Electric-field-tunable intervalley excitons and phonon replicas in bilayer WSe_2
Mashael M. Altaiary, Erfu Liu, Ching-Tarng Liang, Fu-Chen Hsiao, Jeremiah van Baren, Takashi Taniguchi, Kenji Watanabe, Nathaniel M. Gabor, Yia-Chung Chang, Chun Hung Lui
11 Electric-field-tunable intervalley excitons and phonon replicas in bilayer WSe Mashael M. Altaiary , Erfu Liu , Ching-Tarng Liang , Fu-Chen Hsiao , Jeremiah van Baren , Takashi Taniguchi , Kenji Watanabe , Nathaniel M. Gabor , Yia-Chung Chang , Chun Hung Lui Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA. Research Center for Applied Sciences, Academia Sinica, Taipei 11529, Taiwan. Advanced Semiconductor Device and Integration Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science, 1-1 Namiki Tsukuba, Ibaraki 305-0044, Japan. Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. Canadian Institute for Advanced Research, MaRS Centre West Tower, 661 University Avenue,
Toronto, Ontario ON M5G 1M1, Canada. § Equal contribution. * Corresponding authors. Email: [email protected]; [email protected]
Abstract:
We report the direct observation of intervalley exciton between the Q conduction valley and Γ valence valley in bilayer WSe by photoluminescence. Τ he Q Γ exciton lies at ~18 meV below the QK exciton and dominates the luminescence of bilayer WSe . By measuring the exciton spectra at gate-tunable electric field, we reveal different interlayer electric dipole moments and Stark shifts between Q Γ and QK excitons. Notably, we can use the electric field to switch the energy order and dominant luminescence between Q Γ and QK excitons. Both Q Γ and QK excitons exhibit pronounced phonon replicas, in which two-phonon replicas outshine the one-phonon replicas due to the existence of (nearly) resonant exciton-phonon scatterings and numerous two-phonon scattering paths. We can simulate the replica spectra by comprehensive theoretical modeling and calculations. The good agreement between theory and experiment for the Stark shifts and phonon replicas strongly supports our assignment of Q Γ and QK excitons. Transitional metal dichalcogenides (TMDs, e.g. MoS , WSe ) are semiconductors with multiple electronic valleys and remarkable excitonic states [1-9]. They exhibit intravalley direct band gap in the monolayer limit, but transit to intervalley indirect band gap in the bilayer and multilayers [1,2]. While much research has focused on the intravalley excitons in monolayer TMDs, intervalley excitons in bilayers are also interesting. Compared to the monolayer excitons, the bilayer excitons possess longer lifetime, different valley configurations, and extra layer degree of freedom, all of which can be harnessed for novel excitonics and valleytronics [10-23]. Of the many TMDs, bilayer WSe is a model system to explore the bilayer intervalley excitons [21-25]. Natural bilayer WSe crystals possess the 2H stacking order with inversion and time-reversal symmetry [Fig. 1 (a, b)] [11,12,26]. Its electronic band structure exhibits valleys at the zone center ( Γ), two zone corners ( Κ , K’), and six interior locations (Q) in the Brillouin zone [Fig. 1(c)] [27,28]. In monolayer WSe , the Q conduction valleys reside at ~30 meV above the K conduction valleys, and the Γ valence valley resides at ~600 meV below the K valence valleys [29,30]; the Q and Γ valleys play no significant role in the excitonic emission. By contrast, in bilayer WSe , the Q valleys lie at ~200 meV below the K conduction valley, and the Γ valley is only ~140 meV below the K valence valley [Fig. 1(d)] [28-31]; their contribution to the excitonic emission increases significantly compared to the monolayer case. In particular, while the electronic band structure suggests that the QK intervalley exciton ( 𝑋𝑋 𝑄𝑄𝐾𝐾 ) is the lowest and dominant exciton in bilayer WSe , the Q Γ intervalley exciton ( 𝑋𝑋 𝑄𝑄𝛤𝛤 ) can compete with 𝑋𝑋 𝑄𝑄𝐾𝐾 due to its possible stronger excitonic effect [25]. The competition between 𝑋𝑋 𝑄𝑄𝐾𝐾 and 𝑋𝑋 𝑄𝑄𝛤𝛤 becomes even more intriguing under an electric field. Unlike monolayer, bilayer excitons can possess interlayer electric dipole [13,16,21,26,32-37]. By using the different interlayer Stark effect of 𝑋𝑋 𝑄𝑄𝐾𝐾 and 𝑋𝑋 𝑄𝑄𝛤𝛤 , one may tune their energy separation and control their population and luminescence. Their different valley configurations can also induce different exciton-phonon coupling [38], further enriching the exciton physics. Given such an interesting theoretical prospect, it is intriguing to explore bilayer TMDs. But experimental studies have been challenging due to the generally poor quality of bilayer TMD samples. Particularly, although prior research has shown some theoretical and experimental evidence of Q Γ exciton ( e.g. in quantum-dot emission) [25], direct observation of the Q Γ exciton in pristine bilayer TMDs has never been reported. In this Letter, we directly resolve the Q Γ exciton emission in bilayer WSe and conduct comparative electric-field-dependent studies between Q Γ and QK excitons. The Q Γ exciton lies at ~18 meV below the QK exciton, revealing stronger binding of the Q Γ exciton than QK exciton. Under an out-of-plane electric field, both Q Γ and QK excitons exhibit strong Stark effect, but the Q Γ Stark shift is considerably weaker than the QK shift. By tuning the electric field strength between 0 and 0.22 V/nm, we can switch the Q Γ and QK exciton to be the lowest exciton at different field strength. In addition, we resolve a series of phonon replicas for both excitons. Notably, the phonon replicas outshine the primary excitonic emission, and the two-phonon replicas are stronger than the one-phonon replicas. We can simulate the replica spectra by comprehensive theoretical calculation. Our results reveal the existence of numerous two-phonon scattering processes and (nearly) resonant exciton-phonon scatterings in bilayer WSe , which cause unusually strong two-phonon replicas in the system. In our experiment, we fabricate dual-gate bilayer WSe devices encapsulated by hexagonal boron nitride (BN) on Si/SiO substrates [Fig. 2(a)]. The BN encapsulation significantly improves the sample quality and allows for the observation of Q Γ exciton. Thin graphite flakes are used as the contact and gate electrodes to further enhance the device performance. By applying voltages of opposite signs and appropriate ratio on the top gate ( V tg ) and bottom gate ( V bg ), we can generate a vertical electric field across bilayer WSe while keeping the sample charge neutral (See Supplemental Materials for details of device fabrication and experiment [39]). Fig. 2(b) displays the photoluminescence (PL) map at varying voltage difference ( Δ V = V tg – V bg ) between the top and bottom gates at temperature T ~ 15 K. At low Δ V , we observe multiple cross-shape emission features; such field-dependent cross-shape features have never been reported. At large Δ V , several redshifting lines appear; they should be associated with the QK exciton according to prior research [21]. We further perform second-order energy derivative on the PL map to resolve the fine features [Fig. 2(c)]. From the map of the second-order derivative, we extract the energies of different emission features and plot them as a function of electric field in Fig. 2(d). We first consider the two sets of highest-energy features, which consist of one cross and two redshifting lines in 1.60 – 1.63 eV [Fig 2(b-d)]. They are significantly weaker than the replica features at lower energy. We tentatively assign them as the primary 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 emission assisted by defect scattering. Defect scattering is considered because direct optical recombination of intervalley excitons is forbidden by the momentum conservation [40,41]. To confirm our assignment, we conduct first-principles calculations on the band structure of bilayer WSe under different vertical electric fields [Fig. 1(d-e)] [39]. At zero field, each band is doubly degenerate due to the inversion and time-reversal symmetry of 2H-stacked bilayer WSe [Fig. 1(d)]. At the finite electric field, the inversion symmetry is broken, and each band is split into two bands with opposite spin and layer polarizations (except at the Γ point) [Fig. 1(e)]. Consequently, the 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 emission each split into two lines and exhibit different Stark shifts. We have calculated the Q Γ and QK interband transition energies (with no excitonic effect) at different electric fields via the density functional theory (DFT) [dashed lines in Fig. 2(d)] [39]. We offset the theoretical transition energies to match the experimental range at zero field and focus our comparison only on the Stark shifts. The theoretical Stark shifts agree decently with our data (the small deviation may come from the change of exciton binding under electric field). Our theory thus supports our assignment of Q Γ and QK excitons. The Q Γ exciton exhibit two remarkable characteristics. First, 𝑋𝑋 𝑄𝑄𝛤𝛤 is ~18 meV lower than 𝑋𝑋 𝑄𝑄𝐾𝐾 and is significantly brighter than 𝑋𝑋 𝑄𝑄𝐾𝐾 . This result is surprising because the Γ valence valley is ~140 meV lower than the K valence valley in the electronic band structure [Fig. 1(d)] [29]; one would naturally expect that the Q Γ transition has higher energy than the QK transition. However, further examination of the band structure reveals that the Γ valley has a much larger hole effective mass (~1.48 m ) than the K valence valley (~0.43 m ; m is the free electron mass) [30]. Hence, the Q Γ exciton binds more strongly than the QK exciton. The larger exciton binding energy causes 𝑋𝑋 𝑄𝑄𝛤𝛤 to have lower energy than 𝑋𝑋 𝑄𝑄𝐾𝐾 . This result is significant because the lowest excitonic state plays a paramount role in the excitonic dynamics and luminescence. Second, the 𝑋𝑋 𝑄𝑄𝛤𝛤
Stark shift (~70 meV per 1 V/nm field) is 1.7 ~ 2.7 times weaker than the 𝑋𝑋 𝑄𝑄𝐾𝐾
Stark shift (120 ~ 190 meV per 1 V/nm field). This can be qualitatively understood from their different charge density distribution. Fig. 3 shows our calculated spin-dependent charge density along the vertical direction for electron states at the Q, K, and Γ points. We consider an infinitesimal electric field to break the inversion symmetry. The separated states show strong spin polarization – the electric field essentially splits them into spin-up and spin-down states. The K, Q, and Γ states show different layer polarization. The Q conduction states show medium layer polarization with an electric dipole moment 𝑝𝑝 𝑄𝑄 = 2.02 atomic unit (a.u.) [Fig. 3(a, b)]. In contrast, the K valence states are strongly localized in opposite layers, giving rise to a large dipole moment 𝑝𝑝 𝐾𝐾 = 5.80 a.u. [Fig. 3(c, d)]. In the other extreme, the two Γ valence states show symmetric distribution on the two layers, which produces no layer polarization and zero electric dipole moment ( 𝑝𝑝 𝛤𝛤 = 0) [Fig. 3(e, f)]. By combining these dipole moments, we estimate that a Q Γ (QK) electron-hole pair has a dipole moment of 𝑝𝑝 𝑄𝑄𝛤𝛤 = 𝑝𝑝 𝑄𝑄 = 𝑝𝑝 𝑄𝑄𝐾𝐾 = 𝑝𝑝 𝑄𝑄 + 𝑝𝑝 𝐾𝐾 = 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 (Quantitative deviation can occur due to excitonic effect and mixing with other bands under electric field). The different Stark shifts between 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 have an interesting consequence – they allow us to use the electric field to switch the energy order and dominant luminescence between 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 . When the electric field is weak ( e.g.
E < 0.02 V/nm), 𝑋𝑋 𝑄𝑄𝛤𝛤 lies well below 𝑋𝑋 𝑄𝑄𝐾𝐾 and dominates the emission due to its higher population. As the field increases, 𝑋𝑋 𝑄𝑄𝐾𝐾 gradually redshifts to become lower than 𝑋𝑋 𝑄𝑄𝛤𝛤 , and the exciton population transfers from 𝑋𝑋 𝑄𝑄𝛤𝛤 to 𝑋𝑋 𝑄𝑄𝐾𝐾 ; this brightens 𝑋𝑋 𝑄𝑄𝐾𝐾 and suppresses 𝑋𝑋 𝑄𝑄𝛤𝛤 . At strong electric field ( e.g.
E > 0.2 V/nm), 𝑋𝑋 𝑄𝑄𝐾𝐾 dominates the luminescence. Such electric-field switching effect of excitonic emission is not found in monolayer TMDs and may find novel applications for bilayer TMDs. After we address the primary 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 emission, we turn to their replica emission at lower energy. Fig. 2 shows five 𝑋𝑋 𝑄𝑄𝛤𝛤 replicas ( 𝑋𝑋 𝑄𝑄𝛤𝛤1 − 𝑋𝑋
𝑄𝑄𝛤𝛤5 ) with respective redshift energies 13.0, 27.6, 41.7, 45.7, and 57.7 meV from 𝑋𝑋 𝑄𝑄𝛤𝛤 and six 𝑋𝑋 𝑄𝑄𝐾𝐾 replicas ( 𝑋𝑋 𝑄𝑄𝐾𝐾1 − 𝑋𝑋
𝑄𝑄𝐾𝐾6 ) with respective redshift energies 28.8, 32.4, 41.9, 46.2, 49.1, and 58.3 meV from 𝑋𝑋 𝑄𝑄𝐾𝐾 (the uncertainty of these energies is from ± 1 to ±2 ; see Table S6 [39]). The replicas exhibit identical Stark shifts to the primary emission lines. We tentatively attribute them to excitonic luminescence assisted by phonon emission, as phonon replicas have been reported in monolayer WSe [40-44] and bilayer WSe [24]. In our results, the replicas are considerably brighter than the primary emission, indicating that phonon-assisted emission is more efficient than defect-assisted emission in our system. To identify the phonon replicas, we have calculated the phonon band structure of bilayer WSe with a rigid-ion model [39]. Afterward, we calculate the one-phonon replicas for 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 by the perturbation theory. In our theory, the initial excitonic state is scattered to a second excitonic state by emitting a phonon, then the second state decays to emit a photon. The second state is not necessarily a bound exciton, but it must be a momentum-direct exciton in order to emit the photon. Momentum conservation restricts the one-phonon scattering processes to occur only through two paths. For 𝑋𝑋 𝑄𝑄𝛤𝛤 , the electron may be scattered from Q to Γ by emitting a phonon ( 𝑋𝑋 𝑄𝑄𝛤𝛤 → 𝑋𝑋
𝛤𝛤𝛤𝛤 ), or the hole may be scattered from Γ to Q by emitting a phonon ( 𝑋𝑋 𝑄𝑄𝛤𝛤 → 𝑋𝑋
𝑄𝑄𝑄𝑄 ). Similarly, for 𝑋𝑋 𝑄𝑄𝐾𝐾 , the electron can be scattered from Q to K ( 𝑋𝑋 𝑄𝑄𝐾𝐾 → 𝑋𝑋
𝐾𝐾𝐾𝐾 ), or the hole from K to Q ( 𝑋𝑋 𝑄𝑄𝐾𝐾 → 𝑋𝑋
𝑄𝑄𝑄𝑄 ). As the initial 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 excitons can be associated with any of the six Q valleys and two K valleys, the momentum of the emitted phonons will be near one of the six Q points for 𝑋𝑋 𝑄𝑄𝛤𝛤 and near one of the Q or M points for 𝑋𝑋 𝑄𝑄𝐾𝐾 . The emitted phonon can come from any phonon branch, so the one-phonon replica spectra can exhibit multiple peaks (see Table S3 and Fig. S10 for details [39]). These one-phonon scattering processes are all non-resonant, so the one-phonon replicas are generally weak.
Fig. 4 compares our calculated one-phonon spectra for 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 with the experimental spectra. We phenomenologically broaden the theoretical spectra by 2 meV to match the experimental peak width. The theoretical one-phonon spectra can account for the 𝑋𝑋 𝑄𝑄𝛤𝛤1 peak, which is contributed dominantly by the LA and TA acoustic phonons near the Q points. It can also partially explain 𝑋𝑋 𝑄𝑄𝛤𝛤2 and 𝑋𝑋 𝑄𝑄𝐾𝐾1 , . But other replicas ( 𝑋𝑋 𝑄𝑄𝛤𝛤3−5 and 𝑋𝑋 𝑄𝑄𝐾𝐾3−6 ) exceed the range of single phonon energy (~37 meV) in WSe . We would need to consider higher-order scattering processes to explain these replicas. We have conducted comprehensive calculations on the two-phonon replicas of 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 [39]. In our theory, the initial exciton is scattered to a second excitonic state by emitting a phonon, and afterward scattered to a third excitonic state by emitting another phonon, and the third state decays to emit a photon. The second and third states are not necessarily bound excitons, but the third state may be a momentum-direct exciton in order to emit the photon. In contrast to the one-phonon processes with only two scattering paths, the two-phonon processes have numerous scattering paths. In our calculation, we only consider those scattering paths involving a resonant or nearly resonant exciton-phonon scattering process. In bilayer WSe , the six Q conduction valleys are energy-degenerate, and 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 are close in energy (~18 meV). So, carrier-phonon scattering between different Q valleys, between the Γ and K valence valleys, or within the same valley, is resonant or nearly resonant. Two-phonon processes involving such (nearly) resonant scattering are expected to contribute dominantly to the replica spectra. In our survey, there are totally 16 two-phonon scattering paths with a (nearly) resonant component for Q Γ exciton. They can be separated into four groups: 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋
𝑄𝑄′Γ → X Q′Q′ ; 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄′Γ → X ΓΓ ; 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝐾𝐾𝐾𝐾 ; 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋
𝑄𝑄K → 𝑋𝑋
𝑄𝑄Q (here Q′ denotes any of the six Q valleys). Similarly, there are 14 dominant paths for the QK exciton, separated into four groups: 𝑋𝑋 𝑄𝑄K →𝑋𝑋 𝑄𝑄 ′ K → X KK ; 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄 ′ K → X 𝑄𝑄 ′ 𝑄𝑄 ′ ; 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤 → 𝑋𝑋
𝑄𝑄𝑄𝑄 ; 𝑋𝑋 𝑄𝑄𝐾𝐾 → 𝑋𝑋
𝑄𝑄𝛤𝛤 → 𝑋𝑋
𝛤𝛤𝛤𝛤 (see the Supplemental Materials for the explanation of these paths [39]). Momentum conservation restricts the emitted phonons to be near the Γ, K, Q, and M points (Tables S3, S5 [39]). The emitted phonons can come from any phonon branch, so the two-phonon replica spectra are rather complicated. Fig. 4(c-d) display our calculated and broadened two-phonon replica spectra, which include contributions from all the above-mentioned scattering paths and all phonon branches. By summing the one-phonon and two-phonon spectra, our total theoretical spectra match decently the experimental spectra, including the energy and relative intensity of different replica peaks. From the theoretical results, we can attribute 𝑋𝑋 𝑄𝑄𝛤𝛤1 to the one-phonon replica of Q-point acoustic phonons, 𝑋𝑋 𝑄𝑄𝛤𝛤2 to a superposition of one-phonon and two-phonon replicas, 𝑋𝑋 𝑄𝑄𝛤𝛤2−5 and 𝑋𝑋 𝑄𝑄𝐾𝐾1−6 to two-phonon replicas (see Table S6 for detailed assignments of each replica peak [39]). In particular, 𝑋𝑋 𝑄𝑄𝐾𝐾1−5 are dominantly contributed by two-phonon paths involving the 𝑋𝑋 𝑄𝑄𝐾𝐾 → 𝑋𝑋
𝑄𝑄𝛤𝛤 transition with acoustic phonon emission, because this transition is strongly resonant when the emitted phonon energy is close to the 18-meV difference between 𝑋𝑋 𝑄𝑄𝐾𝐾 and 𝑋𝑋 𝑄𝑄𝛤𝛤 (Table S5 [39]). We remark that the two-phonon replicas are considerably stronger than the one-phonon replicas. This characteristic is counter-intuitive because second-order processes are usually much weaker than first-order processes. There are two reasons for this unusual phenomenon. First, one of the scattering components in the two-phonon processes is (nearly) resonant. This makes each considered two-phonon process as strong as a non-resonant one-phonon process. Second, the number of (nearly) resonant two-phonon scattering paths ( ≥
14) considerably exceeds the number of one-phonon scattering paths (only 2). After summing the contributions of these two-phonon paths, the total two-phonon spectra become considerably stronger than the one-phonon spectra. Such a characteristic is not found in monolayer WSe due to its different band structure. It makes bilayer WSe a distinctive system to study novel exciton-phonon phenomena. In sum, we have measured the electric-field-dependent spectra of Q Γ and QK exciton and show their different interlayer Stark shifts. Both excitons exhibit unusually strong two-phonon replicas. In the Supplemental Materials, we also show the Zeeman splitting effect of 𝑋𝑋 𝑄𝑄𝛤𝛤2 under magnetic field, from which we deduce a g-factor of 9.1 ± Γ exciton [25]. The interesting interplay between excitons, phonons, and electric and magnetic fields, as demonstrated in our research, makes bilayer WSe a distinctive excitonic system with potential applications in excitonics and valleytronics. Acknowledgment:
We thank Fatemeh Barati and Matthew Wilson for help in the project, Ao Shi for assistance in manuscript preparation, Kin Fai Mak and Jie Shan for discussion, Yong-Tao Cui for supporting E.L., and Harry Tom for equipment support. C.H.L. acknowledges support from the National Science Foundation Division of Materials Research CAREER Award No.1945660. YCC acknowledges support by the Ministry of Science and Technology (Taiwan) under Grant No. MOST 109-2112-M-001-046. N.M.G acknowledges support from the National Science Foundation Division of Materials Research CAREER Award No. 1651247, and from the United States Department of the Navy Historically Black Colleges, Universities, and Minority Serving Institutions (HBCU/MI) Award No. N00014-19-1-2574. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST.
FIG.1. (a) Top view and (b) side view of the crystal structure of 2H-stacked bilayer WSe . (c) The Brillouin zone of bilayer WSe . (d-e) Calculated electronic band structure along the K- Γ line at zero (d) and 0.3 V/nm (e) vertical electric field. Each band is doubly degenerate at zero field, but split at finite field (except at the Γ point). We denote the 𝑋𝑋 𝑄𝑄𝛤𝛤 and 𝑋𝑋 𝑄𝑄𝐾𝐾 emission and the dominant spin-up (spin-down) polarization by red (blue) color.
FIG. 2. (a) Schematic of our dual-gate BN-encapsulated bilayer WSe devices. (b) Photoluminescence (PL) map of a bilayer WSe device at varying voltage difference Δ V = V tg – V bg between the top and bottom gates while keeping the sample charge neutral. The measurements were conducted at sample temperature T ~ 15 K under 730-nm continuous laser excitation (incident power ~ 50 µW). (c) Second energy derivative of panel (b). (d) Extracted energy of emission features in (c). The primary Q Γ and QK exciton emission ( 𝑋𝑋 𝑄𝑄𝛤𝛤 , 𝑋𝑋 𝑄𝑄𝛫𝛫 ) and phonon replicas ( 𝑋𝑋 𝑄𝑄𝛤𝛤1−5 , 𝑋𝑋 𝑄𝑄𝛫𝛫1−6 ) are denoted. The dashed lines are theoretical electric-field-dependent energy shifts of Q Γ and QK interband transitions. FIG. 3. (a-b) Calculated spin-dependent in-plane-averaged charge density along the out-of-plane direction for the electron states at the Q point of the conduction bands c and c at an infinitesimal vertical electric field. (c-f) Similar plots for the states at the K and Γ points of the valence bands v and v . The states at c (v ) and c (v ) have opposite spin and layer polarization. The vertical dashed lines denote the central position of the atoms. FIG. 4. (a) PL spectrum at Δ V = 0 in Fig. 2(b), offset by the 𝛸𝛸 𝑄𝑄𝛤𝛤 energy. (b) PL spectrum at Δ V = -10 V in Fig. 2(b), offset by the 𝛸𝛸 𝑄𝑄𝐾𝐾 energy. (c-d) Simulated spectra of one-phonon and two-phonon replicas for 𝛸𝛸 𝑄𝑄𝛤𝛤 (c) and 𝛸𝛸 𝑄𝑄𝐾𝐾 (d). Both theoretical spectra are broadened for 2 meV to match the experimental peak widths. References: [1] K. F. Mak, C. Lee, J. Hone, J. Shan and T. F. Heinz, Atomically thin MoS : A new direct-gap semiconductor, Phys. Rev. Lett. , 136805 (2010). [2] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli and F. Wang, Emerging photoluminescence in monolayer MoS , Nano Lett. , 1271 (2010). [3] H. Yu, Y. Wang, Q. Tong, X. Xu and W. 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Toronto, Ontario ON M5G 1M1, Canada. * Corresponding authors. Email: [email protected]; [email protected]
Table of Contents
1. Device fabrication and characterization .................................................................................................... 2 2. Determination of the electric field across bilayer WSe ........................................................................... 3 3. Zeeman splitting of QΓ excitons in bilayer WSe ..................................................................................... 4 4. Photoluminescence results of bilayer WSe Device ................................................................ 6 5.2 Comparison of QΓ and QK exciton emission energies ...................................................................... 9 5.3 Density distribution and electric dipole of band-edge states in bilayer WSe ................................. 10 5.4 Calculation of phonon modes by a rigid-ion model for bilayer WSe ............................................. 11 5.5 Calculation of one-phonon replica spectra ....................................................................................... 13 5.6 Calculation of two-phonon replica spectra ...................................................................................... 18 Device fabrication and characterization
We fabricate dual-gated bilayer WSe devices with hexagonal boron nitride (BN) encapsulation by micro- mechanical exfoliation and co-lamination of thin crystals. We use bulk WSe from HQ Graphene Inc.. We first exfoliate bilayer WSe , thin graphite flakes, and thin BN flakes from their bulk crystals onto silicon substrates with 285-nm-thick SiO epilayer. Afterward, we apply a dry-transfer technique to stack the different thin crystals together. We use a polycarbonate stamp to sequentially pick up the top-gate electrode (thin graphite), top-gate dielectric (thin BN), bilayer WSe , contact electrode (thin graphite), bottom-gate dielectric (thin BN), and bottom-gate electrode (thin graphite). Afterward, we transfer the stack of materials onto a Si/SiO substrate and apply the standard electron beam lithography to deposit 100-nm-thick gold electrodes. Finally, the devices are annealed at 360 ℃ for six hours in an argon environment to cleanse the interfaces. Fig S1(a) displays the optical image of Device with the top gate grounded. We observe the emission of trions (or exciton polarons) on the electron and hole sides. In the main paper, we focus on the exciton PL spectra at the charge neutrality at varying vertical electric field.
Fig. S1. (a) Optical image of BN-encapsulated bilayer WSe Device T ~ 15 K under 633- nm continuous laser excitation with incident power of 100 µW. The top gate is grounded in the measurement. Determination of the electric field across bilayer WSe In our experiment, we have applied gate voltages of opposite signs on the bottom gate ( V bg ) and top gate ( V tg ) to induce a vertical electric field across bilayer WSe . We adjust the ratio between V tg and V bg so that the sample remains charge neutral in the experiment . We solve the electrostatic problem in our devices to determine the strength of the electric field ( E ) across bilayer WSe from Δ V = V tg – V bg . Fig. S2 displays the electrostatic geometry of our BN-encapsulated devices, which consists of a WSe bilayer with thickness d sandwiched between two BN layers with thickness d and d . The static dielectric constant of bilayer WSe and BN along the vertical direction are denoted as 𝜀𝜀 and 𝜀𝜀 𝐵𝐵𝐵𝐵 , respectively. The electric fields across the top BN, 2L WSe , and bottom BN are denoted as E , E , E , respectively. The voltage difference ( Δ V ) between the top and bottom gates is related to the electric fields as 𝛥𝛥𝑉𝑉 = 𝑑𝑑 𝐸𝐸 + 𝑑𝑑𝐸𝐸 + 𝑑𝑑 𝐸𝐸 (1) As the electrostatic gating injects no net charge into the WSe bilayer, the boundary conditions at the two interfaces are 𝜀𝜀 𝐵𝐵𝐵𝐵 𝐸𝐸 = 𝜀𝜀 𝐸𝐸 = 𝜀𝜀 𝐵𝐵𝐵𝐵 𝐸𝐸 (2) Combining Eq. (1) and (2), we obtain 𝐸𝐸 = 𝛥𝛥𝛥𝛥 𝜀𝜀2𝐿𝐿𝜀𝜀𝐵𝐵𝐵𝐵 ( 𝑑𝑑 +𝑑𝑑 ) +𝑑𝑑 (3) In our calculation, we use d = 1.3 nm for the WSe bilayer, d = 26 nm and d = 23.7 nm for the top and bottom BN, respectively, for Device 𝜀𝜀 = 5.51 for bilayer WSe , which is obtained from our theoretical calculation in Section 5.1 below. We use dielectric constant 𝜀𝜀 𝐵𝐵𝐵𝐵 = 2.69 for BN, which is consistent with the literature [1-3]. The calculated electric field for varying Δ V is shown in Fig. 2(d) of the main paper. Fig. S2.
Electrostatic geometry of our BN-encapsulated bilayer WSe devices Zeeman splitting of the Q Γ exciton emission in bilayer WSe The Q Γ exciton state is formed by the hole states in the Γ valley and the electron states in six energy-degenerate Q valleys. The six Q valleys can be separated into two groups – the 𝑄𝑄 , 𝑄𝑄 , 𝑄𝑄 valleys and the 𝑄𝑄 , 𝑄𝑄 , 𝑄𝑄 valleys [see the inset of Fig. 1(b) in the main paper]. The three valleys in each group are related to each other by three-fold rotational symmetry. The two groups of valleys are related to each other by time-reversal symmetry. The application of a vertical magnetic field preserves the three-fold rotational symmetry but breaks the time-reversal symmetry. As a result, the degeneracy between the 𝑄𝑄 , 𝑄𝑄 , 𝑄𝑄 valleys and the 𝑄𝑄 , 𝑄𝑄 , 𝑄𝑄 valleys is lifted. Correspondingly, the Q Γ exciton state will be split into two states, each associated with either the 𝑄𝑄 , 𝑄𝑄 , 𝑄𝑄 valleys or the 𝑄𝑄 , 𝑄𝑄 , 𝑄𝑄 valleys. Moreover, the two-fold generate states at the Γ point will also be split under magnetic field and contribute to the splitting of the Q Γ exciton . Similar valley Zeeman splitting effect has been studied extensively for the Κ - valley excitons in monolayer TMDs [4-7]. We have measured the PL spectra of Q Γ exciton and replicas under finite vertical magnetic field and electric field (Fig. S3). We cannot see clearly the primary 𝑋𝑋 𝑄𝑄𝛤𝛤 line due to its weak signal, but we can observe the 𝑋𝑋 𝑄𝑄𝛤𝛤2 replica, which are split into two lines by the Stark effect under electric field. The 𝑋𝑋 𝑄𝑄𝛤𝛤2 replica is further split into four lines under magnetic field. Two lines show right-handed circular polarization and redshift with increasing magnetic field; the other two lines show left-handed circular polarization and blueshift with increasing magnetic field. We attribute such magnetic-field-dependent splitting to the valley Zeeman effect, in which emission with opposite helicity comes from excitons associated with the 𝑄𝑄 , 𝑄𝑄 , 𝑄𝑄 valleys and the 𝑄𝑄 , 𝑄𝑄 , 𝑄𝑄 valleys. Fig. S3. (a-b) Magnetic-field-dependent PL maps of bilayer WSe at right-handed (a) and left-handed (b) circular polarization at Δ V = -1 V under linearly polarized 532- nm laser excitation . The 𝑋𝑋 𝑄𝑄𝛤𝛤2 replica is split into two lines by the Stark effect at finite Δ V . Each line is further split into two lines with opposite helicity and exhibits linear Zeeman shift under magnetic field. The energy separation (
𝛥𝛥𝐸𝐸 ) between 𝑋𝑋 𝑄𝑄𝛤𝛤2 lines with opposite helicity exhibit linear dependence on the magnetic field, which can be described by
𝛥𝛥𝐸𝐸 = 𝑔𝑔𝜇𝜇 𝐵𝐵 𝐵𝐵 , where 𝜇𝜇 𝐵𝐵 is the Bohr magneton, 𝐵𝐵 is the magnetic field, and 𝑔𝑔 is the effective g-factor. From our data, we extract a g -factor of ~9. This value is consistent with the g-factor (g = 9.5) obtained by a prior experiment that measured quantum-dot emission associated with the Q
Γ exciton in bilayer WSe [8]. Further research is merited to understand the optical selection rules and g-factor of 𝑋𝑋 𝑄𝑄𝛤𝛤2 . Photoluminescence results of bilayer WSe Device
We have obtained similar PL maps in another device (Device
Fig. S4 . (a) Gate-dependent photoluminescence (PL) map of bilayer WSe Device he extracted peak energy of emission features in (a, b). The measurements were conducted at sample temperature T ~ 15 K under 532- nm continuous laser excitation with incident power ~50 µW . Theoretical calculations 5.1
Electric-field effect on the band energies of bilayer WSe We have calculated the shift of band-edge energies and transition energies of various intervalley excitons of bilayer WSe under the application of an electric field ( E ) along the z - axis (perpendicular to the plane). The applied electric field gives rise to a perturbation term 𝐻𝐻� = eEz , which is included in the density-functional theory (DFT) to calculate the effect of electric field on the band structure. The calculation is carried out by using the WIEN2K package [9]. We adopted t he generalized gradient approximation (GGA) with a sufficiently large supercell of size L z = 5.21 nm along the z - axis . We have included the spin-orbit interaction and the van der Waals interaction between the two WSe layers in our calculation. We first calculate the band structure of unstrained bilayer WSe by adopting a lattice constant of the experimental valu e [10]. The calculated band structure displays the conduction band minimum (CBM) at the Q point (near the middle point between K and Γ) and the valance band maximum ( VBM) at the K point (Fig. S1). The energy of the top valence band (v1) at the Γ point is 132 meV below that of the VBM at the K point. Such an energy separation is smaller than the Γ -K separation of 140 meV measured by a recent experiment of angle-resolved photoemission spectroscopy (ARPES) on a BN-encapsulated bilayer WSe sample [11]. This slight deviation suggests that the BN-encapsulated WSe may be slightly strained. Indeed, we find that, by reducing the lattice constant of bilayer WSe for 0.098%, our calculated band structure [Fig. S1(b)] can exhibit a Γ -K separation of 140 meV, which matches the ARPES result. Therefore, in all of the following calculations, we consider strained bilayer WSe with 0.098% reduced lattice constant. Fig. S5 . (a) Calculated band structure of unstrained bilayer WSe . (b) Calculated band structure of strained bilayer WSe with 0.098% reduced lattice constant. The calculations are done by the WIEN2K package with GGA.
Fig. S6 . Band structure of bilayer WSe with 0.098% reduced lattice constant calculated by using WIEN2k under four different vertical electric fields: (a) E = 0.05 V/nm, (b) E = 0.1 V/nm, (c) E = 0.2 V/nm, and (d) E = 0.3 V/nm. We then consider the effect of a perpendicular electric field on the strained bilayer WSe . Fig. S6 displays the calculated band structure of the strained bilayer WSe under perpendicular electric fields E = 0.05, 0.1, 0.2, and 0.3 V/nm. These electric fields are the net fields seen by an electron in the material under the screening of the valence electrons. Our results show that the two-fold degenerate conduction band at Q and valence band at K split because the applied electric field breaks the inversion symmetry of bilayer WSe . However, the valence band at Γ remains two-fold degenerate due to the time-reversal symmetry (Kramer’s degeneracy). In our consideration below, we will focus on the conduction band at the Q point and the valence band at the K and Γ points. We denote the conduction band with lower (higher) energy at Q as c1 (c2), and the valence band with higher (lower) energy at K as v1 (v2). Fig. S7 displays the calculated interband transition energies of the 𝑄𝑄 𝑐𝑐1 − 𝐾𝐾 𝑣𝑣1 , 𝑄𝑄 𝑐𝑐1 − 𝛤𝛤 𝑣𝑣1 , and 𝑄𝑄 𝑐𝑐2 − 𝛤𝛤 𝑣𝑣2 transitions as a function of screened electric field. The 𝑄𝑄 𝑐𝑐1 − 𝛤𝛤 𝑣𝑣1 and 𝑄𝑄 𝑐𝑐2 − 𝛤𝛤 𝑣𝑣2 transition energies are found to split with increasing electric field. Here we have not considered the excitonic effect, which should only produce a constant energy offset with insignificant influence on the field-dependent energy shift. In Fig. S7, the Q- Γ transition energies are the interband transition energies obtained by DFT, whereas the Q- K transition energy is shifted to match the experimental separation between the Q Γ and QK exciton emission lines at zero field in Fig. 2 (d). Fig. S7.
Calculated interband transition energy for the 𝑄𝑄 𝑐𝑐1 − 𝐾𝐾 𝑣𝑣1 , 𝑄𝑄 𝑐𝑐1 − 𝛤𝛤 𝑣𝑣1 , and 𝑄𝑄 𝑐𝑐2 − 𝛤𝛤 𝑣𝑣2 transitions as a function of screened electric field. The solid (dashed) curves are results of self-consistent (non-self-consistent) calculation as described in the text . Here the Q- Γ transition energies are the interband transition energies obtained by DFT, whereas the Q-K transition energy is shifted to match the experiment al separation between the Q Γ and QK exciton s in Fig. 2(d). We have used two different approaches to calculate the band energy shift due to electric field. In the first approach, we carry out self-consistent calculation, which considers the response of valence electrons and the atomic relaxation to an external electric fie ld ( i.e. the screening effect); we obtain the band energy shift as a function of external electric field . In the second approach, we carry out non-self-consistent calculation, in which the applied electric field is taken as the net field after all the screening effects. Afterward, we compare the result of the two calculations and find that their electric-field dependence is off by a ratio of 5.51; this ratio is interpreted as the dielectric constant of bilayer WSe (i.e. 𝜀𝜀 = 5.51). Fig. S7 shows the calculated transition energy as a function of screened electric field for both calculations (solid and dashed lines). They appear to match each other well after we use the dielectric constant 𝜀𝜀 = 5.51 to convert the applied field into screened field in the self-consistent result. We have used the self- consistent results to compare with our experimental data in Fig. 2(d) of the main paper. We find decent agreement on the field-dependent energy shift between experiment and theory . We note that the absolute transition energies in our calculation differ from the experimental emission energy because DFT cannot predict the band gap accurately and we have not considered the excitonic effect . But such deviation should have no significant influence on the field-dependent energy shift. So, in our comparison, we shift the energies of the Q Γ and QK excitons to match the experiment energies at zero electric field and focus our comparison on the field-dependent shift of the transition energy. Comparison of Q Γ and QK exciton emission energies From the theory- experiment comparison of the electric -field-dependent energy shift [Fig. 2(d)], we can identify the higher-energy emission lines at 1.62 – the lower-energy emission lines at 1.605 – 1.62 eV as the Q Γ exciton . However, in the band structure obtained by DFT, the valence band is 140 meV higher at the K point than at the Γ point. That is, the Q- Γ transition energy exceeds the Q-K transition energy by 140 meV. This apparently contradicts against our result that Q Γ exciton lies below the QK exciton. This implies that Q
Γ excitons has much larger binding energy than the QK exciton . There are three factors that contribute to the stronger binding of the Q
Γ exciton . First, the Q Γ exciton has larger effective mass than the QK exciton . Table S1 lists the effective carrier mass at the K, Γ , and Q points calculated by WIEN2k. From the orientation-averaged effective mass at the K, Γ , and Q points, we deduce the reduced mass ( 𝜇𝜇 X ) for the QK and Q Γ excitons to be 0.247 𝑚𝑚 and 0.417 𝑚𝑚 respectively, where 𝑚𝑚 is the free electron mass. If we assume that the QK exciton binding energy is ~150 meV, then the Q Γ exciton binding energy can reach ~253 meV by considering their ratio of effective mass. Such enhanced binding energy can partially counter the 140 meV difference in the Q Γ ad QK interband transition energies. Second, in the Brillouin zone of bilayer WSe , there are six inequivalent Q valleys, all of which contribute equally to the formation of Q Γ exciton . In contrast, only three of the six Q valleys contribute significantly to the
QK exciton formation, whereas the other three are further away and have negligible contribution. The double number of Q valleys in the Q Γ exciton formation can significantly enhance its binding energy compared to the QK exciton. Third, the larger effective mass of Q Γ exciton can cause a larger spread of the exciton envelop function in the k-space, whi ch can lead to significant intervalley mixing and further enhancement of the excitonic binding . For instance, previous studies of donor states in silicon shows that the intervalley mixing effect can increase the donor binding energy by 50% [12]. Similar effect is expected in the QΓ intervalley exciton here.
By combining the three factors above, it is reasonable that the Q Γ exciton binding energy can exceed the QK exciton binding energy by more than 140 meV, thus making the QΓ exciton emission energy lower than the QK exciton energy . K v1 Γ v1 Q v1 Q c1 K c1 K c2 𝑚𝑚 || ∞ 𝑚𝑚 ┴ Table S1.
Band-edge effective masses (in unit of 𝑚𝑚 ) at K, Γ , and Q points. We note that the Q v1 band has a linear dispersion along the Γ -K axis at the Q point; thus, we take 1/ 𝑚𝑚 || = 0 . Density distribution and electric dipole of band-edge states in bilayer WSe To understand the physical mechanism for the electric-field-induced energy splitting, we calculate the z -dependent charge density averaged over the x - y plane for the Bloch states in the bottom conduction band (c1) at Q and the top valence band (v1) at K and Γ in bilayer WSe via DFT by using the relation 𝜌𝜌 𝜇𝜇 ( 𝑧𝑧 𝑒𝑒 ) = �𝜓𝜓 𝜇𝜇 , 𝐾𝐾 � 𝑐𝑐 ∑ 𝑒𝑒 𝑖𝑖𝑔𝑔 𝑧𝑧 ( 𝑧𝑧−𝑧𝑧 𝑒𝑒 ) 𝑔𝑔 𝑧𝑧 �𝜓𝜓 𝜇𝜇 , 𝐾𝐾 � , (4) where μ labels the Block states (K v1 , Γ v1 , Q c1 , and Q c2 ) of concern. Since 𝜌𝜌 𝜇𝜇 is not provided by the WIEN2k package, we use the wave functions obtained from the LASTO (linear augmented Slater-type orbitals) package [13-15] to calculate the in-plane averaged charge density according to Eq. (4). Here the Bloch states are calculated via a supercell method and L c is the length of the supercell used. The band structures obtained by LASTO package are nearly the same as those by WIEN2k with suitable choice of LASTO basis functions. Both approaches are based on all-electron (both core and valence electrons) DFT with augmented basis functions. The LASTO basis functions are properly selected linear combinations of the augmented plane waves (APWs). Fig. 3 in the main paper displays our calculated charge densities 𝜌𝜌 𝜇𝜇 ( 𝑧𝑧 ) for the K v1 , Γ v1 , Q c1 , and Q c2 states in bilayer WSe under a small electric field ( E = 0.001 V/nm). The purpose of using a small electric field is to break the inversion symmetry so that we can see the spin and layer dependence of the wavefunctions when their degeneracy is lifted. The results allow us to calculate the dipole moment and the spin angular momentum of each state. The calculated dipole moment for the K v1 , Γ v1 , and Q c1 states are 5.81, 0, and 2.02 atomic unit (a.u.), respectively. Calculation of phonon modes by a rigid-ion model for bilayer WSe Besides the field- dependent emission energy shift discussed above, our experiment also reveals phonon replicas of the QΓ and QK excitons . We have carried out comprehensive theoretical calculations on the phonon replica spectra. In this section, we will first describe our calculation of the phonon band structure and phonon polarization vectors of bilayer WSe . In the following sections, we will present our theory to calculate the phonon replica spectra. We adopt a rigid-ion model (RIM) [16] to obtain the phonon polarization vectors. In this model, we consider only the short-range ion-ion Coulomb interaction. We neglect the long-range ion-ion Coulomb interaction (due to charge transfer between W and Se atoms), which should be insignificant given the very small energy splitting between the LO and TO phonon modes at the zone center [17]. The weak long-range ion-ion interaction indicates that WSe is a highly covalent material with little inter-atomic charge transfer. Fig. S8 displays the configuration and labels of the atoms in the bilayer WSe unit cell, which includes one W atom and two Se atoms in both the bottom and top WSe monolayers. In the unit cell, we define the position of the W atom in the bottom WSe monolayer as the origin of our coordinate system. The two nearby Se atoms are located at ( 𝑎𝑎 √ ⁄ , 0, ± 𝑑𝑑 /2) , where a is the in-plane lattice constant and d is the vertical distance between the lower and upper Se atoms in the bottom WSe monolayer. Below we will define a set of dynamic matrices to obtain the phonon dispersion. The form of these matrices is determined by symmetry analysis. The dynamic matrix for the nearest -neighbor interaction between the W atom at the origin (labeled as 1) and the Se atom at ( 𝑎𝑎 √ ⁄ , 0, −𝑑𝑑 /2) (labeled as 2) has the form of 𝐷𝐷 = �𝐴𝐴 𝐷𝐷 𝐵𝐵 𝐸𝐸 𝐶𝐶 � , where 𝐴𝐴 − 𝐸𝐸 denote the interaction parameters for the nearest-neighbor interaction. Fig. S8. (a) The side view of the unit cell of bilayer WSe . (b) The bottom view of the lower WSe layer in the unit cell. The numerical labels for different atoms are shown. The dynamic matrix between the Se atom at ( 𝑎𝑎 √ ⁄ , 0, −𝑑𝑑 /2) (labeled as ) and the Se atom at ( −𝑎𝑎 √ ⁄ , −𝑎𝑎 , −𝑑𝑑 /2) (labeled as ′ ) has the form: 𝐷𝐷 ′ = �𝐴𝐴 𝐷𝐷 𝐵𝐵 𝐸𝐸 𝐷𝐷 −𝐸𝐸 𝐶𝐶 � . The dynamic matrix between the W atom at the origin (labeled by ) and the W atom at (0, −𝑎𝑎 , 0) (labeled by ′ ) has the form: 𝐷𝐷 ′ = �𝐴𝐴 𝐵𝐵
00 0 𝐶𝐶 � . The dynamic matrix for the next -neighbor interaction between the Se atom at ( 𝑎𝑎 √ ⁄ , 0, −𝑑𝑑 /2) (labeled as ) and the Se atom at ( 𝑎𝑎 √ ⁄ , 0, 𝑑𝑑 /2) has the form: 𝐷𝐷 = �𝐴𝐴 𝐴𝐴
00 0 𝐶𝐶 � . The dynamic matrix for the second -nearest-neighbor interaction between the W atom at the origin (labeled as 1) and the Se atom at ( − 𝑎𝑎 √ ⁄ , 0, −𝑑𝑑 /2) (labeled as � ) has the form: D = �𝐴𝐴 𝐷𝐷 𝐵𝐵 𝐸𝐸 𝐶𝐶 � . The above five dynamic matrices describe the phonons of monolayer WSe . In bilayer WSe , a second WSe layer sits on top of the bottom WSe layer according to the 2H stacking order [Fig. S8(a)]. In the top WSe monolayer, we label the W atom as 4, the lower (upper) Se atom as 6 (5) [Fig. S8(a)]. The dynamic matric for the interlayer interaction between Se atom 6 and Se atom 2 has the form: 𝐷𝐷 = �𝐴𝐴 𝐵𝐵
00 0 𝐶𝐶 � . The dynamic matrices between other atoms can be related to the above six matrices by symmetry. We have adjusted the interaction parameters in these dynamic matrices to fit the DFT results of phonon dispersion [17] and the phonon replica PL data of monolayer WSe [18,19]. The best-fit force constants are listed in Table S2. Afterward, we use these force constants to generate the bilayer phonon dispersion and polarization vector in our rigid-ion model. The calculated phonon band structure is presented in Fig. S9(a). The polarization vectors of several relevant phonon modes at the M, K, and Q points are shown in Fig. S9(b). 𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐷𝐷 𝐸𝐸 A B C D E A B -10.306 -1.276 -9.823 7.901 7.105 -0.207 -1.849 0.835 0.158 -0.152 -1.712 0.078 C A C A B C D E 𝐴𝐴 𝐵𝐵 C Table S2.
Interaction parameters (in unit of 𝑒𝑒 / 𝑎𝑎 𝑑𝑑 ) used in our rigid-ion model for bilayer WSe . a and d are defined in the text. Fig. S9. (a) Phonon dispersion of bilayer WSe calculated by our rigid-ion model. There are nine pairs of phonon branches (labeled as 1-9 from low to high energy) ; each pair is nearly degenerate except the interlayer shear and breathing modes. (b) Schematic polarization vectors of representative phonon modes at the K, M, Q points. Corresponding labels of the phonon modes are denoted in both panels (a) and (b). The straight (curved) arrows denote linear (circular) polarization. The arrow size is proportional to the displacement magnitude. Some atoms at the M and Q points have both linear and chiral polarization components. For simplicity, we only show the polarization configurations of the lower WSe layer, while the configurations of the upper layer are related by inversion symmetry. Calculation of one-phonon replica spectra
Besides the field-dependent emission energy shift discussed above, our experiment also reveal phonon replica of the Q
Γ and QK excitons, which exhibit the same field dependence but are redshifted from the primary emission lines. Here we will present our theoretical calculation of the photon replica spectra. The phonon replicas can come from one-phonon or multi-phonon emission processes. We first consider the one-phonon process. The emission strength of one-phonon-assisted intervalley exciton recomination is proportional to the transition rate ( 𝑃𝑃 ( ) ) according to Fermi’s golden rule and the perturbation theory: 𝑃𝑃 𝜇𝜇 ( ) ( ω ) = ∑ �∑ � ω ; Ω α� 𝐻𝐻� 𝑒𝑒𝑒𝑒� 𝑋𝑋 𝜈𝜈 ; Ω α�� 𝑋𝑋 𝜈𝜈 ; Ω α� 𝐻𝐻� 𝑒𝑒𝑒𝑒� 𝑋𝑋 𝜇𝜇 �𝐸𝐸 𝜇𝜇 −𝐸𝐸 𝜈𝜈 −ℏΩ α +𝑖𝑖𝑖𝑖𝜈𝜈 � α 2 𝛿𝛿�𝐸𝐸 𝜇𝜇 − ℏΩ α − ℏω� . (5) Here 𝐻𝐻� 𝑒𝑒𝑒𝑒 ( 𝐻𝐻� 𝑒𝑒𝑒𝑒 ) is the electron-light (electron-phonon) interaction Hamiltonian; ω denotes the frequency of emitted photon; Ω α denotes the frequency of the emitted phonon in mode α ; 𝑋𝑋 𝜇𝜇 with μ = Q Γ or QK denotes the initial intervalley exciton state with energy 𝐸𝐸 𝜇𝜇 ; 𝑋𝑋 𝜈𝜈 denotes the mediating exciton state with energy 𝐸𝐸 𝜈𝜈 after the emission of one phonon; γ = 2 meV is a phenomenological broadening energy due to finite carrier lifetime; The associated exciton -phonon scattering strength is defined as: 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 = 𝑁𝑁 � � 𝑋𝑋 𝜈𝜈 ; Ω α� 𝐻𝐻� 𝑒𝑒𝑒𝑒� 𝑋𝑋 𝜇𝜇 �𝐸𝐸 𝜇𝜇 −𝐸𝐸 𝜈𝜈 −ℏΩ α +𝑖𝑖𝑖𝑖 � . (6) Here N is the number of unit cells in the sample. For bilayer WSe , each electronic band is doubly degenerate at zero electric field; here we have summed over the contributions of the two degenerate states associated with level 𝜈𝜈 . Moreover, bilayer WSe shows nine pairs of phonon branches, which are here labeled by 1 – 9 from low to high energy; each pair is nearly degenerate (except the interlayer shear and breathing modes ) due to the weak interlayer lattice coupling [Fig. S9(a)]. We have summed over the contributions from the two nearly degenerate phonon modes in our results of 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 . With 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 defined in Eq. (6), the transition rate in Eq. (5) can be simplified as: 𝑃𝑃 𝜇𝜇 ( ) ( ω ) = ∑ ��ω�𝐻𝐻� 𝑒𝑒𝑒𝑒 �𝑋𝑋 𝜈𝜈 �� α𝜈𝜈 2 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 𝛿𝛿�𝐸𝐸 𝜇𝜇 − ℏΩ α − ℏω� . (7) To obtain 𝑃𝑃 𝜇𝜇 ( ) , we need to first calculate 𝑆𝑆 𝜇𝜇𝜈𝜈𝛼𝛼 for the relevant mediating states. We note that the mediating 𝑋𝑋 𝜈𝜈 states don’t need to bound excitonic states, but they must be momentum-direct exciton ic states to have finite optical transitions. For the 𝑋𝑋 𝑄𝑄𝐾𝐾 initial state, the possible mediating 𝑋𝑋 𝜈𝜈 states include the direct exciton ic states in the K valley or Q valley (denoted as 𝑋𝑋 𝐾𝐾𝐾𝐾 and 𝑋𝑋 𝑄𝑄𝑄𝑄 here). For the 𝑋𝑋 𝑄𝑄𝛤𝛤 initial state, the possible mediating states 𝑋𝑋 𝜈𝜈 include 𝑋𝑋 𝑄𝑄𝑄𝑄 and 𝑋𝑋 𝛤𝛤𝛤𝛤 . Here K and Q denote any of the two K points and six
Q points; the former (latter) subscript refers to the electron (hole) state. We have calculated 𝑆𝑆 𝜇𝜇𝜈𝜈𝛼𝛼 by DFT for different exciton -phonon scattering processes that emit a phonon. The results are presented in Table S3. The calculated 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 values are overall quite small, because all of these one-phonon scattering processes are non-resonant. For all of the listed scattering processes for Q Γ and QK initial excitons, the emitted phonons are near the Q, M, and K points (and their symmetry-related points). Exction-phonon transitions Phonon momentum Phonon branch pairs from low to high energy 1 2 3 4 5 6 7 8 9 𝑋𝑋 𝑄𝑄 K → X KK 𝑋𝑋 𝑄𝑄 K → X KK 𝑋𝑋 𝑄𝑄 K → X KK M 0.093 (13.7) 0. 080 (14.9) 0.053 (16.8) 0.021 (24.1) 𝑋𝑋 𝑄𝑄 K → 𝑋𝑋 𝐾𝐾𝐾𝐾 𝑋𝑋 𝑄𝑄 K → 𝑋𝑋 𝐾𝐾𝐾𝐾 𝑋𝑋 𝑄𝑄 K → 𝑋𝑋 𝐾𝐾𝐾𝐾 Q 𝑋𝑋 𝑄𝑄 K → 𝑋𝑋 𝑄𝑄 Q 𝑋𝑋 𝑄𝑄 K → 𝑋𝑋 𝑄𝑄 Q 𝑋𝑋 𝑄𝑄 K → 𝑋𝑋 𝑄𝑄 Q Q 𝑋𝑋 𝑄𝑄 K → X 𝑄𝑄 𝑄𝑄 𝑋𝑋 𝑄𝑄 K → X 𝑄𝑄 𝑄𝑄 𝑋𝑋 𝑄𝑄 K → X 𝑄𝑄 𝑄𝑄 M 𝑋𝑋 𝑄𝑄Γ → X QQ for any Q valley Q 𝑋𝑋 𝑄𝑄Γ → X ΓΓ for any Q valley Q Table S3.
The 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 values calculated by DFT for different exciton -phonon scattering processes that emits a phonon. These scattering processes are non-resonant with small | 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 | values; the listed | 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 | values are multiplied by a factor of 100. The approximate momentum of the emitted phonon is shown in the second column, where K, Q, M refer respectively to one of the K, Q, M (and symmetry-related) points. At each of these points, there are nine pairs of phonon modes, labeled by 1 – 9 from low to high energy; each pair is nearly degenerate. We have summed over the contributions of the two nearly degenerate phonon modes for the 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 values listed here. The values for strong transitions are bolded. The number in the parenthese denote the energy of the emitted phonon (in unit of meV). Below we will give some details of our calculation of 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 . The calculation involves the electron-phonon interaction, which we describe by the following operator: [20] 𝐻𝐻� 𝑒𝑒𝑒𝑒 = ∑ � ℏ2𝐵𝐵𝑀𝑀 𝑗𝑗 Ω 𝛼𝛼𝐪𝐪 𝑛𝑛𝑛𝑛𝛼𝛼𝐪𝐪 ∇ 𝑛𝑛 𝑉𝑉�𝐫𝐫 − 𝐑𝐑 𝒏𝒏 − 𝐑𝐑 𝒋𝒋 � ⋅ 𝛜𝛜 𝑛𝑛𝛼𝛼 ( 𝑎𝑎 𝛼𝛼 , −𝐪𝐪+ + 𝑎𝑎 𝛼𝛼𝐪𝐪 ) 𝑒𝑒 𝑖𝑖𝐪𝐪⋅𝐑𝐑 𝒏𝒏 . (8) Here N denotes the number of unit cells in the sample; 𝐑𝐑 𝑛𝑛 denotes the position of a unit cell; 𝐑𝐑 𝑛𝑛 and 𝑀𝑀 𝑛𝑛 denote the position and mass of an atom at site j in a unit cell; 𝑉𝑉�𝐫𝐫 − 𝐑𝐑 𝑛𝑛 − 𝐑𝐑 𝑛𝑛 � denotes the electron-ion interaction potential (screened by valence electrons) centered at 𝐑𝐑 𝑛𝑛 + 𝐑𝐑 𝑛𝑛 ; 𝑎𝑎 𝛼𝛼−𝐪𝐪+ ( 𝑎𝑎 𝛼𝛼𝐪𝐪 ) creates (annihilates) a phonon of mode α and wave-vector - q ( q ); 𝛜𝛜 𝑛𝑛𝛼𝛼 is the corresponding phonon polarization vector ( i.e. the displacement unit vector of the atom at site j ). By using Eq. (8), we obtain � 𝑋𝑋 𝜈𝜈 ; Ω α �𝐻𝐻� 𝑒𝑒𝑒𝑒 � 𝑋𝑋 𝜇𝜇 � = 𝑂𝑂 𝜈𝜈𝜇𝜇 ∑ � ℏ 𝑁𝑁𝑀𝑀 𝑗𝑗 Ω 𝜎𝜎 𝚵𝚵 𝐪𝐪𝑗𝑗 𝑗𝑗 ⋅ 𝛜𝛜 𝑗𝑗𝛼𝛼 ( 𝐪𝐪 ) . (9) Here 𝑂𝑂 𝜈𝜈𝜇𝜇 = ∑ 𝜑𝜑 𝜈𝜈∗ ( 𝐤𝐤 ) 𝐤𝐤 𝜑𝜑 𝜇𝜇 ( 𝐤𝐤 ) denotes the overlap between the initial and mediating exciton state with respective k-space envelope functions 𝜑𝜑 𝜇𝜇 ( 𝐤𝐤 ) and 𝜑𝜑 𝜈𝜈 ( 𝐤𝐤 ) , which satisfy the normalization condition ∑ | 𝜑𝜑 𝜇𝜇 , 𝜈𝜈 ( 𝐤𝐤 )| = 1 . Using a trial wave function of the form 𝑒𝑒 −𝜁𝜁𝜁𝜁 to describe the exciton real-space envelope function, we obtain 𝑂𝑂 𝜈𝜈𝜇𝜇 = 4 𝜁𝜁 𝜈𝜈 𝜁𝜁 𝜇𝜇 � 𝜁𝜁 𝜈𝜈 + 𝜁𝜁 𝜇𝜇 � � , where 𝜁𝜁 𝜈𝜈 and 𝜁𝜁 𝜇𝜇 are the exponents of the corresponding ν and μ excitons . Since the exciton binding energy is proportional to 𝜁𝜁 in a hydrogenic model, we have 𝑂𝑂 𝜈𝜈𝜇𝜇 = 4 𝑟𝑟 𝜈𝜈𝜇𝜇 � 𝑟𝑟 𝜈𝜈𝜇𝜇 � ⁄ , where 𝑟𝑟 𝜈𝜈𝜇𝜇 = 𝐸𝐸 𝑏𝑏 , 𝜇𝜇 / 𝐸𝐸 𝑏𝑏 , 𝜈𝜈 is the ratio of binding energies between the μ and ν excitons. Based on this formula, we estimate the 𝑂𝑂 𝜈𝜈𝜇𝜇 values for various exciton transitions ( Table S4). 𝚵𝚵 𝐪𝐪𝑛𝑛 = ∑ 𝑛𝑛 �𝑐𝑐 , 𝐐𝐐 +𝐪𝐪 �∇ 𝑛𝑛 𝑉𝑉�𝐫𝐫 − 𝐑𝐑 𝑛𝑛 − 𝐑𝐑 𝑛𝑛 �𝑒𝑒 𝑖𝑖𝐪𝐪∙𝐑𝐑 𝑛𝑛 �𝑐𝑐 , 𝐐𝐐 � = ∑ 𝐔𝐔 𝐪𝐪𝑛𝑛 ( 𝐆𝐆 ) �𝑢𝑢 𝑐𝑐1 , 𝐐𝐐+𝐪𝐪 �𝑒𝑒 𝑖𝑖 ( 𝐪𝐪+𝐆𝐆 ) ∙𝐫𝐫 �𝑢𝑢 𝑐𝑐1 , 𝐐𝐐 � 𝐆𝐆 (10) is the deformation vector induced by an atomic displacement at site 𝐑𝐑 𝑛𝑛 in each unit cell for the case of electron scattering ( 𝑐𝑐 is replaced by 𝑣𝑣 for hole scattering). Here 𝐔𝐔 𝐪𝐪𝑛𝑛 ( 𝐆𝐆 ) = 𝑐𝑐 𝐿𝐿 𝑐𝑐 ∑ 𝑖𝑖 ( 𝐪𝐪 + 𝐆𝐆 ) 𝐆𝐆 𝑉𝑉� ( 𝐪𝐪 + 𝐆𝐆 ) 𝑒𝑒 −𝑖𝑖𝐆𝐆⋅𝐑𝐑 𝑗𝑗 . (11) 𝐆𝐆 denotes the reciprocal lattice vectors of the crystal; 𝑉𝑉� 𝑛𝑛 ( 𝐪𝐪 ) is the Fourier transform of the crystal potential 𝑉𝑉 ( 𝐫𝐫 ) ; | 𝑢𝑢 𝑐𝑐1 , 𝐐𝐐 � denotes the periodic part of the Bloch state at Q . In our semi-emipircal calculation, we approximate 𝑉𝑉� 𝑛𝑛 ( 𝐪𝐪 ) as 𝑉𝑉� 𝑛𝑛 ( 𝐪𝐪 ) = − 𝑞𝑞 �𝑒𝑒 −�𝑠𝑠𝑞𝑞𝑅𝑅 𝑀𝑀𝑀𝑀𝑗𝑗 � − 𝑒𝑒 −�𝑠𝑠 ′ 𝑞𝑞𝑅𝑅 𝑀𝑀𝑀𝑀𝑗𝑗 � � . (12) Here Z denotes the valency number (6 for both W and Se); 𝑅𝑅 𝑀𝑀𝑀𝑀𝑛𝑛 denotes the muffintin radius of the atom j ; and s denotes an empirical parameter. The first term in Eq. (12) denotes the Coulomb potentential due to the ion with a gaussian charge distribution of radius 𝑠𝑠𝑅𝑅 𝑀𝑀𝑀𝑀𝑛𝑛 ; the second term denotes the screening contribtion of valence electrons described by the same form but with radius 𝑠𝑠 ′ 𝑅𝑅 𝑀𝑀𝑀𝑀𝑛𝑛 . We determine the empirical parameters s and 𝑠𝑠 ′ by fitting the deformation potential constant 𝐷𝐷 = ∑ 𝚵𝚵 ⋅ 𝛜𝛜 𝑛𝑛 𝑛𝑛 for electrons coupled to the Γ -point optical phonons. Prior DFT calculations have obtained 𝐷𝐷 = 2.2 𝑒𝑒𝑉𝑉 Å ⁄ ( 𝑒𝑒𝑉𝑉 Å ⁄ ) for the electrons in the top valance band at the Γ (K) point of monolayer WSe [21]. By fitting these prior results, we obtain 𝑠𝑠 =0.41 and 𝑠𝑠 ′ = 1.1. We have used the LASTO codes to compute the integrals in Eq. (10) to obtain the deformation vector 𝚵𝚵 𝐪𝐪 . Afterward, by using the phonon polarization vectors 𝛜𝛜 𝑛𝑛 obtained by our rigid-ion model, we can calculate the exciton -phonon scattering strength 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 according to Eqs. (6) and (9). The caluclated 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 values for the relevant transitions are listed in Table S3. After we obtain 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 , we move on to calculate the one-phonon replica spectra by Eq. (7). The calculation involes the electron-light interaction strength ��ω�𝐻𝐻� 𝑒𝑒𝑒𝑒 �𝛸𝛸 𝜎𝜎 �� for the three exciton states ( 𝑋𝑋 𝐾𝐾𝐾𝐾 , 𝑋𝑋 𝑄𝑄𝑄𝑄 , 𝑋𝑋 𝛤𝛤𝛤𝛤 ), averaged over their doubly energy-degenerate states (as illustrated in Fig. 3). We assume that the interaction strength is proportional to the absorption strength of the corrsponding transitions measured by spectroscopic ellipsometry as reported in Ref. [22]. The ratios of the 𝑋𝑋 𝐾𝐾𝐾𝐾 , 𝑋𝑋 𝑄𝑄𝑄𝑄 , 𝑋𝑋 𝛤𝛤𝛤𝛤 oscillator strengths are estimated to be 1:2:4. By using these ratios of electron-light interaction strength, combined with the 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 values, we obtain the spectra of the one-phonon replica (Fig. S10). In our calculated replica spectra, the lowest energy peaks are only ~33 meV below the primrary exciton peak due to the limited energy range of single phonon in bilayer WSe . This cannot account for the replica peaks that reach redshift energy >40 meV in our expeirment. Therefore, we need to consider higher- order processes to explain our dat a. Transition 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄 ′ Γ 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄′K 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄Γ 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝐾𝐾𝐾𝐾 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝑄𝑄 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄𝑄𝑄 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 ΓΓ | 𝑂𝑂 𝜈𝜈𝜇𝜇 |
1 0.89 0.99 0.83 0.99 0.83
Table S4.
The absolute value of the overlap integral 𝑂𝑂 𝜈𝜈𝜇𝜇 for various transitions between two excition states (denoted by μ and ν ). Fig. S10.
Calculated one-phonon replica emission spectra for bilayer WSe . (a) One-phonon replica of 𝑋𝑋 𝑄𝑄𝛤𝛤 . (b) One-phonon replica of 𝑋𝑋 𝑄𝑄𝐾𝐾 . We broaden the spectral lines by replacing the delta function in Eq. (7) with a Lorentizian function with half width of 0.5 meV. Fig. S10(a) displays the calculated one-phonon replica spectrum of Q Γ exciton . In our plot, we appropriately broaden the spectral lines by replacing the delta function in Eq. (7) with a Lorentizian function with half width of 0.5 meV. The spectrum is contributed by the X QΓ → X QQ and X QΓ → X ΓΓ transitions, whose constituent spectra are also shown for comparison. Four prominent peaks are found at ~10, ~14.0, 23.6, and 25.4 meV below the zero-phonon line. The two peaks at ~10 and ~14 meV are close to the broad X QΓ1 replica at 13.0 meV in our experiment . The two peaks at 23.6 and 25.4 meV are close to the X QΓ2 replica at 27.6 meV in experiment.
Fig. S10(b) displays the one-phonon replica spectrum of the QK exciton, which is contributed by the X QK → X QQ and X QK → X KK transitions. Calculation of two-phonon replica spectra
The emission strength of two-phonon- assisted intervalley exciton recomination is proportional to the transition rate ( 𝑃𝑃 ( ) ) according to Fermi’s golden rule and the perturbation theory: 𝑃𝑃 𝜇𝜇 ( ) ( ω ) = 2 𝜋𝜋ℏ � �� �ω ; Ω 𝛼𝛼𝐪𝐪 , Ω 𝛽𝛽 �𝐻𝐻� 𝑒𝑒𝑒𝑒 �𝛸𝛸 𝜎𝜎 ; Ω 𝛼𝛼𝐪𝐪 , Ω 𝛽𝛽 ��𝛸𝛸 𝜎𝜎 ; Ω 𝛼𝛼𝐪𝐪 , Ω 𝛽𝛽 �𝐻𝐻� 𝑒𝑒𝑒𝑒 �𝛸𝛸 𝜈𝜈 ; Ω 𝛼𝛼𝐪𝐪 ��𝛸𝛸 𝜈𝜈 ; Ω 𝛼𝛼𝐪𝐪 �𝐻𝐻� 𝑒𝑒𝑒𝑒 �𝑋𝑋 𝜇𝜇 � �𝐸𝐸 𝜈𝜈 − 𝐸𝐸 𝜎𝜎 − ℏΩ β + 𝑖𝑖𝑖𝑖��𝐸𝐸 𝜇𝜇 − 𝐸𝐸 𝜈𝜈 − ℏΩ α𝐪𝐪 + 𝑖𝑖𝑖𝑖� ν , σ � , α , β ∙ 𝛿𝛿�𝐸𝐸 𝜇𝜇 − ℏΩ α 𝐪𝐪 − ℏΩ β − ℏω� . (13) Here 𝐻𝐻� 𝑒𝑒𝑒𝑒 ( 𝐻𝐻� 𝑒𝑒𝑒𝑒 ) is the electron-light (electron-phonon) interaction Hamiltonian; ω denotes the frequency of emitted photon; 𝑋𝑋 𝜇𝜇 with μ = Q Γ or QK denotes the initial intervalley exciton state with energy 𝐸𝐸 𝜇𝜇 ; 𝑋𝑋 𝜈𝜈 ( 𝛸𝛸 𝜎𝜎 ) denotes the first (second) m ediating exciton state with energy 𝐸𝐸 𝜈𝜈 ( 𝐸𝐸 𝜎𝜎 ) ; Ω α 𝐪𝐪 and Ω β denotes the frequencies of the two emitted phonons in modes α and β ; γ = q of the first emitted phonon ( Ω α 𝐪𝐪 ) can vary; therefore, we need to sum over states corresponding to different q . However, once the wave vector of the first phonon is given, the wave vector of the second phonon ( Ω β ) is determined by the conservation of momentum in the two-phonon scattering process. Therefore Eq. (13) does not contain the sum over the wave vector of the second phonon. For the dominant two-phonon processes, the first phonon is coupled (nearly) resonantly to the exciton states, hence the scattering strength is sensitive to a small variation of the phonon wave vector q ; but the second phonon is coupled non- resonantly to the exciton states, hence the scattering strength is insensitive to a small variation of the phonon wave vector. In Eq. (13), the 𝐻𝐻� 𝑒𝑒𝑒𝑒 matrix elements in the numerator are insensitive to q , but the inverse energy factor in the denominator is sensitive to q for the (nearly) resonant process. Therefore, we can simplify the expression by approximating the numerator at a fixed q corresponding to the K, Q or Γ point, while suming over the inverse energy factor over q . By using this approach and summing over energy-degenerate excitonic transitions, Eq. (13) can be simplified as 𝑃𝑃 𝜇𝜇 ( ) ( ω ) = ∑ ∑ 𝑓𝑓 𝜎𝜎𝜈𝜈𝜇𝜇𝛼𝛼 ��ω�𝐻𝐻� 𝑒𝑒𝑒𝑒 �𝛸𝛸 𝜎𝜎 �� 𝑆𝑆 𝜎𝜎𝜈𝜈𝛽𝛽 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 𝛿𝛿�𝐸𝐸 𝜇𝜇 − ℏΩ α − ℏΩ β − ℏω� ν , σα , β . (14) Here 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 and 𝑆𝑆 𝜎𝜎𝜈𝜈𝛽𝛽 are the exciton -phonon scattering strengths defined by Eq. (6), evalulated at a fixed q that corresponds to the K, Q, or Γ point. The integral over q is included in the factor 𝑓𝑓 𝛼𝛼 , which is given as: 𝑓𝑓 𝜎𝜎𝜈𝜈𝜇𝜇𝛼𝛼 ≈ 𝐴𝐴 𝑐𝑐 𝑔𝑔 𝜎𝜎 𝜈𝜈𝜇𝜇 ( ) ∫ 𝑑𝑑 𝑞𝑞 ∞0 � ∆ α +𝑖𝑖𝑖𝑖∆ α − ℏ 𝑞𝑞22𝑚𝑚𝜈𝜈 +𝑖𝑖𝑖𝑖 � = 𝐴𝐴 𝑐𝑐 𝑔𝑔 𝜎𝜎 𝜈𝜈𝜇𝜇 | ∆ α +𝑖𝑖𝑖𝑖 | 𝑚𝑚 𝜈𝜈 ℏ 𝑖𝑖 � 𝜋𝜋2 + tan −1 � ∆ α 𝑖𝑖 �� (15) Here 𝐴𝐴 𝑐𝑐 = 32.75 (in atomic unit) is the area of the primitive cell; 𝑚𝑚 𝜈𝜈 is the effective mass of the exction state ν (in unit of free eletron mass); ∆ α = 𝐸𝐸 𝜇𝜇 − 𝐸𝐸 𝜈𝜈 − ℏΩ α ; 𝑔𝑔 𝜎𝜎𝜈𝜈𝜇𝜇 is a degeneracy factor, which is the number of degenrate transitions emitting the same phonons. In order to obtain 𝑃𝑃 𝜇𝜇 ( ) in Eq. (14), we need to calculate 𝑆𝑆 for different transitions. Below we will consider the dominant two-phonon transition processes. For both Q Γ and QK initial exciton state, the second mediating state should be a momentum- direct exciton state that couples to light. The dominating direct exciton state should be at either K, Q, or Γ valley (or region) (denoted as 𝛸𝛸 𝛫𝛫𝛫𝛫 , 𝑋𝑋 𝑄𝑄𝑄𝑄 , 𝑋𝑋 𝛤𝛤𝛤𝛤 , respectively), because they are all strongly coupled to light. For the Q Γ initial exciton, there are four dominating groups of two-phonon pathsways to reach such a direct exciton state . These pathways dominante because they all invovle one nearly resonant mediating state. Below we will describe each group of pathways. (1) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋
𝑄𝑄′Γ → X Q′Q′
In the first group of pathways, 𝑋𝑋 𝑄𝑄Γ first transits to 𝑋𝑋 𝑄𝑄′Γ through electron scattering from Q to Q’ valley (this is a nearly resonant transition because Q and Q’ valleys are energy degenerate); a phonon with momentum close to Q – Q’ is emitted in this process (this phonon can be near the Q, K or M point, or their symmetry-related points). Afterward, 𝑋𝑋 𝑄𝑄 ′ Γ transits to X 𝑄𝑄 ′ 𝑄𝑄 ′ through hole scattering from Γ valley to Q’ region; a second phonon with momentum close to – Q’ is emitted in this process. Here Q’ denotes any of the six inequivalent Q valleys ; therefore this group of pathways includes six different paths via six different Q’ valleys. We note that when Q’ is Q, the 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋
𝑄𝑄Γ transition is an intravalley transition that emits a phonon near the Γ point. (2) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄′Γ → X ΓΓ In the second group of pathways, the 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄′Γ transition is the same as in the first group. In the second transition, 𝑋𝑋 𝑄𝑄′Γ transits to X ΓΓ through electron scattering from Q’ valley to Γ region in the conduction band; a second phonon with momentum close to Q’ is emitted in this process. This group includes six pathways via six different Q’ valleys. (3) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝐾𝐾𝐾𝐾
In the third group of pathways, 𝑋𝑋 𝑄𝑄Γ first transits to 𝑋𝑋 𝑄𝑄𝐾𝐾 through hole scattering from Γ to Κ valley; a phonon with momentum close to – Κ is emitted in this process (this is a resonant or nearly resonant transition because of the samll 𝑋𝑋 𝑄𝑄Γ − 𝑋𝑋
𝑄𝑄𝐾𝐾 energy separation). Afterward, 𝑋𝑋 𝑄𝑄𝐾𝐾 transits to 𝑋𝑋 𝐾𝐾𝐾𝐾 through electron scattering from Q to K valley; a second phonon with momentum close to Q – K ≈ – Q is emitted. We note that there are two inquivalent K valleys ( i.e. K and K’ valleys); so, this group includes two pathways via the K or K’ valley. (4) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋
𝑄𝑄K → 𝑋𝑋
𝑄𝑄Q
In the fourth group of pathways, the 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋
𝑄𝑄K transition is the same as in the third group. In the second transition, 𝑋𝑋 𝑄𝑄K transits to X QQ through hole scattering from K valley to Q region in the valence band; a second phonon with momentum close to K – Q ≈ Q is emitted in this process. This group includes two pathways via the K or K’ valley. By including all of the four types of pathways above, there are totally 16 (nearly) resonant pathways in the two-phonon process of 𝑋𝑋 𝑄𝑄Γ . For the
QK initial exciton, there are also four dominating groups of two -phonon pathways to reach a momentum- direct exciton state . These pathways dominante because they all invovle one (nearly) resonant mediating state. Below we will describe each group of pathways. (1) 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄 ′ K → X KK In the first group of pathways, 𝑋𝑋 𝑄𝑄Κ first transits to 𝑋𝑋 𝑄𝑄 ′ K through electron scattering from Q to Q’ valley; a phonon with momentum close to Q – Q’ is emitted in this process (this is a nearly resonant transition because Q and Q’ valleys are energy degenerate). Afterward, 𝑋𝑋 𝑄𝑄 ′ K transits to X ΚΚ through electron scattering from Q’ to K valley; a second phonon with momentum close to Q’ – K is emitted in this process. This group includes six different paths via six different Q’ valleys. When Q’ is Q, the 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝑄𝑄K transition is an intravalley transition that emits a phonon near the Γ point. (2) 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄 ′ K → X 𝑄𝑄 ′ 𝑄𝑄 ′ In the second group of pathways, the 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄 ′ K transition is the same as in the first group. In the second transition, 𝑋𝑋 𝑄𝑄 ′ 𝐾𝐾 transits to X 𝑄𝑄 ′ 𝑄𝑄 ′ through hole scattering from K valley to Q’ region in the valence band; a second phonon with momentum close to K – Q’ is emitted in this process. This group includes six pathways via six different Q’ valleys. (3) 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤 → 𝑋𝑋
𝑄𝑄𝑄𝑄
In the third group of pathways, 𝑋𝑋 𝑄𝑄K first transits to 𝑋𝑋 𝑄𝑄𝛤𝛤 through hole scattering from Κ to Γ valley; a phonon with momentum close to Κ is emitted in this process (this is a resonant or nearly resonant transition because the 𝑋𝑋 𝑄𝑄Γ − 𝑋𝑋
𝑄𝑄𝐾𝐾 energy separation is close to the energy of a phonon). Afterward, 𝑋𝑋 𝑄𝑄𝛤𝛤 transits to 𝑋𝑋 𝑄𝑄𝑄𝑄 through hole scattering from Γ to Q valley; a second phonon with momentum – Q is emitted. There is only one pathway in this group. (4) 𝑋𝑋 𝑄𝑄𝐾𝐾 → 𝑋𝑋
𝑄𝑄𝛤𝛤 → 𝑋𝑋
𝛤𝛤𝛤𝛤
In the fourth group of pathways, the 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝑄𝑄𝛤𝛤 transition is the same as in the third group. In the second transition, 𝑋𝑋 𝑄𝑄𝛤𝛤 transits to 𝑋𝑋 𝛤𝛤𝛤𝛤 through electron scattering from Q valley to Γ region in the conduction band; a second phonon with momentum close to Q is emitted. This group includes only one pathway. By including all of the four types of pathways above, there are totally 14 dominating pathways in the two-phonon process of 𝑋𝑋 𝑄𝑄𝐾𝐾 . We note that for all of the above pathways for Q
Γ and QK initial excitons, the emitted phonons are all near the Γ, Q, M, and K points (and their symmetry-related points). These faciliate our assignment of the phonon replicas.
Exction-phonon transitions
Phonon momentum Phonon branches from low to high energy 1 2 3 4 5 6 7 8 9 X Q Γ → X Q Γ Γ X Q K → X Q K Γ X Q Γ → X Q Γ X Q K → X Q K K (26.6) X Q Γ → X Q Γ X Q Γ → X Q Γ X Q K → X Q K X Q K → X Q K Q X Q Γ → X Q Γ X Q Γ → X Q Γ X Q K → X Q K X Q K → X Q K M X QK → X QΓ for any Q valley K X QΓ → X QK for any Q valley K Table S5.
The 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 values calculated by DFT for different exciton -phonon scattering processes that emit the first phonon in the two-phonon replica. These scattering processes are all (nearly) resonant. The approximate momentum of the emitted phonon is listed in the second column, where K denotes the K or K’ point, Q denotes any of the six Q points, and M denotes any of the three M points (M , M , M ) in the Brillouin zone. At each of these points, there are nine pairs of phonon modes, labeled by 1 – 9 from low to high energy; each pair is nearly degenerate. We have summed over the contributions from the two nearly degenearte phonon modes for the 𝑆𝑆 𝜇𝜇𝜈𝜈𝛼𝛼 values listed here. The values for strong transitions are bolded. The number in the parenthese denote the energy of the emitted phonon (in unit of meV). For all of the two-phonon pathways listed above, they consist of two transitions – the first transition emits the first phonon through a (nearly) resonant scattering process, and the second transition emits the second phonon through a non-resonant scattering process. We have calculated the exciton -phonon scattering strength for both transitions. The scattering strength of the first (resonant or nearly resonant) process is listed in Table S5 for all the relevant pathways. The second (non-resonant) process is the same as those in the single-phonon replicas; the scattering strength has been listed in Table S3. By combining Tables S3 and S5, we can obtain the strength of the two-phonon processes as the product of the strengths of its two constituent one-phonon processes. After we obtain the two-phonon scattering strength 𝑆𝑆 𝜎𝜎𝜈𝜈𝛽𝛽 𝑆𝑆 𝜈𝜈𝜇𝜇𝛼𝛼 , we calculate the two-phonon replica spectra by using Eq. (14) and the 1:2:4 ratio of ��ω�𝐻𝐻� 𝑒𝑒𝑒𝑒 �𝛸𝛸 𝜎𝜎 �� for 𝑋𝑋 𝐾𝐾𝐾𝐾 , 𝑋𝑋 𝑄𝑄𝑄𝑄 , 𝑋𝑋 𝛤𝛤𝛤𝛤 , similar to our calculation of single-phonon replica. The calculated two-phonon replica spectra for 𝛸𝛸 𝑄𝑄𝛤𝛤 and 𝛸𝛸 𝑄𝑄𝐾𝐾 are displayed in Fig. S11. In Fig. S11, we broaden the spectral lines by replacing the delta function in Eq. (14) with a Lorentizian function with half width of 0.5 meV. For the two-phonon replicas of Q Γ exciton , our calculated spectrum shows multiple peaks, which can somewhat account for 𝑋𝑋 𝑄𝑄𝛤𝛤1−5 observed in our experiment [see the labels in Fig. S11(a)] . In our theoretical results, the two-phonon replicas for 𝑋𝑋 𝑄𝑄𝛤𝛤 are mainly contributed by the 𝑋𝑋 𝑄𝑄Γ →𝑋𝑋
𝑄𝑄′Γ → X Q′Q′ and 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄′Γ → X ΓΓ paths, while the other two types of paths ( 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄K →𝑋𝑋
𝐾𝐾𝐾𝐾 ; 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋
𝑄𝑄K → 𝑋𝑋
𝑄𝑄Q ) only contribute weakly. Notably, 𝑋𝑋 𝑄𝑄𝛤𝛤5 is contributed significantly by two-phonon paths involving the 𝑋𝑋 𝑄𝑄𝛤𝛤 → 𝑋𝑋
𝑄𝑄𝛤𝛤 intravalley transition, which emits an optical phonon near the Γ point. Fig. S11.
Calculated emission spectra of two-phonon replicas for (a)
QΓ exciton and (b) QK exciton in bilayer WSe . The figure displays the spectra associated with different types of two-phonon pathways as well as the total spectra as their sum. For the two- phonon replicas of QK exciton, our calculated spectrum shows multiple peaks, which can somewhat account for 𝑋𝑋 𝑄𝑄𝐾𝐾1−6 observed in our experiment [see the labels in Fig. S11(b)] . In our theoretical results, the two-phonon replicas for 𝑋𝑋 𝑄𝑄𝐾𝐾 are mainly contributed by the 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤 → 𝑋𝑋
𝑄𝑄𝑄𝑄 and 𝑋𝑋 𝑄𝑄𝐾𝐾 → 𝑋𝑋
𝑄𝑄𝛤𝛤 → 𝑋𝑋
𝛤𝛤𝛤𝛤 pathways. Notably, the 𝑋𝑋 𝑄𝑄𝐾𝐾 → 𝑋𝑋
𝑄𝑄𝛤𝛤 transition with acoustic phonon emission is strongly resonant when the emitted phonon energy is close to the 18-meV difference between 𝑋𝑋 𝑄𝑄𝐾𝐾 and 𝑋𝑋 𝑄𝑄𝛤𝛤 (Table S5). Two-phonon processes involving such strongly resonant transitions contribute dominantly to 𝑋𝑋 𝑄𝑄𝐾𝐾1−5 in our result. 𝑋𝑋 𝑄𝑄𝐾𝐾6 is contributed significantly by two-phonon pathways involving the 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝑄𝑄K intravalley transition, which emits an optical phonon near the Γ point. Fig. S12 displays the calculated total one-phonon and two-phonon replica spectra for both 𝛸𝛸 𝑄𝑄𝛤𝛤 and 𝛸𝛸 𝑄𝑄𝐾𝐾 . The two-phonon replicas are overall stronger than the one-phonon replicas. There are two factors for this behavior. First, the additional transition in the two-phonon process is a (nearly) resonant process, which has a transition rate close to 100%. Second, there are many more pathways in the two-phonon processes ( ≥
14 paths) than in the one-phonon processes (only two paths); thus, the total transition rate is proprotionally larger in the two-phonon processes. In Figs. S10-12, we only mildly broaden the spectra by replacing the delta function in Eqs. (7) and (14) with a Lorentizian function of half width 0.5 meV. The theorectical spectra hence display sharp lines.
In our experiment, however, the replica peaks are relatively broad. To compare with the experimental results, we plo t the theorectial one-phonon, two-phonon, and total replica spectra by using a Lorentizian broadening function of half width 2 meV. The results, displayed in Fig. 4 of the main paper, agree decently with our experimental spectra.
Our theoretical simulation enables us to give detailed assignments of the observed phonon replicas.
In Table S6, we list the relevant exciton -phonon scattering processes and phonon modes for different replica peaks for both Q Γ and QK excitons . Further research is merited to resolve replica s associated with individual scattering process and further understand the rich exciton -phonon coupling physics in bilayer WSe . Fig. S12.
Comparison of relative emission strength of one-phonon replica and two-phonon replica for (a)
QΓ exciton and (b) Q K exciton in bilayer WSe . Phonon replicas Redshift (Expt.) (meV) Redshift (Theory) (meV) First transition Second transition Transition Phonon mode (energy in meV) Transition Phonon mode (energy in meV) X QΓ1 𝑋𝑋 𝑄𝑄Γ → X QQ Q2 (13.5) Q3 (14.0) – – X Q𝛤𝛤2 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄K
K2 (15.4) 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝑄𝑄𝑄𝑄
Q1 (8.57) 26.3 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄K
K3 (17.7) 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝑄𝑄𝑄𝑄
Q1 (8.57) 27.2 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 Q Γ 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 Q Γ M1 (13.8) 𝑋𝑋 𝑄𝑄 Γ → X 𝑄𝑄 𝑄𝑄 𝑋𝑋 𝑄𝑄 Γ → X 𝑄𝑄 𝑄𝑄 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 ΓΓ 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 ΓΓ Q2 (13.5) X QΓ3 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 Q Γ 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 Q Γ M1 (13.8) 𝑋𝑋 𝑄𝑄 Γ → X 𝑄𝑄 𝑄𝑄 𝑋𝑋 𝑄𝑄 Γ → X 𝑄𝑄 𝑄𝑄 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 ΓΓ 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 ΓΓ Q5 (26.6) 40.4 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄K
K2 (15.4) 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝑄𝑄𝑄𝑄
Q5 (25.0) 42.2 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄K
K2 (15.4) 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝑄𝑄𝑄𝑄
Q6 (26.8) X QΓ4 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄K
K3 (17.7) 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝑄𝑄𝑄𝑄
Q6 (26.8) 45.9 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 Q Γ K1 (13.1) 𝑋𝑋 Q Γ → 𝑋𝑋 Q Q Q9 (32.8) 46.6 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 Q Γ 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 Q Γ M1 (13.8) 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 𝑄𝑄 𝑄𝑄 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 𝑄𝑄 𝑄𝑄 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 ΓΓ 𝑋𝑋 𝑄𝑄 Γ → 𝑋𝑋 ΓΓ Q9 (32.8) X QΓ5 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 𝑄𝑄Γ Γ 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋
𝑄𝑄𝑄𝑄
Q6 (26.6) Q7(29.3)
Phonon replicas Redshift (Expt.) (meV) Redshift (Theory) (meV) First transition Second transition Transition Phonon mode (energy in meV) Transition Phonon mode (energy in meV) X QK1 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤
K2 (15.4) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 ΓΓ Q1 (10.2) 27.9 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤
K3 (17.7) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 ΓΓ Q1 (10.2) 28.9 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤
K2 (15.4) 𝑋𝑋 𝑄𝑄Γ → X QQ Q2 (13.5) X QK2 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤
K3 (17.7) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 ΓΓ Q2 (13.5) X QK3 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤
K2 (15.4) 𝑋𝑋 𝑄𝑄Γ → X QQ Q6(26.6) X QK4 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤
K2 (15.4) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 ΓΓ Q8 (32.0) 44.3 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤
K3 (17.7) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 ΓΓ Q6 (26.6) X QK5 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤
K2 (15.4) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 ΓΓ Q9 (32.8) 50.5 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤
K3 (17.7) 𝑋𝑋 𝑄𝑄Γ → 𝑋𝑋 ΓΓ Q9 (32.8) X QK6 𝑋𝑋 𝑄𝑄K → 𝑋𝑋
𝑄𝑄K 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 𝑄𝑄𝛤𝛤 Γ 𝑋𝑋 𝑄𝑄K → 𝑋𝑋 KK 𝑋𝑋 𝑄𝑄Γ → X QQ M7 (29.0) Q9 (32.8)
Table S6.
Dominant transitions and phonon modes involved in different replica features for Q Γ and QK excitons in bilayer WSe All the listed energies are in unit of meV. The errors of the experimental energies represent a confidence interval of about 95%. Supplemental
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