BL luminescence, respec-tively, as well as distinct BL regions of unintentional disorderwith PL brightening due to local strain [41, 42].

Characteristic PL and differential reflectivity (DR) spectraof BL WSe2, recorded at 4.2 K on a representative positionaway from disorder, are shown in Fig. 1c. The PL exhibits aweak peak around 1.71 eV and a stronger peak around 1.55 eV

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FIG. 1: a, Optical micrograph of a WSe2 flake exfoliated ontoSi/SiO2 with monolayer (ML) and bilayer (BL) regions indicatedby the arrows (the scale bar is 15 µm). b, Cryogenic raster-scanphotoluminescence map of the upper corner indicated by the dashedsquare in a. False-color plot of the photoluminescence maximain the spectral range of 1.43 − 1.59 eV. The bilayer exhibits ex-tended and punctual regions of brightening attributed to strain at localfolds. c, Differential reflectivity (black) and normalized photolumi-nescence spectrum (brown, magnified by a factor of 20 in the rangeof 1.62 − 1.82 eV) at a representative bilayer position away fromdefects with model fit shown as red solid line. The energy positionsof momentum-bright (X and D) and momentum-dark BL excitons(Q↓Γ , Q↓K, K↑Γ , K′↓K, K↓Γ , and K′↑K, labelled by the capi-tal letters of electron and hole valleys and the electron out-of-planespin as subscript) are indicated by dashed lines. All spectroscopymeasurements were performed at 4.2 K with excitation at 1.95 eV.

consistent with previous PL studies of BL WSe2 [35, 37–39].The peak in DR, defined as the difference in broad-band re-flectivities on and off the flake normalized to the off-flake re-flectivity, confirms the direct nature of the higher-energy peakin PL. Remarkably, DR remains at finite values towards lowerenergies all the way down to the blue shoulder of the sec-ond PL peak. These observations, combined with theory andcomprehensive experiments, allow us to interpret the opticalresponse of BL WSe2 as originating from both momentum-direct and momentum-indirect excitons with energy positionsindicated by the dashed lines in Fig. 1c and elaborated in thefollowing.

To identify all relevant exciton states involved in absorptionand emission, and to interpret the model fit to the lower-energyPL peak shown as the red solid line in Fig. 1c, it is instructiveto consider first the single-particle band structure of BL WSe2

in Fig. 2a and the associated exciton dispersions plotted inFig. 2b. The relevant states for the construction of excitonswith a hole located at the VB maxima in K or Γ valley (indi-

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FIG. 2: a, Schematic single-particle band diagram of the conductionand the valence bands of bilayer WSe2 along high symmetry lines ofthe hexagonal Brillouin zone shown on the right. Zero-momentumspin-bright (X) and spin-dark (D) excitons are formed in the K val-ley by electrons from spin-down and spin-up conduction sub-bandsindicated in black and grey, respectively, paired with a spin-downvalence band hole. Momentum-indirect excitons with electrons andholes in dissimilar valleys are indicated by dashed ellipses. b, Cal-culated dispersions of lowest-energy exciton manifolds in BL WSe2with energy minima given in eV with respect to the bright excitonX .

cated by ellipses in Fig. 2a) are the spin-polarized sub-bandminima near K, Q and K′ valleys of the CB, with out-of-planespin projections indicated by the arrows. We take the spin-degenerate VB maximum at Γ to be 40 meV below the energyof the spin-polarized band-edge at K [32], and the energies ofthe CB at K, Q and K′ from density functional theory calcula-tions [44, 45].

The excitonic dispersions, shown in Fig. 2b, were computedusing the Wannier equation [46, 47] within the Keldysh for-malism [48–50], taking explicitly the dielectric environmentof the TMD material into account. The corresponding exci-tons, all of which have their counterparts with the hole at K′,can be separated into the class of zero-momentum excitonswith spin-allowed and spin-forbidden configuration (labelledas X and D, respectively), and finite-momentum excitons in-volving Coulomb-correlated electrons and holes from dissim-ilar valleys (labelled according to the electron and hole val-leys as capital letters with the electron out-of-plane spin assubscript Fig. 2b). All excitons but X are dipole-forbidden,either due to spin or momentum conservation constraints.

Energy minima of the branches are given in eV with re-

3

spect to the bright exciton X (see Supplementary Informationfor the details of theoretical calculations). Consistent withthe downshift (upshift) of the Q (Γ) valley in the CB (VB)of BL WSe2, we found the smallest exciton gap for finite-momentumQ↓Γ andQ↓K excitons, followed by six branchesinvolving an electron in K or K′ (two energy-degeneratebranches of D and K ′↓K and X and K ′↑K excitons with thehole at K, as well as exciton branches K↓Γ and K↑Γ with thehole in at Γ), and two branches of excitons with electrons inthe spin-up polarized Q valley Q↑Γ and Q↑K with the holein Γ and K, respectively, at highest energies.

Out of these excitons, spin-bright X states emit PL alongthe detection axis of our microscope, and the PL from spin-dark D excitons with in-plane emission is detected due tothe high numerical of the objective as well [14]. In contrast,all momentum-indirect excitons appear exclusively as phononreplicas of their optically dark zero phonon line (ZPL) as theyemit photons only with the assistance of acoustic or opticalphonons. With this constraint in mind, we note that the higher-energy peak of the BL spectrum in Fig. 1c is dominated bythe ZPL of X (in accord with the onset of strong DR) with aweak contribution from D to the red wing, while the lower-energy PL peak is a superposition of phonon sidebands ofmomentum-dark excitons Q↓Γ , Q↓K and K↑Γ .

Postponing a detailed explanation for the energy ladder ofall relevant exciton states indicated by the dashed lines inFig. 1c, we first discuss the model fit of the lower-energy peakin the BL spectrum. For the decomposition of the peak (redsolid line in Fig. 1c) into the PL contributions from Q↓Γ ,Q↓K and K↑Γ , we set the energy positions of the respec-tive dark ZPLs to the experimentally determined values andmodelled the phonon replicas by inhomogeneously broadenedGaussians with a full-width at half-maximum linewdith γ. Forsimplicity, we involved only one branch of acoustic and opti-cal phonons (the longitudinal acoustic (LA) and optical (A1)phonon branch) with energies given in Ref. 51. Best fit tothe spectrum was obtained with the inhomogeneous linewidthγ = 21 meV. The inclusion of up to sixth order scattering pro-cesses was required to reproduce the extended low energy tailof the spectrum.

At the level of theory, the energetic ordering of Q↓Γ andQ↓K states is ambiguous given the small difference of 2 meVin the energy minima of the two branches (Fig. 2b). How-ever, complementary spectroscopy experiments on strainedBL regions and quantum dots (QDs) discussed in the follow-ing remove this ambiguity and establish the energy scale hi-erarchy for all excitons responsible for the elementary opticalresponse of BL WSe2 with Q↓Γ as the lowest-energy exci-ton branch, followed byQ↓K, Q↓K, K↑Γ , degenerateD andK ′↓K states, K↓Γ , and degenerate X and K ′↑K manifolds.

The first input to the experimental determination of the ex-citon energies is provided by the PL spectroscopy of QDs dis-tributed randomly along the lines of disorder as in Fig. 1b.BL QDs, with intense and spectrally narrow PL emission asin Fig. 3a, emerge as a result of local strain [41, 42]. Akin toML QDs [41, 52–57], the QDs in disordered BL regions were

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FIG. 3: a, Photoluminescence from a strained bilayer region without(red) and with (orange) spectrally narrow and intense quantum dot(QD) emission recorded at a factor of 1000 lower excitation power.The bilayer spectrum away from strained regions (brown) is repro-duced from Fig. 1c for reference. Note the strain-induced emergenceof the shoulder at 1.615 eV labelled as Q↓K. Inset: typical second-order coherence of a single quantum dot with pronounced antibun-ching on ∼ 10 ns time scale. b, Distribution of quantum dot inten-sities as a function of their peak emission energies (filled circles, ex-tracted from the map of Fig. 1b), and photoluminescence excitationspectra of the quantum dot from Fig. 3a and strained bilayer emission(orange and red open circles, respectively). The dashed lines markthe energy positions of the relevant exciton states.

characterized by strong antibunching signatures in the second-order correlation function of their PL emission [41, 42], asdemonstrated exemplarily by the inset data of Fig. 3a recordedon a different QD with a dip of 0.2 at τ = 0 and an exponentialrise to 1 on a timescale of ∼ 10 ns. By plotting the PL inten-sity as a function of the respective energy maximum for allQDs of the hyperspectral map of Fig. 1b, we identify a sharpcut-off to the QD emission energy at 1.584 eV (indicated bythe leftmost dashed line in Fig. 3b) which we assign to thestate Q↓Γ (see Supplementary Information for assignment).

The energy position of the next higher-energy momentum-dark state is revealed by the PL spectroscopy of strained BLregions. The PL spectrum on a strained position features char-acteristic blue- and red-shifts of a few meV for the upper andlower PL peaks (compare red and brown traces in Fig. 3a)consistent with ∼ 0.1% of tensile strain which lowers (raises)the CB energy minimum at K (Q) [58]. In addition, a shoulderat 1.615 eV, indicated by the dashed line in Fig. 3a, becomesapparent due to strain-induced brightening of this momentum-

4

dark transition [59]. The energy position of this shoulderreappears as a resonance in the photoluminescence excita-tion (PLE) spectrum of a strained BL spot (open red circlesin Fig. 3b). The resonance, marked by the dashed line and as-signed toQ↓K, is even more pronounced in the PLE spectrumof the QD from the same spot position (with the spectrum inFig. 3a) shown by open orange circles in Fig. 3b. We notethat the PLE spectrum is not QD specific, it rather representsgeneric BL resonances in the PLE of QDs emitting at differentobservation sites with different energies (see SupplementaryInformation for a PLE spectrum of another QD).

The third successive energy level of momentum-dark states,identified at 1.624 eV by the resonance and the shoulder ofthe QD and BL PLE spectra of Fig. 3b, respectively, is as-cribed to K↑Γ . With this energy, the experimental valuesof the three lowest-energy momentum-dark exciton states cannow be hierarchically ordered with respect to the energy ofthe bright exciton X at 1.710 eV deduced from the peaks ofboth PLE spectra of Fig. 3b and from PL and DR maxima inFig. 1c. Referencing all energies to that of X , we note firstthat the lowest momentum-forbidden state Q↓Γ is red-shiftedby 126 meV instead of the calculated value of 163 meV, whilethe second-lowest state Q↓K exhibits a red-shift of 95 meVinstead of 161 meV expected from theory. Provided that theeffective masses used in the calculations of exciton energieswere correct, these quantitative discrepancies between theoryand experiment convert into an upshift of the CB minimum atthe Q valley by 66 meV and a downshift of the VB at the Γpoint by 29 meV. Given the uncertainties in band structure cal-culations [44, 45] and angle-resolved photoemission [32] usedto calculate the exciton dispersion minima, these correctionsof a few tens of meV seem reasonable.

Finally, with the energies of X and K↑Γ at hand, we es-timate the energies of D and K↓Γ in Fig. 3b by using therespective spin-orbit splittings of 51 meV and 57 meV fromFig. 2b. While the energy level of K↓Γ has no compellingsignature in the PLE spectra of Fig. 3b, the D state coincideswith a clearly pronounced shoulder in the PLE spectrum of theQD. To complete the energetic ordering of all lowest-lying ex-citons in BL WSe2, the states K ′↓K and K ′↑K are placed inresonance with D and X by omitting electron-hole exchange.With this complete set of exciton energies used to explain thePL signatures in the spectrum of Fig. 1c, we can also inter-pret the onset of DR around 1.550 eV as stemming from thefirst optical sideband of Q↓Γ . Absorption due to higher-orderphonon processes are frozen out at the cryogenic temperatureof our experiment, but we expect the absorption onset to be-come more and more pronounced towards lower energies withan increasing population of optical phonons at higher temper-atures.

This notion of momentum-dark exciton states facilitatesa more insightful perspective on the elementary optical re-sponses of other bilayer TMD materials and heterobilayer sys-tems [60]. Moreover, it provides additional insight into theorigin of QDs in ML [41, 52–57] and BL [41, 42] TMDs. Inaddition to spectrally narrow and bright PL with antibunched

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FIG. 4: a, False-color plot of quantum dot magnetoluminescenceunder σ+ (σ−) polarized excitation for positive (negative) magneticfields in Faraday geometry. b, The quantum dot emission doublet(upper panel) is characterized by linearly polarized peaks with or-thogonal polarization axes (lower panel; note the anti-correlation inthe intensities of the higher and lower energy peaks shown in red andblue together with squared sine and cosine fits). c, Energy dispersionof the doublet splitting in magnetic field. Best fit to the data witha hyperbolic function (solid line) was obtained for a zero-field fine-structure splitting ∆0 of 500 µeV and an exciton g-factor of 9.5. d,Distribution of exciton g-factors around the mean value of 9.5 plottedfor ten quantum dots with respect to their zero-field splitting.

photon emission statistics discussed above, BL QDs share allmain signatures of localized exciton states with ML QDs. Inhigh-resolution micro-PL spectroscopy, they exhibit a doubletof states with orthogonal linear polarization (Fig. 4a and b)which evolves into a pair of circularly polarized Zeeman-splitpeaks with increasing magnetic field (Fig. 4a and c). The dis-persion of the Zeeman splitting ∆ between the blue and redQD branches with out-of-plane magnetic field B according tothe hyperbolic function ∆ =

√(gµBB)2 + ∆2

0 (solid line inFig. 4c) is a hallmark of QDs with anisotropic fine-structuresplitting ∆0 [61]. At large enough fields, the linear asymptoteof the Zeeman splitting is determined by the exciton g-factor(g) and the Bohr magneton (µB).

By applying this analysis to ten randomly selected QDs onstrained BL positions, we extracted g and ∆0 from the hyper-bolic fit to the Zeeman splitting as for the QD of Fig. 4a and cwith g = 9.5 and ∆0 = 500 µeV. Remarkably, as evidencedfrom Fig. 4d, the g-factor of all ten QDs shows only minorvariations around the average value of 9.5±0.2 independent ofthe QD PL energy and despite the spread in the fine-structuresplittings that are typical for ML and BL QDs in WSe2 in therange of ∼ 400− 900 µeV [41, 42, 52–55, 57]. This observa-

5

tion suggests that QD excitons are localized momentum-darkexcitons that inherit their g-factor from the delocalized con-tinuum state (i.e. Q↓Γ in the case of BL WSe2) and exhibitsignificant brightening due to their spread in momentum spaceupon spatial localization. This picture is further supported bythe sharp cut-off to the emission energy of BL QDs at the en-ergy of Q↓Γ momentum-dark excitons in Fig. 3b as well as inprevious studies [41, 42].

For QDs in ML WSe2 with similarly sharp cut-off ener-gies at ∼ 20 − 25 meV below the bright state X [41, 42, 52,54, 55, 57] and surprisingly large g-factors in the range of6 − 12 [41, 42, 52–55] this insight suggests the presence of amomentum-dark reservoir with energy in between the brightand dark ML excitonsX andD as discussed in Ref. 28. In MLMoSe2 void of momentum-dark states below the bright exci-ton, on the other hand, no cut-off energy to the QD emissionwas observed and similar values for the the g-factors of QDexcitons and the bright exciton X were found [56]. To lever-age this speculation, theoretical estimates of exciton g-factorswould be required for all excitons constructed from electronsand holes in CB and VB valleys other than K.

Acknowledgments: We thank G. Cassabois (UniversiteMontpellier) and A. Knorr (TU Berlin) for fruitful discus-sions. This research was funded by the European ResearchCouncil under the ERC Grant Agreement No. 336749, theVolkswagen Foundation, and the German Excellence Initia-tive via the Nanosystems Initiative Munich (NIM). A. H.also acknowledges support from the Center for NanoScience(CeNS) and LMUinnovativ. E. M. acknowledges fundingfrom the European Unions Horizon 2020 research programunder grant agreement No. 696656 - Graphene Flagship aswell as from the Swedish Research Council. M. S. is thank-ful to the DFG through SFB 951. J. K. and F. W. acknowl-edge support from National Science Foundation EFRI pro-gram (EFMA-1542741).

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SUPPLEMENTARY INFORMATION

Experimental setup

Confocal spectroscopy studies were performed in liq-uid helium or a closed-cycle cryostat (attocube systems,attoDRY1000) with base temperatures of 4.2 K and 3.1 K,respectively. The latter cryostat was equipped with a solenoidproviding magnetic fields up to ±9 T. The sample was po-sitioned with piezo actuators and scanners (attocube sys-tems, ANP101 series and ANSxy100/lr) into the diffrac-tion limited spot of a low-temperature apochromatic objec-tive with a numerical aperture of 0.82 (attocube systems, LT-APO/VISIR/0.82) and a spot size of 0.6 µm. A fiber-basedhome-built microscope, coupled in back scattering geome-try to a standard spectrometer (PI, Acton SP-2558) with anitrogen-cooled silicon CCD (PI, Spec-10:100BR/LN) and aresolution of 0.26 meV (0.05 meV in Fig. 4 of the main textand in Fig. S2, bottom panel) was used for photoluminescence(PL) and differential reflectivity (DR) measurements. A su-percontinuum laser (NKT Photonics, SuperK EXW-12) wasused for DR measurements. The PL was excited with a con-tinuous wave (cw) diode laser at 639 nm (New Focus, Veloc-ity TLB-6704), a ps-pulsed diode laser at 630 nm (PicoQuant,LDH-P-C-630), or a Ti:sapphire laser (Coherent, Mira 900)operated either in cw or ps-pulsed mode. Photoluminescenceexcitation (PLE) was performed with the Ti:sapphire laserin cw mode. Two single photon counting avalanche photo-diodes (PicoQuant, τ -SPAD) were used in a Hanburry Brown-Twiss configuration for measurements of photon statistics.

Theoretical calculations

To compute excitonic binding energies for direct as well asfor indirect excitons, we solve the Wannier equation [46, 47]

~2q2

2mϕµq −

∑k

V exck−qϕ

µk + Egapϕ

µq = Eµϕµq. (1)

Here,m = memh/(me+mh) denotes the reduced mass withthe electron (hole) mass me (mh). The latter were obtainedfrom density functional theory calculations for electrons[44] and angle-resolved photoemission spectroscopy mea-surements for holes [32]. The electronic bandgapEgap includ-ing band separation for different valleys and spin bands wasobtained from density functional theory calculations (elec-trons) [45] and angle-resolved photoemission spectroscopyexperiments (holes) [32]. The appearing Coulomb matrix el-ement V exc

k−q was treated within the Keldysh formalism [48–50, 62], where we can take explicitly into account the dielec-tric screening from the environment and the width of the in-vestigated TMD material. Solving the Wannier equation, wehave microscopic access to exciton eigenenergies Eµ and ex-citonic wavefunctions ϕµq. Note that we do not include ex-change coupling, which is known to lift the degeneracy be-tween spin-like and spin-unlike exciton states [24].

7

0

1

0

50

100

150

1.50 1.55 1.60 1.65 1.70 1.75 1.800

50

100

150

QD

inte

nsity (

arb

. units)

Energy (eV)

0

1

Norm

aliz

ed P

LE

inte

nsity

Q↓Γ

D X

QD

Q↓K

Q↓Γ K↓Γ K↑Γ

D X

QD

strained

BL

Q↓K

K↓Γ K↑Γ

strained

BL

Figure S1: Data duplicated from Fig. 3b of the main text with excitonlabelling same as in the main text. Upper panel: energy positions andpeak assignments of dark-exciton states according to the theoreticallypredicted energy scale hierarchy. Lower panel: energy positions ofmomentum-dark excitons according to the reversed ordering of thetwo lowest-energy states. Note the missing peak assignment in thelower panel indicated by the arrow.

0

1

0

1

1.50 1.55 1.60 1.65 1.70 1.75 1.800

1

Norm

aliz

ed inte

nsity

Energy (eV)

0

1

Norm

aliz

ed P

LE

inte

nsity

Q↓Γ

D X

QD

strained

BL

Q↓K

K↓Γ K↑Γ

QD1

strained BL

QD2

QD3

QD1

QD2

QD3

pos. 2

pos. 3

Figure S2: Same as in Fig. S1 but recorded on a different positionof disorder-strained bilayer WSe2 with quantum dot emission. Theexciton energies marked by the dashed lines are the same as in theupper panel of Fig. S1 and in Fig. 3b of the main text.

Discussion of PLE spectra and peak ordering

In order to rationalize the energetic ordering of momentum-dark excitons, the data Fig. 3b of the main text is reproduced inboth panels of Fig. S1. The energies identified from the cut-offto the quantum dot (QD) PL as well as from QD and bilayer(BL) PLE spectra were referenced to the energy of the brightexciton X at 1.710 eV that lies 51 meV above its spin-orbitsplit momentum-direct spin-dark counterpart D according totheory. The energies of the two lowest-energy dark-excitonstates were identified as E1 = 1.584 eV and E2 = 1.615 eV,respectively, yielding experimental shifts of ∆exp

1−X = E1 −EX = −126 meV and ∆exp

2−X = E2 − EX = −95 meV withrespect to the energy of X . According to theoretical calcula-tions, the two lowest-energy states Q↓Γ and Q↓K, separatedby ∆th

QΓ−X = −163 meV and ∆thQK−X = −161 meV, re-

spectively, compete for the assignment to the lowest-energystates.

First, we test the scenario of preserved energy scale hierar-chy with Q↓Γ state being lowest in energy (with energy E1),followed by the stateQ↓K (with energyE2). We note that thestatesQ↓K andX share their hole in the K valley and thus theenergy difference can be entirely attributed to the electron inthe CB minimum at Q. With ∆exp

2−X−∆thQK−X = 66 meV, we

thus directly obtain the upshift of the exciton energy with thehole in K and the electron in Q. For the state Q↓Γ we obtainthe difference between the experimental and theoretical valuesas ∆exp

1−X −∆thQΓ−X = 37 meV, which implies a downshift of

the VB maximum at Γ by 29 meV by using the upshift of theCB minimum at Q calculated above. The energy of the stateK↓Γ , which shares with X the electron in the spin-up polar-ized sub-band at K, computes by including the upshift of Γ to1.681 eV. Finally, the energy of the state K↑Γ is obtained as1.624 eV by taking into account the theoretically calculatedspin-orbit splitting of 57 meV between K↓Γ and K↑Γ .

The second scenario probes the reversed ordering where thestate Q↓K is lowest (with energy E1) and Q↓Γ is second-lowest state (with energy E2). Calculations of the respectiveenergies for all relevant momentum-dark states along the linesof arguments given above yields an upshift of the Q valley by35 meV and an upshift of the Γ valley by 33 meV. Accord-ingly, the energies of K↓Γ and K↑Γ states are obtained as1.743 eV and 1.686 eV, respectively.

The energy positions for all relevant excitons obtained fromthe two competing assignment scenarios are plotted as dashedlines in Fig. S1. The upper panel shows the energetic order-ing in accord with preserved hierarchy, while the lower panelshows the results of reversed ordering. The failure of the latterto predict the resonance in PLE at 1.624 eV (indicated by theblack arrow in the lower panel) which is consistently ascribedin the framework of the former to the state K↑Γ , providesstrong evidence for Q↓Γ being the lowest and Q↓K being thesecond-lowest state.

Remarkably, all PLE resonances appear at the same energypositions for a different BL region subjected to unintentionalstrain (red PLE spectrum in Fig. S2) with confocal PL from a

8

QD emitting at a different PL energy (orange PLE spectrum inFig. S2). This observation indicates that the PLE resonancesare not QD-specific (e.g. due to excited QD states that would

differ from dot to dot because of different confinement poten-tials) but indeed probe the absorption of BL WSe2.