Dark exciton-exciton annihilation in monolayer transition-metal dichalcogenides

Daniel Erkensten1,∗ , Samuel Brem2, Koloman Wagner3, Roland Gillen4, Raul Perea-Causın1, Jonas D.Ziegler3, Takashi Taniguchi5, Kenji Watanabe6, Janina Maultzsch4, Alexey Chernikov3,7, and Ermin Malic1,2

1 Department of Physics, Chalmers University of Technology, Gothenburg, Sweden2Department of Physics, Philipps-Universitat, 35037 Marburg, Germany

3 Department of Physics, University of Regensburg, D-93040 Regensburg, Germany4 Department of Physics, Friedrich-Alexander-Universitat Erlangen-Nurnberg, Erlangen-Nurnberg, Germany

5 International Center for Materials Nanoarchitectonics,National Institute for Materials Science, Tsukuba, Ibaraki 305-004, Japan

6 Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-004, Japan7 Institute for Applied Physics, Dresden University of Technology, Dresden, 01187, Germany

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. Atelevated densities, excitons can interact through exciton-exciton annihilation (EEA), an Auger-likerecombination process limiting the efficiency of optoelectronic applications. Although EEA is a well-known and particularly important process in atomically thin semiconductors determining excitonlifetimes and affecting transport at elevated densities, its microscopic origin has remained elusive.In this joint theory-experiment study combining microscopic and material-specific theory with time-and temperature-resolved photoluminescence measurements, we demonstrate the key role of darkintervalley states that are found to dominate the EEA rate in monolayer WSe2. We reveal anintriguing, characteristic temperature dependence of Auger scattering in this class of materials withan excellent agreement between theory and experiment. Our study provides microscopic insights intothe efficiency of technologically relevant Auger scattering channels within the remarkable excitonlandscape of atomically thin semiconductors.

Atomically thin nanomaterials, such as transition-metal dichalcogenides (TMDs), offer an unprecedentedplatform to study intriguing many-particle phenomenain a broad range of external conditions [1–4]. The weakdielectric screening and the resulting strong Coulomb in-teraction in these materials give rise to the formation oftightly bound excitons and promote efficient interactionsbetween charge carriers at elevated densities. In particu-lar, excitons can interact through exciton-exciton annihi-lation (EEA), an Auger recombination process shown tobe very efficient in TMDs [5–8]. EEA is a non-radiativescattering process, in which one exciton recombines non-radiatively by transferring its energy and momentum toanother exciton, resulting in a highly-excited electron-hole pair (HX), cf. Fig.1 (a). The inverse process of im-pact excitation resulting in charge carrier multiplicationhas also been recently observed [9]. Auger recombina-tion leads to an effective saturation of exciton densitiesand is thus of crucial importance for the performance ofmany technological applications, such as photodetectorsand solar cells [1].

Auger scattering has previously been shown to be ex-tremely efficient in graphene [10–13], but was initiallyconsidered to be inefficient in TMDs due to the diffi-culty to simultaneously conserve energy and momentumin parabolic band structures and the lack of resonantfinal states. However, recent up-converted photolumi-nescence (PL) measurements and ab-initio calculationshave confirmed the existence of a higher energetic exci-

∗ Corresponding author: daniel.erkensten@chalmers.se

FIG. 1. Schematic illustration of exciton-exciton annihila-tion (EEA) channels in WSe2. (a) The annihilation of Aexcitons (purple) gives rise to a higher-lying HX exciton state(green). (b) Regular intravalley (I blue) and additional inter-valley Auger recombination processes (II orange and III red)involving momentum-dark KK’, KΛ and KΓ excitons, respec-tively. The spin-split conduction bands are distinguished byblack and grey lines, respectively.

ton state appearing at approximately twice the A excitonresonance both in monolayer and bilayer WSe2 [14, 15].This can be attributed to the existence of higher-lyingconduction bands, enabling a particular type of resonantAuger scattering [16, 17], cf. Fig.1. In the regular in-travalley Auger recombination process discussed so farin literature, an optically excited carrier recombines withthe hole at the K point and induces the excitation of an-other carrier into a higher conduction band (process I inFig.1 (b)).

A microscopic understanding of Auger-like exciton-exciton annihilation in atomically thin semiconductors isstill lacking. Including just the regular intravalley Augerprocesses turns out to be far from sufficient to explain

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the large EEA rates measured across different TMD ma-terials [5–8, 18]. Phonon-assisted exciton-electron Augerrecombination including dark states was also shown to bestrongly suppressed [19] and can not be responsible forthe EEA efficiency seen in experiments. Furthermore,the recent observations of a strong substrate dependenceof Auger scattering in TMD monolayers [20, 21] still re-quire a consistent explanation. Finally, even the boundor free nature of the highly-excited electron-hole pair thatremains in the systems after EEA is not clear. Impor-tantly, the rich excitonic landscape of TMDs consists notonly of bright intravalley excitons but also of momentum-dark intervalley excitons with non-zero center-of-massmomenta [22, 23]. These are expected to open up ad-ditional channels for Auger scattering that would satisfymomentum and energy conservation requirements. More-over, since the energetically lowest states of tungsten-based TMDs are dark [22, 24–26], one would expect in-tervalley exciton-exciton Auger recombination processes(II, III in Fig. 1(b)) to be particularly relevant.

In this joint theory-experiment study, we addressthe nature of Auger-like exciton-exciton annihilation inatomically thin semiconductors by combining time- andtemperature-resolved PL measurements with material-specific microscopic modeling including density matrixand density functional theory methods. In particu-lar, we investigate intra- and intervalley Auger recom-bination channels in monolayer WSe2 for different sub-strates and temperatures. Crucially, we show that darkintervalley Auger recombination clearly dominates theexciton-exciton annihilation. We reveal an intriguingtemperature dependence, characteristic for the impactof dark states - in excellent agreement between the-ory and experiment. Moreover, our calculations explainthe previously observed decrease of Auger scattering forhBN-encapsulated WSe2 monolayers [18, 20, 21] with thechanged resonance condition within the excitonic band-structure.

Modeling of exciton-exciton annihilation rates. Todevelop a realistic and material-specific approach provid-ing microscopic insights into exciton-exciton annihilationprocesses in TMD monolayers, we combine first-principlecalculations with the excitonic density matrix formalism[26, 27]. First, we define the many-particle Hamilton op-erator

Hx−x =1

2

∑

µνρQ,Q′

WµνρQ,Q′Y

†µ,Q+Q′Xν,QXρ,Q′ + h.c. (1)

describing the annihilation of two excitons in the statesν and ρ, which gives rise to the formation of a singlehigher energetic exciton in the state µ. The latter canbe generally considered to be either a ground or an ex-cited state with a principal quantum number n that couldalso include unbound electron-hole pairs in the limit ofn → ∞. We distinguish between excitons formed fromthe higher-lying conduction band c′ and the valence band(in the following denoted as HX excitons) and regular

FIG. 2. Auger matrix elements W and valley-specific ex-citon densities in hBN-encapsulated WSe2. (a) Intravalleyand intervalley exciton Auger matrix elements |W |2 evalu-ated at Q = Q′ = 0 and their dependence on the final (HX)state quantum index n. The matrix elements decrease rapidlywith n, due to the small overlap between initial 1s states andn > 1s HX states. (b) Temperature-dependent valley-specificexciton densities in thermal equilibrium illustrating the cru-cial impact of intervalley KΛ (orange) and KK’ (red) excitons.

spin-allowed A excitons through the exciton creation op-erators X† = c†v and Y † = c′†v, respectively, cf. Fig.1(a). Here we emphasize that HX excitons can be gen-erally formed by electrons and holes located at differenthigh symmetry points of the hexagonal Brillouin zone.The compound indices µ, ν and ρ include the excitonicspin, the principal quantum numbers n = 1s, 2s..., andthe excitonic valley ξ = (ξhξe) = KK(′),KΛ,KΓ, wherethe first (second) letter describes the valley in which theCoulomb-bound hole (electron) is localized. In this work,we consider the hole to be at the K point, but allow theelectron to be at the K, K’, Λ or Γ point, cf. Fig. 1(b).

The Auger matrix element WµνρQ,Q′ = Dµνρ

Q,Q′ − EµνρQ,Q′

appearing in Eq. (1) determines the efficiency of theexciton-exciton annihilation process and consists of a di-rect Dµνρ

Q,Q′ and an exchange term EµνρQ,Q′ reading

DµνρQ,Q′ = Wµeνhρeνe

el,Q′ Gµνρ +Wµeρhνeρeel,Q Gµρν ,

EµνρQ,Q′ =∑

q,q′

(Wµeνhρeνeel,q−q′ Gµνρq,q′ +Wµeρhνeρe

el,q−q′ Gµρνq,q′) . (2)

Here, we have introduced Gµνρ =∑

q,q′ Gµνρq,q′ , G

µνρq,q′ ≡

φ∗µ,qϕν,qϕρ,q′ with φµ,q and ϕν,q being HX and A exci-ton wave functions, respectively. The latter are micro-scopically obtained by solving the Wannier equation [28–30]. Note that Eq. (2) holds for excitonic wave functionswith a weak center-of-mass momentum dependence. Inthis case the exchange matrix element EµνρQ,Q′ becomes ap-proximately constant and non-vanishing in momentum.

The appearing electronic Auger matrix elements Wel,q

depend on the valley-specific overlap of Bloch wave func-

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tions and are obtained in a k · p framework [31, 32] withparameters extracted from first-principle G0W0 calcula-tions. Fig. 2(a) illustrates the excitonic Auger matrixelements as a function of the principal quantum numbern of the final HX exciton state for the intra- and inter-valley processes I, II and III in hBN-encapsulated WSe2depticed in Fig. 1(b). Here, we evaluated the matrix ele-ments for Q = Q′ = 0, while the full momentum depen-dent matrix elements are provided in the SupplementaryMaterial (SM). We find that due to the large momentumtransfer, the intervalley (ξν = ξρ =KK’, ξµ =KΓ) matrixelement is up to two orders of magnitude smaller thanthe corresponding intravalley (ξν = ξρ = ξµ=KK) ma-trix element (note the logarithmic scale). Moreover, formomentum transfers of the order of the reciprocal lat-tice vector, the electronic matrix elements are approxi-mately constant, resulting in a cancellation of direct andexchange terms, cf. Eq. (2) and SM for more details.Note that we restrict our calculations to A excitons inthe ground state n = 1s, and vary the principal quan-tum number of the HX exciton in Fig. 2(a). In principle,higher order A exciton states could also be included, butEEA processes involving n > 1s A excitons are negligi-bly small, since these states are only weakly populated inthermal equilibrium. We find that the matrix elementsdecrease rapidly with increasing quantum index n dueto a shrinking momentum-space overlap with the initial1s state. The only exception is the intravalley n = 2HX state with a slightly larger coupling than for n = 1(see SM). Most importantly, these results imply that thecoupling to unbound electron-hole-pairs in the limit ofn → ∞ should be negligible. Based on our microscopicmodel for the excitonic Auger matrix element, we now de-termine the exciton-exciton annihilation coefficient RA.

The resulting experimentally accessible recombinationrates lead to an effective saturation of exciton densitiesand are thus important for many technological devicesbased on TMDs. We exploit the Heisenberg’s equationof motion to determine the temporal evolution of the den-sity of A excitons nx =

∑νQN

νA,Q with the momentum-

dependent exciton occupation NνA,Q = 〈X†ν,QXν,Q〉 that

we estimate by a thermal Boltzmann distribution in thiswork. The temperature-dependent valley-specific densi-ties nνx(T ) are illustrated in Fig. 2(b), revealing that KΛexcitons dominate the density at room temperature formonolayer WSe2. This reflects the energetic separationbetween bright and dark states and the three-fold degen-eracy of the Λ valley (cf. SM).

Applying the second-order Born-Markov approxima-tion [29] (cf. SM for more details), we find nx = −RAn2xwith the exciton-exciton annihilation rate coefficient orbriefly Auger coefficient RA reading

RA =2π

~∑

µ,ν,ρQ,Q′

|WµνρQ,Q′ |2Nν

A,Q(T )NρA,Q′(T )δ(∆ε) (3)

with N = N/nx. The appearing delta function δ(∆ε)ensures that energy is conserved during the scattering

process with ∆ε = ∆ + εµHX,Q+Q′ − ενA,Q − ερA,Q′ . Thedetuning ∆ determines the resonance condition for theAuger scattering process that is strongly enhanced for∆ ≤ kBT . For the intravalley Auger process (ξµ = ξν =ξρ=KK), the detuning it is defined as ∆ = EHX − 2EA,where EHX and EA are exciton resonance energies ofthe final and initial state, respectively (cf. Fig. 1(b)).These energies can be directly obtained from recent up-converted photoluminescence measurements for mono-layer WSe2 with EHX = 3.35 eV and EA = 1.734 eV[14, 15]. In the case of intervalley EEA processes, thedetuning requires the knowledge of binding energies ofHX and A excitons, which are microscopically calculatedby solving the Wannier equation (see SM for details).Furthermore, we approximate the exciton dispersion asparabolic at the considered high-symmetry points, i.e.

εµQ = ~2Q2

2Mµ with the total exciton mass Mµ = mµee +mµh

h

and the effective electron (hole) masses mµee (mµh

h ) beingextracted from first-principle calculations [33]. Finally,we take into account that excitonic resonances becomered-shifted with increasing temperature [34–36]. As a re-sult, we obtain a temperature-dependent detuning, i.e.

∆→ ∆(T ) = ∆ + ∆v(T ) with the shift of ∆v(T ) = αT 2

T+β

described by the Varshni model, where the constants αand β are extracted from temperature-dependent photo-luminescence measurements [36, 37].

We evaluate the exciton-exciton annihilation coeffi-cient RA from Eq. (3) for an hBN-encapsulated WSe2monolayer. We take explicitly into account bright anddark A excitons (KK, KK’ and KΛ) as initial states andHX excitons (KK, KK’ and KΓ) up to n = 3s as finalstates for the Auger scattering process. The overlap ofhigher-lying states (n > 3s) is negligibly small and thusneglected in the following, cf. Fig. 2 (a). In addition tomatrix elements and resonance conditions, the efficiencyof the Auger process is strongly determined by the dis-tribution of excitons across lower-lying states in thermalequilibrium. Due to the energy splitting between KK,KK’, and KΛ excitons being on the order of tens of meVs,the valley-specific densities strongly change with temper-ature, as illustrated in Fig. 2 (b). The resulting calcu-lated Auger coefficients RA are presented in Fig. 3 (a) asfunction of temperature.

Measurement of exciton-exciton annihilation rates.To test the theoretically predicted Auger-recombinationmechanism we take advantage of density-dependenttemporally resolved photoluminescence on hBN-encapsulated WSe2 monolayers. At all studied temper-atures, the PL transients exhibit a density-dependentincrease of the initial decay rate that is well described bythe bimolecular recombination law. Being accompaniedby the saturation of the total PL intensity, this behavioris characteristic for exciton-exciton annihilation, aspreviously demonstrated in the literature [5–8]. Ateach temperature, we consider the density-dependentcomponent 1/τD of the initial recombination rate as afunction of injected exciton density nx. We then extract

4

FIG. 3. Temperature dependence of the Auger coefficient RA for hBN-encapsulated WSe2. (a) Theoretically calculated RA

illustrating the separate contributions of intravalley and intervalley Auger processes (KK-, KK’- and KΛ, respectively), cf.Fig. 1(b). We reveal that the recombination of two KΛ excitons is the dominant process for temperatures above 30 K. (b)Experimentally extracted Auger coefficients from time-resolved PL measurements. The coefficient is obtained from the slopefrom the recombination rate per excitons, as shown in the inset. We find an excellent agreement between microscopic theoryand experiment (without fitting of parameters).

the bimolecular coefficient attributed to Auger recom-bination RA from the slope, as illustrated in the insetof Fig. 3 (b) (see Supplementary Material for additionaldetails). We note that only at T=5 K the PL intensitydoes not saturate at elevated pump densities, indicatingadditional complexity in the exciton dynamics, so thatthe extracted bimolecular rate coefficient should beconsidered as an upper value in that case. The potentialcontributions to the bimolecular recombination ratefrom biexciton formation [38, 39] can be excluded, sincethe latter form much faster [40, 41] than the observeddensity-dependent decay.

Experimentally obtained, temperature-dependentAuger coefficients are presented in Fig. 3 (b) in directcomparison to the theoretical predictions in Fig. 3 (a).Both, in theory and experiment the Auger coefficientincreases by an order of magnitude when increasing thetemperature in the range of 50 to 100 K and remainsnearly constant up to room temperature. From micro-scopic calculations we obtain RA = 0.005 cm2/s and0.05 cm2/s at T = 10 K and T = 300 K, respectively.These values match (without fitting) the experimentallydetermined coefficients of 0.001–0.004 cm2/s at T ≤ 50 Kand 0.04–0.1 cm2/s at 300 K. The obtained quantitativeagreement between theory and experiment stronglysupports both the predominant role of the dark excitonsand the n = 1 HX final states for Auger recombination.

To understand the microscopic origin of the drasticincrease of the Auger coefficients as a function of tem-perature, we explicitly separate the contributions fromintra- and intervalley Auger scattering in theory (corre-sponding to the processes I, II and III illustrated in Fig.1(b)). We find that the intervalley KK’ and KΛ Augerscattering involving the momentum-dark KK’ and KΛ

excitons clearly dominate the Auger coefficient RA forlow and high temperatures respectively (red and orangeregion in Fig. 3(a)). The intravalley Auger scattering in-volving the bright KK excitons is negligible (blue region,note the logarithmic scale). This is a consequence of thespectral ordering of exciton states in WSe2 monolayers,where the momentum-dark KK’ excitons are the ener-getically lowest states followed by the dark KΛ excitonsand finally the bright KK states. The relative positionof these exciton states has been determined microscop-ically by solving the exciton Wannier equation (see SMfor more details). As a result, the dark states carry byfar the largest occupation and thus dominate the Augerscattering processes, despite smaller values of the Augermatrix elements.

The obtained temperature dependence of the Augercoefficient thus strongly reflects the changes in the valley-specific exciton densities nνx(T ) with ν = KK,KK′,KΛ,cf. Fig. 2 (b). At low temperatures up to 20 K theKK’ intervalley Auger process (red area in Fig. 3(a))is the predominant scattering channel. The KΛ Augerscattering takes over from 20 K and quickly becomes byfar the most prominent contribution (orange area in Fig.3(a)). The sharp increase in the Auger coefficient be-tween 50-100 K reflects the predominant population ofthe three-fold degenerate KΛ state at intermediate tem-peratures (cf. Fig. 2(a)). Furthermore, the Auger matrixelement is much more efficient for KΛ than KK’ inter-valley scattering due to the larger momentum transferinvolved in the latter process suppressing the interaction(cf. Fig. 2(b). For T > 100 K, only minor changes inthe relative exciton distributions result in approximatelyconstant Auger coefficients at higher temperatures.

Similar to the influence of temperature, the overall

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FIG. 4. Substrate and temperature dependence of the Augercoefficient RA in monolayer WSe2 from microscopic theory.(a) RA as a function of screening and temperature, revealinga non-monotonic temperature behaviour. (b) Screening de-pendence of RA for different temperatures T = 50, 100 and300 K. (c) Temperature-dependent Auger coefficients in thecase of hBN-encapsulation (εs = 4.5) and for the SiO2 sub-strate (εs = 2.45). As illustrated in the inset for the KΛ Augerprocess, the increase in screening from εs = 2.45 to εs = 4.5makes the Auger scattering more off-resonant leading to a lessefficient RA in hBN-encapsulated TMDs.

efficiency of the excitonic Auger scattering should alsostrongly depend on the dielectric environment that mod-ifies the relative energies of the involved excitonic states.Recently, a strong suppression of the Auger recombi-nation in hBN-encapsulated TMD monolayers was ex-perimentally demonstrated [20, 42]. The theoretical ap-proach outlined above now allows us to analyze and re-veal the microscopic origin of this effect. In the following,we investigate the impact of the dielectric environmenton the exciton-exciton annihilation rate and study in par-ticular the case of hBN-encapsulated samples versus sam-ples 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 temper-ature, Auger scattering becomes less efficient with thedielectric screening, and (ii) the Auger coefficients showa maximum at a certain substrate-dependent tempera-ture. The first trend is further shown in Fig. 4(b),where we consider the screening dependence of Augercoefficients for three different temperatures. The reduc-tion of Auger scattering with screening is observed at all

temperatures. Comparing the two most common dielec-tric environments, SiO2 (εs ≈ 3.9+1

2 = 2.45) and hBN-

encapsulation (εs = 4.5), we find RA,SiO2 = 0.13 cm2/svs RA,hBN = 0.05 cm2/s at room temperature, i.e. wepredict a reduction of the Auger coefficient by approx-imately 60 % in the case of hBN-encapsulated WSe2.This reduction can partly be attributed to a weakenedCoulomb interaction with screening, but importantly itis also a consequence of quenched resonance conditionsdetermined by the detuning ∆, cf. Eq. (3). To furtherquantify these effects, we determine the decrease in theAuger matrix element between samples on a SiO2 sub-strate and hBN-encapsulated samples for the predomi-nant KΛ Auger scattering channel to be approximately30 % in hBN-encapsulated WSe2 monolayers.

To understand how the resonance conditions changewith the substrate we investigate the screening depen-dence of the individual components entering the detun-ing. The A resonance energy (initial state) is known toonly weakly vary with dielectric screening due to the si-multaneous reduction of the band gap renormalizationand the excitonic binding energy [43, 44] and thereforeit can be assumed to be approximately constant. More-over, the energy splittings between different conductionbands are expected to be to a large extent independentof screening, since the Coulomb renormalization only af-fects the occupied (valence) bands [45]. The detuningfor the dominant KΛ-Auger channel at room tempera-ture acquires a weak screening dependence (cf. the in-set of Fig. 4c), stemming from the different screening-induced changes in intra- and intervalley binding ener-gies. We predict ∆hBN ≈ 2∆SiO2

= 30 meV for theKΛ Auger scattering channel with the dark KΓ HX ex-citon as final state (cf. Fig. 1(b)) resulting in weakerAuger coefficients for hBN-encapsulated samples. This incombination with the weakened Auger matrix elementsis the origin of the previous experimental observationsshowing strongly quenched Auger scattering for hBN-encapsulated TMDs [20, 42]. Note that defect-assistedAuger scattering might also play a role for TMDs on aSiO2 substrate due to disorder, while it is expected to benegligible in the case of hBN-encapsulation.

Finally, we demonstrate an intriguing non-monotonictemperature dependence of Auger scattering for fixeddielectric screening in the case of SiO2 and hBN-encapsulation, cf. Fig. 4(c). Interestingly, we find a clearmaximum in the Auger coefficient for WSe2 on SiO2 ataround 80 K. The Auger coefficient RA can be approxi-mated by RA ≈ 1

kBTexp(−|∆|/(kBT )), displaying a max-

imum at Tmax = |∆|/kB [17]. This approximate expres-sion becomes exact in the limit of a constant Auger ma-trix element and when a single exciton species ν = ρ ≡ ν0is dominating the Auger coefficient. The approximationallows us to understand the temperature dependence ofthe Auger coefficients, which is determined by the ini-tial A exciton distribution and the availability of initialand final states fulfilling the conservation of energy andmomentum, cf. Eq. (3). For very low temperatures the

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exciton distribution is strongly localized at vanishing ki-netic energies and hence, the EEA is inefficient due to thenon-zero energy detuning of initial and final states. Withincreasing temperature the momentum-dependent distri-bution becomes broadened facilitating energy and mo-mentum conservation and thus offering additional scat-tering channels leading to an enhanced RA. A furtherincrease in temperature leads to a redistribution of exci-tons and an overall decrease in the initial population ofexciton states, resulting in a reduction of the Auger coef-ficients. The interplay of those two effects is the origin ofthe observed maximum in the RA at certain intermediatetemperatures. In the case of hBN-encapsulated sampleswe do not observe a pronounced maximum of the Augercoefficient as a function of temperature, as the resonanceenergy can not be reached in the considered tempera-ture range, i.e. |∆| > kBT at all temperatures. Here,the large scattering efficiency at higher temperatures issolely determined by the KΛ exciton occupation.

Conclusion. In this joint theory-experiment studycombining microscopic, material-specific modelling withtime-resolved photoluminescence measurements, we es-tablish a fundamental, microscopic understanding ofAuger-like exciton-exciton annihilation processes inatomically thin semiconductors. We demonstrate thekey importance of dark intervalley excitons in the pro-totypical WSe2 material resulting in an intriguing tem-perature dependence of Auger processes. We find an ex-cellent qualitative and quantitative agreement betweentheory and experiment without any adjusted free param-eters. We also resolve the open question in literatureregarding the origin of consistently observed suppressionof exciton-exciton annihilation upon hBN-encapsulation.Overall, our work provides microscopic insights into themany-particle processes behind the technologically im-portant exciton-exciton annihilation channels in atomi-cally thin semiconductors. The developed approach canbe further generalized to van der Waals heterostructuresand twisted moire exciton systems.

METHODS

Microscopic model To be able to study exciton-exciton annihilation on microscopic footing we need ac-cess to the exciton-exciton Auger Hamilton operator andin particular the excitonic Auger matrix elements. Thelatter are determined by evaluating the Heisenberg equa-tion of motion for the microscopic polarisation in theelectron-hole picture, performing an ansatz for the AugerHamiltonian in the excitonic picture and evaluating thesame equation of motion but now with the excitonicHamiltonian. The matrix elements are then read offby demanding that the resulting equations of motion inthe two pictures coincide. More details are provided inthe Supplementary Material. We take into account bothintra- and intervalley Auger scattering processes. Theeigenenergies and wave functions of the initial and fi-

nal states are obtained by solving the Wannier equationwithin the effective mass approximation [29]. We explic-itly take into account the impact of the dielectric environ-ment and finite thickness effects of the TMD layer [46],when evaluating the matrix elements. In particular, dueto the large momentum transfers involved in the inter-valley processes, we do not employ the long-wave limitRytova-Keldysh screening [47–49], but instead make useof a non-linear screening function [50, 51], - in agree-ment with first-principle studies [46, 52]. We obtain theelectronic matrix elements from k·p theory, with param-eters extracted from ab-initio calculations. For the lat-ter, we used density functional theory (DFT) as imple-mented in the Quantum Espresso suite [53]. We obtainthe converged electronic ground state and the matrix el-ements of the momentum operator for the relevant tran-sitions, with full inclusion of spin-orbit coupling effects.Here, the exchange-correlation interaction was treated bya combination of the PBE approximation and dispersiveinteractions from the D3 model [54]. The electron-ioninteraction was modeled by norm-conserving pseudopo-tentials and the electronic wave functions were expandedin plane waves up to a cutoff energy of 90 Ry. The in-put matrix elements for the k·p theory were then di-rectly derived from the DFT momentum operator ma-trix elements. Quasi-particle band gaps and effectivemasses were extracted from G0W0 calculations using theYAMBO code [55]. To guarantee a high accuracy of thederived quasi-particle properties, we combined an extrap-olation procedure for the inclusion of correlation effects,which we previously showed to perform well for verticallystacked MoSe2/WSe2 heterostructures [56, 57], with therecently proposed Neck sub-sampling method [58] for theproper inclusion of non-local screening effects, cf. Sup-plementary Material for more details.

Sample preparation and photoluminescence measure-ments The encapsulated WSe2 monolayers consideredin this study are fabricated by subsequently stackingpolymer-assisted mechanical exfoliated bulk crys-tal WSe2 and high-quality hBN (NIMS) on a SiO2

substrate[59]. For photoluminescence measurements, thesample is placed in a helium-cooled microscopy cryostatand excited by a 80 MHz, 140 fs-pulsed Ti:sapphirelaser. The energy of the incident photons is tuned intoresonance with the bright KK exciton at T = 5 K and fo-cused to a 1 µm spot. The corresponding exciton densityis estimated taking into account effective temperature-dependent absorption by performing spectrally-resolvedreflectance contrast measurements at each studiedtemperature, analyzed by the transfer-matrix formalism[60] with model dielectric functions [61, 62]. Afterexcitation the emission is collected from the center ofthe spot and guided through an imaging spectrometerinto a streak camera for time-resolved detection. Atlow temperatures the emission is spectrally filteredto detect only phonon side bands of dark excitons[26, 63–65] and exclude contributions from residualtrions / Fermi-polarons and biexcitons. At elevated

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temperatures, the bright neutral KK exciton becomesthe dominant emission channel. We note, however, thatalso at these conditions the time-resolved PL reflects thedynamics of the total equilibrated exciton population,including both dark and bright states. More details aregiven in the Supplementary Material.

Author contributions D.E, S.B, R.P.C, and E.Mcarried out the theoretical modeling, while R.G andJ.M provided the DFT input. K. Wagner, J.D.Z andA.C performed the experiments, while K.Watanabe andT.T provided the high-quality hBN. The main bulk ofthe manuscript was written by D.E with major inputfrom E.M and S.B and additional input provided by K.Wagner and A.C.

Acknowledgments. We thank Maja Feierabend(Chalmers) and Paulo Eduardo de Faria Junior andKai-Qiang Lin (University of Regensburg) for fruit-ful discussions. This project has received fundingfrom Deutsche Forschungsgemeinschaft via CRC 1083(project B09), Emmy Noether Initiative (CH 1672/1),CRC 1277 (project B05), CRC 953 (project B13), andthe European Unions Horizon 2020 research and inno-vation programme under grant agreement no. 881603(Graphene Flagship). Furthermore, we are thankful toVinnova for the support via the 2D-TECH competencecenter. K. Watanabe and T.T. acknowledge supportfrom the Elemental Strategy Initiative, conducted bythe MEXT, Japan, Grant Number JPMXP0112101001,JSPS KAKENHI Grant Numbers JP20H00354 and theCREST (JPMJCR15F3), JST.

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Supplementary Materials forDark exciton-exciton annihilation in monolayer transition-metal dichalcogenides

Daniel Erkensten1, Samuel Brem2, Koloman Wagner3, Roland Gillen4, Raul Perea-Causın1, Jonas D. Ziegler3,

Takashi Taniguchi5, Kenji Watanabe6, Janina Maultzsch4, Alexey Chernikov3,7, and Ermin Malic1,2

1Department of Physics, Chalmers University of Technology, Gothenburg, Sweden2Department of Physics, Philipps-Universitat, 35037 Marburg, Germany

3Department of Physics, University of Regensburg, D-93040 Regensburg, Germany4Department of Physics, Friedrich-Alexander-Universitat Erlangen-Nurnberg, Erlangen-Nurnberg, Germany

5International Center for Materials Nanoarchitectonics,National Institute for Materials Science, Tsukuba, Ibaraki 305-004, Japan

6Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-004, Japan7Institute for Applied Physics, Dresden University of Technology, Dresden, 01187, Germany

I. MEASUREMENTS OF AUGER COEFFICIENTS

The Auger coefficients shown in the main manuscript are obtained from analyzing both transient luminescenceas well as the total yield as function of injected exciton density. Fig. 1 shows a detailed overview of this analysis,including PL and absorbance spectra extracted from reflectance, emission transients and total PL intensity at selectedtemperatures. The individual steps of the applied measurement procedure are outlined in the following.

1

0

1

0

0.4

00.2

t=0total

300K

RA=0.10cm2/s RA=0.05cm2/s

RA=0.027cm2/s RA=0.03cm2/s

RA=0.003cm2/s RA=0.0015cm2/s

6.1011 cm-2

2.1010 cm-2

4.1012 cm-2

2.1011 cm-2

1.6 1.7

PL

Abs.

filter

excit.

102

100

101

103

PL

0 100 0 100

100

0

200

0 2 4 0 2 4

0.8

00.4

1

0

PL

Ab

s.

0.8

00.4

PL

Abs.

100K

50K

PL

1.6 1.7 0 1000 100

5.1011 cm-2

1.1010 cm-2

102

100

101

103

PL

1.6 1.7 0 500 0 500

Energy (eV) Time (ps) Time (ps) nx (1011 cm-2)

PL

102

100

101

103

10

0

20

20

0

40

10

0

20

2

0

4

1/t

PL–

1/t

0(n

s-1)

0 2 4 6

0 20 40 0 20 40

nx (1011 cm-2)

0 2 4 6

1/t

PL–

1/t

0(n

s-1)

10

0

20

1/t

PL–

1/t

0(n

s-1)

(a) Spectra (b) As-meas. PL (c) Norm. PL (d) Norm. PL (e) Bimol. rate

exp. fit

FIG. 1. Analysis of temperature-dependent Auger coefficients. Top, middle and bottom panels correspond to T = 300 K,100 K and 50 K, respectively. (a) Normalized time-integrated PL spectra after continuous-wave excitation at 2.326 eV (redline) and simulated absorbance from reflectance contrast measurements (blue line). Red arrows indicate excitation energy fortime-resolved measurements. Grey areas mark spectrally filtered regions by applying a long pass filter. (b) PL transients afterpulsed excitation and spectral filtering as indicated in (a). The emission is collected from the center of the PL spot. Dashedblack lines indicate mono exponential fits of the initial decay. (c) Data shown in (b) normalized to electron-hole pair densityneh (rescaled by a constant factor). (d) Total photoluminescence IPL(total) and initial photoluminescence IPL(t = 0) asfunction of density, extracted from (b). Solid red lines correspond to phenomenological fits with Auger coefficient RA. Dashedred lines emphasize linear density dependence. (e) Density-dependent component of the recombination rate as function ofexciton density extracted from the initial photoluminescence decay. 1/τPL is estimated by monoexponential fits in (b) andsubtracted by the zero-density recombination constant 1/τ0. The Auger coefficient RA is determined by linear fits. Solid bluelines correspond to best fit, dashed lines to min. and max. fit.

arX

iv:2

106.

0503

5v1

[co

nd-m

at.m

es-h

all]

9 J

un 2

021

2

A. Density estimation

The sample is excited by a pulsed 140 fs Ti:sapphire laser with repetition rate frep = 80 MHz and photon energyEph = 1.726 eV (resonant to the bright KK exciton at T = 5 K) focused to a spot with a diameter of FWHM = 1µm. Effective, temperature dependent absorption is taken into account by performing spectrally-resolved reflectancecontrast measurements at each studied temperature, analyzed by the transfer-matrix formalism [1] with model dielec-tric functions [2, 3]. Accordingly, the blue curves in Fig. 1(a) show the extracted absorbance as function of photonenergy. The average exciton density nx is then calculated by

nx =Pα

frepEphA(S1)

with excitation power P and spot area A=FWHM2π/4. The effective absorption α is determined by additionallyweighting the extracted absorbance with the spectral overlap of the excitation laser.

B. Spectral filtering

To match the theoretical scenario as close as possible we take advantage of spectrally filtering individual emissionsin the experiment. At T ≤ 50 K only phonon-assisted emissions from neutral dark excitons are considered [4–6].Particularly, emissions related to density-induced many body complexes, such as neutral and charged biexcitons, areexcluded. At elevated temperatures weak emissions from charged excitons contribute to the signal, cf. Fig. 1(a). Dueto the weak contribution, their impact to the overall behavior of the bimolecular recombination rate is expected tobe negligible. Here, we note that regardless of the predominant emission channel, the majority of exciton populationin WSe2 monolayers should reside in lower-lying KK’ and KΛ dark states at all studied temperatures.

C. Auger coefficients extracted from total PL

First, Auger coefficients RA are determined from the total luminescence yield IPL. Considering bimolecular re-combination processes the total intensity IPL is expected to saturate with increasing density. For a quantitativedescription we consider [7]

IPL =1

τ0

ln(1 + nxRτ0)

RA(S2)

with average exciton density nx and zero-density recombination time constant τ0. Filled dots in Fig. 1(d) show thetotal time-integrated PL from the spot center as function of electron-hole pair density. Fitting the data with Eq. (S2)and fixed τ0, corresponding to the independently determined zero-density decay time (determined in the low densityregime where the decay time is constant), results in Auger coefficients between 0.027 and 0.1 cm2/s for T = 100−300K and an order of magnitude lower Auger coefficient of 0.003 cm2/s at 50 K. It should be noted that at T = 5 K thetotal luminescence does not show saturation and IPL(t = 0) exhibits a super linear behavior, potentially indicatingeither non-linear dependence of the exciton population on excitation power or increasing radiative rates.

D. Auger coefficients extracted from emission decay

For a more accurate determination of the temperature-dependent Auger rates we take advantage of the density-induced PL decay. Considering time-resolved measurements the density-induced emission decay rate 1/τPL is esti-mated by mono exponential fits as indicated in Fig. 1(b) by black dashed lines. Fig. 1(e) shows the correspondingdensity-induced recombination rates after subtracting the zero-density rate 1/τ0. The Auger coefficients RA are thenextracted from linear fits to the data (blue lines). Estimated limits of RA are indicated by dashed lines. Consistentwith the values determined from the total PL yield, the Auger coefficients are between 0.03 and 0.05 cm2/s at ≥ 100K, whereas at T = 50 K we obtain a by more than an order of magnitude smaller Auger coefficient of 0.0015 cm2/s.At T = 5 K a bimolecular recombination rate of 0.0022 cm2/s would be obtained by accounting for the inferreddensity-induced increase of the radiative recombination rate, determined from IPL(t = 0).

3

II. MICROSCOPIC MODELING

A. Excitonic Auger matrix elements

Here, we show how the excitonic Auger Hamilton operator Hx−x can be extracted from the conventional electron-hole Hamiltonian. Our strategy is to first derive the Heiseberg equation of motion for the microscopic polarization inelectron-hole picture and rewrite the result in an excitonic basis, and second to assume an ansatz for a Hamiltonianin the excitonic picture and construct the corresponding excitonic matrix element such that it corresponds to the firstresult. In the electron-hole picture we can straightforward write down the Auger Hamilton operator reading

HAug =∑

ξ1...ξ4kc,kc,q

W ξ1ξ2ξ3ξ4el,q c′†

ξ1,kc+qv†ξ2,kc−qcξ3,kccξ4,kc + h.c. , (S3)

where ξ1...ξ4 are electronic valley indices and kc, kc and q are momenta and W ξ1ξ2ξ3ξ4el,q is the electronic Auger matrix

element determining the strength of the recombination process. For simplicity, we omit the dependence on spin butnote that it plays a similar role as the electronic valley index in the following calculation. Here, c(†) and v(†) describethe creation or annihilation operators for electrons and holes in the upper spin-split valence bands v and the firstspin-allowed conduction bands c, respectively. However, c′(†) describes a creation or annihilation of an electron ina higher conduction band c′. We now consider the Heisenberg equation of motion for the microscopic interband

polarisation P †ξeξhp+Q,p = c†ξe,p+Qvξh,p yielding

i~∂tP †ξeξhp+Q,p = [P †ξeξhp+Q,p, HAug] =∑

ξ1...ξ4,kc,kc,q

W ξ1ξ2ξ3ξ4el,q

(c′†ξ1,kc+q

cξ4,kcc†ξe,p+Qcξ3,kcδ

p,kc−qξh,ξ2

−

− c′†ξ1,kc+q

vξh,pv†ξ2,kc−qcξ3,kcδ

kc,p+Qξ4,ξe

− c′†ξ1,kc+q

cξ4,kcδkc,p+Qξ3,ξe

δkc−q,pξ2,ξh+ c′†

ξ1,kc+qvξh,pv

†ξ2,kc−qcξ4,kcδ

kc,p+Qξ3,ξe

).

(S4)

Next, we expand the electronic operators in terms of polarisations P † = c†v, P ′† = c′†v. Here, we use the unitoperator method introduced by Ivanov and Haug [8] to transfer electronic operators to electron-hole pairs, such thatc†c ∼ P †P (neglecting higher order terms), with details provided in [9, 10]. This yields the following equation ofmotion

i~∂tP †ξeξhp+Q,p =∑

ξ1...ξ3,k,q

(W ξ1ξhξ3ξeel,q P ′†ξ1ξ2p+Q+q,kP

ξ3ξ2p+q,k +W ξ1ξ2ξeξ3

el,q P ′†ξ1ξhk+q,pPξ3ξ2k,p+Q−q

−W ξ1ξ2ξ3ξeel,q P ′†ξ1ξhp+Q+q,pP

ξ3ξ2k,k−q −W

ξ1ξhξeξ3el,q

∑

k′P ′†ξ1ξ2k′+q,kP

ξ3ξ2k′,k δq,Q

).

(S5)

Now, we express the polarisations in the excitonic basis yielding

P †ξ1ξ2k1,k2=∑

n

X†ξ1ξ2n,k1−k2ϕ∗ξ1ξ2n,βξ1ξ2k1+αξ1ξ2k2

, P ′†ξ1ξ2k1,k2=∑

n

Y †ξ1ξ2n,k1−k2φ∗ξ1ξ2n,βξ1ξ2k1+αξ1ξ2k2

(S6)

Note that we introduced different excitonic operators (X(†) and Y (†)) and wave functions (ϕ and φ) for the (c,v) and(c′,v) excitons, respectively. In the following we will refer to (c,v) as the A exciton and (c′,v) as the HX (higher-lyingexciton). Furthermore, to simplify our calculations, we fix the hole valley index ξ2 ≡ ξh. Then, we can immediately

rewrite the left-hand side in (S5) as i~∂tP †ξeξhp+Q,p = i~∂t∑nX

†ξeξhn,Q ϕ∗ξeξh

n,βξeξhQ+p. To simplify notation, we introduce

the compound index µ = (ξ, n). Moreover, we multiply the entire equation of motion written in exciton basis by

ϕσ,q2and sum over q2 = βµQ + p. Then the left-hand side converts to i~∂t

∑µX

†µ,Qϕ

∗µ,q2→ i~∂tX†σ,Q due to the

orthogonality between excitonic wave functions. By substituting (S6) in the right-hand side of (S5), multiplying byϕσ,q2

and summing over q2, we find the equation of motion in excitonic basis

i~∂tX†σ,Q =∑

µ,ν,Q′(EµνσQ,Q′ −Dµνσ

Q,Q′)Y†µ,Q′+QXν,Q′ , (S7)

with the abbreviations

EµνσQ,Q′ ≡∑

q,q′

(Wµeνhσeνeel,q−q′ φ∗µ,βµνQ+q−αµQϕν,q−βνQ−ασQ′ϕσ,q′ +Wµeσhνeσe

el,q−q′ φ∗µ,βµσQ′+q−αµQ′ϕσ,q−βσQ′−ανQϕν,q′

),

DµνσQ,Q′ ≡Wµeνhσeνe

el,Q′

∑

q,q′

φ∗µ,βµσQ′+βµQ+qϕσ,qϕν,q′ +Wµeσhνeσeel,Q

∑

q,q′

φ∗µ,βµνQ+βµQ′+qϕν,qϕσ,q′

4

Here, D and E constitute direct and exchange interaction terms, respectively. Note that the matrix elements aresymmetric with respect to the exchanges of initial A exciton states ν ↔ σ and momenta Q↔ Q′.

Now, use directly an excitonic Hamiltonian by performing the following ansatz:

Hx−x =1

2

∑

µ,ν,σ,Q,Q′

WµνσQ,Q′Y

†µ,Q+Q′Xν,Q′Xσ,Q + h.c. , (S8)

This way, we are able to read off the excitonic Auger matrix element WµνσQ,Q′ from the corresponding Heisenberg

equation of motion for the excitonic polarisation

i~∂tX†σ,Q = [X†σ,Q, Hx−x] = −∑

µ,ν,Q′Wµνσ

Q,Q′Y†µ,Q′+QXν,Q′ . (S9)

Note that here we have treated the operators X(†) and Y (†) as commuting bosonic operators. By comparison with(S7) we immediately find the expersion for the excitonic Auger matrix elements

WµνσQ,Q′ = (Dµνσ

Q,Q′ − EµνσQ,Q′) , (S10)

with the direct and exchange matrix elements being provided in (S8). In the following, we neglect the center-of-massmomentum dependence in the excitonic wave functions. This leads to the following simplified expressions for thedirect and exchange matrix elements

EµνσQ,Q′ ≈ Eµνσ0,0 =∑

q,q′

(Wµeνhσeνeel,q−q′ φ∗µ,qϕν,qϕσ,q′ +Wµeσhνeσe

el,q−q′ φ∗µ,qϕσ,qϕν,q′)

DµνσQ,Q′ ≈Wµeνhσeνe

el,Q′ Fµσϕν,r=0 +Wµeσhνeσeel,Q Fµνϕσ,r=0 ,

(S11)

with the form factor Fµν =∑

q φ∗µ,qϕν,q and with

∑q ϕν,q = ϕν,r=0, where ϕν,r is the real-space representation of

the A exciton wave function. In the main text we have introduced Gµνσ =∑

q,q′ Gµνσq,q′ and Gµνσq,q′ = φ∗µ,qϕν,qϕσ,q′ in

the definition of the matrix elements to simplify notation. Having microscopic access to the excitonic Auger matrixelement, we can now make use of the simpler Hamiltonian in (S8) when evaluating the exciton-exciton annihilationrate.

Finally, we define the electronic Auger matrix element Wel,q entering (S11). Depending on the size of the momentumtransfer q, different matrix elements are obtained. For scattering processes with small momentum transfer q, i.e.ξµ=KK, ξν , ξρ=KK(′) we make use of k · p theory [11, 12] to find

Wµeνhρeνeel,q = Vqq

2γρe−νhc−v γνe−µec−c′ δµe,νeρe,νh(intravalley) . (S12)

FIG. 2. Momentum-dependent (a) intravalley and (b) intervalley Auger matrix element. (c) Separation of direct and exchangecontributions in the Auger matrix elements. The dashed lines include only the direct term, while the solid lines include the fullinteraction.

5

The matrix element is determined by the screened Coulomb interaction Vq and the electronic dipole matrix element

γη−η′

λ−λ′ = 〈λ, η|r|λ′, η′〉 describing transitions between different bands λ, λ′ = v, c, c′ and valleys η, η′ = K, K’. Forscattering processes with large momentum transfer q on the order of the reciprocal lattice vector, we decomposeq = q + ∆η,η′ , where q is small and ∆η,η′ denotes the momentum-difference between different high symmetry pointsη 6= η′ (i.e. |∆K,Λ| = 2π

3a0or |∆K,K′ | = 4π

3a0, a0 being the lattice constant) and find

Wµeνhρeνeel,q = Vq+∆νh,ρe

Sρe−νhc−v (q)Sνe−µec−c′ (−q) (intervalley), (S13)

with Sη−η′

λ−λ′(q) = (γη−η′

0,λ−λ′ + iqγη−η′

1,λ−λ′), where γη−η′

0,λ−λ′ = 〈λ, η|λ′, η′〉 and γη−η′

1,λ−λ′ = 〈λ, η|ei∆η,η′rr|λ′, η′〉, η, η′ =

K,K′,Λ,Γ. Note that (S13) reduces to (S12) for η = η′. The band- and valley-specific γ-parameters are extractedfrom GW (DFT) calculations described in detail in section III. Due to the large momentum transfer involved in theintervalley processes we do not employ the conventional Keldysh screening [13–15] entering the screened Coulombpotential, as it is seen to diverge in the short wave-length limit. Instead we make use of a non-linear screening function[16, 17] in good agreement with ab-initio calculations for all momenta [18, 19].

In Fig. 2 (a)-(b) we illustrate the excitonic Auger matrix elements WQ,Q′ defined in Eq. (S10) for intra- andintervalley processes. We find that due to the large momentum transfer, the intervalley (ξν = ξρ =KK’, ξµ =KΓ)matrix element is up to two orders of magnitude smaller than the corresponding intravalley (ξν = ξρ = ξµ=KK)matrix element. For large momentum transfer the electronic matrix elements are close to constant in momentum,resulting in a large cancellation of direct and exchange terms (cf. Eq. (S11)), which explains the large dark-bluearea in Fig.2(b). The direct and exchange contributions are further separated in Fig.2 (c), where a cut of the matrixelements at Q = Q′ is shown. Considering only the direct contribution (dashed) we note a small monotonous increasewith momentum. Including also the exchange interaction (solid), we observe a pronounced suppression of the matrixelements further reflecting the cancellation of the direct and the exchange term.

B. Modeling of Auger coefficients

In this section we provide a detailed derivation of the exciton-exciton annihilation rate expressed in the Augercoefficient RA. We start from the density of A excitons nx =

∑ν,QN

νA,Q, where the exciton occupation Nν

A,Q =

〈X†ν,QXν,Q〉 is estimated by a thermalized Boltzmann distribution. Here, Q is the center-of-mass momentum and ν

is a compound index incorporating excitonic spin, valley (incl. valley degeneracy) and principal quantum number(which we restrict to be n = 1s). Now, the goal is to derive an equation of motion for the density, which is non-linearin nA,x. By commuting the excitonic Auger Hamiltonian Hx−x in (S8) with the occupation Nν

A,Q we obtain thefollowing Heisenberg equation of motion

∂tNνA,Q = −2

~∑

µ,ρ,Q′

Im

(Wµνρ

Q,Q′〈Y †µ,Q+Q′Xν,QXρ,Q′〉). (S14)

Then, we write 〈Y †µ,Q+Q′Xν,QXρ,Q′〉 = HF + 〈Y †µ,Q+Q′Xν,QXρ,Q′〉c. We go beyond the Hartree-Fock approximation

(HF) [20, 21] and find an equation of motion for the correlated part CµνρQ,Q′ ≡ 〈Y †µ,Q+Q′Xν,QXρ,Q′〉c reading

∂tCµνρQ,Q′ ≈ i

~(∆ε)CµνρQ,Q′ +

i

~(Wµνρ

Q,Q′)∗Nν

A,QNρA,Q′ . (S15)

Here, we have introduced ∆ε = εµHX,Q+Q′ − ενA,Q − ερA,Q′ . The first part of the equation stems from the com-

mutation with the free part H0 =∑µ,Q ε

µHX,QY

†µ,QYµ,Q +

∑ν,Q ε

νA,QX

†ν,QXν,Q. The second part of the equation

follows from the commutation with the excitonic Auger Hamiltonian. Here, we also performed the mean-field

and random phase approximation according to 〈X†1X†1X3X4〉 ≈ 〈X†1X3〉〈X†2X4〉δ1,32,4 + 〈X†1X4〉〈X†2X3〉δ1,4

2,3 . We as-

sumed the HX occupation NµHX,Q = 〈Y †µ,QYµ,Q〉 to be negligible and only kept terms non-linear in the A exciton

density. Next, we solve the equation of motion (S15) within a Markov approximation [21] yielding CµνρQ,Q′(t) ≈(Wµνρ

Q,Q′)∗NνA,QN

ρA,Q′(iπδ(∆ε) + P(1/∆ε)) with the delta function δ(∆ε) reflecting energy conversation and the real-

valued principal-value integral P(1/∆ε) leading to many-body energy renormalizations. The latter plays no role inscattering processes and cancels out when plugging in CµνρQ,Q′(t) into (S14). Summing (S14) over Q and ν leads to an

equation of motion for the density nA,x in the incoherent limit reading ∂tnA,x = −RAn2A,x with the Auger coefficient

RA =2π

~∑

µνρQQ′

|WµνρQ,Q′ |2Nν

A,QNρA,Q′δ(∆ε) , (S16)

6

with N = N/nx. Similarly, it is possible to find an equation of motion for the HX density nHX,x =∑µ,Q〈Y

†µ,QYµ,Q〉

according to ∂tnHX,x = RA2 n2

A,x [22].

C. Excitonic binding energies

We microscopically calculate excitonic binding energies and wave functions by solving the Wannier equation

~2k2

2mξνϕν,k −

∑

q

Wqϕν,k+q = Ebνϕν,k , (S17)

with mξν = mνemνhmνe+mνh being the reduced mass in valley ξν =KK, KK’, KΛ, KΓ. Here, Ebν is the excitonic binding

energy for the state ν = (ξν , n), n = 1s, 2s... being the principal quantum number. To obtain the energetic position ofexcitonic energies, Eν , we also include the single-particle separation E0

ξνbetween the top of the valence band at the

K-point and the bottom of the first spin-allowed conduction bands at K, K’ and Λ, such that Eν = E0ξν−Ebν . For the

higher-lying exciton (HX) we also include the single-particle separation between the first conduction band (c) and thehigher-lying conduction band (c′). However, when computing the detuning we eliminate the quasiparticle band gaps

at the K-point by using EA = Eg +EAbind(KK) and EHX = Eg +EHXbind(KK) and adapting the A and HX resonancesfrom recent up-converted PL measurements. In Table I we provide the relevant A and HX exciton energies. Note

Exciton energies WSe2 (εs = 4.5) l = A (meV) l = HX (meV)

Elbind(KK) -160 -500

∆lKK,KΛ -39 -

∆lKK,KK′ -53 331

∆lKK,KΓ1

- -6

∆lKK,KΓ2

- 56

TABLE I. Exciton energies in hBN-encapsulated monolayer WSe2 for A (c, v) and HX (c′, v) excitons. Here, ∆lKK,µ denotes

spectral exciton separations with the absolute energy given by Eµl = Elbind(KK) + ∆lKK,µ (relative to the single-particle band

gap).

that conduction bands at the Γ-point are degenerate and have no preferred spin polarisation. However, the bandshave different curvature (cf. Table II) and therefore both bands are explicitly included in the calculations.

III. FIRST-PRINCIPLE PARAMETERS FOR MICROSCOPIC THEORY

The effective masses and k·p matrix elements necessary for the microscopic theory were derived using a modern abinitio approach. The relevant quasiparticle energies were derived from one-shot G0W0 calculations as implemented inthe YAMBO code [23], using 2500 unoccupied bands in combination with the effective energy technique (EET) [24]to account for higher-energy interband transitions. The band energies were extrapolated to infinite cutoff energy forthe correlation contribution with the method we employed previously in Ref.25 and calculations for cutoff energies of250 eV, 300 eV and 350 eV, for which the variation of band energies is linear with the inverse of the number of includedreciprocal lattice vectors. The effect of spin-orbit interaction was explictly accounted for. A homogeneous grid of9x9x1 k-points in the Brillouin zone was found sufficient to capture the dispersion of the electronic quasiparticles.The quasiparticle band gap was then corrected by applying a scissor shift to the conduction bands. For this step,we derived the band gap correction without explicit inclusion of spin-orbit interaction, using the ”Nonuniform NeckSubsampling” method [26] as implemented in the BerkeleyGW package [27]. We found that this method was sufficientto converge the electronic band gap to less than 0.05 eV. The quasiparticle band structures were then obtained throughWannier interpolation using the Wannier90 [28] code. The input electronic energies and spinors and the groundstatedensity for the GW and Wannier calculations were computed with the Quantum Espresso code [29], optimized norm-conserving Vanderbilt pseudopotentials [30], a cutoff energy of 90 Ry, a 12x12x1 k-point grid (for the groundstatecalculations) and the Perdew-Burke-Ernzerhof (PBE) [31] exchange-correlation functional. We fully optimized theatomic positions and cell parameters until the residual forces between atoms and the cell stress were smaller than0.01 eV/A and 0.01 GPa, respectively. Long-range non-covalent interactions were included through the semi-empiricalDFT-D3 correction with Becke-Johnson damping [32]. The resulting quasiparticle bandstructures for monolayer WSe2

and also MoSe2 are shown in Fig. 3.

7

Using the obtained Wannier Hamiltonian, we interpolated the G0W0 quasiparticle energies on a dense regular gridwith within a radius of 0.15 1/A around the high symmetry points. We then derived the effective masses of thevalence and conduction bands valleys of interest by fitting a parabolic function to the obtained quasiparticles energiesof WSe2. The resulting effective masses are shown in Table II and generally in good agreement with the previouslyreported masses using the G0W0 approximation [33] or hybrid functionals within the DFT framework [34].

The electronic spinors of the valence and conduction bands, which are readily available from our DFT calculations,further allowed us to extract the form factors for interband transitions from an initial state with band index mand quasi-crystal momentum to a final state k and a state n,k+Q. Assuming that we can neglect the momentumdependence of the Bloch wavefunctions for a small perturbation in momentum q, i.e. |n,k + Q + q〉 ≈ |n,k + Q〉, wecan further expand the form factor in q, yielding the expression

Γk+Q,kn,m (q)

= 〈n,k + Q|ei(Q+q)·r|m,k〉≈ 〈n,k + Q|eiQ·r(1 + iq · r)|m,k〉≈ 〈n,k + Q|eiQ·r|m,k〉+ iq · 〈n,k + Q|eiQ·rr|m,k〉≈ γk+Q,k

0,nm + iq · γk+Q,k1,nm

For sufficiently small momentum q, the γ parameters should be nearly isotropic. As an additional simplification, wehence decided to use the average of the x- and y-components of γ instead of the vectorial form:

Γk+Q,kn,m (q) ≈ γk+Q,k

0,nm + i |q| · γk+Q,k1,nm,ave

Using the completeness relation I =∑l |l,k 〉〈 l,k|, we recast γk+Q,k

1,nm into the form

γk+Q,k1,nm =

⟨n,k + Q

∣∣eiQ·rr∣∣m,k

⟩

= 〈n,k + Q| eiQ·r(∑

l

|l,k 〉〈 l,k|)r |m,k〉

=∑

l

⟨n,k + Q

∣∣eiQ·r |l,k〉 〈l,k| r∣∣m,k

⟩

=∑

l

γk+Q,k0,nl 〈l,k |r|m,k〉 ,

which only contains terms that are straight-forwardly derived from the quasiparticle bandstructure and the calculatedDFT spinors. In practice, we truncate the sum over l after the first 250 bands, which we found to yield sufficientlyconverged values. Following the typical procedure, the matrix elements of the position operator can be approximated

- 3- 2- 1012345

W S e 2

Γ K M

Energ

y (eV

)

M Λ

c ' c '

c

v

c

- 2

- 1

0

1

2

3

4

5

Λ

M o S e 2

Γ K M

Energ

y (eV

)

M

FIG. 3. Quasiparticle bandstructures of monolayer WSe2 and MoSe2 from our G0W0 calculations. The inserted labels v, c andc′ indicate the bands relevant for the calculation of the Auger rates in WSe2.

8

by the matrix elements of the momentum operator p = −i~∇, with 〈η |r| η′〉 ≈ −i ~m(εζ−εζ′)

〈ζ |p| ζ ′〉. The momentum

operator matrix elements are easily calculated from the Fourier representation of the electronic spinors. For thecalculation of the γ0 parameters, we make use of the relation

Ank,mk′

(G)

=⟨n,k

∣∣∣ei(Q+G)·r∣∣∣m,k′

⟩= F

[u∗n,k (r)um,k′ (r)

], (S18)

where un,k are the lattice-periodic parts of the Bloch functions of band n and crystal momentum k. With this,

γk,k0,nl = Ank,l,k′

(G = 0

). The resulting γ parameters for monolayer WSe2 are listed in Table III.

Eff. mass (×m0) K K’ Λ Γmνec 0.31 0.4 0.6 -

mµec′ -0.52 (-0.46 [35]) -0.46 - -0.84 (-1.74)

mνhv 0.4 - - -

TABLE II. Effective masses extracted from GW (DFT) calculations for the upper spin-split band c and higher-lying c′ (alsoreferred to as the c+ 2-band [35]) in monolayer WSe2. We note a good agreement with previous ab-initio studies [34].

γ-parameters |γ| (a.u)

γK−Kc−v 0.32

γK(′)−K(′)c−c′ 0.058

γΛ−K0,c−v 0.344

γΛ−K1,c−v 0.036

γΛ−Γ0,c−c′ 0.244 (0.208)

γΛ−Γ1,c−c′ 0.232 (0.198)

γK′−K0,c−v 0.252

γK′−K1,c−v 0.0139

γK′−Γ0,c−c′ 0.029 (0.034)

γK′−Γ1,c−c′ 0.105 (0.124)

TABLE III. kp γ-parameters extracted from ab-initio calculations for different intra-and interband transitions in monolayerWSe2. We distinguish intravalley and intervalley by the parameters γ and γ respectively. Note that we only provide theabsolute magnitude of the parameters, since the parameters can be determined up to a phase. Moreover, note that γ1 isformally a two-dimensional vector, but we find that it can be well approximated by the average of its x and y components whenevaluating the electronic Auger matrix elements. For the transitions to the Γ-point we provide two numbers: one number foreach spin-split band. Note that, due to time-reversal symmetry, bands at the Γ-point have no preferred spin polarisation andtherefore we include both bands in the calculation of the Auger rates.

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