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Daniel Erkensten, Samuel Brem, Koloman Wagner, Roland Gillen, Raül Perea-Causín, Jonas D. Ziegler, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Janina Maultzsch, Alexey Chernikov, Ermin Malic

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[Dark exciton-exciton annihilation in monolayer <math>  <msub>    <mrow>      <mi>WSe</mi>    </mrow>    <mn>2</mn>  </msub></math>](https://mdr.nims.go.jp/datasets/ad1afb89-a388-40b6-88e0-6ebd8125071c)

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Dark exciton-exciton annihilation in monolayer WSe$_{2}$PHYSICAL REVIEW B 104, L241406 (2021)Letter Editors’ SuggestionDark exciton-exciton annihilation in monolayer WSe2Daniel Erkensten ,1 Samuel Brem ,2 Koloman Wagner ,3,7 Roland Gillen ,4 Raül Perea-Causín ,1 Jonas D. Ziegler ,3,7Takashi Taniguchi ,5 Kenji Watanabe ,6 Janina Maultzsch ,4 Alexey Chernikov ,3,7 and Ermin Malic 2,11Department of Physics, Chalmers University of Technology, 41296 Gothenburg, Sweden2Department of Physics, Philipps-Universität Marburg, 35037 Marburg, Germany3Department of Physics, University of Regensburg, D-93040 Regensburg, Germany4Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen-Nürnberg, Germany5International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-004, Japan6Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, 1-1 Namiki, Tsukuba 305-004, Japan7Dresden Integrated Center for Applied Physics and Photonic Materials (IAPP) and Würzburg-Dresden Cluster of Excellence ct.qmat,Technische Universität Dresden, 01062 Dresden, Germany(Received 16 June 2021; revised 25 November 2021; accepted 1 December 2021; published 17 December 2021)The exceptionally strong Coulomb interaction in semiconducting transition-metal dichalcogenides (TMDs)gives rise to a rich exciton landscape consisting of bright and dark exciton states. At elevated densities, excitonscan interact through exciton-exciton annihilation (EEA), an Auger-like recombination process limiting theefficiency of optoelectronic applications. Although EEA is a well-known and particularly important processin atomically thin semiconductors determining exciton lifetimes and affecting transport at elevated densities,its microscopic origin has remained elusive. In this joint theory-experiment study combining microscopic andmaterial-specific theory with time- and temperature-resolved photoluminescence measurements, we demonstratethe key role of dark intervalley states that are found to dominate the EEA rate in monolayer WSe2. We revealan intriguing, characteristic temperature dependence of Auger scattering in this class of materials with anexcellent agreement between theory and experiment. Our study provides microscopic insights into the efficiencyof technologically relevant Auger scattering channels within the remarkable exciton landscape of atomically thinsemiconductors.DOI: 10.1103/PhysRevB.104.L241406I. INTRODUCTIONAtomically thin nanomaterials, such as transition-metaldichalcogenides (TMDs), offer an unprecedented platform tostudy intriguing many-particle phenomena in a broad rangeof external conditions [1–4]. The weak dielectric screeningand the resulting strong Coulomb interaction in these ma-terials give rise to the formation of tightly bound excitonsand promote efficient interactions between charge carriers atelevated densities. In particular, excitons can interact throughexciton-exciton annihilation (EEA), an Auger recombinationprocess shown to be very efficient in TMDs [5–8]. EEA is anonradiative scattering process, in which one exciton recom-bines nonradiatively by transferring its energy and momentumto another exciton, resulting in a highly excited electron-holepair (HX) [9–11], cf. Fig. 1(a). The inverse process of impactexcitation resulting in charge carrier multiplication has alsobeen recently observed [12]. Auger recombination leads toan effective saturation of exciton densities [1] and is thusof crucial importance for the performance of many techno-logical applications, such as photodetectors and solar cells.Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI. Fundedby Bibsam.Auger scattering has previously been shown to be extremelyefficient in graphene [13–16], but was initially consideredto be inefficient in TMDs due to the difficulty to simulta-neously conserve energy and momentum in parabolic bandstructures and the lack of resonant final states. However,recent up-converted photoluminescence (PL) measurementsand ab initio calculations confirmed the existence of a higherenergetic exciton state appearing at approximately twice theA exciton resonance both in monolayer and bilayer WSe2[10,17]. This can be attributed to the existence of higher-lying conduction bands, enabling a particular type of resonantAuger scattering [9,18,19], cf. Fig. 1. In the regular intravalleyAuger recombination process discussed so far in the literature,an optically excited carrier recombines with the hole at theK point and induces the excitation of another carrier into ahigher conduction band [process I in Fig. 1(b)].A microscopic understanding of Auger-like exciton-exciton annihilation in atomically thin semiconductors in theentire exciton landscape is still lacking. Including just theregular intravalley Auger processes turns out to be far fromsufficient to explain the large EEA rates measured acrossdifferent TMD materials [5–8,20]. Phonon-assisted exciton-electron Auger recombination including dark states was alsoshown to be strongly suppressed [21] and cannot be responsi-ble for the EEA efficiency seen in experiments. Furthermore,the recent observations of a strong substrate dependence ofAuger scattering in TMD monolayers [22,23] still require aconsistent explanation. Finally, even the bound or free nature2469-9950/2021/104(24)/L241406(7) L241406-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-9632-6376https://orcid.org/0000-0001-8823-1302https://orcid.org/0000-0001-5561-5318https://orcid.org/0000-0002-7913-0953https://orcid.org/0000-0002-2229-0147https://orcid.org/0000-0002-2683-5124https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-6088-2442https://orcid.org/0000-0002-9213-2777https://orcid.org/0000-0003-1434-9003http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.104.L241406&domain=pdf&date_stamp=2021-12-17https://doi.org/10.1103/PhysRevB.104.L241406https://creativecommons.org/licenses/by/4.0/https://www.kb.se/samverkan-och-utveckling/oppen-tillgang-och-bibsamkonsortiet/bibsamkonsortiet.htmlDANIEL ERKENSTEN et al. PHYSICAL REVIEW B 104, L241406 (2021)FIG. 1. Schematic illustration of exciton-exciton annihilation(EEA) channels in WSe2. (a) The annihilation of A excitons (purple)gives rise to a higher-lying HX exciton state (green). (b) Regularintravalley (I blue) and additional intervalley Auger recombinationprocesses (II orange and III red) involving momentum-dark KK′,K�, and K� excitons, respectively. The spin-split conduction bandsare distinguished by black and grey lines, respectively.of the highly excited electron-hole pair that remains in thesystems after EEA is not clear. Importantly, the rich excitoniclandscape of TMDs consists not only of bright intravalleyexcitons but also of momentum-dark intervalley excitonswith nonzero center-of-mass momenta [24,25]. These are ex-pected to open up additional channels for Auger scatteringthat would satisfy momentum and energy conservation re-quirements. Moreover, since the energetically lowest states oftungsten-based TMDs are dark [24,26–28], one would expectintervalley exciton-exciton Auger recombination processes[II, III in Fig. 1(b)] to be particularly relevant.In this joint theory-experiment study, we address the natureof Auger-like exciton-exciton annihilation in atomically thinsemiconductors by combining time- and temperature-resolvedPL measurements with material-specific microscopic model-ing including density matrix and density functional theorymethods. In particular, we investigate intra and intervalleyAuger recombination channels in monolayer WSe2 for dif-ferent substrates and temperatures. Crucially, we show thatdark intervalley Auger recombination clearly dominates theexciton-exciton annihilation. We reveal an intriguing tem-perature dependence, characteristic for the impact of darkstates—in excellent agreement between theory and exper-iment. Moreover, our calculations provide insight into thepreviously observed decrease of Auger scattering for hBN-encapsulated WSe2 and WS2 monolayers [20,22,23] and weexplain this effect with the changed resonance conditionwithin the excitonic bandstructure.II. MODELING OF EXCITON-EXCITONANNIHILATION RATESTo develop a realistic and material-specific approach pro-viding microscopic insights into exciton-exciton annihilationprocesses in TMD monolayers, we combine first-principlecalculations with the excitonic density matrix formalism[28,29]. First, we define the many-particle Hamilton operatorHx−x = 12∑μνρQ,Q′W μνρQ,Q′Y†μ,Q+Q′Xν,QXρ,Q′ + H.c. (1)describing the annihilation of two excitons in the states νand ρ with center-of-mass momenta Q and Q′, respectively,which gives rise to the formation of a single higher ener-getic exciton in the state μ. The later state can be generallyconsidered to be either a ground or an excited state with aprincipal quantum number n that could also include unboundelectron-hole pairs in the limit of n → ∞. We distinguishbetween excitons formed from the higher-lying conductionband c′ and the valence band (in the following denoted asHX excitons) and regular spin-allowed A excitons through theexciton creation operators X † = c†v and Y † = c′†v, respec-tively, cf. Fig. 1(a). Here we emphasize that HX excitons canbe generally formed by electrons and holes located at differenthigh-symmetry points of the hexagonal Brillouin zone. Thecompound indices μ, ν, and ρ include the excitonic spin, theprincipal quantum numbers n = 1s, 2s, . . ., and the excitonicvalley ξ = (ξhξe) = KK(′ ), K�, K�, where the first (second)letter describes the valley in which the Coulomb-bound hole(electron) is localized. In this work, we consider the hole to beat the K point, but allow the electron to be at the K, K′, � or� point, cf. Fig. 1(b).The Auger matrix element W μνρQ,Q′ appearing in Eq. (1)determines the efficiency of the exciton-exciton annihilationprocess and consists of a direct and an exchange term, theexchange terms reflects the fermionic character of the ex-citons. The direct and exchange components of the matrixelement, which is provided in Eq. (S11) in the SupplementalMaterial (SM) [30], crucially depend on the A and HX excitonwave functions (obtained from the Wannier equation [31–33])as well as the screened Coulomb interaction and electronicdipole matrix elements, with the matrix elements being ex-tacted from ab initio G0W0 calculations. Additional details onthe microscopic derivation of the Auger matrix elements andthe ab initio modeling are found in Secs. II A and III (SM),respectively.Figure 2(a) illustrates the excitonic Auger matrix elementsas a function of the principal quantum number n of the finalHX exciton state for the intra and intervalley processes I, II,and III in hBN-encapsulated WSe2 depicted in Fig. 1(b). Here,we evaluated the matrix elements for Q = Q′ = 0, while thefull momentum-dependent matrix elements are provided inSM, cf. Fig. S2. We find that, due to the large momentumtransfer, the intervalley (ξν = ξρ = KK′, ξμ = K�) matrixelement is up to two orders of magnitude smaller than the cor-responding intravalley (ξν = ξρ = ξμ = KK) matrix element(note the logarithmic scale). Note that we restrict our calcu-lations to A excitons in the ground state n = 1s, and vary theprincipal quantum number of the HX exciton in Fig. 2(a). Wefind that the matrix elements decrease rapidly with increasingquantum index n due to a shrinking momentum-space overlapwith the initial 1s state. The only exception is the intravalleyn = 2 HX state with a slightly larger coupling than for n = 1.Most importantly, these results imply that the coupling tounbound electron-hole pairs in the limit of n → ∞ shouldbe small. Based on our microscopic model for the excitonicAuger matrix element, we now determine the exciton-excitonannihilation coefficient RA.The resulting experimentally accessible recombinationrates lead to an effective saturation of exciton densities andare thus important for many technological devices basedon TMDs. We exploit the Heisenberg’s equation of mo-tion to determine the temporal evolution of the density ofL241406-2DARK EXCITON-EXCITON ANNIHILATION IN … PHYSICAL REVIEW B 104, L241406 (2021)FIG. 2. Auger matrix elements W and valley-specific excitondensities in hBN-encapsulated WSe2. (a) Intravalley and intervalleyexciton Auger matrix elements |W |2 evaluated at Q = Q′ = 0 andtheir dependence on the final (HX) state quantum index n. The matrixelements decrease rapidly with n due to the small overlap betweeninitial 1s states and n > 1s HX states. (b) Temperature-dependentvalley-specific exciton densities in thermal equilibrium illustratingthe crucial impact of intervalley K� (orange) and KK′ (red) excitonsand weak impact of intravalley (blue) excitons.A excitons nx = ∑νQ NνA,Q with the momentum-dependentexciton occupation NνA,Q = 〈X †ν,QXν,Q〉 that we estimate by athermal Boltzmann distribution in this work. The temperature-dependent valley-specific densities nνx (T ) = ∑Q NνA,Q areillustrated in Fig. 2(b), revealing that K� excitons dominatethe density at room temperature for monolayer WSe2. Thisreflects the energetic separation between bright and dark states(cf. Table I in SM) and the three-fold degeneracy of the �valley [34]. The three-fold degeneracy explicitly enters theexciton distribution, crucially enhancing the occupation ofK� excitons relative to KK and KK′ excitons.Applying the second-order Born-Markov approximation[32] (cf. Sec. II B in SM), we find ṅx = −RAn2x with theexciton-exciton annihilation rate coefficient or briefly Augercoefficient RA readingRA = 2πh̄∑μ,ν,ρQ,Q′∣∣W μνρQ,Q′∣∣2N̄νA,Q(T )N̄ρA,Q′ (T )δ( ε), (2)with N̄ = N/nx. The appearing delta function δ( ε) en-sures that energy is conserved during the scattering processwith  ε =   + εμHX,Q+Q′ − ενA,Q − ερA,Q′ . The detuning   de-termines the resonance condition for the Auger scatteringprocess that is strongly enhanced for   � kBT . For the in-travalley Auger process (ξμ = ξν = ξρ=KK), the detuning isdefined as   = EHX − 2EA, where EHX and EA are excitonresonance energies of the final and initial states, respec-tively [cf. Fig. 1(b)]. These energies can be directly obtainedfrom recent up-converted photoluminescence measurementsfor monolayer WSe2 with EHX = 3.35 eV and EA = 1.734 eV[10,17]. In the case of intervalley EEA processes, the detun-ing requires the knowledge of binding energies of HX andA excitons, which are microscopically calculated by solvingthe Wannier equation (see Sec. II C in SM). Furthermore,we approximate the exciton dispersion as parabolic at theconsidered high-symmetry points, i.e., εμQ = h̄2Q22Mμ with thetotal exciton mass Mμ = mμee + mμhh and the effective electron(hole) masses mμee (mμhh ) being extracted from first-principlecalculations [34]. In particular, we note that the effectivemass approximation is expected to hold for the final KKHX state as shown by recent ab initio calculations [10].Finally, we take into account that excitonic resonances be-come red-shifted with increasing temperature [35–37]. Asa result, we obtain a temperature-dependent detuning, i.e.,  →  (T ) =   +  v (T ) with the shift of  v (T ) = αT 2T +βdescribed by the Varshni model, where the constants α and βare extracted from temperature-dependent photoluminescencemeasurements [37,38].III. EXCITON-EXCITON ANNIHILATION RATESWe evaluate the exciton-exciton annihilation coefficient RAfrom Eq. (2) for an hBN-encapsulated WSe2 monolayer. Wetake explicitly into account bright and dark A excitons (KK,KK′, and K�) as initial states and HX excitons (KK, KK′,and K�) up to n = 3s as final states for the Auger scatteringprocess. The overlap of higher-lying states (n > 3s) is negli-gibly small and thus neglected in the following, cf. Fig. 2(a).In addition to matrix elements and resonance conditions, theefficiency of the Auger process is strongly determined by thedistribution of excitons across lower-lying states in thermalequilibrium. Due to the energy splitting between KK, KK′,and K� excitons being on the order of tens of meVs, thevalley-specific densities strongly change with temperature,as illustrated in Fig. 2(b). The resulting calculated Augercoefficients RA are presented in Fig. 3(a) as a function oftemperature.To test the theoretically predicted Auger-recombinationmechanism we take advantage of density-dependent tem-porally resolved photoluminescence on hBN-encapsulatedWSe2 monolayers. At all studied temperatures, the PL tran-sients exhibit a density-dependent increase of the initial decayrate (cf. Fig. S1 in SM) that is well described by the bi-molecular recombination law. Being accompanied by thesaturation of the total PL intensity, this behavior is charac-teristic for exciton-exciton annihilation, as was previouslydemonstrated in the literature [5–8]. At each temperature,we extract the density-induced recombination rate rexpA as afunction of injected exciton density nx at early times afterthe excitation. The bimolecular coefficient attributed to Augerrecombination RA is then determined from the slope, as illus-trated in the inset in Fig. 3(b) (cf. Sec. I in SM for details).The potential contributions to the bimolecular recombinationrate from biexciton formation [39,40] can be excluded sincethe biexcitons form much faster [41,42] than the observeddensity-dependent decay. Furthermore, we emphasize that atthe considered densities the average interparticle separation isalways smaller than the diffusion length, which should be amandatory condition for Auger scattering. In the investigatedsamples, long-lived dark excitons exhibit diffusion lengths onthe order of hundreds of nm [43], whereas the exciton-excitonL241406-3DANIEL ERKENSTEN et al. PHYSICAL REVIEW B 104, L241406 (2021)FIG. 3. Temperature dependence of the Auger coefficient RA for hBN-encapsulated WSe2. (a) Theoretically calculated RA illustratingthe separate contributions of intravalley and intervalley Auger processes (KK, KK′, and K�, respectively), cf. Fig. 1(b). We reveal that therecombination of two K� excitons is the dominant process for temperatures above 30 K. (b) Experimentally extracted Auger coefficients fromtime-resolved PL measurements. Error bars represent lower and upper limits for the extracted values. The inset illustrates the density-inducedrecombination rate rexpA , from which the Auger coefficient can be extracted from the slope.separation ranges within 30 and 10 nm for the studied densi-ties between 1011 and 1012 cm−2.Experimentally obtained, temperature-dependent Augercoefficients are presented in Fig. 3(b) in direct comparison tothe theoretical predictions in Fig. 3(a). Both, in theory andexperiment the Auger coefficient increases by an order ofmagnitude when increasing the temperature in the range of 50to 100 K and remains nearly constant up to room temperature.From microscopic calculations we obtain RA = 0.005 cm2/sand 0.05 cm2/s at T = 10 K and T = 300 K, respectively.These values match (without fitting) the experimentally de-termined coefficients of 0.001–0.006 cm2/s at T � 50 K and0.04–0.1 cm2/s at 300 K. The obtained quantitative agree-ment between theory and experiment strongly supports boththe predominant role of the dark excitons and the n = 1 HXfinal states for Auger recombination.To understand the microscopic origin of the drastic in-crease of the Auger coefficients as a function of temperature,we explicitly separate the contributions from intra and in-tervalley Auger scattering in theory [corresponding to theprocesses I, II, and III illustrated in Fig. 1(b)]. We find thatthe intervalley KK′ and K� Auger scattering involving themomentum-dark KK′ and K� excitons clearly dominate theAuger coefficient RA for low and high temperatures, respec-tively [red and orange region in Fig. 3(a)]. The intravalleyAuger scattering involving the bright KK excitons is neg-ligible (blue region, note the logarithmic scale). This is aconsequence of the spectral ordering of exciton states in WSe2monolayers, where the momentum-dark KK′ excitons are theenergetically lowest states followed by the dark K� excitonsand finally the bright KK states. The relative position of theseexciton states has been determined microscopically by solvingthe exciton Wannier equation (see SM for more details). As aresult, the dark states carry by far the largest occupation andthus dominate the Auger scattering processes, despite smallervalues of the Auger matrix elements.The obtained temperature dependence of the Auger coeffi-cient thus strongly reflects the changes in the valley-specificexciton densities nνx (T ) with ν = KK, KK′, K�, cf. Fig. 2(b).At low temperatures up to 20 K the KK′ intervalley Augerprocess [red area in Fig. 3(a)] is the predominant scatteringchannel. The K� Auger scattering takes over from 20 Kand quickly becomes, by far, the most prominent contribution[orange area in Fig. 3(a)]. The sharp increase in the Augercoefficient between 50–100 K reflects the predominant pop-ulation of the three-fold degenerate K� state at intermediatetemperatures [cf. Fig. 2(a)]. Furthermore, the Auger matrixelement is much more efficient for K� than KK′ intervalleyscattering due to the larger momentum transfer involved in thelater process suppressing the interaction [cf. Fig. 2(b)]. ForT > 100 K, only minor changes in the relative exciton distri-butions result in approximately constant Auger coefficients athigher temperatures.Similar to the influence of temperature, the overall effi-ciency of the excitonic Auger scattering should also stronglydepend on the dielectric environment that modifies the relativeenergies of the involved excitonic states. Recently, a strongsuppression of the Auger recombination in hBN-encapsulatedTMD monolayers was experimentally demonstrated [22,44].The theoretical approach outlined above now allows us toanalyze and reveal the microscopic origin of this effect. Inthe following, we investigate the impact of the dielectric en-vironment on the exciton-exciton annihilation rate and study,in particular, the case of hBN-encapsulated samples versussamples placed on the standard SiO2 substrate.In Fig. 4(a), we illustrate the combined temperatureand substrate dependence of the Auger coefficient RA.There are two distinct trends: (i) For a fixed temperature,Auger scattering becomes less efficient with the dielectricscreening and (ii) the Auger coefficients show a maximum ata certain substrate-dependent temperature. The first trend isfurther shown in Fig. 4(b), where we consider the screeningdependence of Auger coefficients for three different tem-peratures. The reduction of Auger scattering with screeningis observed at all temperatures. Comparing the two mostcommon dielectric environments, SiO2 (εs ≈ 3.9+12 = 2.45)and hBN-encapsulation (εs = 4.5), we find RA,SiO2 =0.13 cm2/s vs RA,hBN = 0.05 cm2/s at room temperature,i.e., we predict a reduction of the Auger coefficient byapproximately 60% in the case of hBN-encapsulated WSe2.This reduction can partly be attributed to a weakenedCoulomb interaction with screening, but importantly itL241406-4DARK EXCITON-EXCITON ANNIHILATION IN … PHYSICAL REVIEW B 104, L241406 (2021)FIG. 4. Substrate and temperature dependence of the Auger co-efficient RA in monolayer WSe2 from microscopic theory. (a) RA asa function of screening and temperature, revealing a nonmonotonictemperature behavior. (b) Screening dependence of RA for differenttemperatures T = 50, 100, and 300 K. (c) Temperature-dependentAuger coefficients in the case of hBN-encapsulation (εs = 4.5) andfor the SiO2 substrate (εs = 2.45). As illustrated in the inset forthe K� Auger process, the increase in screening from εs = 2.45 toεs = 4.5 makes the Auger scattering more off-resonant leading to aless efficient RA in hBN-encapsulated TMDs.is also a consequence of quenched resonance conditionsdetermined by the detuning  , cf. Eq. (2). To furtherquantify these effects, we determine the decrease in theAuger matrix element between samples on a SiO2 substrateand hBN-encapsulated samples for the predominant K�Auger scattering channel to be approximately 30% inhBN-encapsulated WSe2 monolayers.To understand how the resonance conditions change withthe substrate we investigate the screening dependence of theindividual components entering the detuning. The A reso-nance energy (initial state) is known to only weakly varywith dielectric screening due to the simultaneous reductionof the band-gap and the exciton binding energy [45,46] andtherefore it can be assumed to be approximately constant.Moreover, the energy splittings between different conductionbands are expected to be to a large extent independent ofscreening since the Coulomb renormalization only affects theabsolute energies of the bands [47]. The detuning for thedominant K�-Auger channel at room temperature acquiresa weak screening dependence [cf. the inset of Fig. 4(c)],stemming from the different screening-induced changes inintra and intervalley binding energies. We predict  hBN ≈2 SiO2 = 30 meV for the K� Auger scattering channel withthe dark 1s K� HX exciton as final state [cf. Fig. 1(b)]resulting in weaker Auger coefficients for hBN-encapsulatedsamples. The contribution from higher-order exciton states(n > 1s) to the Auger rates is seen to be suppressed due tolarge detunings (e.g., | |2s,hBN ≈ 200 meV). This in com-bination with the weakened Auger matrix elements is theorigin of the previous experimental observations showingstrongly quenched Auger scattering for hBN-encapsulatedTMDs [22,44]. Note that defect-assisted Auger scatteringmight also play a role for TMDs on a SiO2 substrate due todisorder, while it is expected to be negligible in the case ofhBN-encapsulation.Finally, we demonstrate an intriguing nonmonotonic tem-perature dependence of Auger scattering for fixed dielectricscreening in the case of SiO2 and hBN-encapsulation,cf. Fig. 4(c). Interestingly, we find a clear maximumin the Auger coefficient for WSe2 on SiO2 at around80 K. The Auger coefficient RA can be approximated byRA ≈ 1kBT exp[−| |/(kBT )], displaying a maximum atTmax = | |/kB [18]. This approximate expression becomesexact in the limit of a constant Auger matrix element andwhen a single exciton species ν = ρ ≡ ν0 dominates theAuger coefficient. The approximation allows us to understandthe temperature dependence of the Auger coefficients,which is determined by the initial A exciton distributionand the availability of initial and final states fulfilling theconservation of energy and momentum, cf. Eq. (2). For verylow temperatures the exciton distribution is strongly localizedat vanishing kinetic energies and hence, the EEA is inefficientdue to the nonzero energy detuning of initial and final states.With increasing temperature the momentum-dependentdistribution becomes broadened, facilitating energy andmomentum conservation, and thus offering additionalscattering channels leading to an enhanced RA. A furtherincrease in temperature leads to a redistribution of excitonsand an overall decrease in the initial population of excitonstates, resulting in a reduction of the Auger coefficients. Theinterplay of those two effects is the origin of the observedmaximum in the RA at certain intermediate temperatures. Inthe case of hBN-encapsulated samples we do not observe apronounced maximum of the Auger coefficient as a functionof temperature, as the resonance energy cannot be reachedin the considered temperature range, i.e., | | > kBT atall temperatures. Here, the large scattering efficiency athigher temperatures is solely determined by the K� excitonoccupation.IV. CONCLUSIONIn this joint theory-experiment study combining micro-scopic, material-specific modeling with time-resolved pho-toluminescence measurements, we establish a fundamentalunderstanding of Auger-like exciton-exciton annihilation pro-cesses in atomically thin semiconductors. We demonstratethe key importance of dark intervalley excitons in the proto-typical WSe2 material resulting in an intriguing temperaturedependence of Auger processes. We find an excellentqualitative and quantitative agreement between theory andexperiment without any adjusted free parameters. Our resultsalso contribute to resolving an open question in the litera-ture regarding the origin of consistently observed suppressionL241406-5DANIEL ERKENSTEN et al. PHYSICAL REVIEW B 104, L241406 (2021)of exciton-exciton annihilation upon hBN-encapsulation.Overall, our work provides microscopic insights into themany-particle processes behind the technologically impor-tant exciton-exciton annihilation channels in atomically thinsemiconductors. The developed approach can be further gen-eralized to van der Waals heterostructures and twisted moiréexciton systems.ACKNOWLEDGMENTSWe thank Maja Feierabend (Chalmers) and Paulo Eduardode Faria Junior and Kai-Qiang Lin (University of Regensburg)for fruitful discussions. This project received funding fromDeutsche Forschungsgemeinschaft via CRC 1083 (ProjectNo. B09), Emmy Noether Initiative (CH 1672/1, Project-ID 287022282), CRC 1277 (Project-ID 314695032, ProjectNo. B05), CRC 953 (Project No. B13), the EuropeanUnions Horizon 2020 research and innovation program un-der Grant agreement No. 881603 (Graphene Flagship) andthe Würzburg-Dresden Cluster of Excellence on Complex-ity and Topology in Quantum Matter ct.qmat (EXC 2147,Project-ID: 390858490). 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