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

Monolayer transition metal dichalcogenides (TMDs) undergo substantial changes in the single-particle band structure and excitonic optical response upon the addition of just one layer. As opposed to the single-layer limit, the bandgap of bilayer (BL) TMD semiconductors is indirect which results in reduced photoluminescence with richly structured spectra that have eluded a detailed understanding to date. Here, we provide a closed interpretation of cryogenic emission from BL WSe2 as a representative material for the wider class of TMD semiconductors. By combining theoretical calculations with comprehensive spectroscopy experiments, we identify the crucial role of momentum-indirect excitons for the understanding of BL TMD emission. Our results shed light on the origin of quantum dot formation in BL crystals and will facilitate further advances directed at opto-electronic applications of layered TMD semiconductors in van der Waals heterostructures and devices.

In BL TMDs, the single-particle bandgap is indirect because of a downshift of the conduction band (CB) energy at 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][30][31][32] . In the specific case of BL WSe 2 crystals, the lowest CB minimum is located at Q, while the VB maximum at K exceeds the one at Γ only by 40 ± 30 meV according to angle-resolved photoemission spectroscopy 33 . The associated photoluminescence (PL) spectra are thus dominated by momentum-indirect transitions interconnecting CB electrons and empty VB states in dissimilar valleys 30,[34][35][36][37][38][39][40] . The BL emission is consistently less efficient, with PL from short-lived direct excitons 39 redshifted by a few tens of meV from the ML peak emission, and a second peak with larger redshift and longer lifetimes 39 attributed to momentum-indirect excitons composed of electrons in the K or Q valleys and unoccupied states in the K or Γ valleys [36][37][38][39][40] . A detailed understanding of both peaks, however, has remained elusive 41 despite the significance of BL TMDs as hosts of novel single-photon sources 42,43 , finite valley polarization 39 , or potential utilization of the spin-layer locking effect in charged BLs 44 .
Here, we present a comprehensive study of exciton manifolds in BL WSe 2 carried out both in experiment and theory. Using cryogenic spectroscopy of BL regions subjected to strain at unintentional disorder, we identify brightening of momentumindirect excitons that in some cases is accompanied by the formation of quantum dots (QDs) with intense emission of nonclassical light. Complementary experiments reveal the energylevel hierarchy of all excitons involved in determining the optical response of BL WSe 2 . These findings, in good quantitative agreement with theoretical calculations, not only explain the intricate details of the BL PL spectra and the origin of the QD PL, they can be also generalized to other representatives of TMD materials to facilitate a detailed understanding of opto-electronic properties of BL and multilayer semiconductors.

Results
Photoluminescence spectroscopy of bilayer WSe 2 . Cryogenic spectroscopy of ML and BL WSe 2 was carried out on a flake shown in Fig. 1a obtained by standard exfoliation onto an Si/SiO 2 substrate (see Supplementary Notes 1 and 2 for details). Extended ML and BL regions (marked with arrows) were identified by their respective contrast in the optical micrograph of Fig. 1a and by Raman spectroscopy. The dashed square indicates the region of the cryogenic hyperspectral raster-scan PL map recorded with a home-built confocal microscope. The false-color map in Fig. 1b shows PL peak maxima in the spectral range of 1.43-1.59 eV, highlighting extended homogeneous regions of bright ML and dim BL luminescence, as well as distinct BL regions of unintentional disorder with PL brightening due to local strain 42,43 .
A characteristic PL spectrum of BL WSe 2 on SiO 2 , recorded at 4.2 K on a representative position away from disorder, is shown in Fig. 1c (see Supplementary Fig. 1 for spectra from other BL WSe 2 samples). The PL exhibits a weak peak around 1.71 eV and a stronger peak around 1.55 eV consistent with previous PL studies of BL WSe 2 36,38-40 . Based on comprehensive experiments described in the following, we develop a model to interpret the PL of WSe 2 BLs as originating from both momentum-direct and momentum-indirect excitons with energy positions indicated by the dashed lines in Fig. 1c.
Excitons in bilayer WSe 2 . To identify all relevant excitons that contribute to cryogenic PL and to interpret the model fit to the lower-energy PL peak shown as the red solid line in Fig. 1c, it is instructive to consider first the single-particle band structure of BL WSe 2 in Fig. 2a and the associated exciton dispersions plotted in Fig. 2b. The relevant states for the construction of excitons (indicated by ellipses in Fig. 2a) with an empty state located at the VB maxima in the K or Γ valley are the spin-polarized subband minima near K, Q, and K′ valleys of the CB, with out-ofplane spin projections indicated by the arrows. We take the spin degenerate VB maximum at Γ to be 40 meV below the energy of  The energy positions of momentum-bright (X and D) and momentum-dark BL excitons (Q ↑ Γ, Q ↑ K, K ↓ Γ, K 0 " K, K ↑ Γ, and K 0 # K, labeled by the capital letters of electron and empty state valleys and the electron out-of-plane spin as subscript) are indicated by dashed lines. All spectroscopy measurements were performed at 4.2 K with excitation at 1.95 eV the spin-polarized band edge at K 33 , and the energies of the CB at K, Q, and K′ from density functional theory calculations 45,46 .
The excitonic dispersions, shown in Fig. 2b, were computed using the Wannier equation 47,48 within the Keldysh formalism 49-51 , taking explicitly the dielectric environment of the TMD material into account. The corresponding excitons, all of which have their counterparts with the empty VB state at K′, can be separated into the class of zero-momentum excitons with spinallowed and spin-forbidden configuration (labeled as X and D, respectively), and finite-momentum excitons involving Coulombcorrelated electrons and unoccupied VB states from dissimilar valleys (labeled in Fig. 2b according to the valleys of the CB electron and the VB unoccupied state as capital letters with the electron out-of-plane spin as subscript). 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 respect to the bright exciton X (see Supplementary Note 3 for the details of theoretical calculations). Consistent with the downshift (upshift) of the Q (Γ) valley in the CB (VB) of BL WSe 2 , we found the smallest exciton gap for finite-momentum Q ↑ Γ and Q ↑ K excitons, followed by six branches involving an electron in K or K′ (two energy degenerate branches of D and K ′ " K, and X and K ′ # K excitons with the unoccupied state at K, as well as exciton branches K ↑ Γ and K ↓ Γ with the empty VB state at Γ), and two branches of excitons at highest energies with electrons in one of the three spin-down polarized Q valleys forming Q ↓ Γ and Q ↓ K with the unoccupied states in Γ and K valleys, respectively. We emphasize that although we use a spin-tagged notation for Qmomentum excitons consistent with the schematics in Fig. 2a, there are energetically degenerate excitons composed of electrons from the other three inequivalent Q pockets with opposite spin.
Out of this set of excitons, spin-bright X states emit PL along the detection axis of our microscope, and the PL from spin-dark D excitons with in-plane emission is detected due to the high numerical aperture of the objective as well 14 . In contrast, all momentum-indirect excitons appear exclusively as phonon replicas of their optically dark zero-phonon line (ZPL) as they emit photons only with the assistance of acoustic or optical phonons. With this constraint in mind, we note that the higherenergy peak of the BL spectrum in Fig. 1c is dominated by the ZPL of X (in accord with the onset of differential reflectivity, shown in Supplementary Fig. 1a) with a weak contribution from D to the red wing, while the lower-energy PL peak is a superposition of phonon sidebands of momentum-dark excitons Postponing a detailed explanation for the energy ladder of all relevant exciton states indicated by the dashed lines in Fig. 1c, we first discuss the model fit of the lower-energy peak in the BL spectrum. For the decomposition of the peak (red solid line in Fig. 1c) into the PL contributions from Q ↑ Γ, Q ↑ K, and K ↓ Γ, we set the energy positions of the respective dark ZPLs to the experimentally determined values and modeled the phonon replicas by inhomogeneously broadened Gaussians with a fullwidth at half-maximum linewidth γ. For simplicity, we involved only one branch of acoustic and optical phonons (the longitudinal acoustic and optical phonon branch) with energies given in ref. 52 (see Supplementary Table 1 for phonon energies and Supplementary Note 4 for details of the fit procedure). Best fit to the spectrum was obtained with the inhomogeneous linewidth γ = 21 meV. The inclusion of up to sixth-order scattering processes was required to reproduce the extended low-energy tail of the spectrum.
At the level of theory, the energetic ordering of Q ↑ Γ and Q ↑ K states is ambiguous given the small difference of 2 meV in the energy minima of the two branches (Fig. 2b). However, complementary spectroscopy experiments on strained BL regions and QDs discussed in the following remove this ambiguity and establish the energy scale hierarchy for all excitons responsible for the elementary optical response of BL WSe 2 with Q ↑ Γ as the lowest-energy exciton branch, followed by Q ↑ K, K ↓ Γ, degenerate D and K ′ " K states, K ↑ Γ, and degenerate X and K ′ # K manifolds.
Effects of strain and local disorder. The first input to the experimental determination of the exciton energies is provided by the PL spectroscopy of QDs distributed randomly along the lines of disorder as in Fig. 1b. BL QDs, with intense and spectrally narrow PL emission as in Fig. 3a, emerge as a result of local strain 42,43 . Akin to ML QDs 42,53-58 , the QDs in disordered BL regions were characterized by strong antibunching signatures in the second-order correlation function g (2) (τ) of their PL emission 42,43 , as demonstrated exemplarily by the inset data of Fig. 3a recorded on a different QD with a dip of 0.2 at τ = 0 and an exponential rise to 1 on a timescale of~10 ns. By plotting the PL intensity as a function of the respective energy maximum for all QDs of the hyperspectral map of Fig. 1b, we identify a sharp cutoff to the QD emission energy at 1.584 eV (indicated by the leftmost dashed line in Fig. 3b), which we assign to the state Q ↑ Γ (see Supplementary Note 5 and Supplementary Fig. 2 for assignment). The energy position of the next higher-energy momentumdark state is revealed by the PL spectroscopy of strained BL   Fig. 3a) consistent with~0.1% of tensile strain which lowers (raises) the CB energy minimum at K (Q) 59 . In addition, a shoulder at 1.615 eV, indicated by the dashed line in Fig. 3a, becomes apparent due to strain-induced brightening of this momentum-dark transition 60 . The energy position of this shoulder reappears as a resonance in the photoluminescence excitation (PLE) spectrum of a strained BL spot (open red circles in Fig. 3b). The resonance, marked by the dashed line and assigned to Q ↑ K, is even more pronounced in the PLE spectrum of the QD from the same spot position (with the spectrum in Fig. 3a) shown by open orange circles in Fig. 3b. We note that the PLE spectrum is not QD specific, it rather represents generic BL resonances in the PLE of QDs emitting at different observation sites with different energies (see Supplementary Fig. 3 for PLE spectra of other QDs).
The third successive energy level of momentum-dark states, identified at 1.624 eV by the resonance and the shoulder of the QD and strained BL PLE spectra of Fig. 3b, respectively, is ascribed to K ↓ Γ. With this energy, the experimental values of the three lowest-energy momentum-dark exciton states can now be hierarchically ordered with respect to the energy of the bright exciton X at 1.710 eV deduced from the peaks of both PLE spectra of Fig. 3b and from PL in Fig. 1c (and differential reflectivity shown in Supplementary Fig. 1a). Referencing all energies to that of X, we note first that the lowest momentum-forbidden state Q ↑ Γ is redshifted by 126 meV instead of the calculated value of 163 meV, while the second lowest state Q ↑ K exhibits a redshift of 95 meV instead of 161 meV expected from theory. Provided that the effective masses used in the calculations of exciton energies were correct, these quantitative discrepancies between theory and experiment convert into an upshift of the CB minimum at the Q valley by 66 meV and a downshift of the VB at the Γ point by 29 meV. Given the uncertainties in band structure calculations 45,46 and angle-resolved photoemission 33 used to calculate the exciton dispersion minima, these corrections of a few tens of meV seem reasonable.
Finally, with the energies of X and K ↓ Γ at hand, we estimate the energies of D and K ↑ Γ in Fig. 3b by using the respective spin-orbit splittings of 51 and 57 meV from Fig. 2b. While the energy level of K ↑ Γ has no compelling signature in the PLE spectra of Fig. 3b, the D state coincides with a clearly pronounced shoulder in the PLE spectrum of the QD. To complete the energetic ordering of all lowest-lying excitons in BL WSe 2 , the states K ′ " K and K ′ # K are placed in resonance with D and X by omitting electron-hole exchange.
Field-effect control of doping in bilayer WSe 2 embedded in hexagonal boron nitride. We tested this set of exciton energies obtained from the analysis of PL signatures of BL WSe 2 on SiO 2 in Fig. 1c on narrow spectra of a gate-tunable WSe 2 bilayer encapsulated in hexagonal boron nitride (hBN). Figure 4a shows the normalized PL spectrum (brown) under charge neutrality conditions with the corresponding model fit. Remarkably, the intricate spectral features of the spectrum can be reproduced with   1.7 1.5 Fig. 4 Field-effect control of bilayer WSe 2 photoluminescence. a Normalized photoluminescence (brown) of a bilayer WSe 2 encapsulated in hexagonal boron nitride and tuned toward charge neutrality with positive gate voltages. Best fit to the spectrum (red) was obtained with the same set of excitons energies as in Fig. 1c and an overall redshift of 2 meV. b Evolution of the photoluminescence with gate voltage from −30 to 30 V. Note the cross-over to the charged regime below −20 V signified by a simultaneous redshift of 22 meV for all peaks as indicated by the red arrows below the dashed line the set of exciton energies established above with an overall redshift of 2 meV. The spectrally narrow inhomogeneous linewidths down to~8 meV required a refined model fit with all phonon modes included and variations of ±2 meV around the values calculated for ML WSe 2 in ref. 52 (see Supplementary Note 4 for details of the fit procedure).
Including first-order scattering from K ↓ Γ and Q ↑ K exciton manifolds, and up to second-order scattering from Q ↑ Γ, our model identifies the blue-most peak at~1.6 eV out of the three strong peaks in the spectrum of Fig. 4a in the range 1.61-1.57 eV as optical and acoustic phonon replica of K ↓ Γ and Q ↑ K, respectively, the central peak as an optical sideband of Q ↑ K, and the red-most peak as an acoustic sideband of Q ↑ Γ. The consecutive peak toward lower energies around 1.56 eV is an optical sideband of Q ↑ Γ that merges with the extended red wing composed of higher-order phonon replica from all the abovementioned momentum-dark states.
Interestingly, the ambiguity with respect to the origin of the blue-most peak as being composed of acoustic or optical sidebands of K ↓ Γ and Q ↑ K momentum-dark states is removed by the observation of a simultaneous shift of all three peaks upon negative doping at gate voltages below −20 V in Fig. 4b. At this voltage, finite electron population of the Q pockets favors the formation of BL trions with electrons forced to reside in different Q valleys by the Pauli exclusion principle. Thus, all sideband replica associated with Q ↑ K and Q ↑ Γ sates are expected to shift simultaneously. This is exactly what we observe upon negative doping with a redshift by the trion binding energy of~22 meV for all three peaks. The shift of the blue-most peak implies that it originates from the Q ↑ K rather than the K ↓ Γ state, as the latter is insensitive to the increasing doping level at Q. Data at higher doping levels and with better signal-to-noise ratio have not been recorded prior to a fatal breakdown of the device and thus a more detailed analysis of charge control of BL WSe 2 PL must be postponed to future work.
Quantum dots in bilayer WSe 2 . The notion of momentum-dark exciton states provides a new perspective on the origin of QDs in ML 42,53-58 and BL 42,43 TMDs. In addition to spectrally narrow and bright PL with antibunched photon emission statistics discussed above, BL QDs share all main signatures of localized excitons with ML QDs. In high-resolution micro-PL spectroscopy, they exhibit a doublet of states with orthogonal linear polarization (Fig. 5a, b), which evolves into a pair of circularly polarized Zeeman-split peaks with increasing magnetic field (Fig. 5a, c). The dispersion of the Zeeman splitting Δ between the blue and red QD branches with out-of-plane magnetic field B according to the hyperbolic function Δ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Fig. 5c) is a hallmark of QDs with anisotropic fine-structure splitting Δ 0 61 . At large enough fields, the linear asymptote of the Zeeman splitting is determined by the exciton g-factor scaled by the Bohr magneton μ B .
By applying this analysis to ten randomly selected QDs on strained BL positions, we extracted g and Δ 0 from the hyperbolic fit to the Zeeman splitting as for the QD of Fig. 5a, c with g = 9.5 ± 0.1 and Δ 0 = 500 ± 10 μeV. Remarkably, as evidenced from Fig. 5d, the g-factor of all ten QDs shows only minor variations around the average value of 9.5 independent of the QD PL energy and despite the spread in the fine-structure splittings in the range of~400-900 μeV 42,43,[53][54][55][56]58 . This observation suggests that QD excitons relate to momentum-dark excitons that inherit their gfactor from the delocalized continuum state (i.e., Q ↑ Γ in the case of BL WSe 2 ) and exhibit significant brightening due to their spread in momentum space upon spatial localization. This picture is further supported by the sharp cutoff to the emission energy of BL QDs at the energy of Q ↑ Γ momentum-dark excitons in Fig. 3b as well as in previous studies 42,43 .
For QDs in ML WSe 2 with similarly sharp cutoff energies at 20-25 meV below the bright state X 42,43,53,55,56,58 and surprisingly large g-factors in the range of 6-12 42,43,[53][54][55][56] , this insight suggests the presence of a momentum-dark reservoir with energy in between the bright and dark ML excitons X and D as discussed in ref. 28 . In ML MoSe 2 void of momentum-dark states below the bright exciton, on the other hand, no cutoff energy to the QD emission was observed and similar values for the the g-factors of QD excitons and the bright exciton X were found 57 . To leverage this speculation, theoretical calculations of exciton g-factors are required for all excitons constructed from CB electrons and VB unoccupied states in valleys other than K.
Data availability. The data that support the findings of this study are available from the corresponding author on reasonable request.