Enhancement of Exciton−Phonon Scattering from Monolayer toBilayer WS2Archana 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 Energy NanoScience Institute, Berkeley, California 94720, United States‡Department of Applied Physics, 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 Chemistry, 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 limitalong with pronounced excitonic effects. In these materials,interaction with phonons is the dominant mechanism thatlimits the exciton coherence lifetime. Exciton-phononinteraction also facilitates energy and momentum relaxation,and influences exciton diffusion under most experimentalconditions. However, the fundamental changes in theexciton−phonon interaction are not well understood as thematerial undergoes the transition from a direct to an indirectbandgap semiconductor. Here, we address this questionthrough optical spectroscopy and microscopic theory. In theexperiment, we study room-temperature statistics of the exciton line width for a large number of mono- and bilayer WS2samples. We observe a systematic increase in the room-temperature line width of the bilayer compared to the monolayer of 50meV, corresponding to an additional scattering rate of ∼0.1 fs−1. We further address both phonon emission and absorptionprocesses by examining the temperature dependence of the width of the exciton resonances. Using a theoretical approach basedon many-body formalism, we are able to explain the experimental results and establish a microscopic framework for exciton−phonon interactions that can be applied to naturally occurring and artificially prepared multilayer structures.

KEYWORDS: 2D materials, excitons, exciton−phonon interaction, scattering lifetime

Layered transition metal dichalcogenides (TMDCs) in theMX2 family (M = Mo, W and X = S, Se, Te) have been

the subject of intense investigations over the past decade dueto their intriguing optical and electronic properties in theultrathin, quasi two-dimensional (2D) limit. The semi-conducting TMDCs are characterized by the emergence of adirect bandgap at monolayer (1L) thickness,1,2 strongabsorption and emission of light with dominant excitoniceffects,3−6 and the ability to access and control the spin-valleydegree of freedom.7 Recently, it has become possible to createatomically thin heterostructures by stacking one layer uponanother,8 with further broad technological appeal resultingfrom the materials’ mechanical robustness and chemicalflexibility. This situation has led to the rapid development ofa variety of nanoscale structures to explore both fundamentalscientific questions and device applications.9−11

The impact of additional layers on many-particle inter-actions remains a topic of considerable importance for 2Dmaterials and their heterostructures. It is related to bothelectronic properties, such as charge transfer,12−14 bandgaprenormalization,15,16 and hybridization of electronic statesbetween layers,17 as well as to the phonons18,19 that governcarrier relaxation and heat transport. In particular, theinteraction between electronic and vibrational excitations inTMDC layers was found to affect a number of fundamentalproperties, including the temperature-dependent bandgaprenormalization,20 carrier transport,21−23 optical heating ofthe lattice,24 and electron and exciton coherence and

Received: May 3, 2018Revised: July 8, 2018Published: August 10, 2018

Letter

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

© 2018 American Chemical Society 6135 DOI: 10.1021/acs.nanolett.8b01793Nano Lett. 2018, 18, 6135−6143

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intervalley scattering.25,26 In addition, the strength of theCoulomb interaction in the 2D limit, combined with efficientphonon scattering, makes exciton−phonon interactions andtransport particularly interesting in these systems.To advance fundamental understanding of exciton−phonon

coupling in artificial multilayer materials, it is key to addressthe physics of natural bilayers (2L) through systematicexperiments and to develop quantitative theoretical modelswith high predictive power. A number of recent reports haveindeed focused on related processes in mono- and few-layerTMDCs by measuring the coherence lifetime27−29 andtemperature dependent line widths.30−32 However, there islimited understanding of the microscopic mechanisms govern-ing exciton−phonon scattering rates during the transition fromthe direct bandgap monolayer to the indirect bandgap bilayersemiconductor. Both the impact of interlayer coupling andhybridization, and that of exciton−phonon scattering throughintra- and intervalley channels are of particular interest in thiscontext.Here we address these topics through systematic optical

measurements of the temperature-dependent exciton peakenergies and line widths in 1L and 2L WS2 samples. We furthertake advantage of the stability of the excitons in both systemswith binding energies on the order of hundreds ofmillielectronvolts,35−37 permitting the observation of thesestates up to room-temperature. At cryogenic temperatureswe observe an increase in the exciton line width from 1L to 2Lof 25−30 meV, corresponding to extra scattering on the 20 fstime scale from phonon emission processes. Temperature

activated scattering with contributions from phonon absorp-tion adds another 20−25 meV at room-temperature. A detailedcomparison of the temperature-dependent line widths andenergies with predictions of many-body theory allows us tointerpret the experimental observations in terms of micro-scopic processes and to identify individual scattering channels.The results provide a fundamental picture of exciton−phononinteractions in multilayer TMDCs and should establish amicroscopic basis for understanding these processes in bothnatural and artificial structures.

Excitons in Monolayer and Bilayer WS2. As previouslydiscussed, most TMDC crystals undergo a transition fromindirect to direct gap semiconductors at the monolayer limit,which is also the case for WS2. An optical micrograph of a WS2sample with regions of mono- and bilayer thickness is shown inFigure 1a. The room-temperature reflectance contrast from thetwo areas is plotted in Figure 1b in the spectral rangecorresponding to the lowest energy direct transition at 2 eV.38

The observed resonance arises from excitons formed at the Kand, equivalently, K′ points, commonly labeled as “A” states.According to the single-particle picture, they can be denoted asK−K (or K′−K′) by the corresponding electron transitionsfrom the valence to conduction band. This notation is usedthroughout the manuscript to indicate the origin of theexcitons. A schematic of the underlying electron states acrossthe hexagonal Brillouin zone is presented in Figure 1c.The electronic states at the K and K′ points are primarily

composed of the transition metal d orbitals. As a consequence,the energy of the corresponding conduction and valence bands

Figure 1. Exciton transitions in monolayer and bilayer WS2. (a) Optical micrograph of a WS2 sample consisting of both monolayer (1L) and bilayer(2L) regions. (b) Room-temperature reflectance contrast of 1L and 2L WS2 around the fundamental optical gap. The peaks correspond to the Aexciton resonance. Inset: room-temperature photoluminescence spectra. An additional lower energy peak for the 2L corresponds to indirect gapemission. (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 Γand Λ points, respectively. (d) Histogram of room-temperature width (full-width at half-maximum - fwhm) of the A exciton for 1L and 2L samplesobtained from PL and reflectance contrast measurements following the procedure described in the text. The difference in the mean fwhms of themonolayer and bilayer samples is denoted by Δfwhm = 51 ± 15 meV. The intrinsic limit of purely homogeneous broadening for the WS2 1L isindicated by the gray area.25,33 (e) Schematic overview of the two-dimensional Brillouin zone with contributing intervalley electronic scatteringprocesses represented by dashed arrows and their associated phonon mode, as identified by the experiment-theory comparison, discussed in themain text. Blue and red arrows indicate possible inelastic intervalley scattering processes in 1L and 2L WS2, respectively; purple arrows apply forboth materials and include intravalley scattering at the K point mediated via Γ phonons.

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exhibits little sensitivity to interlayer interactions due to thelimited overlap of the corresponding wave functions. Inaddition, for the direct gap monolayer, these excitons emit atapproximately the same energy at which they are created bylight absorption, as shown in the photoluminescence (PL)spectra (Figure 1b, inset). For samples thicker than themonolayer, such as the studied bilayers, the lowest energytransition is indirect in momentum-space. Excitons created atthe K/K′ points scatter toward lower-lying states associatedwith the indirect gap. The energy relaxation is usuallyfacilitated by the interaction with phonons. This leads to theobservation of PL at about 1.75 eV in the bilayer, roughly 0.2eV below the direct transition at the K and K′ points.38−40In the following, we examine the effect of the change from

direct to indirect bandgap on exciton−phonon scatteringprocesses. Experimentally, we measure both the room-temperature and temperature-dependent line widths ofmono- and bilayer WS2. Generally, the line width of an opticaltransition is determined by homogeneous contributions fromvarious scattering processes and radiative recombination,convoluted by inhomogeneous broadening from disorder andensemble 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 thecorresponding transition in the monolayer.By measuring the total room-temperature line widths of the

exciton peaks of 62 monolayer and 21 bilayer samples, weshow that despite the presence of inhomogeneous broadening,the greater line width of the exciton peak in bilayer WS2compared to the monolayer is a statistically robust effect. Theresulting histogram of A exciton line widths obtained from acombination of PL and reflectance contrast measurements, isshown in Figure 1d. The exciton widths are deduced directlyfrom the experimental PL spectra. In the case of the reflectancecontrast spectra, we deduce the line width of the excitonfeatures from a fit based on parametrization of the dielectricfunction, as described in the Methods section. The intrinsiclimit of purely homogeneous broadening for the WS2 1L isroughly indicated in Figure 1d for reference.25 This approachallows us to properly address the statistics and establish adifference of 51 ± 15 meV between 1L and 2L WS2 excitonbroadening at room-temperature in the studied samples. Itcorresponds to additional ultrafast scattering processes forexcitons in the bilayer that occur on the 10 fs time scale.A schematic overview of the main scattering pathways for

electrons in the single particle picture in both conduction andvalence bands is presented in Figure 1e, illustrating the mainphonon-assisted processes. It is based on the band structure inFigure 1c and previous work25 on exciton−phonon scatteringin monolayer TMDCs. The hexagonal two-dimensionalBrillouin zone is represented in black with the gray circlesindicating high symmetry points. The dashed arrows illustratepossible electron scattering channels and are labeled by theassociated phonon modes. Blue and red arrows further indicatescattering channels present only for 1L and 2L, respectively;the purple arrow represents processes that are present for both,including K to K′ intervalley scattering and intravalleyscattering processes at K. The latter is indicated by a purplecircular arrow and includes both inelastic absorption of opticalphonons and quasi-elastic scattering with the low-energyacoustic phonons. By symmetry, corresponding processesoccur for electrons at the K′ point as well.

An easily identifiable additional scattering channel in the 2Lwithin the single-particle picture involves transition of carriersbetween K and Γ in the valence band. However, as we discussbelow in greater detail, the efficiency of all scattering processesis strongly affected by considering excitons instead of freecarriers. In particular, the combination of Coulomb inter-actions and interlayer coupling leads to the emergence of lowlying dark excitons formed from an empty electron state at theK or K′ point in the valence band and an electron at Λ or Λ′ inthe conduction band. As a result, the nature of the scatteringprocesses related to the conduction band structure are shownto change in the bilayer, with significant ramifications for theoverall scattering efficiency. In addition, at room-temperaturewe find that processes associated with phonon emission andabsorption contribute roughly in equal measure in the bilayersystem.

Temperature Dependence of the Exciton Resonancein Monolayer and Bilayer WS2. To distinguish thecontributions of temperature-independent and thermallyactivated scattering channels, we performed temperaturedependent reflectance contrast measurements from 4.5 K toroom-temperature on monolayer and bilayer WS2 samples, asshown in Figures 2a and 2b, respectively. As previously shown,the relatively broad distribution of line widths across exfoliatedmonolayers and bilayers (Figure 1d) is attributed to varyingdegrees of inhomogeneous broadening across differentsamples. Therefore, for the temperature dependent study wespecifically chose a monolayer and a bilayer that are part of thesame flake (see highlighted regions in Figure 1a).The two should thus experience a comparable degree ofinhomogeneous broadening. This assumption is further

Figure 2. Temperature dependence of the A-exciton resonance inmonolayer and bilayer WS2. Reflectance contrast spectra from 4.5 to290 K of the (a) monolayer and (b) bilayer shown in Figure 1a. (c)Exciton peak energies relative to the 4.5 K data. (d) Difference in peakline widths between 2L and 1L. The average Δfwhm from thehistogram in Figure 1b is shown by the dashed line.

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supported by the difference in the room-temperature linewidths of these particular 1L and 2L being very close to theaverage obtained from the statistical distributions in Figure 1d,as indicated by the dashed line in Figure 2d. At the same time,the absolute line widths are on the lower end of the ensembledata, indicating a relatively low overall contribution frominhomogeneous broadening. From the reflectance contrastspectra presented in parts a and b of Figure 2, we observe forboth 1L and 2L a red shift in the exciton peak energies and abroadening of the line width with increasing temperatureunder conservation of the total oscillator strength which isproportional to the peak area. The charged exciton feature,observed at low temperatures in the 1L as a weak shoulder 30meV below the exciton peak, indicates relatively low levels ofunintentional doping.Analyzing the data in parts a and b of Figure 2 quantitatively,

we find that the 1L and 2L exciton peak energies follow thesame relative trend with temperature as illustrated in Figure 2c.In general, the shift in the exciton energy with increasingtemperature is a measure of changes in the electronic bandgapdue to lattice expansion and renormalization of the transitionenergy due to interaction with phonons, i.e., the polaronshift.41 In addition to peak energies, the difference in linewidths, Δfwhm, is plotted as a function of temperature in Figure2d. We find that the exciton peak in the 2L is alreadyconsiderably broader than that of the 1L at cryogenictemperatures by about 20 meV from temperature-independentprocesses. The difference in the line widths further increases athigher temperatures, indicating additional thermally activatedscattering.Theoretical Study of Exciton−Phonon Scattering and

Comparison to Experiment. To understand and interpretthe experimental findings, we apply a previously developedmicroscopic theory to quantitatively address individualexciton−phonon scattering channels in mono- and bilayerWS2. We evaluate both the changes in the excitonic bandstructure and exciton−phonon coupling, as discussed in arecent work25 for monolayer TMDCs.As a brief summary of the theoretical approach, we start with

the ab initio quasiparticle band structures established in theliterature;42,43 we then compute the energies of the excitonicstates by numerically solving the Wannier equation.25,41,44 Thespecific nature of the electric field screening between chargecarriers in a 2D sheet is explicitly taken into account using anapproximate thin-film Coulomb potential.45,46 The resultingenergy levels of the spin-allowed exciton transitions relative tothe bright K−K state are summarized in Table 1.A schematic overview of the exciton band structure is

presented in Figure 3 with the radiative, intravalley, andintervalley scattering channels indicated by arrows. Here, onlythe exciton states with the same spin configuration are takeninto account, since phonon-assisted processes also require spinconservation, i.e., Δms = 0 in the first approximation. We alsonote that while there are more complex calculations of theexciton ground states, the main conclusions related to higherbinding energies of the states with higher electron masses(such as at Λ) should be largely independent of thespecific theoretical approach. One of the key results in thecalculated exciton band structure of the W-based materials isthe lower lying K-Λ dark exciton state, primarily due to theincreased effective mass as compared to the direct K−Kexciton. This dark state appears to be necessary to explain thetemperature dependent exciton broadening in W-based

monolayers in contrast to Mo-based ones25 and is proposedas an interpretation of the exciton dynamics in intrabandspectroscopy experiments in monolayer WSe2.

47,48

A closely related aspect of the 1L and 2L band structures arethe specific spin configurations of the conduction band states.In both systems, the upper conduction and valence bands withthe same electron spin at the K and K′ points correspond tothe bright A-exciton transition.49 The energy difference in thebands with opposite spin is essentially determined by thespin−orbit coupling. In monolayers, the splitting at the Λpoint is also given by the spin−orbit interaction. In contrast tothat, the interlayer coupling (or hybridization) from the overlapof the electronic wave functions is the dominant effectdetermining the splitting at the Λ and Λ’ points in bilayerTMDCs.43 As a consequence for bilayer WS2, the Λ′ valleywith the same electron spin as K becomes the lowestconduction band state and the Λ valley with the same electronspin shifts to much higher energies compared to themonolayer. Hence, in addition to the more obvious Γ−Kscattering channel in the valence band of the bilayer, there is asubtle but important change in the conduction band relaxationpathways, where the K−Λ scattering becomes energeticallyunfavorable while the K−Λ′ channel opens up.After the energies of the exciton states are obtained, the

microscopic polarization is derived using the semiconductorBloch equations approach. For the bilayer, the oscillatorstrength of the optical matrix element is adjusted to match thepeak area in the experimentally determined A excitonabsorption. As a basis for the calculation of the exciton−phonon interaction, the underlying electron-phonon matrixelements for optical and acoustic phonon are sourced from abinitio calculations by Jin et al.22 for the monolayer case. For the

Table 1. Theoretical Energy Separation of Momentum−Dark Exciton States Relative to the K−K Bright Excitona

aEnergy of intervalley excitons in 1L and 2L WS2 with respect to thebright K−K exciton. The negative/positive sign means that thecorresponding state is lower/higher in energy than the K−Ktransition.

Figure 3. Exciton scattering channels. Schematic illustration (not toscale) of the parabolic minima of the exciton dispersion as a functionof the center-of-mass momentum (Q) based on Table 1. The blueand red valleys represent 1L and 2L WS2, respectively. The brightexciton at K−K can recombine radiatively as shown by the solidpurple line (γrad.) or scatter nonradiatively via phonon emission (orabsorption) toward dark exciton states (Q ≠ 0) as indicated by thedashed lines.

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bilayer, the matrix elements are assumed to be the same in thefirst approximation. The latter is supported by the similartemperature dependent shift of the 1L and 2L transitionenergies observed in experiment (see Figure 2c). It stronglyimplies that the phonon-renormalization of the transitionenergies and therefore the exciton−phonon interaction isessentially of the same strength over the studied temperaturerange.50,51 The individual rates for exciton scattering pathwaysacross the different states illustrated in Figure 3 are thencomputed under energy and momentum conservation require-ments from either absorption or emission of single phonons.52

We note that we include both inelastic scattering of excitonswith optical and zone-edge acoustic phonons as well as quasi-elastic scattering with low-energy acoustic phonons from thelinear dispersion branches. The spontaneous recombination ofelectrons and holes at the bright K−K exciton is alsoconsidered, leading to radiative dephasing. The sum of theseinteractions yields a total scattering rate (γtotal) that isproportional to the homogeneous line width Γh of the excitonresonance, i.e., γtotal=Γh/ℏ. The inverse of the rate isproportional to the commonly defined characteristic coher-ence time T2 = 2/γtotal in the literature.53

In Figure 4a, the individual contributions from eachscattering channel in the monolayer are presented as a

function of temperature by cumulatively adding them on topof each other. The major contributions are the intravalleyscattering (labeled as “+K−K”) with optical and low-energyacoustic phonons at Γ, relaxation of the excitons to the K−Λvalley (“+K−Λ”), mediated by Λ phonons, and scattering tothe K′ valley through K phonons (“+K−K′”). Figure 4bpresents the corresponding results for the bilayer case. Here,the intravalley +K−K and the intervalley +K−K′ scatteringremain essentially the same as for the monolayer. Thescattering toward the K−Λ′ state, however, mediated by Mphonons, becomes the dominant channel. The additionalrelaxation to the Γ−K exciton via emission of K phonons

provides a much weaker contribution. We also note that thethreefold degeneracy of the Λ and Λ′ states provides a largedensity of available final states, thus strongly contributing tothe overall scattering efficiency towards these states. The resultsof the microscopic theory are largely consistent with the formof the commonly used phenomenological model of temper-ature-induced broadening simplified as a sum of a linear andbosonic terms, as discussed in more detail by Selig et al.52 forthe monolayer case.Interestingly, the carrier-phonon coupling constant for

processes mediated by M phonons is an order of magnitudelarger in comparison to the Λ phonons, as calculated by Jin etal.22 This leads to M-phonon mediated scattering to the K−Λ′valley being significantly stronger in bilayer WS2 than the Λphonon mediated scattering to K−Λ in the monolayer.Phonon-assisted relaxation toward Λ and Λ′ valleys has alsobeen highlighted in studies of double resonance Ramanspectroscopy in MoS2

26 and WS2,54 photoluminescence

excitation measurements in MoSe255,56 and angle-resolved

photoemission pump−probe spectroscopy in bulk WSe2.57

To compare the theoretical findings to experimentalobservations, it is necessary to isolate purely homogeneouscontributions to broadening, i.e., the sum of the radiative andscattering contributions, across a wide temperature range.Recently, techniques based on four-wave mixing have beensuccessfully applied to determine directly the coherencelifetime and thus the intrinsic line width.27−29 However, theresults were typically reported for temperatures below 100 K.On the other hand, reflectance contrast and PL spectroscopyhave been used to extract total line widths also at highertemperatures for bulk TMDCs,58−60 and more recently for theultrathin layers.30−32 For a quantitative analysis, however,potential inhomogeneous contributions to the measuredbroadening need to be considered to obtain the limits forthe homogeneous broadening and place it in the context ofstatistics on a large number of samples, as discussed inconnection to Figure 1d. In the present case we thus estimatethe inhomogeneous contributions to the total line width andinfer corresponding limits for the homogeneous broadening.The procedure is outlined in the Methods section. We furthernote that the relative contribution from inhomogeneousbroadening is more pronounced at low temperatures. Ourexperimental procedure is thus less sensitive to smaller changesin homogeneous line widths at cryogenic temperatures incontrast to direct measurements of the dephasing27−29 andleads to larger variations when estimating the intrinsic linewidths in this regime.The extracted minimum and maximum values of the

temperature dependent homogeneous line widths are plottedin Figure 4c as filled and open circles, respectively. As furthernoted above, the room-temperature line widths of the twosamples fall on the lower side of the measured distributions,suggesting relatively low inhomogeneous broadening of thestudied 1L and 2L flakes exfoliated on SiO2 substrates. Thetotal temperature dependent scattering rates from theory areoverlaid as solid lines. The agreement between theexperimental and theoretical temperature dependence is veryreasonable. We also note that the line width of the bilayer isless sensitive to inhomogeneous broadening because of muchlarger homogeneous contributions, as also predicted by thetheory. The difference in the line widths of mono- and bilayerWS2, presented in Figure 2d is equally captured by the theoryon the same footing, within experimental limits. This further

Figure 4. Temperature dependence of homogeneous line widths.Theoretical fwhm of individual exciton−phonon scattering channelscumulatively added on top of each other for (a) 1L and (b) 2L WS2.(c) Experimentally determined upper and lower bounds for thehomogeneous line widths of 1L and 2L WS2 are marked by open andfilled circles, respectively. The total theoretically computed fwhms arepresented by lines on top of the experimental data. The distributionsof room-temperature line widths for 1L and 2L from Figure 1d arecondensed into box-plots on the right for comparison.

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supports the overall validity of the theoretical approach andhighlights the necessity to consider the dominant K−Λ′scattering channel in the bilayer. This channel is largelyresponsible for the total broadening on the order of 30−40meV at cryogenic temperatures and the additional increase ofthe line width towards 80 meV at room-temperature in 2L.Finally, we note that by deconvoluting the theoreticallyexpected room-temperature homogeneous line widths fromthe sample statistics presented in Figure 1d, we obtaincomparable averages of inhomogeneous line widths for the 1Land 2L of 24 ± 11 meV and 37 ± 14 meV, respectively, in thestudied ensemble.In conclusion, we have systematically studied exciton−

phonon scattering in monolayer and bilayer WS2 throughoptical spectroscopy by aggregating statistics across a largenumber of samples and performing temperature-dependentexperiments on a representative flake containing bothmonolayer and bilayer regions. Using microscopic many-body theory we were able to identify individual exciton−phonon scattering channels in 1L and 2L, including theinterplay of the K−Λ and K−Λ′ transitions to quantitativelyaccount for the experimental results.While radiative recombination is dominant in 1L samples at

low temperatures, phonon-assisted relaxation on the 20−25 fstime scale from K−K toward K−Λ′ and Γ−K states leads toadditional broadening of 25−30 meV in the 2L. At room-temperature, the difference in the 1L and 2L line widthsincreases to about 50 meV due to temperature activatedphonon-absorption. The absolute width of the 2L exciton peakreaches 80 meV, corresponding to an under 10 fs totalscattering lifetime of the K−K exciton. We also note that dueto the theoretical considerations outlined above, the majorchanges in exciton−phonon scattering occur at the 1L to 2Ltransition and the exciton dephasing rate does not significantlychange from the bilayer to thicker layers and bulk. The latterappears to be further supported by earlier experimentalreports.59,60 Finally, the measurements of total line widthsreported by Arora et al. on WSe2

31 and MoSe230 suggest

qualitatively similar behavior in those systems as well. Inparticular, considering the similarities of the exciton bandstructure of WS2 and WSe2 and the effects associated with theΛ and Λ′ valleys, one could expect the changes in the exciton−phonon scattering from mono- to bilayer to be largelycomparable in these materials.In summary, our joint experimental and theoretical work

offers a detailed picture of exciton−phonon scattering in 2Dsemiconductors in the presence of a second layer. Specifically,we have identified individual scattering channels and separateexciton relaxation pathways through phonon emission andabsorption both experimentally and theoretically. We furtheremphasize the dominant role of the renormalized conductionband valleys for the exciton−phonon scattering physics innatural bilayers in contrast to the valence band scatteringtoward the Γ point. The theoretical approach is found to bequantitatively consistent with the experimental observations.These findings should be important for understanding excitonrelaxation and lifetime dominated by phonon-interaction in avariety of natural and artificial multilayer systems, especiallywhen strong electronic hybridization results in significantchanges of the band structure. The overall success of thetheoretical approach should further serve as a basis to addressboth interlayer excitons and higher-lying transitions in

semiconducting 2D materials and their heterostructures infuture research.

Methods and Data Analysis. Experimental Section.Monolayer and bilayer WS2 flakes were prepared on SiO2substrates by micromechanical exfoliation of bulk crystals (2DSemiconductors Inc.). For room-temperature statistics, datafrom 97 individual samples were obtained and analyzed. Fortemperature dependent measurements from 4 to 290 K, arepresentative sample containing mono- and bilayer regionswas mounted in a liquid helium cooled cryostat. Opticalreflectance spectroscopy was used to probe the exciton statesusing a tungsten−halogen white-light source for illumination ofthe samples. Photoluminescence (PL) measurements wereperformed using the 514 nm line of an argon ion laser forexcitation, while keeping the incident power very low, around10 nW. In both cases, the light was focused down to a 1−2 μmspot on the sample using a 40× objective. The reflected lightand PL were spectrally resolved in a grating spectrometer andsubsequently detected using a Peltier-cooled EMCCD.

Analysis of Reflectance Contrast. The reflectance contrast,RC, corresponds to (Rsample − Rsubstrate)/Rsubstrate, whereR denotes the intensity of the reflected signal. For thin filmson transparent substrates such as fused silica, the reflectancecontrast is dominated by the imaginary part of the sample’sdielectric response and is largely proportional to theabsorption.61,62 The dielectric response of a material can beapproximated by a sum of Lorentzian oscillators on a constantbackground.53,63,64 Here, a physically meaningful number ofLorentzians was chosen to represent the main excitonicresonances in the material. By simulating the reflectancecontrast spectra in a standard transfer matrix formalism65 weobtained parameters related to the exciton transitions. eq 1describes the dielectric function ϵ in the thin film approach62 asa function of the photon energy E. ϵb is the backgrounddielectric function, f k is the oscillator strength of the kthresonance (in units of eV2), E0,k is the central frequency of theoscillator and Γk the nonradiative broadening, approximatelycombining both homogeneous (excluding the radiativecoupling) and inhomogeneous components.

Ef

E E i E( ) b

k

k

k k0,2 2∑ϵ = ϵ +

− − Γ (1)

The radiative dephasing rate is proportional to the oscillatorstrength f k and following Glazov et al.66 and comparing thereflection coefficient with bulk-like susceptibility,67 a radiativebroadening of around 5.07 meV is obtained for the studiedWS2 samples on fused silica substrates. The relationshipbetween f k and fwhm radiative broadening, Γ0, is given by

f a

c0 4k s b

s bΓ = ℏ

ϵ + ϵϵ ϵ

, where a is the thickness of the 2D layer and ϵs is

the average dielectric constant of the surroundings. The sum ofthe nonradiative and radiative components yields the total linewidth.In our simulations, the main resonances included are the A,

B, and C excitons, along with a small contribution from thecharged exciton. In the case of PL, which is essentially abackground-free measurement, the line widths are directlyextracted from the emission spectra. We also note, that atroom-temperature, the A-exciton peaks in reflectance contrastand PL are essentially equivalent. At low temperatures,however, additional emission features appear, so that we use

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only reflectance contrast data for the temperature-dependentanalysis.Obtaining Limits for Homogeneous Line Widths from

Reflectance Contrast. To analyze the temperature-dependentdata in our study, we first extract the total broadening(excluding radiative dephasing) from the resonances in parts aand b of Figure 2 via the dielectric function approach, asdescribed above (an alternative method would be to directlyread-out the fwhm from absorption spectra). Then wedetermine the homogeneous limits in the low-temperature1L data by considering the maximum value to be the smallesttotal line width observed across all our samples (11 meV) andthe minimum value naturally being zero. Deconvoluting thesewidths from the total line widths, we obtain for the limitinginhomogeneous broadening in the studied monolayer a valuebetween 10 and 14 meV. We assume that the inhomogeneousbroadening is largely temperature independent and is roughlythe same for the adjoining bilayer of the same flake, asdiscussed in the main text. By deconvoluting theseinhomogeneous line widths from the temperature-dependenttotal values and adding the radiative broadening from themeasured oscillator strengths, the experimentally determinedminimum and maximum limits for the homogeneous broad-ening of the monolayer and bilayer are obtained for all studiedtemperatures. Here, we follow the procedure for thedeconvolution of the Lorentzian line shape broadened by aGaussian.68 We also note, that the deconvolution procedure isessentially equivalent to fitting the line shape of the imaginarypart of the dielectric function or of the peak in the emissionspectrum using a Voigt profile.Theoretical Section. The Wannier equation is evaluated for

all relevant exciton states to obtain exciton wave functions andbinding energies.25,69 With the calculated exciton bindingenergies, the excitonic dispersion is then computed for thedirect and indirect exciton states from the electronicdispersion.25,42 The excitonic line width γQ is obtained byevaluating the excitonic Bloch equation70 under the influenceof exciton−phonon coupling, where the latter is treated in asecond-order Born Markov approximation25,71 as describedbelow in eq 2.

ikjjj

y{zzzg n E E

12

12

( )QK

K K Q Q K K,

2∑γ δ ω= | ± + − ∓ ℏα

α α α+

(2)

The exciton−phonon coupling element given by gKα depends

on electronic coupling elements22 and excitonic wavefunctions, with α being the phonon mode index and K beingthe transferred momentum. The line width further depends ona factor related to the phonon occupation nK

α , where the + termaccounts for phonon emission and the − term accounts forphonon absorption processes. The expression for line widthcontains a Dirac distribution ensuring energy conservationduring a phonon scattering event. Here, EQ and EQ+K denotethe excitonic energies of initial and final states, respectively;and ℏωK

α denotes the phonon energies of the mode α as afunction of the momentum. Note that Q ≈ 0 holds foroptically accessible exciton states. The mode index containsboth acoustic (LA, TA) and optical modes (LO, TO, A1).

22

The momentum sum includes small momenta leading tointravalley scattering as well as larger momenta to take intoaccount intervalley scattering.

Depending on the transferred momentum K, the exciton−phonon scattering can be seen as purely elastic (K = 0) orinelastic. In particular, quasi-elastic scattering due to scatteringwith low-energy acoustic phonons provides a momentumtransfer on the order of some tens of μm−1 being comparableto the radius of the radiative cone in momentum space. Itconstitutes the dominant contribution to the intravalleyscattering even at room-temperature in 1L and 2L WS2 withthe temperature dependent line width broadening coefficientof 15 μeV/K. This coefficient is larger in WSe2 and MoSe2systems27−29 due to more efficient carrier phonon couplingand smaller velocities of sound.22 However, the dominantcontribution to the homogeneous width in 1L and 2L WS2stems from scattering with zone edge phonons, with atransferred exciton momenta exceeding the radiative cone byorders of magnitude.

■ AUTHOR INFORMATION

Corresponding Authors*(A.R.) E-mail: archana.raja@berkeley.edu.*(A.C.) E-mail: alexey.chernikov@ur.de.

ORCIDArchana Raja: 0000-0001-8906-549XMalte Selig: 0000-0003-0022-412XAlbert F. Rigosi: 0000-0002-8189-3829Louis E. Brus: 0000-0002-5337-5776Tony F. Heinz: 0000-0003-1365-9464NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The authors would like to thank Andor Kormanyos, VasiliPerebeinos, and Mikhail Glazov for instructive discussions.A.R. gratefully acknowledges funding through the Heising-Simons Junior Fellowship within the Kavli Energy Nano-Science Institute at the University of California, Berkeley. A.C.gratefully acknowledges funding by the Deutsche Forschungs-gemeinschaft (DFG) via Emmy Noether Grant CH 1672/1-1and Collaborative Research Center SFB 1277 (B05). H.M.H.and A.F.R. acknowledge funding from the National ScienceFoundation through the Integrated Graduate Education andResearch Training Fellowship (DGE-1069240) and theGraduate Research Fellowship Program (DGE-1144155),respectively. Spectroscopic measurements at Columbia Uni-versity were supported by the NSF MRSEC program throughthe Center for Precision Assembly of Superstratic andSuperatomic Solids (DMR-1420634). This work was sup-ported through the AMOS program at SLAC NationalAccelerator Laboratory within the Chemical Sciences, Geo-sciences, and Biosciences Division and through the Gordonand Betty Moore Foundations EPiQS Initiative through GrantNo. GBMF4545 (T.F.H.) for data analysis. M.S. and G.B.acknowledge financial support from the Deutsche Forschungs-gemeinschaft (DFG) through SFB 787, and A.K. acknowledgesfinancial support through SFB 951. Furthermore, G.B.acknowledges support from the Swedish Research Council.E.M. acknowledges support from the European Union’sHorizon 2020 research and innovation programme undergrant agreement No 696656.

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