Long-lived populations of momentum- and spin-indirect excitons in monolayer WSe 2
Shao-Yu Chen, Maciej Pieczarka, Matthias Wurdack, Eliezer Estrecho, Takashi Taniguchi, Kenji Watanabe, Jun Yan, Elena A. Ostrovskaya, Michael S. Fuhrer
11 Long-lived populations of momentum- and spin-indirect excitons in monolayer WSe Shao-Yu Chen,
Maciej Pieczarka,
Matthias Wurdack,
Eliezer Estrecho,
Takashi Taniguchi, Kenji Watanabe, Jun Yan, Elena A. Ostrovskaya, and
Michael S. Fuhrer ARC Centre of Excellence in Future Low-Energy Electronics Technologies School of Physics and Astronomy, Monash University, Clayton, Victoria, 3800, Australia Nonlinear Physics Centre, Research School of Physics, The Australian National University, Canberra, ACT 2601, Australia Department of Experimental Physics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50 -
370 Wrocław, Poland International Center for Materials Nanoarchitectonics, National Institute of Materials Science, 1- -0044, Japan Research Center for Functional Materials, National Institute of Materials Science, 1-1
Namiki, Tsukuba, Ibaraki 305 -0044, Japan Department of Physics, University of Massachusetts, Amherst, MA 01003, USA * Corresponding Author: Michael S. Fuhrer
Tel: +61 3 9905 1353
E-mail: [email protected]
Abstract
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-energ y 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 ↔ 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. Keywords intervalley exciton, dark exciton, Coulomb exchange interaction, time-resolved photoluminescence, excitonic screening, bandgap renormalisation, Coulomb exchange interaction.
Excitons in two- dimensional (2D) hexagonal transition metal dichalcogenides (h -TMDs) feature large binding energy and strong light-matter interaction, while strong spin-orbit coupling and the two-fold valley degeneracy lead to optical, spintronic and valleytronic properties.
Excitons in monolayer h-TMDs have orders of magnitude smaller effective mass compared with e.g. alkal i atoms, and have been predicted to achieve Bose-
Einstein condensation (BEC) at critical temperature s up to K. 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 . One approach to overcoming this issue is to implement a spatially indirect excitonic system in double-layer h-TMD heterostructures.
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. Recently, the signatures of the exciton BEC in such a system were reported through the observation of enhanced tunnelling conductance at 100 K. In the tungsten-based h-TMDs, in contrast to their molybdenum counterparts, the lower- energy dark excitons have 2−3 orders longer population lifetime. 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. Long-lived d ark exciton s 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.
The first step towards these applications is to understand the dynamics of dark exciton s and the exciton-exciton interactions in tungsten-based h-TMDs.
In monolayer tungsten diselenide (1L -WSe ), the strong spin-orbital coupling of d-orbitals of W atoms results in the energy splitting of 38 meV and 460 meV in the conduction and valence band, respectively. The spin and valley configurations of various type of excitons in 1L-WSe 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 -WSe 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. The
D exciton has much weaker but finite oscillator strength with the out -of-plane dipoles and thus can still be accessed optically.
Several studies have revealed the interesting optical and valleytronic properties of the D exciton.
The intervalley X K 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 X K about 10 meV, yielding the estimated energy of 30 meV below X exciton. X K exciton has the centre of mass momentum of K. To be radiatively recombined, it requires the coupling to zone-boundary phonons with momentum K (~31 meV) to scatter the exciton into the light cone (the blue zigzag arrow in Fig ure 1b), further reducing the oscillator strength. As a result, the energy of a photon emitted by the X K exciton is 61 meV below the X exciton emission, even lower than D exciton. Figure 1. Three types of charge-neutral excitons in 1L-WSe . a , the spin-valley configuration of the intravalley X ( S = 0, pink solid), intravalley D ( S = 1, green dashed), and intervalley X K ( S = 0, blue dotted) excitons. b , the schematic of excitonic band structure of X, D , and X K excitons. X and D are the intravalley excitons at k ex = Γ while X K is the intervalley exciton at k ex = K. The energy of X K is estimated around 30 meV below X but the emission photon energy is measured 61 meV below X due to the extra phonon ene rgy about 31 meV. In this work, by measuring the time - resolved photoluminescence (Tr -PL) of the high- quality hexagonal boron nitride (hBN) encapsulated 1L -WSe at cryogenic temperature, we observe dense populations of D and X K excitons with sub-ns lifetime. Notably, we find that the X K exciton exhibits much slower growth rate and superlinear fluence-dependent dynamics. These observations indicate that X K exciton formation and lifetime are strongly governed by a second-order exciton- exciton interaction (D + D ↔ X K + X K ) 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. – 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 X K 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 X K excitons, reflecting the strong inter-state exciton-exciton interaction as well as the da rk exciton -mediated 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-WSe samples are made by the polymer-based dry transfer technique in a nitrogen-filled glovebox (s ee 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. Figure 2. Time-resolved photoluminescence of X, D, and X K excitons. a , the contour plot of Tr-PL spectra at pump fluence of 3.4 μ J/cm -2 at 2.75 eV. The associated PL spectrum is overlaid for comparing the integrated PL intensity. b , the normalized PL intensity of X, D, and X K excitons are fitted by double exponential decay. c , left: The schematic of four-level (X, D, X K , and ground state) rate equation model for describing the exciton dynamics in 1L-WSe . The radiative recombination rates ( 𝛾𝛾 X , 𝛾𝛾 D and 𝛾𝛾 X K ) and the inter-state conversion rates between X, D and X K excitons ( 𝛾𝛾 X−D , 𝛾𝛾 X−X K , and 𝛾𝛾 K ) are labelled accordingly. Right: The spin-valley configurations of the corresponding inter-state conversion. d , the Tr- PL of D and X K exciton under resonant excitation at pump fluence of 0.13 𝜇𝜇 J/cm -2 at 1.72 eV. The data is fitted with the rate equation model described in c . e , the evolution of the exciton density with initial condition 𝑁𝑁 X = 7.5 × 10 cm -2 Inset: a comparison of the contribution from
X to X K and D to X K inter-state conversions. Figure 2a displays a typical contour plot of Tr- PL spectra of our sample taken by the streak camera: the x -axis is the emission photon energy, and the y-axis is the time t . 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 Fig ure 2a to show the integrated peak intensity. Four prominent peaks are denoted from high to low energy: X, XD , D, and X K . Among them, X, D , and X K 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. ( Note that the X K peak here is actually the intervalley exciton phonon replica emission. However, w e 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, sug gesting 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 X K 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 X K 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, t he intensity of X K 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 𝐼𝐼 ( 𝑡𝑡 ) = 𝑦𝑦 + 𝐴𝐴 exp �− 𝑡𝑡𝑡𝑡 � + 𝐴𝐴 exp �− 𝑡𝑡𝑡𝑡 � . 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. However, because the time resolution in our setup is a bout 5 ps, the 𝑡𝑡 =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 𝑡𝑡 = 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. In Table 1, we further estimate the fraction of total intensity emitted by this channel 𝑌𝑌 = 𝐴𝐴 𝑡𝑡 /( 𝐴𝐴 𝑡𝑡 + 𝐴𝐴 𝑡𝑡 ) . We found that 𝑌𝑌 for the X exciton is > %, even higher than the analogous process in carbon nanotubes (CNTs; < , suggesting that the dark exciton in 1L -WSe behaves as a significant reservoir of X exciton emission through efficient inter -state up-conversion. Table 1.
The fitting results in Figure 2b. Note that the t and A of X K exciton are acquired from the fit of growth rate. t (ps) t (ps) A (a.u.) A (a.u.) Y X
215 ± 8 D
24 ± 2.7
258 ± 11 X K
45 ± 1
481 ± 5 − − For the
D and X K excitons, we observe a much slower population decay rate, consistent with the much smaller oscillator strength compared to X. Specifically, the radiative emission of D and X K requires a spin-flip of electrons and defect/phonon scattering , respectively. To capture the dynamics of all the excitons populations, especially the one of X K exciton, we perform a universal fitting with the rate equations, Eq.(1 ) −( (X, D, X K and the ground state) system illustrated in Figure 2c: 𝑑𝑑𝑁𝑁 X ( 𝑡𝑡 ) 𝑑𝑑𝑡𝑡 = −�𝛾𝛾 X + 𝛾𝛾 X−D + 𝛾𝛾 X−X K �𝑁𝑁 X ( 𝑡𝑡 ) + 𝛾𝛾 X−D 𝑁𝑁 D ( 𝑡𝑡 ) + 𝛾𝛾 X−X K 𝑁𝑁 X K ( 𝑡𝑡 ) (1) 𝑑𝑑𝑁𝑁 D ( 𝑡𝑡 ) 𝑑𝑑𝑡𝑡 = − ( 𝛾𝛾 D + 𝛾𝛾 X−D ) 𝑁𝑁 D ( 𝑡𝑡 ) + 𝛾𝛾 X−D 𝑁𝑁 X ( 𝑡𝑡 ) + 𝛾𝛾 K �𝑁𝑁 X K ( 𝑡𝑡 ) − 𝑁𝑁 D ( 𝑡𝑡 ) � (2) 𝑑𝑑𝑁𝑁 XK ( 𝑡𝑡 ) 𝑑𝑑𝑡𝑡 = −�𝛾𝛾 X K + 𝛾𝛾 X−X K �𝑁𝑁 X K ( 𝑡𝑡 ) + 𝛾𝛾 X−X K 𝑁𝑁 X ( 𝑡𝑡 ) + 𝛾𝛾 K �𝑁𝑁 D ( 𝑡𝑡 ) − 𝑁𝑁 X K ( 𝑡𝑡 ) � (3) where 𝑁𝑁 X ( 𝑡𝑡 ) , 𝑁𝑁 D ( 𝑡𝑡 ) , and 𝑁𝑁 X K ( 𝑡𝑡 ) are the densities of X, D , and X K excitons, respectively; 𝛾𝛾 X , 𝛾𝛾 D , and 𝛾𝛾 X K are the corresponding radiative recombination rates; and 𝛾𝛾 X−D , 𝛾𝛾 X−X K , and 𝛾𝛾 K are the inter-state conversion rates. In order to achieve a good description of the experiment, we need to introduce the 𝛾𝛾 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-define d initial condition of the X exciton density.
We resonantly excite the sample with pump fluence of 0.13 μ J·cm − , corresponding to the estimated initial X exciton density of cm − ( see Supplementary S1 for the estimation of exciton density.) Figure 2d shows the resulting Tr-PL along with the fits to our model, for both D and X K excitons. The fits allow us to determine four parameters: 𝛾𝛾 X−1 = 4 ps, 𝛾𝛾 D−1 = 𝛾𝛾 X−D−1 = 150 ps, and 𝛾𝛾 K −1 = 8.3 × 10 ps·cm . We furthermore determine that the times 𝛾𝛾 X K −1 and 𝛾𝛾 X−X K −1 exceed 10,000 ps, indicating that these processes are negligible in our experiment. The negligible 𝛾𝛾 X−X K indicates an inefficient phonon coupling, as expected at cryogenic temperatures. The other two inter-state conversion rates, 𝛾𝛾 X−D and 𝛾𝛾 K , are quite significant. The large 𝛾𝛾 X−D suggests an efficient conversion between X and D excitons due to spin -flip assisted by the spin-orbit interaction, in agreement with the prior discussion of the large 𝑌𝑌 of the X exciton. The negligible 𝛾𝛾 X−X K suggests that the direct process D ↔ X K should also be negligible as it requires a defect or phonon. However , the second-order process 𝛾𝛾 K is comparable in magnitude to the first-order process 𝛾𝛾 X−D , providing a superlinear conversion between D and X K excitons as discussed below. The significant conversion rate 𝛾𝛾 X−D and 𝛾𝛾 K leads to a long-lived and high-density population of D and X K 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 significant ly drops through the radiative recombination. Only a fraction of X excitons 𝛾𝛾 X−1 / 𝛾𝛾 X−D−1 = to D excitons, and then achieve the maximum population of D exciton cm − at t = 12 ps, matching our observation of the fast growth rate of D exciton in Figure 2b. For the X K exciton, as shown in the inset of Figure 2e, the rate 𝛾𝛾 K 𝑁𝑁 D2 is about 3 to 4 orders of magnitude larger than our upper-bound estimate of the rate 𝛾𝛾 X−X K 𝑁𝑁 X during almost the whole time frame, suggesting that the population of X K exciton is created mainly through the upconversion from the D exciton. Our results reveal that the Coulomb exchange interaction from D and X K exciton indeed plays an essential role in the population dynamics. Figure 3. Evolution of the peak energy and the linewidth of X exciton at high excitation fluence. a, t he contour plots (PL intensity in log) of Tr -PL spectra at excitation fluence of , 22 and µJ·cm -2 with near resonant excitation at 1.78 eV (60 meV above E X ) . The vertical dashed line is aligned with the peak energy of X at the lowest fluence. b , the fluence- dependent linewidth of X exciton as a function of time. The dashed curves are the fitting results of exciton-exciton interaction mediated linewidth broadening. c , the fluence- dependent peak energy of X exciton as a function of time. The dashed curv es are fitted with double exponential functions. The grey vertical lines in b and c are aligned with the timing of the maximum intensity at around 30 ps. Taking advantage of the streak camera, we turn to study ing 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 μ J·cm − , the features are similar to Figure 2a. As the fluence increases to μ J·cm − , we observe apparent biexciton features which grow super-linearly with the excitation density. 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 o f 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 to μ J·cm − . 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. Under the near-resonant excitation, the incoming photons couple to the phonons at time zero, resulting in highly- populated hot X excitons out side 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). 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 − as an example, we estimate that the initial exciton density is 𝑁𝑁 X = 6.5 × 10 cm − , given that the absorption at nm is about 0. % 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 𝑁𝑁 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. We instead consider that the linewidth broadening as the result of exciton-exciton interactions, leading to an exciton density-dependent linewidth 𝑤𝑤 ( 𝑁𝑁 X ) = 𝑤𝑤 + 𝑤𝑤 X 𝑁𝑁 X ( 𝑡𝑡 ) , where, 𝑤𝑤 = - excitation density linewidth taken from the linewidth at t = 250 ps where 𝑁𝑁 X is negligible), 𝑤𝑤 X is the coefficient relating the density of X exciton to the linewidth. By plugging in the estimated 𝑁𝑁 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 𝑤𝑤 X = 1.4 × 10 −11 meV·cm , and the time constant of 𝜏𝜏 𝑤𝑤 = 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 t he population of X exciton significant drops via both the efficient radiative recombination and the
X to D inter -state 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 𝐹𝐹 = 15 µJ·cm − , the redshift seems to persist beyond the timeframe of our measurement. Even at the highest fluence 𝐹𝐹 = 37 µJ·cm − , 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 X K exciton population shown in Figure 2e. Figure 4. The dynamics and the excitonic screening effect of the X K exciton. a , the magnitude of redshift extracted in Figure 3c at various excitation fluence. The dashed and dot-dashed lines are corresponding to the power law of 𝛽𝛽 = 1 and 𝛽𝛽 = 2 to the fluence, respectively. b, the fluence- dependent dynamics of the X K exciton. The fluence-dependent decay time t is plotted in the inset. c, left y-axis: The maximum population of the D and X K excitons as function of the initial population of the X exciton ( N X ). Right y- axis: The maximum PL intensity of X K exciton extracted from b , which follows a power law of 𝑁𝑁 X2 . d, The navy solid squares are the timing of the maximum redshift extracted from Figure 3c. The solid curves are the simulation results of the timing of maximum population of X K exciton with various 𝛾𝛾 X−X K . 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, ∆𝐸𝐸 X ∝ 𝐹𝐹 , 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 cm − should give a negligible heating effect, as the estimated lattice temperature increase of <10 K would correspond to < and <0.4 meV redshift in X exciton. 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. We can first rule out the excitonic screening from X excitons because 𝑁𝑁 X drops sharply in 25 ps and keep decreasing exponentially, in contrast to the nonmonoton ic redshift. However, the populations of D and X K excitons persist for a much longer time. Particularly, the X K 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 X K exciton to the excitation density. Figure 4b displays the Tr-PL of the X K exciton at various incident fluence, and the extracted maximum intensities I max are plotted as open red circles in Figure 4c. As can be seen, the trend can be well described by the dependence 𝑁𝑁 X K −max ∝ 𝑁𝑁 X2 . We note that the behaviour of I max should not be confused with the power dependent peak intensity in the continuous- wave (CW) measurement. In Supplementary S3, we show the power- dependent PL measurements with a CW laser at 2.33 eV, where the intensity of X K 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 super- linearly, the lifetime of the exponential decay also decreases (due to more efficient conversion X K → D → X), resulting in a linear dependence of PL intensity. The quadratic dependence 𝑁𝑁 X K −max ∝ 𝑁𝑁 X2 confirms our assertion that the creation of X K is governed by the second-order process 𝛾𝛾 K . It also matches the superlinear dependence of the magnitude of the redshift on fluence in Figure 4a , strongly indicating that the X K population density is responsible for the redshift. To further support our observation, we perform the fluence-dependent simulation of the D and X K exciton populations with the rate equations Eq.(1 ) −( 𝑁𝑁 D−max is mostly linearly proportional to 𝑁𝑁 X in the whole range, as shown in Figure 4b. In contrast, the 𝑁𝑁 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 X K excitons, we can estimate the redshift based on the excitonic screening theory. We find that both the dynamics of the redshift ( Figure
Figure 4a) are qualitatively consistent with the time evolution and fluence dependence of the X K exciton density ( Figure 4b). In Supplementary S4, we estimate the coefficient 𝑎𝑎 X K relating the redshift to the density of X K exciton by 𝑎𝑎 X K = ∆𝐸𝐸 X / 𝑁𝑁 X K = (1.3 ± 0.08) × 10 −9 meV ∙ cm . Our results reveal an essential role of the X K exciton in the excitonic screening. It is of interest to note that, although the D exciton has higher population density than the X K 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 X K excitons. Lastly, we address the issue of whether the direct single-exciton intervalley scattering process X K ↔ X is observable. So far, we have mode lled our results assuming negligible inter-state conversion 𝛾𝛾 X−X K and recombination rates 𝛾𝛾 X K , implying the weak coupling between exciton and K-phonons. In Figure 4d, we plot the corresponding time of the maximum density t max extracted from Figure 3c, along with the simulation results with various scattering rates from 𝛾𝛾 X−X K −1 = 𝛾𝛾 X K −1 = the fits performed in Figure 2d are insensitive to both 𝛾𝛾 X−X K and 𝛾𝛾 X K in this range of magnitude. As demonstrated in Figure 4c, all the curves qualitatively describe the observed t max quite well at the exciton density rang e in our experiments. However, at the low -density regime, 𝛾𝛾 X−X K and 𝛾𝛾 X K become dominant due to the superlinear dependence of the 𝛾𝛾 K on density. This indicates that the direct rates 𝛾𝛾 X−X K and 𝛾𝛾 X K should be measurable experimentally in lower density experiments with sufficient signal-to-noise ratio. Conclusion
In conclusion, through Tr-PL studies of the charge-neutral exciton complexes in 1L-WSe , we observe long-lived populations of the spin- K exciton and the spin- double exponential decay observed in the bright X exciton. Assisted by the rate equations analysis, our results provide comprehensive evidence for the efficient and superlinear conversion between two D and two X K excitons through mutual Coulomb exchange interactions. Furthermore, we demonstrate an appreciable excitonic screening effect from the highly populated X K excitons, which could enable studies of the rich many- body correlated physics for the dark excitons. Our work paves the way to study new phenomena in long-lived indirect excitons in TMDs, including many-body interactions, transport and manipulation of valley and spin, and dark excitonic condensa tes. Methods:
Sample Fabrication.
The bulk WSe crystals are grown by chemical vapour transport method. The 1L-WSe and hBN flakes are first exfoliated on precleaned 300 nm SiO /Si wafers. To assure the sample quality, we employ the typical optical microscope and atomic force microscopy to select the residue-free 1L-WSe and few- layer hBN flakes. The hBN/1L-WSe /hBN sa mples are then stacked using a dry transfer technique with PPC (poly -propylene carbonate) stamps using a home-made micro-alignment system. All the sample preparation processes are completed in a nitrogen-purged dry glovebox to avoid sample degradation. Aft er stacking, the sample is further thermally annealed at 350 ℃ for 1 hour in argon atmosphere (99.999%) to improve the sample quality. Time-resolved photoluminescence measurement.
We mounted the sample in a continuous- flow cryostat (Janis ST - temperature of 4.2 K for spectroscopy measurements. The sample is excited with a linear polarised femtosecond pulsed laser (Coherent: Chameleon Ultra II) with a nominal pulse width of 140 fs. The incident laser is first spectrally cleaned by the short-pass filters (Thorlabs FESH0700 or FESH0750) and then focused by a 50× objective lens (numerical aperture=0.5) . T he PL signal is collected in the back -scattered geometry. To avoid the local heating effect, instead of focusing the laser in a diffraction-limited spot, we added a thin lens in the excitation path to expand the spot size up to 40 µm in diameter. The pulse-width on the sample is estimated at ps due to the dispersion of the optics. In the collection path, we set a pinhole with a 4f optical system to perform the confocal spectroscopy, enabling us to detect the signal in the selected area with a diameter of 3 µm. The collected signal is then spectrally filtered (Thorlabs
FELH0700 or FELH0750) to remove the Rayleigh scattered signal. To measure the Tr-
PL with a streak camera, we first dispersed the PL signal by a monochromator (Andor Kymera 328i) with a
600 groove/mm grating, and then coupled to the s treak camera (Optronis
OptoScope SC-10). The time resolution in all the data shown in this work is about 5 ps, which is determined by the FWHM of IRF function.
Alternatively, we could measure the PL spectra by a high- resolution spectrometer ( coupled devices. (Andor iXon EMCCD). Acknowledgements
This work was supported by the Australian Research Council (ARC) through the Centre of Excellence Grant CE170100039 (FLEET).
JY is supported by NSF DMR- . KW and TT acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, Grant Number JPMXP0112101001, JSPSKAKENHI Grant Numbers JP20H00354, and the CREST (Grant JPMJCR15F3), JST. References: (1)
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In the Figure 2 c−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: 𝑁𝑁 X = 𝐹𝐹𝐸𝐸 photon × 𝐴𝐴 (S1) Where 𝐸𝐸 photon is the photon energy in Joule, A is the wavelength-dependent absorptance of the 1L-WSe . 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 ( 𝑑𝑑𝑑𝑑 / 𝑑𝑑 ) in a back-scattered geometry. Figure S1a shows the 𝑑𝑑𝑑𝑑 / 𝑑𝑑 spectra of our hBN encapsulated 1L-WSe sample at 4.2 K. As can be seen, due to the interference of the reflected light from the multiple interfaces, the 𝑑𝑑𝑑𝑑 / 𝑑𝑑 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-WSe sample. 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-WSe is better. Therefore, our sample should have larger oscillator strength and narrower linewidth. In this work, we estimate the exciton density by employing the following values: For the resonant excitation (1.72 eV), we take 𝐴𝐴 = 0.2 ± 0.1 . For near-resonant excitation (1.78 eV), we take 𝐴𝐴 = 0.005 ± 0.003 . The uncertainty in A is represented as the error bar in x-axis of 𝑁𝑁 X in the corresponding figures. Figure S1. a.
The 𝑑𝑑𝑑𝑑 / 𝑑𝑑 spectra of our hBN-encapsulated 1L-WSe sample at 4 K. b . The absorptance of CVD 1L-WSe at 10 K derived from Ref 3. S2. Heat Capacity of 1L-WSe 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 𝑁𝑁 X = 10 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 ∆𝑄𝑄 = 𝑁𝑁 𝑋𝑋 ∆𝐸𝐸 = 60meV × 𝑁𝑁 𝑋𝑋 , 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 𝐶𝐶 l ( 𝑇𝑇 ) = 𝐶𝐶 o ( 𝑇𝑇 ) + 𝐶𝐶 a ( 𝑇𝑇 ) , where 𝐶𝐶 o ( 𝑇𝑇 ) is the heat capacity contributed by six optical phonon branches, and 𝐶𝐶 a ( 𝑇𝑇 ) is by three acoustic phonon branches. To simplify the calculation, we take the average phonon energy ℏ𝜔𝜔 𝑜𝑜 = 29 meV from 6 optical phonon branches ( 𝐸𝐸′ , 𝐸𝐸′′ , 𝐴𝐴 , 𝐴𝐴 ) and calculate the contribution to 𝐶𝐶 𝑙𝑙 per unit cell by: 𝐶𝐶 o ( 𝑇𝑇 ) = 𝜕𝜕𝜕𝜕𝜕𝜕 � o 𝑒𝑒 ℏ𝜔𝜔o / k𝑇𝑇 −1 � (S2) For estimating the contribution from the acoustic branches per unit cell, we apply the 2D Debye model: 𝐶𝐶 a ( 𝑇𝑇 ) = 6k b � 𝜕𝜕Θ � ∫ 𝑥𝑥 𝑒𝑒 𝑥𝑥 ( 𝑒𝑒 𝑥𝑥 −1 ) Θ / T0 (S3) where Θ is the Debye temperature Θ = ( ℏ𝑣𝑣 / 𝑘𝑘 b ) � 𝜋𝜋 / 𝐴𝐴 Cell . By plugging the average of the group velocity for 3 acoustic phonon branches 𝑣𝑣 ~ 10 cm/s and the area per unit cell 𝐴𝐴 Cell = 9.1 × 10 −16 cm −2 given the lattice constant 𝑎𝑎 = 3.25 Å . We found that Θ =
84 K in 1L-WSe . In our experimental condition, the sample temperature is 4 K, which is much lower than 84 K. Therefore, we can approximate 𝐶𝐶 a ( 𝑇𝑇 ) in the analytic form at the low-temperature limit: 𝐶𝐶 𝑎𝑎 ( 𝑇𝑇 ) ≈ −4 𝑇𝑇 meV ∙ cm −2 ∙ K −1 (S4) 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 𝑁𝑁 𝑋𝑋 × 𝐴𝐴 𝐶𝐶𝑒𝑒𝑙𝑙𝑙𝑙 = ∫ 𝐶𝐶 l ( 𝑇𝑇 ) 𝑑𝑑𝑇𝑇 . We plot the three curves corresponding to different initial lattice temperature 𝑇𝑇 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 ( 𝑁𝑁 X = 4 × 10 cm -2 ), the lattice temperature can only increase 7 K, 3 K, and 1 K at 𝑇𝑇 i = Figure S2. a.
The temperature-dependent lattice heat capacity of 1L-WSe . b . The estimated increase of lattice temperature at different initial temperatures. S3. The Power Dependence of the X K Exciton PL Emission with Continuous-wave Excitation
In the main text, we have demonstrated that the maximum population of the X K exciton, which is superlinear to the excitation fluence. In this section, we show that the PL intensity of X K exciton is, however, linear in the excitation power under the continuous-wave (CW) excitation. As can be seen in Figure S3a, the peak intensity of the X K 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 X K exciton exhibits a linear relationship to the incident power. Figure S3. a.
A selected PL spectrum of hBN-encapsulated 1L-WSe at 4 K. b . The power-dependent PL intensity of X K exciton with CW laser excitation at 2.33 eV. The dashed line shows a power law, 𝐼𝐼 ∝ 𝑃𝑃 . 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 X K exciton to the redshift of X exciton. In Figure S4, the navy squares represent the magnitude of the redshift ∆𝐸𝐸 X shown in Figure 4a in the main text. Here, we plot this data against the density of the maximum population of X K exciton ( 𝑁𝑁 X K ). which is estimated by the simulation results (the X K 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 𝑎𝑎 X K relating the redshift to the density of X K exciton by 𝑎𝑎 X K = ∆𝐸𝐸 X / 𝑁𝑁 X K ≈ (1.3 ± 0.08) × 10 −9 meV ∙ cm . For comparison, we also extract the data from previous studies on the excitonic screening mediated by X exciton in 1L-MoS and 1L-WS . We find that 𝑎𝑎 X K extracted in our experiment is about an order of magnitude larger than the extracted coefficient of X exciton in WS , suggesting the X K excitons in our sample exhibit stronger excitonic screening effect. Figure S4.
The relation of the redshift of the X exciton to the population density of the corresponding type of exciton defined in the legend. The navy squares are from our work. The blue dashed line shows the linear fitting of our data. The orange triangles and the red circles are extracted from the previous studies of the excitonic screening mediated by the X exciton in different materials.
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