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1 In the format provided by the authors and unedited. DOI: /NNANO Magnetic brightening and control of dark excitons in monolayer WSe 2 Xiao-Xiao Zhang 1,2,3, Ting Cao 4,5, Zhengguang Lu 6, 7, Yu-Chuan Lin 8, Fan Zhang 9, Ying Wang 6,7, Zhiqiang Li 6, James C. Hone 9, Joshua A. Robinson 8, Dmitry Smirnov 6, Steven G. Louie 4,5, Tony F. Heinz 2,3 * Affiliations: 1 Department of Physics, Columbia University, New York 10027, USA 2 Department of Applied Physics, Stanford University, Stanford, California 94305, USA 3 SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA 4 Department of Physics, University of California, Berkeley, California 94720, USA 5 Materials Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA 6 National High Magnetic Field Laboratory, Tallahassee, Florida 32312, USA 7 Department of Physics, Florida State University, Tallahassee, Florida 32310, USA 8 Department of Materials Science and Engineering and Center for 2-Dimensional and Layered materials, The Pennsylvania State University, University Park, PA 16802, USA 9 Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA *Corresponding author: tony.heinz@stanford.edu NATURE NANOTECHNOLOGY 1

2 1. Additional description of materials and experimental methods a. Grown WSe 2 monolayers WSe 2 monolayers were grown directly on a polished sapphire substrate by a powder vaporization technique (1). To this end, WO 3 powder (0.5g) was placed in a quartz crucible located at the center of a 2-inch tube furnace. Another quartz crucible loaded with Se powder was positioned upstream in the tube and maintained at a temperature of 270 C during synthesis. Vapor from the WO 3 and Se powders was transported downstream to the sapphire substrate by means of a carrier gas of Ar/H 2 (160 sccm/40 sccm) at 1 Torr pressure. Film growth occurred over a period of an hour at a temperature of 950 C; the furnace was then allowed to cool back to room temperature. The as-grown films exhibit monolayer thickness and large-area homogeneity, as was confirmed by several distinct characterization techniques 1. b. Optical arrangement for photoluminescence measurements under an in-plane magnetic field (1) Fiber optics setup For measurements at high magnetic fields (31 T), we made use of a fiber-based probe. This unit was designed to be loaded into the 24-mm diameter of the cryogenic insert of the 31T resistive magnet at the National High Magnetic Field Laboratory. As shown in Fig. S1a, both the excitation light for the photoluminescence (PL) measurements and the emitted light from the Fig. S1 Design of probe head for a, fiber-based and b, free-space optics measurements in the Voigt geometry. sample were transported by optical fibers. The excitation, provided by a 532-nm laser, was injected into a polarization maintaining single-mode fiber, which guided the light to the sample stage. The latter was inserted into a helium-flow variable temperature cryostat and focused by a lens (NA= 0.5) on the sample mounted in the Voigt geometry. The sample stage was placed on a 3-dimensional piezoelectric translation stage, providing accurate adjustment of the focal point on NATURE NANOTECHNOLOGY 2

3 the sample. The collected PL signal was directed through a 50/50 beamsplitter into a 200- m core multimode optical fiber, which brought the emitted light to a spectrometer (Princeton Instruments, IsoPLane 320) equipped with a thermoelectrically cooled CCD camera. Appropriate bandpass filters and apertures were installed in the excitation and collection channels to reduce the background signal and achieve an excitation spot size of 3.3 m on the sample. Linear polarized excitation and unpolarized collection were employed in our measurements. (2) Free-space optical setup To achieve a micron-size laser focus within the vertical bore of the superconducting magnet, two sets of aspheric achromatic lenses and a right-angle mirror were employed in our probe, as shown in Fig. S1b. The sample was mounted on a 3-dimensional piezoelectric translation stage for precise positioning. White light could also be coupled into the same optical path to locate the micron-sized exfoliated samples. The PL from the sample was collected in a back-scattering geometry. The PL emission traversed the same path as the excitation, until separated by a dichroic beamsplitter and directed to the detection system. The latter consisted of a spectrometer (Princeton Instruments, 120 lines/mm grating) and a liquid-nitrogen cooled CCD. For polarization-resolved PL measurements, an excitation wavelength of 647 nm was chosen. A Fresnel rhomb was used to produce circularly polarized excitation light. The back-scattered PL signal passed through the same Fresnel rhomb and was analyzed by a linear polarizer before entering spectrometer. Uncoated optical elements (except for dichroic beamsplitter) were used throughout to reduce possible wavelength-dependent distortions of the light polarization. NATURE NANOTECHNOLOGY 3

4 Fig. S2 Complete set of photoluminescence spectra for an exfoliated monolayer of WSe 2 at 30K as a function of the strength of the in-plane magnetic field B. 2. Additional results for photoluminescence from WSe 2 monolayers under an in-plane magnetic field a. Complete set of spectra for different magnetic field strengths In the main text, we compared the PL spectra of several selected values of B for a sample temperature of 30K. Here we provide the complete PL spectra for all measurements as a function of B measured (Fig. S2), including results at intermediate field strengths not shown in the main text. For each B, a full spatial map of the sample was obtained. The spectra shown in the figure correspond to the position with the strongest PL signal. In this fashion, we ensure that we are measuring emission from the same point in the sample. No normalization was performed on the spectra presented in Fig. S2. In Fig. S3, we compare measured PL spectra when the direction of the in-plane magnetic field is reversed. The dark excitonic states emission features remain essentially unchanged. The difference in emission strength for the bright trion is ascribed to lateral inhomogeneity in the doping level of the sample and a slight shift in the spot being probed in the measurement. NATURE NANOTECHNOLOGY 4 Fig. S3 Comparison of PL spectra under positive and negative in-plane magnetic fields. The dark state features are insensitive to the specific direction of the applied in-plane field. The change in the bright trion state (between +30 T and -30 T) is attributed to slightly different doping of the material in the region probed in the two measurements. 4

5 b. Results of sample at different temperatures Beyond the PL measurements on the brightened dark states of WSe 2 presented in the main text at temperatures of 30 K and 100 K, we examined exfoliated samples over a broad range of temperatures. Here we summarize the key findings. Fig. S4a presents the measured energy difference between the bright and dark neutral exciton states for temperatures between 4 and 100 K. The values for the bright-dark splitting are very consistent, yielding mev. Fig. S4b shows the variation of emission linewidths as a function of temperature. We present the linewidths of the dark exciton and bright trion relative to the bright exciton linewidth. The dark exciton always exhibits a comparable or slightly narrower linewidth, while the bright trion linewidth increases significantly at higher temperature. (The determination of the linewidth of the dark excitons at higher temperature is less accurate due to the decreasing strength of the corresponding PL features.) At low temperatures, the dark exciton X D clearly has a narrower linewidth than the bright exciton X 0, with an absolute narrowing of the linewidth by ~2 mev. The increased linewidth of the bright Fig. S4 a, Summary of extracted bright-dark splitting of the neutral exciton E D for two exfoliated monolayer samples at different temperatures. We find mev. b, Spectral linewidth of dark excitons and trions compared to that of the neutral excitons as a function of temperature, extracted from two different samples (a and b). Fig. S5. Photoluminescence from monolayer WSe 2 at a temperature of 140K in the presence and absence of B = 14 T. A weak, but reproducible emission feature from the neutral dark exciton can be seen in this logarithmic plot, with a bright-dark splitting of. The dark exciton has width similar to that of the strong bright exciton feature. In these measurements, we used exfoliated sample c, with a low doping level, and no significant trion emission was observed. NATURE NANOTECHNOLOGY 5

6 exciton can be interpreted as arising from an extra decay channel with a lifetime of ~1.5 ps compared to the dark exciton. This additional decay may arise from the radiative channel, which is significant only for the bright exciton. At a temperature of 140K, without a magnetic field, there was no trace of bright trions or defectrelated states in the PL, further simplifying the spectrum (Fig. S5). Under the presence of an inplane magnetic field of B = 14 T, we observe a weak, but well-defined additional peak from the dark neutral exciton (Fig. S5). The bright-dark splitting is found to be mev, a value similar to that presented above for lower sample temperatures. This set of data further confirms our assignment of the dark neutral exciton, without any complications from localized exciton or trion states. Here we briefly consider the observed trends in the strengths of the features seen in the PL spectra as a function of temperature. As temperature increases, the signal from the dark exciton becomes more and more difficult to resolve. This situation reflects the much stronger emission from thermal activation of the dark excitons, leading to population of the bright excitons, with their much greater radiative rates. In addition, thermal broadening of all emission features makes them more difficult to separate from on another. Although the brightened dark exciton feature in Fig. S5 seems much less prominent, its strength is actually not dramatically weaker than the corresponding feature at a temperature of 100 K (Fig. 2b in the main text), but appears smaller due to line broadening. From Figs. 2b and S5, we estimate for an applied magnetic field of B = 14 T that the intensity of dark exciton emission compared to that of the bright exciton is ~ 3% at T = 100 K and ~1% at T = 140 K. The decreasing relative strength of the dark exciton with increasing temperature is expected based on the thermal activation of the bright exciton state. c. Field-induced behavior for grownwse 2 monolayers We also performed measurements on WSe 2 samples grown on sapphire substrates by the powder vaporization technique described above. The grown WSe 2 monolayers had significantly different doping levels and defect densities compared to the exfoliated layers. In the absence of the applied magnetic field B, the neutral emission feature at a temperature of 4K is more than five times broader than that in the exfoliated samples (Fig. S6a, inset), suggesting the existence more defect sites in these samples. There is no clear spectral signature of the trion. Defect-related emission appears about 100 mev below the neutral exciton peak. A comparison of the PL spectra with and without an in-plane magnetic field of B = 17.5 T is presented in the inset of Fig. S6a for a sample temperature of 4K. In the main part of Fig. S5a, we show the differential PL signal induced by various values of B. As for exfoliated WSe 2 monolayers, there is no observable energy shift in the PL peaks as a function of B, nor is there a change in the intensity of the bright emission features. We further see that the strength of the induced PL increases quadratically with the B (Fig. S6b). In these samples, we could not resolve either a separate bright trion peak or a separate field-induced dark trion peak. The energy separation between the bright and dark exciton peaks is found to be, slightly larger than for the exfoliated monolayers. We note that the extracted bright-dark splitting is less precise, not only because of the broad linewidth of the features, but also because of the emission from defect states of sapphire substrates around 1.78 ev. NATURE NANOTECHNOLOGY 6

7 Fig. S6 Results for photoluminescence from a grown WSe 2 sample at 4K under an applied inplane magnetic field B. a, The differential change in the PL spectra as a function of B. The new feature arises from the magnetically brightened dark exciton. The inset shows PL spectra from the sample with and without the applied field, including the unaltered bright neutral exciton around 1.73 ev. Lower energy peak present in the absence of the applied magnetic field arises from defect emission. No separate trion peak was identified. b, PL intensity of the dark exciton feature as a function of B. The symbols are the experimental data and the red line is a quadratic fit. Fig. S7 PL spectra from an exfoliated WSe 2 monolayer at 10 K under application of outof-plane magnetic fields of different strengths. Although small Zeeman shifts are present in some of the peaks, in contrast to the effect of an in-plane field, no new peaks emerge. NATURE NANOTECHNOLOGY 7

8 3. PL spectra of monolayer WSe 2 under an out-of-plane magnetic field We also examined the influence of the application of an out-of-plane magnetic field on the emission properties of an exfoliated WSe 2 monolayer, as previously investigated in published studies 2-4. The results are presented in Fig. S7 for field strengths up to 14T at a sample temperature of 10 K. The neutral exciton peak is slightly broadened with the applied field because of the (opposite) Zeeman splitting in the K and K valleys. The trion peak also shifts and changes in strength because of Zeeman effects. One of the defect peaks exhibits a large g-factor and splits into two separate peaks, in agreement with previous reports 2-4. Importantly, unlike the case of an in-plane magnetic field, there is no signature of new peaks growing in at fixed emission energy. The comparison of PL for the out-of-plane and in-plane magnetic fields proves that the dark exciton and dark trion states are only activated by an in-plane field, as expected based on spin selection rules. 4. Comparison with spectroscopy of MoSe 2 monolayers To explore the response of a TMDC monolayer with the opposite spin-ordering of the conduction bands, we also performed measurements on the influence of an in-plane magnetic field on exfoliated monolayers of MoSe 2. In monolayer MoSe 2, the CB splitting is calculated to be -20 mev 5, 6, i.e., the conduction band with the same spin as the upper valence band lies below the band with opposite spin. Calculating the difference in exciton binding energies between the dark and bright states for MoSe 2, as was done for WSe 2, we find that the bright exciton state lies at an energy below (or comparable) to that of the dark exciton. Consequently, the ordering of the bright and dark excitons in MoSe 2 is expected to be opposite to that in WSe 2. As shown in Fig. S8, there is no measurable change in the photoluminescence with the application of an in-plane magnetic field. This result is just as expected if the dark exciton lies near or above the bright exciton in energy. Although the dark state will still undergo magnetic brightening, i.e., it will start to have finite oscillator strength just as for the WSe 2 monolayer, its thermal population relative to the bright state will be too low to yield any observable photoluminescence. Since the in-plane field does not alter the energy of the dark and bright states meaningfully, the population of the equilibrium bright states is also not expected to change. Consequently, the observed emission spectra remain unchanged. Our results indicate that the combination of conduction band splitting, which pushes the dark state above the bright state, and the many-body corrections in the relative exciton binding energies, which are expected to push the dark state downwards in energy, lead to a position of the dark state near or above the energy of the bright state. NATURE NANOTECHNOLOGY 8

9 Fig. S8 Photoluminescence from monolayer of MoSe 2 a function of the strength of an applied in-plane magnetic field. The two features correspond to the neutral (1.66 ev) and charged (1.63 ev) excitons. The PL remains unchanged as a function of the strength B.The measurements were performed at a sample temperature of 5 K. These results are expected if the dark exciton state lies near or above the bright state in energy. 5. Energy levels for p-type dark trions In the main text, we discussed the situation for trions formed with an added electron, i.e., n-type trions and dark trions. Here we consider the energy levels of p-type dark trions formed with two holes. The lowest-energy configuration for the p-type dark trion is shown in Fig. S9a, with the one hole in each valley. It is instructive to compare the configuration of dark n- and p- type trions (Fig. S9). The single-particle contribution and the e-h exchange interaction terms are identical for these two trion states. The binding energy difference is related to the effective mass of the carriers. Our calculations show that the effective masses of the lower CB (CB2 mass ~ 0.56 m 0 ) and upper VB (VB mass ~ 0.54 m 0 ) are very similar, which could be understood by considering the electron-hole symmetry in this system. The corresponding difference in the binding energy should be negligible compared to our measurement accuracy (~ 1 mev). Fig. S9 Configuration of p-type dark trions a, and of n-type dark trions b. The red and blue lines indicate spin up and down bands, respectively. 9 NATURE NANOTECHNOLOGY 9

10 6. Time-resolved photoluminescence of the magnetically brightened dark trion state We have applied time-resolved single-photon counting to measure the emission dynamics from the magnetically brightened dark trion X DT. Fig. S9 shows the time-resolved emission traces in the presence and absence of an in-plane magnetic field of B = 17.5 T when measured in a spectral window capturing the X DT feature for a sample temperature at 4K. In the absence of the brightening magnetic field, we observe background emission from the L1 defect peak, with a decay time of ~100 ps (blue curve in Fig. S10). With increasing in-plane magnetic field, the dark trion feature emerges and the observed emission time increases. The results for an applied field of 17.5T, shown as the red curve in Fig. S10, can be described by a decay with a ~150 ps time constant. Here and below, we fit the experimental data to the predictions of a model convoluted with the measured instrument response function (IFR). For accurate comparison of the absolute scale of the TRPL intensities, we normalize the measured decay curves to match the relative strengths of the time-integrated PL measured under the same experimental conditions. To extract the contribution from the brightened dark trion in the presence of the background signal from the L1 defect, we subtract the response for B = 0 T from the measured response for B = 17.5 T. The resulting differential signal, shown in Fig. 2d of the main text, can be described by a single-exponential decay with a decay time of ~240 ps. The above analysis, although simple, does rely on the assumption that the emission from the L1 defect state is not changed meaningfully by the presence of the magnetic field. This hypothesis is strongly supported by the experimental fact that the lower-energy L2 defect peak remains unchanged for all applied magnetic fields (see Fig. S2 in main text). We further substantiate this analysis and improve our estimate of the decay time of the magnetically brightened dark trion by simultaneously fitting the full set of data for applied fields of B = 5T, 9T, 12T, 14T, 16T, and 17.5T. We do so by assuming that the decay of the L1 state remains unaltered by the presence of the applied magnetic field, that the decay time of the dark trion is independent of the magnetic field strength, and that the strength of the brightened dark trion emission varies quadratically in B. This model is consistent with the quadratic variation in oscillator strength of the magnetically brightened dark trion and the existence of a dominant nonradiative decay channel (that is independent of B ). The results of this global fitting procedure are shown in Fig. S11. We find that we can simultaneously reproduce the experimental decay traces to high accuracy for all values of B, as illustrated explicitly for B = 0, 12T, 16T, 17.5T. The inferred lifetime of the magnetically brightened dark trion is given by We stress that this time constant reflects the nature of non-radiative decay channels present in our sample and should not be considered as a fundamental limit for the lifetime of magnetically brightened dark exciton. The quoted time constant does, however, provide a rigorous lower bound for the effective radiative lifetime of the magnetically brightened dark exciton. NATURE NANOTECHNOLOGY 10

11 We are unable to present emission dynamics for the magnetically brightened neutral exciton X D because of the difficulty in extracting accurate data. Under our excitation conditions (using a femtosecond pulsed source for the time-resolved PL measurements), we observed strong biexciton emission 7. This feature lies near the X D peak in energy and prevents an accurate analysis of the dark exciton emission. Fig. S10 Measured time-resolved photoluminescence data at an emission wavelength around 740 nm for the determination of the dynamics of the magnetically brightened dark trion state. Traces are shown in the presence and absence of B = 17.5 T. The background signal for B = 0 arises from emission from the L1 defect feature. The difference of the two traces, shown in Fig. 2D of the main text, yields the decay of the brightened dark trion. Fig. S11 Results of global fitting of the magnetic-field dependent TRPL. a, Complete fitted curves (normalized) for different magnetic field strength. b-e, Comparison of the experimental PL decay traces and the model for magnetic fields of B = 0, 12T, 16T, 17.5T, respectively. The global fit reproduces the full set of measured experimental curves to high accuracy. 11 NATURE NANOTECHNOLOGY 11

12 7. Details of theoretical studies a. First-principles calculations of electronic structure Using the approach described in the Methods section of main text, we performed first-principles calculations of the electronic structure of monolayer WSe 2. Our calculated quasiparticle band structure yields effective masses of CB1 and CB2 K valleys of ~ 0.41 m 0 and ~ 0.56 m 0, respectively. The effective mass for the higher VB is ~0.54 m 0. Fig. S12 Modulus squared of the real-space wavefunction for the bright and dark neutral excitons. The plot shows the electron density variation as a function of its distance to the hole. The bright exciton is slightly more spatially extended than the dark exciton. The energy levels and wavefunction of two-particle electron-hole excitonic states have the form where denotes a free electron-hole pair configuration, with the excited electron in the conduction band and the hole in the valence band at the point of the Brillouin zone; is the corresponding exciton amplitude. We have suppressed the band indices in the summation because the first few excited states in monolayer WS 2 only involve electron-hole configurations from the bottom conduction band and the top valence band. The exciton amplitude is determined by the Bethe-Salpeter equation (BSE), which takes the form Here is the electron-hole interaction kernel; and correspond, respectively, to the quasiparticle energies of states and ; and is the energy of the excitonic state. NATURE NANOTECHNOLOGY 12

13 In our GW-BSE method, the wavefunctions and and the matrix elements of the electronhole kernel, and hence, are calculated consistently at each individual k point. In a real space representation, the wavefunction of the excitonic state is To compare the character of dark and bright excitons, we plot the density of the real-space exciton wavefunctions for both the bright and dark neutral excitons (Fig. S12). The hole is placed near a tungsten atom at. b. Two-level model for brightening of the dark exciton Under the application of an in-plane magnetic field B the excited states of the system are modified. We approximate this behavior by considering as the dominant effect the mixing of the bright and dark excitons in each valley. We can then model the system by an effective two-level Hamiltonian: =. The diagonal terms and are the energies of the bright and dark states in the absence of the magnetic field, and the off-diagonal terms describe the field-induced coupling between the bright and dark states. Since our applied magnetic field lies in the plane of the sample, it does not couple to the orbital motion of the electrons. Accordingly, it is given simply by, where is the Bohr magneton. Within this model, we can predict the change in energy of the bright and dark states under application of an in-plane magnetic field. In the (relevant) weak field limit, we have energy shifts of (μ B B /ΔE D ) 2. For the maximum field strength used in our investigations, the shift (μ B B /ΔE D ) 2 only reaches ~ 0.1 mev. This value is below our experimental resolution and is compatible with the negligible exciton energy shifts observed in our measurements. NATURE NANOTECHNOLOGY 13

14 The mixing of the dark and bright states under B also leads to the emergence of a finite optical oscillator strength for transition from the ground state to the dark state. The predicted oscillator strength of the dark state relative to the bright state is f D / f 0 = (μ B B /ΔE D ) 2. Fig. S13 displays the predicted brightening of the dark state as a function for the magnetic field for the experimental bright-dark splitting of ΔE D = 47 mev. In this analysis, we neglect any difference in the spatial Fig. S13 Quadratic increase in the dark exciton oscillator strength f D with in-plane magnetic field strength relative to the bright oscillator strength f 0, as predicted by the two-level model wavefunction of the bright and dark states so that the radiative rate of brightened dark states can be directly compared to that of the bright state. The validity of this approximation has been justified above by our ab-initio calculations of the excitonic spatial wavefunctions of the bright and dark states (Fig. S12). c. Estimate of oscillator strength of the magnetically brightened dark exciton from experiment observations To estimate the oscillator strength of the magnetically brightened dark exciton from the experimental data, we need to know the relative occupation of the bright and dark states. To this end, we analyze data obtained at a sample temperature of 100K where previous measurements of the time-resolved photoluminescence show the lack of any non-thermalized decay component. We assume accordingly that the bright and dark states remain in thermal equilibrium under these conditions. Using the experimentally determined dark-bright splitting of ΔE D = 47 mev, we can then estimate using a single Boltzmann factor for the relative population of bright and dark excitons. For 100K, the bright excitons population is expected to be 0.4% of the total exciton population. NATURE NANOTECHNOLOGY 14

15 For an applied magnetic field strength of B = 31 T, integrated PL intensity of the X D feature is found to be ~10% of that of the X 0 peak. Given the population ratio cited above, we then infer that the relative oscillator strength of the magnetically brightened dark exciton f D /f 0 = 0.04%. In comparison, the two-level model predicts f D /f 0 = 0.14%. d. Theoretical analysis of radiative lifetimes of excitons We calculate the radiative transition rate of the bright excitons using a thermal ensemble average, where is the transition rate of each individual excitonic state, the Boltzmann distribution function for excitons in state. We first consider the radiative lifetime of bright excitons. In transition metal dichalcogenide monolayers, the optically bright excitons have two branches. The lower energy one has a parabolic energy dispersion with exciton momentum, (with ), while the upper branch has a linear dispersion, ( estimated from MoS 8 2 ). Taking both the parabolic and linear branch into account, the average transition rate is given by where is the radiative recombination rate for excitons, is the momentum of light, and is the emission angle dictated by momentum conservation in the photoluminescence process, i.e.,. The term comes from the phase space of the photons and the in-plane projection of polarization vector. The rate at, and thus, is deduced from a previously reported experimental estimate of the bright exciton lifetime of 150 fs 9. To determine the total effective radiative lifetime of the system with magnetic brightening, we must generally also include emission from the dark excitons. As there is no intervalley exchange coupling between the dark exciton states, which leads to the linearly dispersive exciton branch, we have At low temperatures, the radiative lifetime is dominated by the dark excitons. For thermal equilibrium at and, we have and obtain NATURE NANOTECHNOLOGY 15

16 For and,, we find. However, at this higher temperature, emission from the bright state also begins to make a contribution. This yields a slightly reduced effective radiative lifetime of e. Modeling the bright-dark exciton splitting As shown previously, our calculations show that the dark and bright excitons have very similar in wavefunctions (Fig. S12). We can therefore apply first-order perturbation theory to identify the origin of the bright-dark exciton splitting determined in the ab initio GW-BSE calculations. The effective Hamiltonian for the dark exciton is where and are electron and hole momentum operators, respectively; and are effective masses of CB2 and VB as shown in Fig. 1 in the main text; denotes the screened attractive electron-hole Coulomb interaction; and is the band gap between the VB maximum and the CB2 minimum. The corresponding effective Hamiltonian for the bright exciton is given by where is the effective mass of CB1 (Fig. 1 in the main text). denotes the unscreened repulsive electron-hole exchange interaction and is the splitting between CB1 and CB2. The perturbation Hamiltonian, defined as the difference between and, is Denoting the dark exciton wavefunction as perturbation, we have for the energy shift induced by the The three terms,, and correspond, respectively, to the conduction band splitting, exchange energy, and effective mass-induced shift in the exciton binding energy, respectively. For our problem, they have values of 40 mev, 6 mev, and 11 mev, respectively. The sum of the three terms obtained by perturbation theory is in good agreement with the ab initio GW-BSE results, which also yield an energy difference of 57 mev. For the case of trions, the bright and dark states differ due to the electron configuration in the conduction band (Fig. 3 in the main text). Like the case of the neutral exciton, the energy difference between bright and dark states can be also attributed to three factors, the conduction band splitting, the difference in the exchange energy, and an effective mass induced difference in energy. The conduction band splitting is the main cause of the bright-dark trion splitting, which has a value of 40 mev in single-layer WSe 2 from our calculation. The exchange interactions for the NATURE NANOTECHNOLOGY 16

17 bright and dark trions are shown in of Fig. 3b. These two exchange energies are almost identical (within 1 mev) from our theory. So the exchange energy difference between the two trions does not contribute meaningfully to their energy difference. The effective mass related energy difference between the two trions can be estimated by assuming δe T = 0.5 δe 0, a first-order approximation based on averaging the electron band masses for the trion states. Therefore, from our theory, we estimate a total bright-dark splitting of 47 mev ( = 40 mev, < 1 mev, and mev ) for the bright and dark trion states. f. Effective mass dependence of exciton binding energies To investigate the dependence of exciton binding energy on electron effective mass, we artificially vary the effective mass of the lower energy conduction band, while keeping the valence band unchanged. As shown in Fig. S14, the calculated dark exciton energy increases as the mass is reduced. To compare the effective mass related binding energy change between CB1- VB to CB2-VB, we note that the calculated effective mass of CB1 is ~27% smaller than that of CB2, corresponding, according to Fig. S14, to a binding energy reduction of ~11 mev. Fig. S14 Calculated dark exciton energy as a function of the effective mass of the lower conduction band effective mass. The x-axis corresponds to the percentage reduction of conduction band effective mass (from its original value of 0.56 m 0 ). NATURE NANOTECHNOLOGY 17

18 References: 1. Huang, J.-K., et al. Large-Area Synthesis of Highly Crystalline WSe2 Monolayers and Device Applications. ACS Nano 2014, 8(1): Koperski, M., et al. Single photon emitters in exfoliated WSe2 structures. Nat Nanotechnol 2015, 10(6): Srivastava, A., et al. Valley Zeeman effect in elementary optical excitations of monolayer WSe2. Nat Phys Aivazian, G., et al. Magnetic control of valley pseudospin in monolayer WSe2. Nat Phys Liu, G.-B., Shan, W.-Y., Yao, Y., Yao, W., Xiao, D. Three-band tight-binding model for monolayers of group-vib transition metal dichalcogenides. Phys Rev B: Condens Matter Mater Phys 2013, 88(8): Kormányos, A., et al. k p theory for two-dimensional transition metal dichalcogenide semiconductors. 2D Materials 2015, 2(2): You, Y., et al. Observation of biexcitons in monolayer WSe2. Nat Phys 2015, 11(6): Qiu, D. Y., Cao, T., Louie, S. G. Nonanalyticity, Valley Quantum Phases, and Lightlike Exciton Dispersion in Monolayer Transition Metal Dichalcogenides: Theory and First- Principles Calculations. Phys Rev Lett 2015, 115(17): Pöllmann, C., et al. Resonant internal quantum transitions and femtosecond radiative decay of excitons in monolayer WSe2. Nat Mater 2015, 14(9): NATURE NANOTECHNOLOGY 18