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Cite This: Nano Lett. 2018, 18, 6135−6143 pubs.acs.org/NanoLett

Enhancement of − Scattering from Monolayer to Bilayer WS2 † ‡ ¶ § ∥ ‡ ⊥ Archana Raja,*, , Malte Selig, Gunnar Berghauser,̈ Jaeeun Yu, Heather M. Hill, , ‡ ⊥ ∥ ¶ ‡ # § Albert F. Rigosi, , Louis E. Brus, Andreas Knorr, Tony F. Heinz, , Ermin Malic, ○ and Alexey Chernikov*, † Kavli NanoScience Institute, Berkeley, California 94720, United States ‡ Department of Applied , Stanford University, Stanford, California 94305, United States ¶ Department of Theoretical Physics, Technical University of Berlin, Hardenbergstraße 36, 10623 Berlin, Germany § Department of Physics, Chalmers University of Technology, Fysikgården 1, 41258 Gothenburg, Sweden ∥ Department of , Columbia University, New York, New York 10027, United States ⊥ Departments of Physics and Electrical Engineering, Columbia University, New York, New York 10027, United States # SLAC National Accelerator Laboratory, Menlo Park, California 94025, United States ○ Department of Physics, University of Regensburg, Regensburg D-93040, Germany

ABSTRACT: Layered transition metal dichalcogenides ex- hibit the emergence of a direct bandgap at the monolayer limit along with pronounced excitonic effects. In these materials, interaction with is the dominant mechanism that limits the exciton coherence lifetime. Exciton-phonon interaction also facilitates energy and relaxation, and influences exciton diffusion under most experimental conditions. However, the fundamental changes in the exciton−phonon interaction are not well understood as the material undergoes the transition from a direct to an indirect bandgap . Here, we address this question through optical spectroscopy and microscopic theory. In the experiment, we study room- statistics of the exciton line width for a large number of mono- and bilayer WS2 samples. We observe a systematic increase in the room-temperature line width of the bilayer compared to the monolayer of 50 meV, corresponding to an additional scattering rate of ∼0.1 fs−1. We further address both phonon emission and absorption processes by examining the temperature dependence of the width of the exciton resonances. Using a theoretical approach based on many-body formalism, we are able to explain the experimental results and establish a microscopic framework for exciton− fi Downloaded via STANFORD UNIV on February 7, 2019 at 23:40:34 (UTC). phonon interactions that can be applied to naturally occurring and arti cially prepared multilayer structures. KEYWORDS: 2D materials, , exciton−phonon interaction, scattering lifetime See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. ayered transition metal dichalcogenides (TMDCs) in the The impact of additional layers on many- inter- L MX2 family (M = Mo, W and X = S, Se, Te) have been actions remains a topic of considerable importance for 2D the subject of intense investigations over the past decade due materials and their heterostructures. It is related to both − to their intriguing optical and electronic properties in the electronic properties, such as charge transfer,12 14 bandgap ultrathin, quasi two-dimensional (2D) limit. The semi- ,15,16 and hybridization of electronic states conducting TMDCs are characterized by the emergence of a between layers,17 as well as to the phonons18,19 that govern direct bandgap at monolayer (1L) thickness,1,2 strong carrier relaxation and heat transport. In particular, the absorption and emission of with dominant excitonic interaction between electronic and vibrational excitations in − effects,3 6 and the ability to access and control the -valley TMDC layers was found to affect a number of fundamental degree of freedom.7 Recently, it has become possible to create properties, including the temperature-dependent bandgap 20 21−23 atomically thin heterostructures by stacking one layer upon renormalization, carrier transport, optical heating of 24 another,8 with further broad technological appeal resulting the lattice, and and exciton coherence and from the materials’ mechanical robustness and chemical flexibility. This situation has led to the rapid development of Received: May 3, 2018 a variety of nanoscale structures to explore both fundamental Revised: July 8, 2018 − scientific questions and device applications.9 11 Published: August 10, 2018

© 2018 American Chemical Society 6135 DOI: 10.1021/acs.nanolett.8b01793 Nano Lett. 2018, 18, 6135−6143 Nano Letters Letter

Figure 1. Exciton transitions in monolayer and bilayer WS2. (a) Optical micrograph of a WS2 sample consisting of both monolayer (1L) and bilayer fl (2L) regions. (b) Room-temperature re ectance contrast of 1L and 2L WS2 around the fundamental optical gap. The peaks correspond to the A exciton resonance. Inset: room-temperature spectra. An additional lower energy peak for the 2L corresponds to indirect gap emission. (c) Schematic illustration of the single particle band structure of 1L and 2L WS2 showing valence and conduction bands (without spin- orbit splitting) adapted from Zeng et al.34 The arrows indicate the electronic direct and indirect gap transitions at the K point and between the Γ and Λ points, respectively. (d) Histogram of room-temperature width (full-width at half-maximum - fwhm) of the A exciton for 1L and 2L samples obtained from PL and reflectance contrast measurements following the procedure described in the text. The difference in the mean fwhms of the Δ ± monolayer and bilayer samples is denoted by fwhm =51 15 meV. The intrinsic limit of purely homogeneous broadening for the WS2 1L is indicated by the gray area.25,33 (e) Schematic overview of the two-dimensional with contributing intervalley electronic scattering processes represented by dashed arrows and their associated phonon mode, as identified by the experiment-theory comparison, discussed in the main text. Blue and red arrows indicate possible inelastic intervalley scattering processes in 1L and 2L WS2, respectively; purple arrows apply for both materials and include intravalley scattering at the K point mediated via Γ phonons. intervalley scattering.25,26 In addition, the strength of the activated scattering with contributions from phonon absorp- Coulomb interaction in the 2D limit, combined with efficient tion adds another 20−25 meV at room-temperature. A detailed phonon scattering, makes exciton−phonon interactions and comparison of the temperature-dependent line widths and transport particularly interesting in these systems. with predictions of many-body theory allows us to To advance fundamental understanding of exciton−phonon interpret the experimental observations in terms of micro- coupling in artificial multilayer materials, it is key to address scopic processes and to identify individual scattering channels. the physics of natural bilayers (2L) through systematic The results provide a fundamental picture of exciton−phonon experiments and to develop quantitative theoretical models interactions in multilayer TMDCs and should establish a with high predictive power. A number of recent reports have microscopic basis for understanding these processes in both indeed focused on related processes in mono- and few-layer natural and artificial structures. 27−29 TMDCs by measuring the coherence lifetime and Excitons in Monolayer and Bilayer WS2. As previously − temperature dependent line widths.30 32 However, there is discussed, most TMDC undergo a transition from limited understanding of the microscopic mechanisms govern- indirect to direct gap at the monolayer limit, − ing exciton phonon scattering rates during the transition from which is also the case for WS2. An optical micrograph of a WS2 the direct bandgap monolayer to the indirect bandgap bilayer sample with regions of mono- and bilayer thickness is shown in semiconductor. Both the impact of interlayer coupling and Figure 1a. The room-temperature reflectance contrast from the hybridization, and that of exciton−phonon scattering through two areas is plotted in Figure 1b in the spectral range intra- and intervalley channels are of particular interest in this corresponding to the lowest energy direct transition at 2 eV.38 context. The observed resonance arises from excitons formed at the K Here we address these topics through systematic optical and, equivalently, K′ points, commonly labeled as “A” states. measurements of the temperature-dependent exciton peak According to the single-particle picture, they can be denoted as − ′− ′ energies and line widths in 1L and 2L WS2 samples. We further K K (or K K ) by the corresponding electron transitions take advantage of the stability of the excitons in both systems from the valence to conduction band. This notation is used with binding energies on the order of hundreds of throughout the manuscript to indicate the origin of the − millielectronvolts,35 37 permitting the observation of these excitons. A schematic of the underlying electron states across states up to room-temperature. At cryogenic the hexagonal Brillouin zone is presented in Figure 1c. we observe an increase in the exciton line width from 1L to 2L The electronic states at the K and K′ points are primarily of 25−30 meV, corresponding to extra scattering on the 20 fs composed of the transition metal d orbitals. As a consequence, time scale from phonon emission processes. Temperature the energy of the corresponding conduction and valence bands

6136 DOI: 10.1021/acs.nanolett.8b01793 Nano Lett. 2018, 18, 6135−6143 Nano Letters Letter exhibits little sensitivity to interlayer interactions due to the An easily identifiable additional scattering channel in the 2L limited overlap of the corresponding functions. In within the single-particle picture involves transition of carriers addition, for the direct gap monolayer, these excitons emit at between K and Γ in the valence band. However, as we discuss approximately the same energy at which they are created by below in greater detail, the efficiency of all scattering processes light absorption, as shown in the photoluminescence (PL) is strongly affected by considering excitons instead of free spectra (Figure 1b, inset). For samples thicker than the carriers. In particular, the combination of Coulomb inter- monolayer, such as the studied bilayers, the lowest energy actions and interlayer coupling leads to the emergence of low transition is indirect in momentum-. Excitons created at lying dark excitons formed from an empty electron state at the the K/K′ points scatter toward lower-lying states associated KorK′ point in the valence band and an electron at Λ or Λ′ in with the indirect gap. The energy relaxation is usually the conduction band. As a result, the of the scattering facilitated by the interaction with phonons. This leads to the processes related to the conduction band structure are shown observation of PL at about 1.75 eV in the bilayer, roughly 0.2 to change in the bilayer, with significant ramifications for the − eV below the direct transition at the K and K′ points.38 40 overall scattering efficiency. In addition, at room-temperature In the following, we examine the effect of the change from we find that processes associated with phonon emission and direct to indirect bandgap on exciton−phonon scattering absorption contribute roughly in equal measure in the bilayer processes. Experimentally,wemeasureboththeroom- system. temperature and temperature-dependent line widths of mono- and bilayer WS2. Generally, the line width of an optical transition is determined by homogeneous contributions from various scattering processes and radiative recombination, convoluted by inhomogeneous broadening from disorder and ensemble effects. Both components will be discussed through- out the mansucript. As it can be already observed in Figure 1b, the exciton peak in the bilayer is significantly broader than the corresponding transition in the monolayer. By measuring the total room-temperature line widths of the exciton peaks of 62 monolayer and 21 bilayer samples, we show that despite the presence of inhomogeneous broadening, the greater line width of the exciton peak in bilayer WS2 compared to the monolayer is a statistically robust effect. The resulting histogram of A exciton line widths obtained from a combination of PL and reflectance contrast measurements, is shown in Figure 1d. The exciton widths are deduced directly from the experimental PL spectra. In the case of the reflectance contrast spectra, we deduce the line width of the exciton features from a fit based on parametrization of the dielectric function, as described in the Methods section. The intrinsic limit of purely homogeneous broadening for the WS2 1L is roughly indicated in Figure 1d for reference.25 This approach allows us to properly address the statistics and establish a difference of 51 ± 15 meV between 1L and 2L WS exciton 2 Figure 2. Temperature dependence of the A-exciton resonance in broadening at room-temperature in the studied samples. It fl monolayer and bilayer WS2.Re ectance contrast spectra from 4.5 to corresponds to additional ultrafast scattering processes for 290 K of the (a) monolayer and (b) bilayer shown in Figure 1a. (c) excitons in the bilayer that occur on the 10 fs time scale. Exciton peak energies relative to the 4.5 K data. (d) Difference in peak Δ A schematic overview of the main scattering pathways for line widths between 2L and 1L. The average fwhm from the in the single particle picture in both conduction and histogram in Figure 1b is shown by the dashed line. valence bands is presented in Figure 1e, illustrating the main phonon-assisted processes. It is based on the band structure in Temperature Dependence of the Exciton Resonance Figure 1c and previous work25 on exciton−phonon scattering in Monolayer and Bilayer WS2. To distinguish the in monolayer TMDCs. The hexagonal two-dimensional contributions of temperature-independent and thermally Brillouin zone is represented in black with the gray circles activated scattering channels, we performed temperature indicating high symmetry points. The dashed arrows illustrate dependent reflectance contrast measurements from 4.5 K to possible electron scattering channels and are labeled by the room-temperature on monolayer and bilayer WS2 samples, as associated phonon modes. Blue and red arrows further indicate shown in Figures 2a and 2b, respectively. As previously shown, scattering channels present only for 1L and 2L, respectively; the relatively broad distribution of line widths across exfoliated the purple arrow represents processes that are present for both, monolayers and bilayers (Figure 1d) is attributed to varying including K to K′ intervalley scattering and intravalley degrees of inhomogeneous broadening across different scattering processes at K. The latter is indicated by a purple samples. Therefore, for the temperature dependent study we circular arrow and includes both inelastic absorption of optical specifically chose a monolayer and a bilayer that are part of the phonons and quasi-elastic scattering with the low-energy same flake(seehighlightedregionsinFigure 1a). acoustic phonons. By symmetry, corresponding processes The two should thus experience a comparable degree of occur for electrons at the K′ point as well. inhomogeneous broadening. This assumption is further

6137 DOI: 10.1021/acs.nanolett.8b01793 Nano Lett. 2018, 18, 6135−6143 Nano Letters Letter supported by the difference in the room-temperature line monolayers in contrast to Mo-based ones25 and is proposed widths of these particular 1L and 2L being very close to the as an interpretation of the exciton dynamics in intraband 47,48 average obtained from the statistical distributions in Figure 1d, spectroscopy experiments in monolayer WSe2. as indicated by the dashed line in Figure 2d. At the same time, the absolute line widths are on the lower end of the ensemble Table 1. Theoretical Energy Separation of Momentum− a data, indicating a relatively low overall contribution from Dark Exciton States Relative to the K−K Bright Exciton inhomogeneous broadening. From the reflectance contrast spectra presented in parts a and b of Figure 2, we observe for both 1L and 2L a red shift in the exciton peak energies and a broadening of the line width with increasing temperature under conservation of the total which is proportional to the peak area. The charged exciton feature, observed at low temperatures in the 1L as a weak shoulder 30 a Energy of intervalley excitons in 1L and 2L WS2 with respect to the meV below the exciton peak, indicates relatively low levels of bright K−K exciton. The negative/positive sign means that the unintentional doping. corresponding state is lower/higher in energy than the K−K Analyzing the data in parts a and b of Figure 2 quantitatively, transition. we find that the 1L and 2L exciton peak energies follow the same relative trend with temperature as illustrated in Figure 2c. In general, the shift in the exciton energy with increasing temperature is a measure of changes in the electronic bandgap due to lattice expansion and renormalization of the transition energy due to interaction with phonons, i.e., the shift.41 In addition to peak energies, the difference in line Δ widths, fwhm, is plotted as a function of temperature in Figure 2d. We find that the exciton peak in the 2L is already considerably broader than that of the 1L at cryogenic Figure 3. Exciton scattering channels. Schematic illustration (not to temperatures by about 20 meV from temperature-independent scale) of the parabolic minima of the exciton dispersion as a function processes. The difference in the line widths further increases at of the center-of-mass momentum (Q) based on Table 1. The blue higher temperatures, indicating additional thermally activated and red valleys represent 1L and 2L WS2, respectively. The bright exciton at K−K can recombine radiatively as shown by the scattering. γ Theoretical Study of Exciton−Phonon Scattering and purple line ( rad.) or scatter nonradiatively via phonon emission (or absorption) toward dark exciton states (Q ≠ 0) as indicated by the Comparison to Experiment. To understand and interpret dashed lines. the experimental findings, we apply a previously developed microscopic theory to quantitatively address individual exciton−phonon scattering channels in mono- and bilayer A closely related aspect of the 1L and 2L band structures are WS2. We evaluate both the changes in the excitonic band the specific spin configurations of the conduction band states. structure and exciton−phonon coupling, as discussed in a In both systems, the upper conduction and valence bands with recent work25 for monolayer TMDCs. the same electron spin at the K and K′ points correspond to As a brief summary of the theoretical approach, we start with the bright A-exciton transition.49 The energy difference in the the ab initio band structures established in the bands with opposite spin is essentially determined by the literature;42,43 we then compute the energies of the excitonic spin−orbit coupling. In monolayers, the splitting at the Λ states by numerically solving the .25,41,44 The point is also given by the spin−orbit interaction. In contrast to specific nature of the electric field screening between charge that, the interlayer coupling (or hybridization) from the overlap carriers in a 2D sheet is explicitly taken into account using an of the electronic wave functions is the dominant effect approximate thin-film Coulomb potential.45,46 The resulting determining the splitting at the Λ and Λ’ points in bilayer 43 Λ′ energy levels of the spin-allowed exciton transitions relative to TMDCs. As a consequence for bilayer WS2, the valley the bright K−K state are summarized in Table 1. with the same electron spin as K becomes the lowest A schematic overview of the exciton band structure is conduction band state and the Λ valley with the same electron presented in Figure 3 with the radiative, intravalley, and spin shifts to much higher energies compared to the intervalley scattering channels indicated by arrows. Here, only monolayer. Hence, in addition to the more obvious Γ−K the exciton states with the same spin configuration are taken scattering channel in the valence band of the bilayer, there is a into account, since phonon-assisted processes also require spin subtle but important change in the conduction band relaxation Δ fi conservation, i.e., ms = 0 in the rst approximation. We also pathways, where the K−Λ scattering becomes energetically note that while there are more complex calculations of the unfavorable while the K−Λ′ channel opens up. exciton ground states, the main conclusions related to higher After the energies of the exciton states are obtained, the binding energies of the states with higher electron masses microscopic is derived using the semiconductor (such as at Λ)shouldbelargelyindependentofthe Bloch equations approach. For the bilayer, the oscillator specific theoretical approach. One of the key results in the strength of the optical matrix element is adjusted to match the calculated exciton band structure of the W-based materials is peak area in the experimentally determined A exciton the lower lying K-Λ dark exciton state, primarily due to the absorption. As a basis for the calculation of the exciton− increased effective mass as compared to the direct K−K phonon interaction, the underlying electron-phonon matrix exciton. This dark state appears to be necessary to explain the elements for optical and acoustic phonon are sourced from ab temperature dependent exciton broadening in W-based initio calculations by Jin et al.22 for the monolayer case. For the

6138 DOI: 10.1021/acs.nanolett.8b01793 Nano Lett. 2018, 18, 6135−6143 Nano Letters Letter bilayer, the matrix elements are assumed to be the same in the provides a much weaker contribution. We also note that the first approximation. The latter is supported by the similar threefold degeneracy of the Λ and Λ′ states provides a large temperature dependent shift of the 1L and 2L transition density of available final states, thus strongly contributing to energies observed in experiment (see Figure 2c). It strongly the overall scattering efficiency towards these states. The results implies that the phonon-renormalization of the transition of the microscopic theory are largely consistent with the form energies and therefore the exciton−phonon interaction is of the commonly used phenomenological model of temper- essentially of the same strength over the studied temperature ature-induced broadening simplified as a sum of a linear and range.50,51 The individual rates for exciton scattering pathways bosonic terms, as discussed in more detail by Selig et al.52 for across the different states illustrated in Figure 3 are then the monolayer case. computed under energy and momentum conservation require- Interestingly, the carrier-phonon coupling constant for ments from either absorption or emission of single phonons.52 processes mediated by M phonons is an order of magnitude We note that we include both inelastic scattering of excitons larger in comparison to the Λ phonons, as calculated by Jin et with optical and zone-edge acoustic phonons as well as quasi- al.22 This leads to M-phonon mediated scattering to the K−Λ′ fi Λ elastic scattering with low-energy acoustic phonons from the valley being signi cantly stronger in bilayer WS2 than the linear dispersion branches. The spontaneous recombination of phonon mediated scattering to K−Λ in the monolayer. electrons and holes at the bright K−K exciton is also Phonon-assisted relaxation toward Λ and Λ′ valleys has also considered, leading to radiative dephasing. The sum of these been highlighted in studies of double resonance Raman γ 26 54 interactions yields a total scattering rate ( total) that is spectroscopy in MoS2 and WS2, photoluminescence Γ 55,56 proportional to the homogeneous line width h of the exciton excitation measurements in MoSe2 and angle-resolved γ Γ ℏ − 57 resonance, i.e., total= h/ . The inverse of the rate is photoemission pump probe spectroscopy in bulk WSe2. proportional to the commonly defined characteristic coher- To compare the theoretical findings to experimental γ 53 ence time T2 =2/total in the literature. observations, it is necessary to isolate purely homogeneous In Figure 4a, the individual contributions from each contributions to broadening, i.e., the sum of the radiative and scattering channel in the monolayer are presented as a scattering contributions, across a wide temperature range. Recently, techniques based on four-wave mixing have been successfully applied to determine directly the coherence − lifetime and thus the intrinsic line width.27 29 However, the results were typically reported for temperatures below 100 K. On the other hand, reflectance contrast and PL spectroscopy have been used to extract total line widths also at higher − temperatures for bulk TMDCs,58 60 and more recently for the − ultrathin layers.30 32 For a quantitative analysis, however, potential inhomogeneous contributions to the measured broadening need to be considered to obtain the limits for the homogeneous broadening and place it in the context of statistics on a large number of samples, as discussed in connection to Figure 1d. In the present case we thus estimate the inhomogeneous contributions to the total line width and infer corresponding limits for the homogeneous broadening. The procedure is outlined in the Methods section. We further note that the relative contribution from inhomogeneous broadening is more pronounced at low temperatures. Our Figure 4. Temperature dependence of homogeneous line widths. experimental procedure is thus less sensitive to smaller changes Theoretical fwhm of individual exciton−phonon scattering channels in homogeneous line widths at cryogenic temperatures in cumulatively added on top of each other for (a) 1L and (b) 2L WS2. 27−29 (c) Experimentally determined upper and lower bounds for the contrast to direct measurements of the dephasing and homogeneous line widths of 1L and 2L WS2 are marked by open and leads to larger variations when estimating the intrinsic line filled circles, respectively. The total theoretically computed fwhms are widths in this regime. presented by lines on top of the experimental data. The distributions The extracted minimum and maximum values of the of room-temperature line widths for 1L and 2L from Figure 1d are temperature dependent homogeneous line widths are plotted condensed into box-plots on the right for comparison. in Figure 4casfilled and open circles, respectively. As further noted above, the room-temperature line widths of the two function of temperature by cumulatively adding them on top samples fall on the lower side of the measured distributions, of each other. The major contributions are the intravalley suggesting relatively low inhomogeneous broadening of the “ − ” fl scattering (labeled as +K K ) with optical and low-energy studied 1L and 2L akes exfoliated on SiO2 substrates. The acoustic phonons at Γ, relaxation of the excitons to the K−Λ total temperature dependent scattering rates from theory are valley (“+K−Λ”), mediated by Λ phonons, and scattering to overlaid as solid lines. The agreement between the the K′ valley through K phonons (“+K−K′”). Figure 4b experimental and theoretical temperature dependence is very presents the corresponding results for the bilayer case. Here, reasonable. We also note that the line width of the bilayer is the intravalley +K−K and the intervalley +K−K′ scattering less sensitive to inhomogeneous broadening because of much remain essentially the same as for the monolayer. The larger homogeneous contributions, as also predicted by the scattering toward the K−Λ′ state, however, mediated by M theory. The difference in the line widths of mono- and bilayer phonons, becomes the dominant channel. The additional WS2, presented in Figure 2d is equally captured by the theory relaxation to the Γ−K exciton via emission of K phonons on the same footing, within experimental limits. This further

6139 DOI: 10.1021/acs.nanolett.8b01793 Nano Lett. 2018, 18, 6135−6143 Nano Letters Letter supports the overall validity of the theoretical approach and semiconducting 2D materials and their heterostructures in highlights the necessity to consider the dominant K−Λ′ future research. scattering channel in the bilayer. This channel is largely Methods and Data Analysis. Experimental Section. − fl responsible for the total broadening on the order of 30 40 Monolayer and bilayer WS2 akes were prepared on SiO2 meV at cryogenic temperatures and the additional increase of substrates by micromechanical exfoliation of bulk crystals (2D the line width towards 80 meV at room-temperature in 2L. Semiconductors Inc.). For room-temperature statistics, data Finally, we note that by deconvoluting the theoretically from 97 individual samples were obtained and analyzed. For expected room-temperature homogeneous line widths from temperature dependent measurements from 4 to 290 K, a the sample statistics presented in Figure 1d, we obtain representative sample containing mono- and bilayer regions comparable averages of inhomogeneous line widths for the 1L was mounted in a helium cooled cryostat. Optical and 2L of 24 ± 11 meV and 37 ± 14 meV, respectively, in the reflectance spectroscopy was used to probe the exciton states studied ensemble. using a tungsten−halogen white-light source for illumination of In conclusion, we have systematically studied exciton− the samples. Photoluminescence (PL) measurements were phonon scattering in monolayer and bilayer WS2 through performed using the 514 nm line of an argon for optical spectroscopy by aggregating statistics across a large excitation, while keeping the incident power very low, around number of samples and performing temperature-dependent 10 nW. In both cases, the light was focused down to a 1−2 μm experiments on a representative flake containing both spot on the sample using a 40× objective. The reflected light monolayer and bilayer regions. Using microscopic many- and PL were spectrally resolved in a grating spectrometer and body theory we were able to identify individual exciton− subsequently detected using a Peltier-cooled EMCCD. phonon scattering channels in 1L and 2L, including the Analysis of Reflectance Contrast. The reflectance contrast, − interplay of the K−Λ and K−Λ′ transitions to quantitatively RC, corresponds to (Rsample Rsubstrate)/Rsubstrate,where account for the experimental results. R denotes the intensity of the reflected signal. For thin films While radiative recombination is dominant in 1L samples at on transparent substrates such as fused silica, the reflectance low temperatures, phonon-assisted relaxation on the 20−25 fs contrast is dominated by the imaginary part of the sample’s time scale from K−K toward K−Λ′ and Γ−K states leads to dielectric response and is largely proportional to the additional broadening of 25−30 meV in the 2L. At room- absorption.61,62 The dielectric response of a material can be temperature, the difference in the 1L and 2L line widths approximated by a sum of Lorentzian oscillators on a constant 53,63,64 increases to about 50 meV due to temperature activated background. Here, a physically meaningful number of phonon-absorption. The absolute width of the 2L exciton peak Lorentzians was chosen to represent the main excitonic reaches 80 meV, corresponding to an under 10 fs total resonances in the material. By simulating the reflectance 65 scattering lifetime of the K−K exciton. We also note that due contrast spectra in a standard transfer matrix formalism we to the theoretical considerations outlined above, the major obtained parameters related to the exciton transitions. eq 1 62 changes in exciton−phonon scattering occur at the 1L to 2L describes the dielectric function ϵ in the thin film approach as ϵ transition and the exciton dephasing rate does not significantly a function of the energy E. b is the background change from the bilayer to thicker layers and bulk. The latter dielectric function, f k is the oscillator strength of the kth 2 appears to be further supported by earlier experimental resonance (in units of eV ), E0,k is the central of the 59,60 Γ reports. Finally, the measurements of total line widths oscillator and k the nonradiative broadening, approximately 31 30 reported by Arora et al. on WSe2 and MoSe2 suggest combining both homogeneous (excluding the radiative qualitatively similar behavior in those systems as well. In coupling) and inhomogeneous components. particular, considering the similarities of the exciton band ff f structure of WS2 and WSe2 and the e ects associated with the ϵ=ϵ+()E ∑ k Λ and Λ′ valleys, one could expect the changes in the exciton− b 22 k EEiE0,kk−−Γ (1) phonon scattering from mono- to bilayer to be largely comparable in these materials. The radiative dephasing rate is proportional to the oscillator 66 In summary, our joint experimental and theoretical work strength f k and following Glazov et al. and comparing the offers a detailed picture of exciton−phonon scattering in 2D reflection coefficient with bulk-like susceptibility,67 a radiative semiconductors in the presence of a second layer. Specifically, broadening of around 5.07 meV is obtained for the studied fi we have identi ed individual scattering channels and separate WS2 samples on fused silica substrates. The relationship Γ exciton relaxation pathways through phonon emission and between f k and fwhm radiative broadening, 0, is given by faϵ+ϵ absorption both experimentally and theoretically. We further ks b ϵ Γ=0 , where a is the thickness of the 2D layer and s is emphasize the dominant role of the renormalized conduction 4ℏc ϵϵsb band valleys for the exciton−phonon scattering physics in the average dielectric constant of the surroundings. The sum of natural bilayers in contrast to the valence band scattering the nonradiative and radiative components yields the total line toward the Γ point. The theoretical approach is found to be width. quantitatively consistent with the experimental observations. In our simulations, the main resonances included are the A, These findings should be important for understanding exciton B, and C excitons, along with a small contribution from the relaxation and lifetime dominated by phonon-interaction in a charged exciton. In the case of PL, which is essentially a variety of natural and artificial multilayer systems, especially background-free measurement, the line widths are directly when strong electronic hybridization results in significant extracted from the emission spectra. We also note, that at changes of the band structure. The overall success of the room-temperature, the A-exciton peaks in reflectance contrast theoretical approach should further serve as a basis to address and PL are essentially equivalent. At low temperatures, both interlayer excitons and higher-lying transitions in however, additional emission features appear, so that we use

6140 DOI: 10.1021/acs.nanolett.8b01793 Nano Lett. 2018, 18, 6135−6143 Nano Letters Letter only reflectance contrast data for the temperature-dependent Depending on the transferred momentum K, the exciton− analysis. phonon scattering can be seen as purely elastic (K =0)or Obtaining Limits for Homogeneous Line Widths from inelastic. In particular, quasi-elastic scattering due to scattering Reflectance Contrast. To analyze the temperature-dependent with low-energy acoustic phonons provides a momentum data in our study, we first extract the total broadening transfer on the order of some tens of μm−1 being comparable (excluding radiative dephasing) from the resonances in parts a to the radius of the radiative cone in momentum space. It and b of Figure 2 via the dielectric function approach, as constitutes the dominant contribution to the intravalley described above (an alternative method would be to directly scattering even at room-temperature in 1L and 2L WS2 with read-out the fwhm from absorption spectra). Then we the temperature dependent line width broadening coefficient determine the homogeneous limits in the low-temperature of 15 μeV/K. This coefficient is larger in WSe and MoSe − 2 2 1L data by considering the maximum value to be the smallest systems27 29 due to more efficient carrier phonon coupling total line width observed across all our samples (11 meV) and and smaller velocities of .22 However, the dominant the minimum value naturally being zero. Deconvoluting these contribution to the homogeneous width in 1L and 2L WS2 widths from the total line widths, we obtain for the limiting stems from scattering with zone edge phonons, with a inhomogeneous broadening in the studied monolayer a value transferred exciton momenta exceeding the radiative cone by between 10 and 14 meV. We assume that the inhomogeneous orders of magnitude. broadening is largely temperature independent and is roughly the same for the adjoining bilayer of the same flake, as AUTHOR INFORMATION discussed in the main text. By deconvoluting these ■ inhomogeneous line widths from the temperature-dependent Corresponding Authors total values and adding the radiative broadening from the *(A.R.) E-mail: [email protected]. measured oscillator strengths, the experimentally determined *(A.C.) E-mail: [email protected]. minimum and maximum limits for the homogeneous broad- ORCID ening of the monolayer and bilayer are obtained for all studied temperatures. Here, we follow the procedure for the Archana Raja: 0000-0001-8906-549X deconvolution of the Lorentzian line shape broadened by a Malte Selig: 0000-0003-0022-412X Gaussian.68 We also note, that the deconvolution procedure is Albert F. Rigosi: 0000-0002-8189-3829 essentially equivalent to fitting the line shape of the imaginary Louis E. Brus: 0000-0002-5337-5776 part of the dielectric function or of the peak in the emission Tony F. Heinz: 0000-0003-1365-9464 fi spectrum using a Voigt pro le. Notes Theoretical Section. The Wannier equation is evaluated for The authors declare no competing financial interest. all relevant exciton states to obtain exciton wave functions and binding energies.25,69 With the calculated exciton binding energies, the excitonic dispersion is then computed for the ■ ACKNOWLEDGMENTS direct and indirect exciton states from the electronic The authors would like to thank Andor Kormanyos,́ Vasili 25,42 γ dispersion. The excitonic line width Q is obtained by Perebeinos, and Mikhail Glazov for instructive discussions. 70 evaluating the excitonic Bloch equation under the influence A.R. gratefully acknowledges funding through the Heising- − of exciton phonon coupling, where the latter is treated in a Simons Junior Fellowship within the Kavli Energy Nano- 25,71 second-order Born Markov approximation as described Science Institute at the University of California, Berkeley. A.C. below in eq 2. gratefully acknowledges funding by the Deutsche Forschungs- gemeinschaft (DFG) via Emmy Noether Grant CH 1672/1-1 1 1 and Collaborative Research Center SFB 1277 (B05). H.M.H. γ =|±+−∓ℏ∑ gnEEαα2 δω() α Q K 2 2 KQ QKK+ and A.F.R. acknowledge funding from the National Science α,K Foundation through the Integrated Graduate Education and (2) Research Training Fellowship (DGE-1069240) and the ji zy − j z α Graduate Research Fellowship Program (DGE-1144155), The exciton phononk coupling{ element given by gK depends on electronic coupling elements22 and excitonic wave respectively. Spectroscopic measurements at Columbia Uni- functions, with α being the phonon mode index and K being versity were supported by the NSF MRSEC program through the transferred momentum. The line width further depends on the Center for Precision Assembly of Superstratic and α Superatomic (DMR-1420634). This work was sup- a factor related to the phonon occupation nK, where the + term accounts for phonon emission and the − term accounts for ported through the AMOS program at SLAC National phonon absorption processes. The expression for line width Accelerator Laboratory within the Chemical Sciences, Geo- contains a Dirac distribution ensuring energy conservation sciences, and Biosciences Division and through the Gordon during a phonon scattering event. Here, EQ and EQ+K denote and Betty Moore Foundations EPiQS Initiative through Grant the excitonic energies of initial and final states, respectively; No. GBMF4545 (T.F.H.) for data analysis. M.S. and G.B. ℏωα α fi and K denotes the phonon energies of the mode as a acknowledge nancial support from the Deutsche Forschungs- function of the momentum. Note that Q ≈ 0 holds for gemeinschaft (DFG) through SFB 787, and A.K. acknowledges optically accessible exciton states. The mode index contains financial support through SFB 951. Furthermore, G.B. 22 both acoustic (LA, TA) and optical modes (LO, TO, A1). acknowledges support from the Swedish Research Council. The momentum sum includes small momenta leading to E.M. acknowledges support from the European Union’s intravalley scattering as well as larger momenta to take into Horizon 2020 research and innovation programme under account intervalley scattering. grant agreement No 696656.

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6143 DOI: 10.1021/acs.nanolett.8b01793 Nano Lett. 2018, 18, 6135−6143