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Signatures of dark excitons in exciton–polariton optics of transition metal dichalcogenides

Signatures of dark excitons in exciton–polariton optics of transition metal dichalcogenides Anyfurtherdistribution Integrating2Dmaterialsintohigh-qualityopticalmicrocavitiesopensthedoortofascinating ofthisworkmust many-particlephenomenaincludingtheformationofexciton–polaritons.Thesearehybrid maintainattributionto theauthor(s)andthetitle quasi-particlesinheritingpropertiesofboththeconstituentphotonsandexcitons.Inthiswork,we ofthework,journal citationandDOI. investigatetheso-faroverlookedimpactofdarkexcitonsonthemomentum-resolvedabsorption spectraofhBN-encapsulatedWSe andMoSe monolayersinthestrong-couplingregime.In 2 2 particular,thankstotheefficientphonon-mediatedscatteringofpolaritonsintoenergeticallylower darkexcitonstates,theabsorptionofthelowerpolaritonbranchinWSe ismuchhigherthanin MoSe .Itshowsuniquestep-likeincreasesinthemomentum-resolvedprofileindicatingopening ofspecificscatteringchannels.Westudyhowdifferentexternallyaccessiblequantities,suchas temperatureormirrorreflectance,changetheopticalresponseofpolaritons.Ourstudycontributes toanimprovedmicroscopicunderstandingofexciton–polaritonsandtheirinteractionwith phonons,potentiallysuggestingexperimentsthatcoulddeterminetheenergyofdarkexcitonstates viamomentum-resolvedpolaritonabsorption. 1.Introduction Sofar,polariton–phononinteractionsinTMDshave not been well studied, leaving many open questions Monolayers of transition metal dichalcogenides on the impact of dark exciton states on polariton (TMDs) show a rich exciton landscape, including absorption. bright and dark exciton states [1, 2]. This class of Inourpreviouswork,westudiedtransportprop- atomically-thin materials exhibits a large oscillator erties of exciton–polaritons in MoSe monolayers strengthandexcitonbindingenergiesintherangeof [18], where dark excitons do not play an import- a few hundreds of meV, hence governing the opto- ant role as they are energetically higher than the 1s electronic properties even at room temperature [1, bright states [1, 13, 15]. Now, the focus lies on the 3–7].TMDshavealreadybeensuccessfullyintegrated optical response of polaritons in WSe monolayers, into optical cavities [8–10], where the coupling of specifically addressing the impact of dark exciton cavity photons with excitons gives rise to the form- states.Inthisregard,thepolaritonabsorptionisespe- ation of exciton–polaritons [11, 12]. It is only the cially informative as it unambiguously demonstrates bright exciton states that can couple to photons to strong coupling via the Rabi splitting [19], and its form these quasi-particles, while momentum-dark magnitudeisdeterminedbythebalancebetweenthe excitons[1,2,13]cannotbedirectlyaccessedbylight. polariton–phononandcavitydecayrates.Wemicro- Nonetheless,thesestatesareavailablescatteringpart- scopicallycalculatethepolaritonabsorptionbycom- nersforpolaritonsviatheinteractionwithphonons. bining the Heisenberg–Langevin equations [20] for The presenceofdark excitonsis expected tosignific- polaritonswiththeexcitondensitymatrixformalism antly change the polariton–phonon scattering rates, [21,22].Wecalculatethefullvalley-andmomentum- especially for tungsten-based TMDs, since here dark dependent polariton–phonon scattering rates that excitons are the energetically lowest states [14–17]. govern the optical response of TMD materials via ©2022TheAuthor(s). PublishedbyIOPPublishingLtd 2DMater.10(2023)015012 BFerreiraetal both spectral linewidths and magnitude. In particu- lar,weexplorethisinthecontextofthecriticalcoup- ling condition [23], where the total cavity decay rate coincideswiththepolariton–phononscatteringrate. We predict that the presence of dark excitons has a large impact on the polariton scattering rates, giv- ing rise to clear signatures in momentum-resolved absorption spectra that could be exploited to meas- uretheenergyofdarkexcitonstates.Furthermore,we predictandexplainasurprisingdifferenceinabsorp- tion intensity between the upper and lower polari- ton branch at zero momentum and zero detuning, despite equal photonic and excitonic contributions. We also study the influence of externally accessible quantities to tune the scattering rates (via temperat- ure) and cavity decay rates (via mirror reflectance). For the latter, we find that the cavity quality factor playsanimportantrolefortheabsorption,inpartic- ularforthelowerpolaritonbranchthathasasmaller photoniccomponent. Figure1.(a)SchematicillustrationofaTMDmonolayerin 2.Theory aFabry–Perotcavitywiththefundamentalcavitymode representedbytheredcurve.TMDexcitonsinteractwith photonsandphononsasindicatedbythecreation We start by describing the theoretical approach to ˆ ˆ (annihilation)operatorsforphotons(c (c))andphonons microscopicallycalculatetheabsorptionspectrumfor ˆ† ˆ (b (b)).Thecavitysysteminteractswiththeoutsideworld polaritons in TMD monolayers integrated into an ˆ ˆ viatheoperators (B (B)).(b)Exciton–polaritonband structure,wherepolaritonscanscatterintodarkexciton opticalcavity.Excitonenergiesandwavefunctionsin statesbyemittingphononsifmomentumandenergycan TMD monolayers are obtained by solving the Wan- beconserved. nier equation [15, 24, 25] including DFT input on single-particleenergies[26].TMDsarecharacterized by regular bright excitons that are directly access- ible in optical spectra, as well as dark exciton states and describes the free energy of excitons E , phon- vk b c thatareknowntobetheenergeticallyloweststatesin ons E as well as photons within (E ) and out- αq k tungsten-based TMDs [14–17, 26, 27]. In this work, side the cavity (ℏω). Here, v is the exciton index we focus on momentum-dark excitons consisting of (we consider only 1s states), α the phonon mode, Coulomb-boundelectronsandholesthatarelocated k and q are the in-plane momentum of excitons/- at different valleys within the Brillouin zone (K, K photons (center-of-mass momentum for excitons) or Λ).Thismeansthattherequiredlargemomentum and phonons, respectively. Furthermore, we have † † † † transfercannotbeprovidedbyphotons,makingthese ˆ ˆ ˆ ˆ ˆ ˆ introduced X (X ), b (b ), ˆc (ˆc ), B (B ) vk αq αq k jkω vk k jkω statesopticallydark[13,16,27–30]. as exciton, phonon, inner-cavity and outer-cavity In this work, we combine the density matrix photon creation (and annihilation) operators, formalismwiththeHopfieldapproach[11],tomodel respectively. the optical response of polaritons. We quantize sep- ˆ The second term in the Hamiltonian, H = X−c ( ) arately a single internal cavity mode of a Fabry– † † ˆ ˆ g ˆc X +ˆc X describes the exciton-light k vk k vk k vk Perot resonator and the external radiation fields, interaction mediated by the exciton-photon coup- which are split into two sets of continuum modes ling matrix element g [15, 31], where photons corresponding to the left and the right of the cav- need to have the same in-plane momentum k ity (figure 1). The internal and external modes are as excitons to fulfill the momentum conserva- weakly coupled via the end mirrors, where the in- tion (hence restricting the coupling only to the plane wavevector is conserved. The starting point is bright exciton states). In general, the out-of-plane the many-particle Hamiltonian in the excitonic pic- component k influences the cavity energy and ˆ ˆ ˆ ˆ ˆ ture H =H +H +H +H . The first term 0 X−c X−b B−c exciton–photon coupling. However, we assume readsinsecondquantization the existence of one resonant photon mode (i.e. X c ∑ ∑ ∑ E =E ). The third contribution in the Hamilto- † † KK,0 0 X c b † ˆ ˆ ˆ ˆ ˆ H = E X X + E ˆc ˆc + E b b ′ 0 vk k αq † † v,k vk k k αq αq vv ˆ ˆ ˆ ˆ ˆ nian H = D X X (b +b ) X−b ′ v k αq αq α,−q vv kαq vk+q vk k q ˆ describes the exciton–phonon interaction [15], ∑ ∑ ˆ ˆ + dω ℏω(k)B B (1) where the coupling strength is determined by the jkω jkω vv j=L,R exciton–phononmatrixelement D .Finally,thelast αq 2 2DMater.10(2023)015012 BFerreiraetal ∑ ∑ ´ dω term, H =iℏ a (ω)[B ˆc − last term in equation (2) describes the polariton– B−c j,k k j=L,R k 0 2π jωk † ˜ phonon interaction. Here, the matrix element D is B ˆc ], provides the interaction between the inner- jωk k ′ n n related to the exciton–phonon coupling via D = and outer-cavity photons [20, 32]. The free photons kαq ′ ′ n ∗ n n n interact with the cavity with a coupling parameter, h D h and depends on the excitonic Hop- αq X,k+q X,k a (ω). Assuming broadband end mirrors, it is field coefficients h [33], since phonons only couple j,k appropriate to take the first Markov approxima- totheexcitonicpartofpolaritons. tion and approximate this parameter as frequency Toobtainanexpressionforthepolaritonabsorp- independent[20].ThiscontributionintheHamilto- tion, we exploit the Heisenberg equations of motion nian leads to a consistent description of both the forthecoherentpopulationofpolaritonandexternal radiative decay rate within the cavity as well as the radiation field (cf the supplementary information). couplingofpolaritonstoinputandoutputfields. For this we make a correlation expansion includ- Now, we investigate the strong-coupling regime, ing the dynamics of the phonon-assisted polariza- wheretheexciton–photoncouplingstrengthg islar- tion.Weusetheinput–outputmethod[20]tocouple ger than (the difference of) cavity and non-radiative thedynamicsbetweenintra-andouter-cavityphoton excitondecayrates[12].Theneweigenmodes,known modesateachport.Wetreatthescatteringwithphon- as exciton–polaritons, can be obtained by applying ons within a Markov approximation and assuming a Hopfield transformation of the excitonic Hamilto- a thermalized reservoir of incoherent phonons [16]. niandiscussedabove,yielding[11,12] The absorption then follows from energy conserva- tion as the difference between incoming fields and n† n n 0 0 the total reflected and transmitted light. To simplify ˆ ˆ ˆ H = E Y Y +H +H k k k b B the resulting expression, we assume that the cavity k,n is symmetric and ignore interference effects between dω +iℏ a (ω) polaritons in different branches. The latter is a good jk 2π k,n,j approximation if the branches are widely spaced in ( ) energy compared to the polaritonic spectral width. n n† n n∗ ˆ ˆ ˆ ˆ × h B Y −h B Y jkω c,k jkω k c,k k We obtain an Elliot-like formula for the polariton ( ) ∑ ′ absorption[32], n † n n n ˆ ˆ ˜ ˆ ˆ + D b +b Y Y . (2) αq kαq α,−q k+q k n n kαqnn 4γ Γ n k k A (ℏω) = , (3) n 2 n n 2 (ℏω −E ) + (2γ + Γ ) k k k Here, the first term provides the free polaritonic n† ˆ ˆ for each polariton branch and momentum n,k. The Hamiltonian with Y (Y ) denoting the polariton k k obtained equation is similar to the expression found creation (annihilation) operator with the polari- in[32],however,thekeydifferenceliesinthemicro- ton mode n and momentum k. The energy of the scopic treatment of polariton–phonon interaction. corresponding polariton, E , includes in particu- Thismeansthatphononscanchangethemomentum lar lower and upper polariton branches (LP, UP) of the excitonic component of the polariton, lead- that are separated in k =0 by the Rabi splitting UP LP ing to a momentum dependent scattering rate. In ℏΩ =E −E . This is a consequence of the mix- 0 0 equation(3)weintroducedthedecayrates ing between excitons and photons (with the same center-of-mass and total momentum), as quantified n 2 2 γ = ℏc(1−|r | )|h | /(4L ), (4) m c,k cav bytheHopfieldcoefficients[12].Weincludealso,for notational convenience, polaritons stemming from ( ) momentum-dark excitons, although these show no 1 1 n n n 2 b Γ =2π |D | ± +n ′ ′ k α,k −k α,k −k exciton–photon mixing. Nevertheless, we will show 2 2 ′ ′ n αk below their crucial role for the polariton absorption ( ) n n b ×L E −E ±E , (5) viaadditionalphonon-inducedscatteringchannelsto ′ ′ γ˜ 0 k k α,k −k the optically active polaritons. Both polariton ener- n n n n gies E and Hopfield coefficients h and h are where γ is the effective cavity decay rate of one k X,k c,k k calculated analytically (with subscript X and c refer- port and Γ is the polariton–phonon scattering rate. ring to exciton and intra-cavity photon component, Here we are summing over all possible scattering respectively)[12]. channels from a polariton n,k to all possible receiv- ′ ′ The second and the third term in equation (2) ing polaritons n ,k via interaction with a phonon are the free phonon and free outer-cavity photon with mode α and momentum q, such that the over- contribution, respectively, which are not affected all momentum is conserved. The quality factor of c 2 by the Hopfield transformation. The fourth term the cavity reads Q =E L /[ℏc(1−|r | )|)], where f cav m describestheinteractionofpolaritonswiththeouter- r is the reflectivity of the cavity. In this work, cavity photons, mediated by the photonic Hopfield we use the default value of r =0.99 if not stated coefficients as only the photonic part of polaritons otherwise. Importantly, we explicitly consider inter- ′ ′ couples to the external radiation field. Finally, the valley scattering by including K and Λ phonons 3 2DMater.10(2023)015012 BFerreiraetal whichallowscatteringintopolaritonscoincidingwith KK and KΛ excitons, respectively. The polariton– phononratesarecalculatedwithintheMarkov–Born approximation [31,34] including effects beyond the completed-collision limit [35] by an energy conser- vationdescribedviaaLorentzianfunction L witha γ˜ broadening γ˜ =0.1meV[18]. Crucially, the polaritonic Elliot formula offers insightintohowunderlyingmicroscopicdecaychan- nelsmanifestintheabsorptionoflightbypolaritons, which would not be possible using the more com- monly used classical transfer-matrix method [19]. Evaluating equation (3) at resonance reveals that absorption is maximized when the two effective polariton decay rates are closest in value. It fol- Figure2.Polaritonabsorption.(a)Surfaceplotof lows that maximum absorption of 0.5 is possible at absorptioninanhBN-encapsulatedWSe monolayerasa functionofmomentumandenergyatatemperatureof the so-called critical coupling condition [23, 36] of 77K,assumingaRabisplittingof ℏΩ =50meVanda n n 2γ = Γ , i.e. when the leakage out of both ports k k cavityqualityfactorofQ =160.Thedashedwhitelines of the cavity is equal to the exciton dissipation rate correspondtothebareexcitonandcavitydispersion,while thesolidblacklinesdescribethepolaritondispersion. within the TMD layer in the cavity. The maximum (b)Absorptioncutsasafunctionofenergyforthree possible absorption of 50% is a well-known con- differentmomenta. straint for mirror-symmetric two-port systems that support a single resonance [37, 38]. We expect the presence of dark excitons to significantly increase polaritonshaveanequalphotonicandexcitoniccon- the polariton–phonon scattering rates in tungsten- tribution at k =0, hence also the cavity decay rate based TMDs (where they are the energetically lowest is the same for both polaritons. As a result, the states).Theopeningofintervalleyscatteringchannels phonon-induced decay rate of polaritons must be isexpectedtostronglyimpactthebalancebetweenthe responsible for the observed difference in the height effectiveradiativecouplingandscatteringloss,which of absorption peaks. Furthermore, we find that shouldtranslateintomeasurablesignaturesinpolari- the absorption is enhanced for increasing momenta LP tonabsorptionspectra. for the lower polariton (A ) up to approximately −1 k =1.6 µm ,whileitisreducedfortheupperpolari- UP 3.Results ton(A ),(cfalsotheabsorptioncutsinfigure2(b)). Moreover, we observe that not only the absorption LP 3.1.PolaritonabsorptionofWSe intensitybutalsothelinewidthofA becomeslarger Now, we evaluate equation (3), using numeric- forincreasingin-planemomentum,beforeitisagain −1 ally calculated polariton-phonon scattering rates, to reducedformomentahigherthank =1.6 µm .The studythepolaritonabsorptioninthestrong-coupling absorption intensity and the spectral linewidth of regime for an hBN-encapsulated WSe monolayer polaritonresonancescanbeascribedtotheinterplay integrated into a Fabry–Perot cavity with a quality ofthecavitydecayandnon-radiativedecayofpolari- factor of Q ≈160 and a Rabi splitting of ℏΩ = tonsviascatteringwithphononsasdiscussedindetail f R 50meV. Note that the choice of the substrate has below. some impact on the polariton-phonon scattering rates and polariton absorption. The main effect is 3.2.Criticalcoupling substrate-induced screening that changes the separ- To explain the different behavior in the absorption ationbetweenbrightanddarkexcitonsandcanopen spectra of the upper and lower polariton branch, orclosescatteringchannelswithphonons.Inthesup- we plot the maximal absorption A of the UP and plementary material, we show a direct comparison the LP branch at 77 K in figure 3(a). The absorp- between hBN-encapsulated and free-standing WSe tion intensity of the UP branch generally decreases monolayers. with the momentum, however, with one excep- −1 Figure 2(a) presents an energy- and in-plane tion at approximately k =1 µm , where we observe momentum-resolved surface plot of the polariton a small increase (blue line). In contrast for the absorption for hBN-encapsulated WSe . Interest- lowerpolaritonbranchwefindanenhancedabsorp- −1 ingly,wefindtheupperpolaritontobemuchhigher tion until approximately k =1.6 µm , where the −1 in intensity than the lower polariton at k =0 µm critical coupling condition with a maximum pos- (cfalsothebluelinesinfigure2(b)).Previousreports sible value of A =0.5, is reached (red line). The in GaAs have shown that in the case of zero detun- increase of the absorption includes several steep ing, the lower and upper polariton peaks intensit- step-like enhancements before the absorption starts −1 ies are similar [39, 40]. In the resonant case, the todecreaseforvalueslargerthank =1.6 µm . 4 2DMater.10(2023)015012 BFerreiraetal Infigure3(c),weplotthelowerpolaritondisper- sion in relation to the bright exciton energy together withthephonondispersionforLA,TAandTOmodes that are responsible for the scattering into the dark KΛ excitons. Whenever a phonon line crosses with the polariton energy, a scattering channel into dark excitonstatesopensup.Thisisclearlyvisibleasastep- likeincreaseinthepolariton–phononscatteringrates showninfigure3(b).Atmomentumk =0,theenergy LP E of the lower polariton is too small to allow scat- teringintotheKΛexcitonviaemissionofphononsas LP X E −E ≈11.2meV,whichisjustsmallerthanthe 0 Λ,0 energy of 11.4 meV of intervalley TA phonons [41]. −1 Whenkreachesthethresholdvalueofk≈0.3 µm , the scattering channel into KΛ states opens, result- ing in the abrupt increase of Γ , cf also figures 1(b) and3(c).Forintervalleyscattering,bothacousticand opticalphononmodeshavefiniteenergiesinthecor- respondingsymmetrypoints[41].However,acoustic phononshavemuchsmallerenergies(12–14meVfor acousticphononsvs27–32meVforopticalphonons Figure3.Criticalcoupling.(a)Maximalabsorptionatthe resonantenergyasafunctionofmomentumforthelower at the Λ point [41]). This results in a more efficient (red,LP)andupper(blue,UP)polaritonat77Kwith scattering with acoustic modes, as the correspond- ℏΩ =50meV,Q ≈160.(b)Polariton–phononscattering R f n n rate Γ (solidlines)andcavitydecayrate2γ (dashedlines) ing rates are inversely proportional to phonon ener- k k asafunctionofmomentumfortheupperandlower gies,see[41].Wealsonotethat,incontrast,thecavity polariton(samecolorsasin(a)).Themaximumvalueof decayrate γ increases/decreasessmoothlywithkfor absorptionofA =0.5identifiesthecriticalcoupling n n conditions Γ =2γ fortherespectivepolaritonanditis the UP/LP branch, cf the dashed lines in figure 3(b). k k markedbyaverticalblackline.Thegreylinesshowthecase Thisincrease/decreaseisdeterminedbythephotonic withoutconsideringdarkstatesandonlytakinginto accountthebrightKKexcitons.(c)Lowerpolariton Hopfield coefficient, which increases for the UP and dispersion(redline)andphononenergies(plustheenergy decreasesfortheLPbranch. ofthedarkKΛexciton)showingtheopeningofemission To illustrate the importance of dark excitons, channelsintothedarkexcitonstatesatk ≈0.3,0.8,2.4and −1 µm .(d)Criticalcouplingmomentumk asafunctionof c we also show the polariton absorption and the temperaturefortheupper(blue)andlowerpolariton(red). polariton-phonon scattering rates without including Theshadedareacorrespondstotherange dark exciton states, i.e. we only take into account 0.5 ⩾A ⩾0.495. the bright KK excitons (grey lines in figures 3(a) and (b)). We find that for the lower polariton the resonant absorption is drastically reduced at small To better understand the change of the absorp- momenta, with the critical coupling condition shif- tion as a function of the in-plane momentum and ted to higher momenta. We also find that the steep the opposite behavior of the upper and the lower increases step-like increases found for these polari- polariton branch observed in figure 3(a), we invest- tons disappear (red vs. lower grey line), as they stem igate in figure 3(b) the momentum-dependent cav- from scattering into dark excitons. For the scatter- itydecayrate γ andpolariton-phononscatteringrate ing rates of the lower polariton, the intravalley scat- Γ ,cfequations(4)and(5).Wefindthatforthelower tering is orders of magnitude smaller than the inter- polariton branch, the critical coupling condition of valley one (grey line is basically 0), due to the for- n n −1 Γ =2γ isreachedatk =1.6 µm ,asdenotedwith bidden optical absorption for low temperatures and k k theblackverticallineinfigure3(b).Thiscorresponds since the scattering of LP polaritons with intraval- exactlytothemomentumwherethemaximalabsorp- ley acoustic modes is energetically forbidden [18]. LP tion of A =0.5 is reached. The microscopic calcu- In the case of the upper polariton, the qualitative lation of polariton-photon scattering rates explains shape of the absorption curve in figure 3(a) is sim- the step-like increase in the absorption of both the ilar to intravalley scattering without dark excitons UP and LP polariton branch. These can be clearly (blue vs upper grey line). In particular, both lines −1 attributed to an increase of the polariton–phonon show a step-like increase at k ≈1 µm , which scattering rates at certain momenta (at k≈0.3, 0.8, stems from the intravalley emission via emission of −1 LP −1 UP 2.4 and 3.1 µm for Γ and at 1 µm for Γ ). optical modes. However, the intensity of the UP k k Importantly, each of the steep increases for the LP absorption is strongly reduced in the absence of absorption/rates is a signature of an opening of an dark excitons. This is due to the overall decrease of intervalleyscatteringchannelintodarkexcitonstates the polariton–phonon scattering rates, figure 3(b), (seediscussionaboutgreylinesbelow). moving the system further away from the critical 5 2DMater.10(2023)015012 BFerreiraetal coupling condition. In a nutshell, the scattering into dark excitons leads to a considerable quantitative as well as qualitative variation of the optical absorp- tion, in particular for LP states (cf colored arrows in figure3(a)). So far, we have only considered the polariton absorption at 77K, where the critical coupling con- dition can only be reached for the lower polariton branch. To further investigate this, we present in figure 3(d) the critical coupling momentum k as a functionoftemperaturefortheupper(blueline)and the lower polariton (red line). The blue- and red- shaded areas correspond to the region 0.5 ⩾A ⩾ 0.495totakeintoaccountuncertaintiesintheexper- imentalmeasurementofthemaximalabsorption.As we increase the temperature, the critical coupling occurs at smaller momenta for the LP branch due to an overall increase of the scattering with phon- ons. Since the cavity decay rates γ are temperature- independent within our model, the overall increase in Γ athighertemperaturesresultsinsmallerk ful- k Figure4.Temperatureandqualityfactorstudy. filling the critical coupling conditions. Interestingly, Polariton–phononscatteringrate Γ forthe(a)upper(UP) and(b)lowerpolaritonbranch(LP)atk =0asafunction for the UP branch, we find that there is no crit- oftemperature(forQ ≈160).Weidentifythe ical coupling for temperatures below approximately contributionsoftheintravalley(KK)aswellasintervalley (KK ,KΛ)scatteringchannels(shadedareas).Notethat 125K. We show in figure 3(b) that at k =0 the cav- 2Γ correspondstothespectrallinewidthoftherespective n n LP 0 itydecayrate γ islargerthan Γ .However,while γ k k k polariton.Thecrossingpointwiththecavitydecayrate2γ decreases for increasing momenta, thus approaching (dashedline)correspondstothecriticalcoupling n condition.(c)Polariton–phonondecayratesandthecavity the smaller values of Γ , the opposite takes place for k n decayrate2γ atk =0asafunctionofthequalityfactor(at UP k γ . Thus, for upper polaritons, the critical coup- T =77K). lingcanonlyoccurathighertemperatures,where Γ is considerably enhanced. Interestingly, we find that at around 200 K two different momenta fulfill the critical coupling condition for UP (blue lines in in the black dashed line shows the temperature- figure 3(d). At these temperatures the cavity decay independent 2γ , indicating at which temperatures rate crosses the polariton–phonon scattering rate in the critical coupling condition is reached at k =0. the region of the opening of the optical emission This is the case for the UP branch at T ≈125K-in (step-likeincrease),wherewecanhavethesamevalue agreement with figure 3(c). In contrast, for the LP ofscatteringandcavity-decayratesfortwo(ormore) branch, the critical coupling at k =0 can only be momenta. reached at temperatures significantly higher than roomtemperature. 3.3.Absorptionengineering Forbothpolaritonbranches,thelargestcontribu- We have demonstrated that the polariton absorp- tiontothelinewidthcomesfromtheintervalleyscat- tion depends on two key quantities: the polariton- tering into dark KΛ excitons reflecting the efficient phonon scattering rate Γ and the cavity decay rate scatteringplusthethree-folddegeneracyofthe Λval- γ . Now, we would like to tune the absorption by ley, similar to the excitonic case [42]. Furthermore, changing these quantities. While Γ is strongly sens- atroomtemperature,intervalleyscatteringwithinthe itive to temperature, γ is determined by the cavity K valley is also important. At 20K, the LP linewidth quality factor (i.e. in particular the reflectance of the is determined to a large extent by scattering into the cavityendmirrors). dark KK excitons. We stress that here we are focus- Infigures4(a)and(b)weshowthetemperature- ing on the scattering from the k =0 polariton state. dependent polariton–phonon scattering rates at There are further possible scattering channels at lar- k =0 for the UP and LP branch, respectively. Note germomenta,asshowninfigure3(b).Increasingthe that this corresponds to the half linewidth of the temperature to 300K increases the LP linewidth by respectivepolaritonresonanceinabsorptionspectra. around one order of magnitude as the absorption We add up different scattering channels including of intervalley phonons becomes possible. At 77 K intravalley scattering within the K valley (KK) as theintravalleycontributiontothephonon-scattering well as intervalley scattering into momentum-dark rates is very small, in accordance with figure 3. exciton states (KK and KΛ). For comparison, Thelinewidthoftheupperpolaritonisat20Kmuch 6 2DMater.10(2023)015012 BFerreiraetal largercomparedtotheLPbranchsinceemissioninto dark excitons is possible even at k =0 thanks to the much higher polariton energy (figure 1(b)). Hence, the increase in UP from 20K to 300 K is not as sub- stantialasintheLPcase.Overall,figures4(a)and(b) illustrate the huge impact of dark exciton states on thepolariton–phononscatteringratesandthusonthe polaritonabsorption. Afterhavingaddressedtheroleoftemperaturein tuning the polariton–phonon scattering rates Γ , we now focus on the change of the cavity decay rate γ as a function of the quality factor Q . In figure 4(c) n n weshow γ (grayline)and Γ (redandbluelines)as k k afunctionofQ .WefindthatUPhasacriticalcoup- lingconditionk =0aroundQ ≈200.Forsmallval- C f uesof the quality factor,the UP absorptionis expec- tedtoincrease,butaswemovefurtherawayfromthe criticalcouplingtheabsorptiondecreases.FortheLP, thecriticalcouplingconditionisonlyreachedathigh valuesofQ . 3.4.PolaritonabsorptionofMoSe So far we have studied the polariton absorption for WSe monolayers, where dark excitons turned out Figure5.AbsorptionofMoSe .(a)Surfaceplotofpolariton to play a crucial role. Now we investigate the MoSe absorptionofahBN-encapsulatedMoSe monolayerasa monolayerexhibitingadifferentenergeticalignment functionofmomentumandenergy(77K, ℏΩ =50meV andQ ≈160).(b)Absorptioncutsasafunctionofenergy of dark and bright states. With the latter being the f forthreedifferentmomenta.(c)Absorptionintensityasa lowest ones in MoSe [1, 15, 27], we expect only functionofmomentumforthelower(LP)andtheupper n n a negligible contribution from dark excitons. Simil- (UP)polaritonbranch.(d)Decayrates Γ and2γ asa k k functionofmomentumfortheLPandUPbranch.Thethin arly to the case of WSe , we show in figure 5(a) the greylinesin(c)and(d)correspondtothecasewithout absorptionofpolaritonsasafunctionofmomentum darkexcitons(i.e.consideringonlythebrightKKexcitons). and energy for the zero-detuning case at T =77K. We find a drastic reduction in absorption as well as in the linewidth of the LP absorption compared to WSe (figure 2(a)). This can be clearly observed in of dark states. For the LP branch, the scattering into the momentum cuts shown in figure 5(b). Although theenergeticallyhigherdarkexcitonstatesisgenerally theintensityoftheresonantabsorptionincreasesfor weakasitisdrivenbyabsorptionofintervalleyphon- larger momenta, similar to the case of WSe , quant- ons from high-momenta states. Since for LP also the itatively the increase is much slower, reaching only a intravalley scattering with acoustic modes is forbid- −1 maximal value of approximately 0.1 at k =1.5µm denduetomomentumandenergyconservation(due (compared to almost 0.5 predicted for WSe ). Inter- to the almost flat phonon dispersion making it diffi- estingly, for larger momenta we find also an increase culttofulfiltheenergyconservation[18]),thelower of the absorption for the UP branch (figure 5(b))— polaritonhasonlyaverysmallscatteringrate(thered opposite to the case of WSe (figure 2(b)). In addi- lineinfigure4(d)isalmostnotvisible).Nevertheless, tion,weobservealargeincreaseinthespectralwidth the decrease of the cavity decay rate γ with increas- ofpolaritonresonancesatlargerin-planemomentak. ing momenta allows for the critical coupling condi- −1 To microscopically understand the qualitative as tionattheveryhighmomentaofk =3.25 µm (cf well as quantitative differences of the momentum- figure5(d)),wheretheLPabsorptionreachesitsmax- resolvedabsorptioninMoSe andWSe ,weinvestig- imum value of A =0.5 (figure 5(c)). Interestingly, 2 2 atetheintensityoftheresonantpolaritonabsorption even though dark excitons have only a small contri- and the underlying polariton–phonon and cavity bution,theirpresenceshiftsthecriticalcouplingcon- decay rates. We assume the same value of reflectiv- ditiontoa smallermomentum(cf grey vsred line in ity r =0.99 as in figure 2, resulting in similar cav- figure5(c)). ity decay rates γ as for WSe As a consequence, the For the upper polariton, the intravalley scatter- observed difference in polariton absorption must be ing contribution is also dominant (only small devi- duetothephonon-scatteringrates.Weshowboththe ations between the grey and blue line), showing −1 absorption and decay rates also for the case without two step-like increases at k≈1 and k ≈1.3 µm dark excitons (grey lines in figures 5(c) and (d)). As due to the emission of intravalley optical modes expected,inMoSe thereisonlyaminorcontribution (LO/TOandA1withenergy36.1/36.6meVand30.3 7 2DMater.10(2023)015012 BFerreiraetal meV, respectively). In contrast to WSe , we observe Acknowledgments a large increase in the phonon-scattering rates for the UP branch, reflecting a more efficient intraval- We thank Marten Richter (TU Berlin) for inspir- leyscatteringwithopticalmodesinMoSe [41].This ing discussions. This project has received funding leads to the much broader spectral width of the res- support from the DFG via SFB 1083 (Project B9), onancesobservedinfigure5(b).Thecontributionof the European Union’s Horizon 2020 Research and dark excitons is minor, however, we still observe an Innovation programme under Grant Agreement No. opening of an emission channel into dark states, cf 881603 (Graphene Flagship) and from the Knut UP −1 the step-like increase of Γ at k ≈0.4 µm . This and Alice Wallenberg Foundation via the Grant opening is important for understanding the increase KAW 2019.0140. The computations were enabled by of the resonant absorption, when going from k =0 resources provided by the Swedish National Infra- −1 tok =0.75µm observedinfigure5(b)(incontrast structureforComputing(SNIC). to the prediction for WSe in figure 1(a)). Without dark states, there would be a decrease of the absorp- ORCIDiDs −1 tion up to approximately 0.9 µm (cf the grey line in figure 5(c)). In MoSe , the upper polariton fulfills BeatrizFerreira https://orcid.org/0000-0002- the critical coupling condition at the four different 5395-345X −1 momenta k ≈ 1, 1.2, 1.3 and 1.6 µm . The lowest RobertoRosati https://orcid.org/0000-0002-2514- two values of k are a consequence of polariton scat- teringintodarkexcitonstates. 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Signatures of dark excitons in exciton–polariton optics of transition metal dichalcogenides

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© 2022 The Author(s). Published by IOP Publishing Ltd
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10.1088/2053-1583/aca211
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Abstract

Anyfurtherdistribution Integrating2Dmaterialsintohigh-qualityopticalmicrocavitiesopensthedoortofascinating ofthisworkmust many-particlephenomenaincludingtheformationofexciton–polaritons.Thesearehybrid maintainattributionto theauthor(s)andthetitle quasi-particlesinheritingpropertiesofboththeconstituentphotonsandexcitons.Inthiswork,we ofthework,journal citationandDOI. investigatetheso-faroverlookedimpactofdarkexcitonsonthemomentum-resolvedabsorption spectraofhBN-encapsulatedWSe andMoSe monolayersinthestrong-couplingregime.In 2 2 particular,thankstotheefficientphonon-mediatedscatteringofpolaritonsintoenergeticallylower darkexcitonstates,theabsorptionofthelowerpolaritonbranchinWSe ismuchhigherthanin MoSe .Itshowsuniquestep-likeincreasesinthemomentum-resolvedprofileindicatingopening ofspecificscatteringchannels.Westudyhowdifferentexternallyaccessiblequantities,suchas temperatureormirrorreflectance,changetheopticalresponseofpolaritons.Ourstudycontributes toanimprovedmicroscopicunderstandingofexciton–polaritonsandtheirinteractionwith phonons,potentiallysuggestingexperimentsthatcoulddeterminetheenergyofdarkexcitonstates viamomentum-resolvedpolaritonabsorption. 1.Introduction Sofar,polariton–phononinteractionsinTMDshave not been well studied, leaving many open questions Monolayers of transition metal dichalcogenides on the impact of dark exciton states on polariton (TMDs) show a rich exciton landscape, including absorption. bright and dark exciton states [1, 2]. This class of Inourpreviouswork,westudiedtransportprop- atomically-thin materials exhibits a large oscillator erties of exciton–polaritons in MoSe monolayers strengthandexcitonbindingenergiesintherangeof [18], where dark excitons do not play an import- a few hundreds of meV, hence governing the opto- ant role as they are energetically higher than the 1s electronic properties even at room temperature [1, bright states [1, 13, 15]. Now, the focus lies on the 3–7].TMDshavealreadybeensuccessfullyintegrated optical response of polaritons in WSe monolayers, into optical cavities [8–10], where the coupling of specifically addressing the impact of dark exciton cavity photons with excitons gives rise to the form- states.Inthisregard,thepolaritonabsorptionisespe- ation of exciton–polaritons [11, 12]. It is only the cially informative as it unambiguously demonstrates bright exciton states that can couple to photons to strong coupling via the Rabi splitting [19], and its form these quasi-particles, while momentum-dark magnitudeisdeterminedbythebalancebetweenthe excitons[1,2,13]cannotbedirectlyaccessedbylight. polariton–phononandcavitydecayrates.Wemicro- Nonetheless,thesestatesareavailablescatteringpart- scopicallycalculatethepolaritonabsorptionbycom- nersforpolaritonsviatheinteractionwithphonons. bining the Heisenberg–Langevin equations [20] for The presenceofdark excitonsis expected tosignific- polaritonswiththeexcitondensitymatrixformalism antly change the polariton–phonon scattering rates, [21,22].Wecalculatethefullvalley-andmomentum- especially for tungsten-based TMDs, since here dark dependent polariton–phonon scattering rates that excitons are the energetically lowest states [14–17]. govern the optical response of TMD materials via ©2022TheAuthor(s). PublishedbyIOPPublishingLtd 2DMater.10(2023)015012 BFerreiraetal both spectral linewidths and magnitude. In particu- lar,weexplorethisinthecontextofthecriticalcoup- ling condition [23], where the total cavity decay rate coincideswiththepolariton–phononscatteringrate. We predict that the presence of dark excitons has a large impact on the polariton scattering rates, giv- ing rise to clear signatures in momentum-resolved absorption spectra that could be exploited to meas- uretheenergyofdarkexcitonstates.Furthermore,we predictandexplainasurprisingdifferenceinabsorp- tion intensity between the upper and lower polari- ton branch at zero momentum and zero detuning, despite equal photonic and excitonic contributions. We also study the influence of externally accessible quantities to tune the scattering rates (via temperat- ure) and cavity decay rates (via mirror reflectance). For the latter, we find that the cavity quality factor playsanimportantrolefortheabsorption,inpartic- ularforthelowerpolaritonbranchthathasasmaller photoniccomponent. Figure1.(a)SchematicillustrationofaTMDmonolayerin 2.Theory aFabry–Perotcavitywiththefundamentalcavitymode representedbytheredcurve.TMDexcitonsinteractwith photonsandphononsasindicatedbythecreation We start by describing the theoretical approach to ˆ ˆ (annihilation)operatorsforphotons(c (c))andphonons microscopicallycalculatetheabsorptionspectrumfor ˆ† ˆ (b (b)).Thecavitysysteminteractswiththeoutsideworld polaritons in TMD monolayers integrated into an ˆ ˆ viatheoperators (B (B)).(b)Exciton–polaritonband structure,wherepolaritonscanscatterintodarkexciton opticalcavity.Excitonenergiesandwavefunctionsin statesbyemittingphononsifmomentumandenergycan TMD monolayers are obtained by solving the Wan- beconserved. nier equation [15, 24, 25] including DFT input on single-particleenergies[26].TMDsarecharacterized by regular bright excitons that are directly access- ible in optical spectra, as well as dark exciton states and describes the free energy of excitons E , phon- vk b c thatareknowntobetheenergeticallyloweststatesin ons E as well as photons within (E ) and out- αq k tungsten-based TMDs [14–17, 26, 27]. In this work, side the cavity (ℏω). Here, v is the exciton index we focus on momentum-dark excitons consisting of (we consider only 1s states), α the phonon mode, Coulomb-boundelectronsandholesthatarelocated k and q are the in-plane momentum of excitons/- at different valleys within the Brillouin zone (K, K photons (center-of-mass momentum for excitons) or Λ).Thismeansthattherequiredlargemomentum and phonons, respectively. Furthermore, we have † † † † transfercannotbeprovidedbyphotons,makingthese ˆ ˆ ˆ ˆ ˆ ˆ introduced X (X ), b (b ), ˆc (ˆc ), B (B ) vk αq αq k jkω vk k jkω statesopticallydark[13,16,27–30]. as exciton, phonon, inner-cavity and outer-cavity In this work, we combine the density matrix photon creation (and annihilation) operators, formalismwiththeHopfieldapproach[11],tomodel respectively. the optical response of polaritons. We quantize sep- ˆ The second term in the Hamiltonian, H = X−c ( ) arately a single internal cavity mode of a Fabry– † † ˆ ˆ g ˆc X +ˆc X describes the exciton-light k vk k vk k vk Perot resonator and the external radiation fields, interaction mediated by the exciton-photon coup- which are split into two sets of continuum modes ling matrix element g [15, 31], where photons corresponding to the left and the right of the cav- need to have the same in-plane momentum k ity (figure 1). The internal and external modes are as excitons to fulfill the momentum conserva- weakly coupled via the end mirrors, where the in- tion (hence restricting the coupling only to the plane wavevector is conserved. The starting point is bright exciton states). In general, the out-of-plane the many-particle Hamiltonian in the excitonic pic- component k influences the cavity energy and ˆ ˆ ˆ ˆ ˆ ture H =H +H +H +H . The first term 0 X−c X−b B−c exciton–photon coupling. However, we assume readsinsecondquantization the existence of one resonant photon mode (i.e. X c ∑ ∑ ∑ E =E ). The third contribution in the Hamilto- † † KK,0 0 X c b † ˆ ˆ ˆ ˆ ˆ H = E X X + E ˆc ˆc + E b b ′ 0 vk k αq † † v,k vk k k αq αq vv ˆ ˆ ˆ ˆ ˆ nian H = D X X (b +b ) X−b ′ v k αq αq α,−q vv kαq vk+q vk k q ˆ describes the exciton–phonon interaction [15], ∑ ∑ ˆ ˆ + dω ℏω(k)B B (1) where the coupling strength is determined by the jkω jkω vv j=L,R exciton–phononmatrixelement D .Finally,thelast αq 2 2DMater.10(2023)015012 BFerreiraetal ∑ ∑ ´ dω term, H =iℏ a (ω)[B ˆc − last term in equation (2) describes the polariton– B−c j,k k j=L,R k 0 2π jωk † ˜ phonon interaction. Here, the matrix element D is B ˆc ], provides the interaction between the inner- jωk k ′ n n related to the exciton–phonon coupling via D = and outer-cavity photons [20, 32]. The free photons kαq ′ ′ n ∗ n n n interact with the cavity with a coupling parameter, h D h and depends on the excitonic Hop- αq X,k+q X,k a (ω). Assuming broadband end mirrors, it is field coefficients h [33], since phonons only couple j,k appropriate to take the first Markov approxima- totheexcitonicpartofpolaritons. tion and approximate this parameter as frequency Toobtainanexpressionforthepolaritonabsorp- independent[20].ThiscontributionintheHamilto- tion, we exploit the Heisenberg equations of motion nian leads to a consistent description of both the forthecoherentpopulationofpolaritonandexternal radiative decay rate within the cavity as well as the radiation field (cf the supplementary information). couplingofpolaritonstoinputandoutputfields. For this we make a correlation expansion includ- Now, we investigate the strong-coupling regime, ing the dynamics of the phonon-assisted polariza- wheretheexciton–photoncouplingstrengthg islar- tion.Weusetheinput–outputmethod[20]tocouple ger than (the difference of) cavity and non-radiative thedynamicsbetweenintra-andouter-cavityphoton excitondecayrates[12].Theneweigenmodes,known modesateachport.Wetreatthescatteringwithphon- as exciton–polaritons, can be obtained by applying ons within a Markov approximation and assuming a Hopfield transformation of the excitonic Hamilto- a thermalized reservoir of incoherent phonons [16]. niandiscussedabove,yielding[11,12] The absorption then follows from energy conserva- tion as the difference between incoming fields and n† n n 0 0 the total reflected and transmitted light. To simplify ˆ ˆ ˆ H = E Y Y +H +H k k k b B the resulting expression, we assume that the cavity k,n is symmetric and ignore interference effects between dω +iℏ a (ω) polaritons in different branches. The latter is a good jk 2π k,n,j approximation if the branches are widely spaced in ( ) energy compared to the polaritonic spectral width. n n† n n∗ ˆ ˆ ˆ ˆ × h B Y −h B Y jkω c,k jkω k c,k k We obtain an Elliot-like formula for the polariton ( ) ∑ ′ absorption[32], n † n n n ˆ ˆ ˜ ˆ ˆ + D b +b Y Y . (2) αq kαq α,−q k+q k n n kαqnn 4γ Γ n k k A (ℏω) = , (3) n 2 n n 2 (ℏω −E ) + (2γ + Γ ) k k k Here, the first term provides the free polaritonic n† ˆ ˆ for each polariton branch and momentum n,k. The Hamiltonian with Y (Y ) denoting the polariton k k obtained equation is similar to the expression found creation (annihilation) operator with the polari- in[32],however,thekeydifferenceliesinthemicro- ton mode n and momentum k. The energy of the scopic treatment of polariton–phonon interaction. corresponding polariton, E , includes in particu- Thismeansthatphononscanchangethemomentum lar lower and upper polariton branches (LP, UP) of the excitonic component of the polariton, lead- that are separated in k =0 by the Rabi splitting UP LP ing to a momentum dependent scattering rate. In ℏΩ =E −E . This is a consequence of the mix- 0 0 equation(3)weintroducedthedecayrates ing between excitons and photons (with the same center-of-mass and total momentum), as quantified n 2 2 γ = ℏc(1−|r | )|h | /(4L ), (4) m c,k cav bytheHopfieldcoefficients[12].Weincludealso,for notational convenience, polaritons stemming from ( ) momentum-dark excitons, although these show no 1 1 n n n 2 b Γ =2π |D | ± +n ′ ′ k α,k −k α,k −k exciton–photon mixing. Nevertheless, we will show 2 2 ′ ′ n αk below their crucial role for the polariton absorption ( ) n n b ×L E −E ±E , (5) viaadditionalphonon-inducedscatteringchannelsto ′ ′ γ˜ 0 k k α,k −k the optically active polaritons. Both polariton ener- n n n n gies E and Hopfield coefficients h and h are where γ is the effective cavity decay rate of one k X,k c,k k calculated analytically (with subscript X and c refer- port and Γ is the polariton–phonon scattering rate. ring to exciton and intra-cavity photon component, Here we are summing over all possible scattering respectively)[12]. channels from a polariton n,k to all possible receiv- ′ ′ The second and the third term in equation (2) ing polaritons n ,k via interaction with a phonon are the free phonon and free outer-cavity photon with mode α and momentum q, such that the over- contribution, respectively, which are not affected all momentum is conserved. The quality factor of c 2 by the Hopfield transformation. The fourth term the cavity reads Q =E L /[ℏc(1−|r | )|)], where f cav m describestheinteractionofpolaritonswiththeouter- r is the reflectivity of the cavity. In this work, cavity photons, mediated by the photonic Hopfield we use the default value of r =0.99 if not stated coefficients as only the photonic part of polaritons otherwise. Importantly, we explicitly consider inter- ′ ′ couples to the external radiation field. Finally, the valley scattering by including K and Λ phonons 3 2DMater.10(2023)015012 BFerreiraetal whichallowscatteringintopolaritonscoincidingwith KK and KΛ excitons, respectively. The polariton– phononratesarecalculatedwithintheMarkov–Born approximation [31,34] including effects beyond the completed-collision limit [35] by an energy conser- vationdescribedviaaLorentzianfunction L witha γ˜ broadening γ˜ =0.1meV[18]. Crucially, the polaritonic Elliot formula offers insightintohowunderlyingmicroscopicdecaychan- nelsmanifestintheabsorptionoflightbypolaritons, which would not be possible using the more com- monly used classical transfer-matrix method [19]. Evaluating equation (3) at resonance reveals that absorption is maximized when the two effective polariton decay rates are closest in value. It fol- Figure2.Polaritonabsorption.(a)Surfaceplotof lows that maximum absorption of 0.5 is possible at absorptioninanhBN-encapsulatedWSe monolayerasa functionofmomentumandenergyatatemperatureof the so-called critical coupling condition [23, 36] of 77K,assumingaRabisplittingof ℏΩ =50meVanda n n 2γ = Γ , i.e. when the leakage out of both ports k k cavityqualityfactorofQ =160.Thedashedwhitelines of the cavity is equal to the exciton dissipation rate correspondtothebareexcitonandcavitydispersion,while thesolidblacklinesdescribethepolaritondispersion. within the TMD layer in the cavity. The maximum (b)Absorptioncutsasafunctionofenergyforthree possible absorption of 50% is a well-known con- differentmomenta. straint for mirror-symmetric two-port systems that support a single resonance [37, 38]. We expect the presence of dark excitons to significantly increase polaritonshaveanequalphotonicandexcitoniccon- the polariton–phonon scattering rates in tungsten- tribution at k =0, hence also the cavity decay rate based TMDs (where they are the energetically lowest is the same for both polaritons. As a result, the states).Theopeningofintervalleyscatteringchannels phonon-induced decay rate of polaritons must be isexpectedtostronglyimpactthebalancebetweenthe responsible for the observed difference in the height effectiveradiativecouplingandscatteringloss,which of absorption peaks. Furthermore, we find that shouldtranslateintomeasurablesignaturesinpolari- the absorption is enhanced for increasing momenta LP tonabsorptionspectra. for the lower polariton (A ) up to approximately −1 k =1.6 µm ,whileitisreducedfortheupperpolari- UP 3.Results ton(A ),(cfalsotheabsorptioncutsinfigure2(b)). Moreover, we observe that not only the absorption LP 3.1.PolaritonabsorptionofWSe intensitybutalsothelinewidthofA becomeslarger Now, we evaluate equation (3), using numeric- forincreasingin-planemomentum,beforeitisagain −1 ally calculated polariton-phonon scattering rates, to reducedformomentahigherthank =1.6 µm .The studythepolaritonabsorptioninthestrong-coupling absorption intensity and the spectral linewidth of regime for an hBN-encapsulated WSe monolayer polaritonresonancescanbeascribedtotheinterplay integrated into a Fabry–Perot cavity with a quality ofthecavitydecayandnon-radiativedecayofpolari- factor of Q ≈160 and a Rabi splitting of ℏΩ = tonsviascatteringwithphononsasdiscussedindetail f R 50meV. Note that the choice of the substrate has below. some impact on the polariton-phonon scattering rates and polariton absorption. The main effect is 3.2.Criticalcoupling substrate-induced screening that changes the separ- To explain the different behavior in the absorption ationbetweenbrightanddarkexcitonsandcanopen spectra of the upper and lower polariton branch, orclosescatteringchannelswithphonons.Inthesup- we plot the maximal absorption A of the UP and plementary material, we show a direct comparison the LP branch at 77 K in figure 3(a). The absorp- between hBN-encapsulated and free-standing WSe tion intensity of the UP branch generally decreases monolayers. with the momentum, however, with one excep- −1 Figure 2(a) presents an energy- and in-plane tion at approximately k =1 µm , where we observe momentum-resolved surface plot of the polariton a small increase (blue line). In contrast for the absorption for hBN-encapsulated WSe . Interest- lowerpolaritonbranchwefindanenhancedabsorp- −1 ingly,wefindtheupperpolaritontobemuchhigher tion until approximately k =1.6 µm , where the −1 in intensity than the lower polariton at k =0 µm critical coupling condition with a maximum pos- (cfalsothebluelinesinfigure2(b)).Previousreports sible value of A =0.5, is reached (red line). The in GaAs have shown that in the case of zero detun- increase of the absorption includes several steep ing, the lower and upper polariton peaks intensit- step-like enhancements before the absorption starts −1 ies are similar [39, 40]. In the resonant case, the todecreaseforvalueslargerthank =1.6 µm . 4 2DMater.10(2023)015012 BFerreiraetal Infigure3(c),weplotthelowerpolaritondisper- sion in relation to the bright exciton energy together withthephonondispersionforLA,TAandTOmodes that are responsible for the scattering into the dark KΛ excitons. Whenever a phonon line crosses with the polariton energy, a scattering channel into dark excitonstatesopensup.Thisisclearlyvisibleasastep- likeincreaseinthepolariton–phononscatteringrates showninfigure3(b).Atmomentumk =0,theenergy LP E of the lower polariton is too small to allow scat- teringintotheKΛexcitonviaemissionofphononsas LP X E −E ≈11.2meV,whichisjustsmallerthanthe 0 Λ,0 energy of 11.4 meV of intervalley TA phonons [41]. −1 Whenkreachesthethresholdvalueofk≈0.3 µm , the scattering channel into KΛ states opens, result- ing in the abrupt increase of Γ , cf also figures 1(b) and3(c).Forintervalleyscattering,bothacousticand opticalphononmodeshavefiniteenergiesinthecor- respondingsymmetrypoints[41].However,acoustic phononshavemuchsmallerenergies(12–14meVfor acousticphononsvs27–32meVforopticalphonons Figure3.Criticalcoupling.(a)Maximalabsorptionatthe resonantenergyasafunctionofmomentumforthelower at the Λ point [41]). This results in a more efficient (red,LP)andupper(blue,UP)polaritonat77Kwith scattering with acoustic modes, as the correspond- ℏΩ =50meV,Q ≈160.(b)Polariton–phononscattering R f n n rate Γ (solidlines)andcavitydecayrate2γ (dashedlines) ing rates are inversely proportional to phonon ener- k k asafunctionofmomentumfortheupperandlower gies,see[41].Wealsonotethat,incontrast,thecavity polariton(samecolorsasin(a)).Themaximumvalueof decayrate γ increases/decreasessmoothlywithkfor absorptionofA =0.5identifiesthecriticalcoupling n n conditions Γ =2γ fortherespectivepolaritonanditis the UP/LP branch, cf the dashed lines in figure 3(b). k k markedbyaverticalblackline.Thegreylinesshowthecase Thisincrease/decreaseisdeterminedbythephotonic withoutconsideringdarkstatesandonlytakinginto accountthebrightKKexcitons.(c)Lowerpolariton Hopfield coefficient, which increases for the UP and dispersion(redline)andphononenergies(plustheenergy decreasesfortheLPbranch. ofthedarkKΛexciton)showingtheopeningofemission To illustrate the importance of dark excitons, channelsintothedarkexcitonstatesatk ≈0.3,0.8,2.4and −1 µm .(d)Criticalcouplingmomentumk asafunctionof c we also show the polariton absorption and the temperaturefortheupper(blue)andlowerpolariton(red). polariton-phonon scattering rates without including Theshadedareacorrespondstotherange dark exciton states, i.e. we only take into account 0.5 ⩾A ⩾0.495. the bright KK excitons (grey lines in figures 3(a) and (b)). We find that for the lower polariton the resonant absorption is drastically reduced at small To better understand the change of the absorp- momenta, with the critical coupling condition shif- tion as a function of the in-plane momentum and ted to higher momenta. We also find that the steep the opposite behavior of the upper and the lower increases step-like increases found for these polari- polariton branch observed in figure 3(a), we invest- tons disappear (red vs. lower grey line), as they stem igate in figure 3(b) the momentum-dependent cav- from scattering into dark excitons. For the scatter- itydecayrate γ andpolariton-phononscatteringrate ing rates of the lower polariton, the intravalley scat- Γ ,cfequations(4)and(5).Wefindthatforthelower tering is orders of magnitude smaller than the inter- polariton branch, the critical coupling condition of valley one (grey line is basically 0), due to the for- n n −1 Γ =2γ isreachedatk =1.6 µm ,asdenotedwith bidden optical absorption for low temperatures and k k theblackverticallineinfigure3(b).Thiscorresponds since the scattering of LP polaritons with intraval- exactlytothemomentumwherethemaximalabsorp- ley acoustic modes is energetically forbidden [18]. LP tion of A =0.5 is reached. The microscopic calcu- In the case of the upper polariton, the qualitative lation of polariton-photon scattering rates explains shape of the absorption curve in figure 3(a) is sim- the step-like increase in the absorption of both the ilar to intravalley scattering without dark excitons UP and LP polariton branch. These can be clearly (blue vs upper grey line). In particular, both lines −1 attributed to an increase of the polariton–phonon show a step-like increase at k ≈1 µm , which scattering rates at certain momenta (at k≈0.3, 0.8, stems from the intravalley emission via emission of −1 LP −1 UP 2.4 and 3.1 µm for Γ and at 1 µm for Γ ). optical modes. However, the intensity of the UP k k Importantly, each of the steep increases for the LP absorption is strongly reduced in the absence of absorption/rates is a signature of an opening of an dark excitons. This is due to the overall decrease of intervalleyscatteringchannelintodarkexcitonstates the polariton–phonon scattering rates, figure 3(b), (seediscussionaboutgreylinesbelow). moving the system further away from the critical 5 2DMater.10(2023)015012 BFerreiraetal coupling condition. In a nutshell, the scattering into dark excitons leads to a considerable quantitative as well as qualitative variation of the optical absorp- tion, in particular for LP states (cf colored arrows in figure3(a)). So far, we have only considered the polariton absorption at 77K, where the critical coupling con- dition can only be reached for the lower polariton branch. To further investigate this, we present in figure 3(d) the critical coupling momentum k as a functionoftemperaturefortheupper(blueline)and the lower polariton (red line). The blue- and red- shaded areas correspond to the region 0.5 ⩾A ⩾ 0.495totakeintoaccountuncertaintiesintheexper- imentalmeasurementofthemaximalabsorption.As we increase the temperature, the critical coupling occurs at smaller momenta for the LP branch due to an overall increase of the scattering with phon- ons. Since the cavity decay rates γ are temperature- independent within our model, the overall increase in Γ athighertemperaturesresultsinsmallerk ful- k Figure4.Temperatureandqualityfactorstudy. filling the critical coupling conditions. Interestingly, Polariton–phononscatteringrate Γ forthe(a)upper(UP) and(b)lowerpolaritonbranch(LP)atk =0asafunction for the UP branch, we find that there is no crit- oftemperature(forQ ≈160).Weidentifythe ical coupling for temperatures below approximately contributionsoftheintravalley(KK)aswellasintervalley (KK ,KΛ)scatteringchannels(shadedareas).Notethat 125K. We show in figure 3(b) that at k =0 the cav- 2Γ correspondstothespectrallinewidthoftherespective n n LP 0 itydecayrate γ islargerthan Γ .However,while γ k k k polariton.Thecrossingpointwiththecavitydecayrate2γ decreases for increasing momenta, thus approaching (dashedline)correspondstothecriticalcoupling n condition.(c)Polariton–phonondecayratesandthecavity the smaller values of Γ , the opposite takes place for k n decayrate2γ atk =0asafunctionofthequalityfactor(at UP k γ . Thus, for upper polaritons, the critical coup- T =77K). lingcanonlyoccurathighertemperatures,where Γ is considerably enhanced. Interestingly, we find that at around 200 K two different momenta fulfill the critical coupling condition for UP (blue lines in in the black dashed line shows the temperature- figure 3(d). At these temperatures the cavity decay independent 2γ , indicating at which temperatures rate crosses the polariton–phonon scattering rate in the critical coupling condition is reached at k =0. the region of the opening of the optical emission This is the case for the UP branch at T ≈125K-in (step-likeincrease),wherewecanhavethesamevalue agreement with figure 3(c). In contrast, for the LP ofscatteringandcavity-decayratesfortwo(ormore) branch, the critical coupling at k =0 can only be momenta. reached at temperatures significantly higher than roomtemperature. 3.3.Absorptionengineering Forbothpolaritonbranches,thelargestcontribu- We have demonstrated that the polariton absorp- tiontothelinewidthcomesfromtheintervalleyscat- tion depends on two key quantities: the polariton- tering into dark KΛ excitons reflecting the efficient phonon scattering rate Γ and the cavity decay rate scatteringplusthethree-folddegeneracyofthe Λval- γ . Now, we would like to tune the absorption by ley, similar to the excitonic case [42]. Furthermore, changing these quantities. While Γ is strongly sens- atroomtemperature,intervalleyscatteringwithinthe itive to temperature, γ is determined by the cavity K valley is also important. At 20K, the LP linewidth quality factor (i.e. in particular the reflectance of the is determined to a large extent by scattering into the cavityendmirrors). dark KK excitons. We stress that here we are focus- Infigures4(a)and(b)weshowthetemperature- ing on the scattering from the k =0 polariton state. dependent polariton–phonon scattering rates at There are further possible scattering channels at lar- k =0 for the UP and LP branch, respectively. Note germomenta,asshowninfigure3(b).Increasingthe that this corresponds to the half linewidth of the temperature to 300K increases the LP linewidth by respectivepolaritonresonanceinabsorptionspectra. around one order of magnitude as the absorption We add up different scattering channels including of intervalley phonons becomes possible. At 77 K intravalley scattering within the K valley (KK) as theintravalleycontributiontothephonon-scattering well as intervalley scattering into momentum-dark rates is very small, in accordance with figure 3. exciton states (KK and KΛ). For comparison, Thelinewidthoftheupperpolaritonisat20Kmuch 6 2DMater.10(2023)015012 BFerreiraetal largercomparedtotheLPbranchsinceemissioninto dark excitons is possible even at k =0 thanks to the much higher polariton energy (figure 1(b)). Hence, the increase in UP from 20K to 300 K is not as sub- stantialasintheLPcase.Overall,figures4(a)and(b) illustrate the huge impact of dark exciton states on thepolariton–phononscatteringratesandthusonthe polaritonabsorption. Afterhavingaddressedtheroleoftemperaturein tuning the polariton–phonon scattering rates Γ , we now focus on the change of the cavity decay rate γ as a function of the quality factor Q . In figure 4(c) n n weshow γ (grayline)and Γ (redandbluelines)as k k afunctionofQ .WefindthatUPhasacriticalcoup- lingconditionk =0aroundQ ≈200.Forsmallval- C f uesof the quality factor,the UP absorptionis expec- tedtoincrease,butaswemovefurtherawayfromthe criticalcouplingtheabsorptiondecreases.FortheLP, thecriticalcouplingconditionisonlyreachedathigh valuesofQ . 3.4.PolaritonabsorptionofMoSe So far we have studied the polariton absorption for WSe monolayers, where dark excitons turned out Figure5.AbsorptionofMoSe .(a)Surfaceplotofpolariton to play a crucial role. Now we investigate the MoSe absorptionofahBN-encapsulatedMoSe monolayerasa monolayerexhibitingadifferentenergeticalignment functionofmomentumandenergy(77K, ℏΩ =50meV andQ ≈160).(b)Absorptioncutsasafunctionofenergy of dark and bright states. With the latter being the f forthreedifferentmomenta.(c)Absorptionintensityasa lowest ones in MoSe [1, 15, 27], we expect only functionofmomentumforthelower(LP)andtheupper n n a negligible contribution from dark excitons. Simil- (UP)polaritonbranch.(d)Decayrates Γ and2γ asa k k functionofmomentumfortheLPandUPbranch.Thethin arly to the case of WSe , we show in figure 5(a) the greylinesin(c)and(d)correspondtothecasewithout absorptionofpolaritonsasafunctionofmomentum darkexcitons(i.e.consideringonlythebrightKKexcitons). and energy for the zero-detuning case at T =77K. We find a drastic reduction in absorption as well as in the linewidth of the LP absorption compared to WSe (figure 2(a)). This can be clearly observed in of dark states. For the LP branch, the scattering into the momentum cuts shown in figure 5(b). Although theenergeticallyhigherdarkexcitonstatesisgenerally theintensityoftheresonantabsorptionincreasesfor weakasitisdrivenbyabsorptionofintervalleyphon- larger momenta, similar to the case of WSe , quant- ons from high-momenta states. Since for LP also the itatively the increase is much slower, reaching only a intravalley scattering with acoustic modes is forbid- −1 maximal value of approximately 0.1 at k =1.5µm denduetomomentumandenergyconservation(due (compared to almost 0.5 predicted for WSe ). Inter- to the almost flat phonon dispersion making it diffi- estingly, for larger momenta we find also an increase culttofulfiltheenergyconservation[18]),thelower of the absorption for the UP branch (figure 5(b))— polaritonhasonlyaverysmallscatteringrate(thered opposite to the case of WSe (figure 2(b)). In addi- lineinfigure4(d)isalmostnotvisible).Nevertheless, tion,weobservealargeincreaseinthespectralwidth the decrease of the cavity decay rate γ with increas- ofpolaritonresonancesatlargerin-planemomentak. ing momenta allows for the critical coupling condi- −1 To microscopically understand the qualitative as tionattheveryhighmomentaofk =3.25 µm (cf well as quantitative differences of the momentum- figure5(d)),wheretheLPabsorptionreachesitsmax- resolvedabsorptioninMoSe andWSe ,weinvestig- imum value of A =0.5 (figure 5(c)). Interestingly, 2 2 atetheintensityoftheresonantpolaritonabsorption even though dark excitons have only a small contri- and the underlying polariton–phonon and cavity bution,theirpresenceshiftsthecriticalcouplingcon- decay rates. We assume the same value of reflectiv- ditiontoa smallermomentum(cf grey vsred line in ity r =0.99 as in figure 2, resulting in similar cav- figure5(c)). ity decay rates γ as for WSe As a consequence, the For the upper polariton, the intravalley scatter- observed difference in polariton absorption must be ing contribution is also dominant (only small devi- duetothephonon-scatteringrates.Weshowboththe ations between the grey and blue line), showing −1 absorption and decay rates also for the case without two step-like increases at k≈1 and k ≈1.3 µm dark excitons (grey lines in figures 5(c) and (d)). As due to the emission of intravalley optical modes expected,inMoSe thereisonlyaminorcontribution (LO/TOandA1withenergy36.1/36.6meVand30.3 7 2DMater.10(2023)015012 BFerreiraetal meV, respectively). In contrast to WSe , we observe Acknowledgments a large increase in the phonon-scattering rates for the UP branch, reflecting a more efficient intraval- We thank Marten Richter (TU Berlin) for inspir- leyscatteringwithopticalmodesinMoSe [41].This ing discussions. This project has received funding leads to the much broader spectral width of the res- support from the DFG via SFB 1083 (Project B9), onancesobservedinfigure5(b).Thecontributionof the European Union’s Horizon 2020 Research and dark excitons is minor, however, we still observe an Innovation programme under Grant Agreement No. opening of an emission channel into dark states, cf 881603 (Graphene Flagship) and from the Knut UP −1 the step-like increase of Γ at k ≈0.4 µm . This and Alice Wallenberg Foundation via the Grant opening is important for understanding the increase KAW 2019.0140. The computations were enabled by of the resonant absorption, when going from k =0 resources provided by the Swedish National Infra- −1 tok =0.75µm observedinfigure5(b)(incontrast structureforComputing(SNIC). to the prediction for WSe in figure 1(a)). Without dark states, there would be a decrease of the absorp- ORCIDiDs −1 tion up to approximately 0.9 µm (cf the grey line in figure 5(c)). In MoSe , the upper polariton fulfills BeatrizFerreira https://orcid.org/0000-0002- the critical coupling condition at the four different 5395-345X −1 momenta k ≈ 1, 1.2, 1.3 and 1.6 µm . The lowest RobertoRosati https://orcid.org/0000-0002-2514- two values of k are a consequence of polariton scat- teringintodarkexcitonstates. 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Journal

2D MaterialsIOP Publishing

Published: Jan 1, 2023

Keywords: exciton–polariton; TMD; dark excitons; 2D materials; absorption

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