Long-lived populations of momentum- and spin-indirect excitons in monolayer WSe$_2$

Monolayer transition metal dichalcogenides are a promising platform to investigate many-body interactions of excitonic complexes. In monolayer tungsten diselenide, the ground-state exciton is dark (spin-indirect), and the valley degeneracy allows low-energy dark momentum-indirect excitons to form. Interactions between the dark exciton species and the optically accessible bright exciton (X) are likely to play significant roles in determining the optical properties of X at high power, as well as limiting the ultimate exciton densities that can be achieved, yet so far little is known about these interactions. Here, we demonstrate long-lived dense populations of momentum-indirect intervalley ($X_K$) and spin-indirect intravalley (D) dark excitons by time-resolved photoluminescence measurements (Tr-PL). Our results uncover an efficient inter-state conversion between X to D excitons through the spin-flip process and the one between D and $X_K$ excitons mediated by the exchange interaction (D + D to $X_K$ + $X_K$). Moreover, we observe a persistent redshift of the X exciton due to strong excitonic screening by $X_K$ exciton with a response time in the timescale of sub-ns, revealing a non-trivial inter-state exciton-exciton interaction. Our results provide a new insight into the interaction between bright and dark excitons, and point to a possibility to employ dark excitons for investigating exciton condensation and the valleytronics.


Excitons in two-dimensional (2D) hexagonal transition metal dichalcogenides (h-
TMDs) feature large binding energy and strong light-matter interaction, while strong spinorbit coupling and the two-fold valley degeneracy lead to optical, spintronic and valleytronic properties. 1,2 Excitons in monolayer h-TMDs have orders of magnitude smaller effective mass compared with e.g. alkali atoms, and have been predicted to achieve Bose-Einstein condensation (BEC) at critical temperatures up to 80−100 K. 3,4 However achieving the high exciton densities and strong exciton-exciton interactions necessary for condensation is challenging. Owing to the giant oscillator strength of the bright exciton in monolayer h-TMD, the ultrashort population lifetime of less than 2 ps leads to a significant drop of exciton density before thermalization takes place. 5 One approach to overcoming this issue is to implement a spatially indirect excitonic system in double-layer h-TMD heterostructures. 3,4,6,7 Having the electrons and holes in the different h-TMD layers, the spatially-indirect exciton exhibits much longer radiative lifetime of the order of ns. 8 Recently, the signatures of the exciton BEC in such a system were reported through the observation of enhanced tunnelling conductance at 100 K. 9 In the tungsten-based h-TMDs, in contrast to their molybdenum counterparts, the lower-energy dark excitons have 2−3 orders longer population lifetime. 10 It is therefore of interest to explore the possibility of achieving high density, strongly interacting dark exciton populations in the tungsten-based h-TMDs, which could host exciton BECs. 7 Long-lived dark excitons could also be useful as robust carriers of spin or valley information in spintronic or valleytronic devices through manipulating the helicity of the emitting photons. 11,12 The first step towards these applications is to understand the dynamics of dark excitons and the exciton-exciton interactions in tungsten-based h-TMDs.
In monolayer tungsten diselenide (1L-WSe2), the strong spin-orbital coupling of dorbitals of W atoms results in the energy splitting of 38 meV and 460 meV in the conduction and valence band, respectively. 13,14 The spin and valley configurations of various type of excitons in 1L-WSe2 are illustrated in Figure 1a, and the corresponding quasiparticle band structures are shown in Figure 1b. To simplify the discussion, we only consider the top valence band and the case of holes in K valley; the exciton with holes in −K valley can be derived by the corresponding time-reversal pairs. The intravalley exciton X (as marked by the solid pink line in Figure 1a), also known as the bright exciton, has a total spin S = 0 (spin ½ electron plus spin -½ hole). As can be seen in Figure 1b, X exciton in 1L-WSe2 is not the two-particle ground state. Upon photon excitation, the hot X excitons thermalise to other energetically favourable states. The spin-forbidden intravalley D exciton (S = 1, marked by the dashed green line) has an energy about 40 meV below the X exciton. 15 The D exciton has much weaker but finite oscillator strength with the out-of-plane dipoles and thus can still be accessed optically. 10,16 Several studies have revealed the interesting optical and valleytronic properties of the D exciton. 17,18 The intervalley XK exciton (S = 0, marked by the dotted blue line) is composed of electrons and holes with the same energy as the D exciton. However, the exchange interaction raises the binding energy of XK about 10 meV, 19,20 yielding the estimated energy of 30 meV below X exciton. XK exciton has the centre of mass momentum of K. To be radiatively recombined, it requires the coupling to zoneboundary phonons with momentum K (~31 meV) 21 to scatter the exciton into the light cone (the blue zigzag arrow in Figure 1b), further reducing the oscillator strength. As a result, the energy of a photon emitted by the XK exciton is 61 meV below the X exciton emission, even lower than D exciton. [21][22][23][24]  In this work, by measuring the time-resolved photoluminescence (Tr-PL) of the high-quality hexagonal boron nitride (hBN) encapsulated 1L-WSe2 at cryogenic temperature, we observe dense populations of D and XK excitons with sub-ns lifetime.
Notably, we find that the XK exciton exhibits much slower growth rate and superlinear fluence-dependent dynamics. These observations indicate that XK exciton formation and lifetime are strongly governed by a second-order exciton-exciton interaction (D + D ↔ XK + XK) via Coulomb exchange. Moreover, as the exciton density increases, we observe a redshift of the X exciton sustained up to sub-ns, distinct from the ultrafast response reported before. [25][26][27] Most interestingly, the magnitude of redshift is proportional to the square of the excitation density, which is highly consistent with the population density of the XK exciton. Assisted by the rate equation analysis, we argue that the long-lived redshift of X exciton is caused by the excitonic screening effect, mainly due to the XK excitons, reflecting the strong inter-state exciton-exciton interaction as well as the dark excitonmediated Coulomb screening. Our findings reveal the capability to create a long-lived, high-density population of momentum-and spin-indirect dark excitons for studies of excitonic many-body physics and exciton BEC.

Results and Discussion
The hBN-encapsulated 1L-WSe2 samples are made by the polymer-based dry transfer technique in a nitrogen-filled glovebox (see Methods for more detail). After stacking, the samples are further thermally annealed in the argon atmosphere at 350 ℃ for 1 hour to remove the polymer residue. The sample is then transferred to a continuous flow cryostat with optical access and cooled down with liquid helium to the base temperature of 4.2 K. The details of the experimental setup of Tr-PL are described in Methods. Briefly, we excite the sample with a linear-polarised pulsed laser with a pulse width of 140 fs at various photon energies. The collected PL signal is filtered by a thin-film long-pass filter and then dispersed spectrally by a monochromator. The signal is detected by a thermoelectric-cooled charge-coupled device (CCD) for measuring the spectra. For measuring Tr-PL, the signal is redirected to a streak camera for acquiring the evolution of the PL emission in both time and frequency domains.  The time zero is approximated by measuring the arrival time of the laser reflection from the sample. We also overlay a PL spectrum taken with CCD in Figure 2a to show the integrated peak intensity. Four prominent peaks are denoted from high to low energy: X, XD, D, and XK. Among them, X, D, and XK are assigned to the three exciton species as illustrated in Figure 1 and XD is the biexciton composed of X and D situated in the opposite valleys. 15 (Note that the XK peak here is actually the intervalley exciton phonon replica emission. However, we anticipate the PL intensity to be proportional to the intervalley exciton population under experimental conditions in this paper.) We note that the emission from the trion (indicated by X − at 30 meV below X) in our PL spectra is ≤ 10% of that from X, suggesting that our sample is quite neutral. In Figure 2a, it can already be seen that the four emission modes behave quite differently in the time domain. For X and XD, the signal becomes strong shortly after t = 0 and decays quickly, indicating a short population lifetime.
The D exciton emission is weak but has much longer lifetime compared with X and XD, reflecting that the ground-state D exciton has fewer decay channels. The emission from the XK exciton, however, behaves dramatically different from the others: the integrated signal is surprisingly more intense than the X exciton and spread out over time, suggesting both slower growth and decay rate.
In Figure 2b, we plot the evolution of PL intensity of the X, D, and XK exciton, with each normalised to its maximum intensity. After performing deconvolution with the instrument response function (IRF), we found that both X and D excitons exhibit similar double exponential decays. In contrast, the intensity of XK emission first grows slowly and reaches its maximum at about 200 ps before decreasing with a slow exponential decay. We extracted the growth and the decay lifetimes by performing a double exponential fitting of the intensity I for all excitons to the function ( The fitting results are plotted along with the experimental data in Figure 2b and summarised in Table 1 as well. For the X exciton, the fast decay is associated with the intrinsic radiative recombination. 5 However, because the time resolution in our setup is about 5 ps, the 1 = 4 ps is subject to having large uncertainty, and likely represents an upper bound of the radiative recombination lifetime. In addition to the initial fast decay, we observe a significant contribution from the second decay component 2 = 215 ps. The prominent two-time-constant behaviour has been previously attributed to the inter-state conversion between the bright X exciton and the underlying dark excitons. 28,29 In Table 1, we further estimate the fraction of total intensity emitted by this channel 2 = 2 2 /( 1 2 + 2 2 ).
We found that 2 for the X exciton is > 74%, even higher than the analogous process in carbon nanotubes (CNTs; < 40%) 29 , suggesting that the dark exciton in 1L-WSe2 behaves as a significant reservoir of X exciton emission through efficient inter-state upconversion. 24 Table 1. The fitting results in Figure 2b. Note that the t1 and A1 of XK exciton are acquired from the fit of growth rate.  : where X ( ), D ( ), and X K ( ) are the densities of X, D, and XK excitons, respectively; X , D , and X K are the corresponding radiative recombination rates; and X−D , X−X K , and 2D−2X K are the inter-state conversion rates. In order to achieve a good description of the experiment, we need to introduce the 2D−2X K terms, which are associated with the second order in density. This is consistent with the efficient conversion through the Coulomb exchange interaction, as illustrated in Figure 2c. As we discuss below, the second-order dependence on density is required to fit the fluence-dependent data.
Resonant excitation of the X exciton in the low-density regime allows us to establish a well-defined initial condition of the X exciton density. We resonantly excite the sample with pump fluence of 0.13 μJ·cm −2 , corresponding to the estimated initial X exciton density of 7.5 × 10 10 cm −2 (see Supplementary S1 for the estimation of exciton density.) We furthermore determine that the times X K −1 and X−X K −1 exceed 10,000 ps, indicating that these processes are negligible in our experiment. The negligible X−X K indicates an inefficient phonon coupling, as expected at cryogenic temperatures. The other two inter-state conversion rates, X−D and 2D−2X K , are quite significant. The large X−D suggests an efficient conversion between X and D excitons due to spin-flip assisted by the spin-orbit interaction, 30 in agreement with the prior discussion of the large 2 of the X exciton. The negligible X−X K suggests that the direct process D ↔ XK should also be negligible as it requires a defect or phonon. However, the second-order process 2D−2X K is comparable in magnitude to the first-order process X−D , providing a superlinear conversion between D and XK excitons as discussed below.
The significant conversion rate X−D and 2D−2X K leads to a long-lived and highdensity population of D and XK excitons. In Figure 2e, we simulate the evolution of the exciton density to reveal the interplay between the three exciton species. In the condition of resonant excitation at 4.2 K, most of the X excitons are within the light cone as generated.
As a result, the population of X exciton significantly drops through the radiative  Taking advantage of the streak camera, we turn to studying the evolution of the spectral shape of PL emission. Here, we perform Tr-PL with a near-resonant excitation at 60 meV above the X exciton. In Figure 3a, we display the contour plots of Tr-PL spectra taken under various excitation fluence. At 7.5 μJ·cm −2 , the features are similar to Figure   2a. As the fluence increases to 37 μJ·cm −2 , we observe apparent biexciton features which grow super-linearly with the excitation density. 15 More interestingly, the time-dependent emission of X exciton shows an asymmetrical shape within the extended tail, indicating an excitation fluence-sensitive interaction between the X exciton and the dark excitonic states.
To resolve the evolution of the PL spectra, we perform a multipeak fitting with Lorentzian functions. The evolution of the linewidth and the peak energy of the X exciton are shown in Figure 3b and 3c, respectively. The origin of the asymmetric shape in Figure   3a is primarily due to a combination of linewidth broadening as well as a persistent redshift of the peak energy. We first discuss the evolution of linewidth (defined by the half-width of the half maximum of the Lorentzian function) at different excitation fluences from 7.5 to 37 μJ·cm −2 . As can be seen in Figure 3b, the dynamics of linewidth can be described by two mechanisms acting at different timescales: 1) an initial linewidth broadening followed by an exponential decay to 4 meV at t > 250 ps, and 2) a sharp linewidth narrowing at around t = 30 ps, correlated with the maximum PL intensity. The first mechanism is mainly due to exciton-exciton interaction induced homogeneous linewidth broadening. 31 Under the near-resonant excitation, the incoming photons couple to the phonons at time zero, resulting in highly-populated hot X excitons outside the light cone. These hot X excitons then thermalise either by intra-band scattering via exciton-exciton and exciton-phonon interaction or inter-band scattering to the D excitons, leading to the exponential decrease of the linewidth. The origin of the peak narrowing at t ≈ 30 ps is not clear, but could be associated with the optical Stark effect at the high photon density, as it appears coincidently with a sharp blue shift (see below). 32 We first examine whether the linewidth broadening due to heating of the lattice mediated by exciton-phonon interactions. Quantitatively, taking the excitation fluence at 37 µJ·cm −2 as an example, we estimate that the initial exciton density is X = 6.5 × 10 11 cm −2 , given that the absorption at 1.78 eV nm is about 0.5% and assuming that the conversion rate from the incident photons to excitons is unity. Under the near-resonant excitation at 60 meV above the X exciton, the excess energy of the hot X excitons is about X × 60 meV. If we assume that the total excess energy is transferred to the lattice, the corresponding lattice temperature increase is less than 10 K (see Supplementary S2 for heat capacity calculation), leading to a nominal linewidth broadening less than 0.5 meV, which is mostly negligible. 15 We instead consider that the linewidth broadening as the result of exciton-exciton interactions, leading to an exciton density-dependent linewidth 31 where, 0 = 4.5 meV is the zero-excitation density linewidth taken from the linewidth at t = 250 ps where X is negligible), X is the coefficient relating the density of X exciton to the linewidth. By plugging in the estimated X at time zero for all different fluences, as shown in the fitting curves in Figure 3b, we can describe the evolution of the linewidth by a single exponential decay with X = 1.4 × 10 −11 meV·cm 2 , and the time constant of = 25 ps. We note that the above fittings are done by excluding the peak narrowing effect at t ≈ 30 ps. Our results suggest that both exciton-phonon and exciton-exciton interaction of X exciton should only contribute in t < 100 ps, where the population of X exciton significant drops via both the efficient radiative recombination and the X to D interstate conversion.
Next, we study the dynamics of the peak energy of the X exciton. As can be seen in Figure 3c, the timescales of the redshift dynamics are dramatically different from the observed linewidth dynamics. The evolution of the peak energy exhibits an initial slow redshift followed by a slow blueshift. (A narrowly timed blueshift around t ≈ 30 ps may be related to the similarly timed linewidth narrowing discussed above.) At the lower fluence = 15 µJ·cm −2 , the redshift seems to persist beyond the timeframe of our measurement.
Even at the highest fluence = 37 µJ·cm −2 , the magnitude of the redshift reaches a maximum at about 100 ps and then decays very slowly. This behaviour is substantially similar to the evolution of XK exciton population shown in Figure 2e. To be quantitative, we perform double exponential fitting to extract the magnitude of the maximum redshift and plot it against the fluence in Figure 4a. Interestingly, the redshift of the X exciton can be approximately described by a power law, ∆ X ∝ 2 , reflecting that the dynamics of the redshift is quite sensitive to the excitation fluence. The superlinear fluence dependence, as well as the slow response time of the redshift, helps us to elucidate the underlying mechanisms. We first consider whether the redshift could be due to the laser heating effect. In our case, the near-resonant excitation creating the exciton density of ~10 12 cm −2 should give a negligible heating effect, as the estimated lattice temperature increase of <10 K would correspond to <2 meV of bandgap narrowing 33 and <0.4 meV redshift in X exciton. 15 Furthermore, heating should be accompanied by a linewidth broadening due to the exciton-phonon interaction, however in our examination of the linewidth above, we already argued that the laser-induced heating, if any, should be fully thermalised within 100 ps, in contrast to the observed slowly evolving and persistent redshift. Lastly, heating should be linear in fluence.
An alternative explanation of the redshift is the excitonic screening effect. The excitonic Coulomb interaction can reduce both the electronic bandgap and the exciton binding energy, resulting in a net redshift of the X exciton state. 33 We can first rule out the excitonic screening from X excitons because X drops sharply in 25 ps and keep decreasing exponentially, in contrast to the nonmonotonic redshift. However, the populations of D and XK excitons persist for a much longer time. Particularly, the XK exciton has a superlinear conversion rate from the D exciton, matching our observation that the maximum of the exciton population is superlinear in the incident fluence.
We then examine the relation of the density of the XK exciton to the excitation density. Figure 4b displays the Tr-PL of the XK exciton at various incident fluence, and the extracted maximum intensities Imax are plotted as open red circles in Figure 4c. As can be seen, the trend can be well described by the dependence X K −max ∝ X 2 . We note that the behaviour of Imax should not be confused with the power dependent peak intensity in the continuous-wave (CW) measurement. 22,23 In Supplementary S3, we show the powerdependent PL measurements with a CW laser at 2.33 eV, where the intensity of XK exhibits a linear dependence on the incident power. This discrepancy can be understood as illustrated in the inset of Figure 4b: Although the maximum intensity is growing superlinearly, the lifetime of the exponential decay also decreases (due to more efficient conversion XK → D → X), resulting in a linear dependence of PL intensity. The quadratic dependence X K −max ∝ X 2 confirms our assertion that the creation of XK is governed by the second-order process 2D−2X K . It also matches the superlinear dependence of the magnitude of the redshift on fluence in Figure 4a, strongly indicating that the XK population density is responsible for the redshift. To further support our observation, we perform the fluence-dependent simulation of the D and XK exciton populations with the rate equations Eq.(1)−(3). Using the fitting parameters acquired from Figure 2d, the D−max is mostly linearly proportional to X in the whole range, as shown in Figure 4b. In contrast, the X K −max behaves super-linearly at the low-density regime and slowly evolves to the linear behaviour at the high-density regime.
With the understanding of the distinct behaviour of D and XK excitons, we can estimate the redshift based on the excitonic screening theory. We find that both the dynamics of the redshift (Figure 3c) and the dependence on fluence (Figure 4a) are qualitatively consistent with the time evolution and fluence dependence of the XK exciton density (Figure 4b). In Supplementary S4, we estimate the coefficient X K relating the redshift to the density of XK exciton by X K = ∆ X / X K = (1.3 ± 0.08) × 10 −9 meV • cm 2 . Our results reveal an essential role of the XK exciton in the excitonic screening. It is of interest to note that, although the D exciton has higher population density than the XK exciton in the initial 100 ps, its contribution to the Coulomb screening is much less. This could be because the D exciton has smaller dipole moment due to its spin configuration: The same spin of electron and hole in the D exciton leads to a larger Bohr radius compared to the X and XK excitons.
Lastly, we address the issue of whether the direct single-exciton intervalley scattering process XK ↔ X is observable. So far, we have modelled our results assuming negligible inter-state conversion X−X K and recombination rates X K , implying the weak coupling between exciton and K-phonons. In Figure 4d, we plot the corresponding time of the maximum density tmax extracted from Figure 3c, along with the simulation results with various scattering rates from X−X K −1 = X K −1 = 10,000 ps to 100,000 ps. We note that the fits performed in Figure 2d are insensitive to both X−X K and X K in this range of magnitude. As demonstrated in Figure 4c, all the curves qualitatively describe the observed tmax quite well at the exciton density range in our experiments. However, at the low-density regime, X−X K and X K become dominant due to the superlinear dependence of the 2D−2X K on density. This indicates that the direct rates X−X K and X K should be measurable experimentally in lower density experiments with sufficient signal-to-noise ratio.

Conclusion
In

S1. Estimation of the Exciton Density
In the Figure 2c−e in the main text, we employed a four-level rate equation model to fit the time-resolved photoluminescence (Tr-PL) data. It is critical to determine the initial condition in our model through estimating how many excitons are generated at time zero. Given an incident fluence F, we can calculate the exciton density by Eq.(S1) as follows: Where photon is the photon energy in Joule, A is the wavelength-dependent absorptance of the 1L-WSe2. This estimation gives the upper bound of the exciton density by assuming that one absorbed photon generates one exciton. A widely employed method to measure the absorptance of the 2D materials is by measuring the differential reflectance ( / ) in a back-scattered geometry. 1,2 Figure S1a shows the / spectra of our hBN encapsulated 1L-WSe2 sample at 4.2 K. As can be seen, due to the interference of the reflected light from the multiple interfaces, the / spectrum is highly distorted. It is therefore difficult to directly estimate the absorptance by this method. Alternatively, we found a report of the measurements for both reflectance and transmittance on a chemical vapor deposition (CVD) grown 1L-WSe2 sample. 3 From the Figure 2 in the Ref. 3, we can derive the absorptance spectra, as plotted in Figure S2. We note that the sample quality of our hBN-encapsulated 1L-WSe2 is better. Therefore, our sample should have larger oscillator strength and narrower linewidth. In this work, we estimate the exciton density by employing the

S2. Heat Capacity of 1L-WSe2 and the Estimation of Laser Heating Effect
In Figure 3, we applied a near-resonant excitation at 60 meV to populate the X exciton with the estimated exciton density around X = 10 12 cm −2 . In this section, we estimate the laser-induced heating effect by considering the heat capacity at cryogenic temperatures. Here, we first assume that every photon-excited hot exciton carries the excess energy of 60 meV. The whole excitonic system thus gains the total excess energy density ∆ = ∆ = 60meV × , yielding the exciton temperature around 700 K.
Next, we consider the thermal energy from the hot excitons is fully thermalised to the lattice through exciton-phonon scattering. The increase of lattice temperature can be estimated by calculating the lattice heat capacity l ( ) = o ( ) + a ( ) , where o ( ) is the heat capacity contributed by six optical phonon branches, and a ( ) is by three acoustic phonon branches. To simplify the calculation, we take the average phonon energy ℏ = 29 meV from 6 optical phonon branches 4 (2 ′, 2 ′′, 1 ′ , 2 ′′ ) and calculate the contribution to per unit cell by: For estimating the contribution from the acoustic branches per unit cell, we apply the 2D Debye model: where Θ is the Debye temperature Θ = (ℏ / b )�4 / Cell . By plugging the average of the group velocity for 3 acoustic phonon branches ~ 10 5 cm/s and the area per unit cell Cell = 9.1 × 10 −16 cm −2 given the lattice constant 0 = 3.25 Å. 5 We found that Θ = 84 K in 1L-WSe2. In our experimental condition, the sample temperature is 4 K, which is much lower than 84 K. Therefore, we can approximate a ( ) in the analytic form at the low-temperature limit: In Figure S2a, we plot the lattice heat capacity as a function of temperature. As can be seen, the lattice heat capacity is mainly contributed by the acoustic phonon. Now we estimate the deviation of lattice temperature at various exciton density by 60meV × × = ∫ l ( ) . We plot the three curves corresponding to different initial lattice temperature i at 4.2 K, 10 K, and 20 K in Figure S2b. The shady area indicates the range of exciton density we estimated in Figure 3. We found even at the highest excitation fluence throughout the experiments ( X = 4 × 10 12 cm -2 ), the lattice temperature can only increase 7 K, 3 K, and 1 K at i = 4.2 K, 10 K, and 20 K, respectively. Our results suggest that the laser heating induced linewidth broadening in our experimental condition is negligible.

S3. The Power Dependence of the XK Exciton PL Emission with Continuous-wave Excitation
In the main text, we have demonstrated that the maximum population of the XK exciton, which is superlinear to the excitation fluence. In this section, we show that the PL intensity of XK exciton is, however, linear in the excitation power under the continuouswave (CW) excitation. As can be seen in Figure S3a, the peak intensity of the XK exciton can be extracted through the multi-peak fitting with Lorentzian functions; a selected fitting result is shown in Figure S3a. Figure S3b shows that PL intensity I of the XK exciton exhibits a linear relationship to the incident power.

S4. The Comparison of the Excitonic Screening Mediated by Different Types of Exciton
In this section, we would like to estimate the relation of the population density of XK exciton to the redshift of X exciton. In Figure S4, the navy squares represent the magnitude of the redshift ∆ X shown in Figure 4a in the main text. Here, we plot this data against the density of the maximum population of XK exciton ( X K ). which is estimated by the simulation results (the XK curve in Figure 4c). Assuming the linear dependence at the low-density regime, we can perform a linear fitting with our data and extract the coefficient X K relating the redshift to the density of XK exciton by X K = ∆ X / X K ≈ (1.3 ± 0.08) × 10 −9 meV • cm 2 . For comparison, we also extract the data from previous studies on the excitonic screening mediated by X exciton in 1L-MoS2 6 and 1L-WS2. 7 We find that X K extracted in our experiment is about an order of magnitude larger than the extracted coefficient of X exciton in WS2, 6 suggesting the XK excitons in our sample exhibit stronger excitonic screening effect.