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Bo Han, Cedric Robert, Emmanuel Courtade, Marco Manca, Shalini Shree, Thierry Amand, Pierre Renucci, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Xavier Marie, Leonid  E. Golub, Mikhai  M. Glazov, Bernhard Urbaszek

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[Exciton States in Monolayer MoSe2 and MoTe2 Probed by Upconversion Spectroscopy](https://mdr.nims.go.jp/datasets/3520c124-d3f8-4eb8-a28d-94cbdb00bb53)

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Exciton States in Monolayer MoSe2 and MoTe2 Probed by Upconversion Spectroscopy Exciton States in Monolayer MoSe2 and MoTe2 Probed by Upconversion SpectroscopyB. Han,1 C. Robert,1 E. Courtade,1 M. Manca,1 S. Shree,1 T. Amand,1 P. Renucci,1 T. Taniguchi,2K. Watanabe,2 X. Marie,1 L. E. Golub,3 M. M. Glazov,3,* and B. Urbaszek1,†1Université de Toulouse, INSA-CNRS-UPS, LPCNO, 135 Avenue Rangueil, 31077 Toulouse, France2National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan3Ioffe Institute, 194021 St. Petersburg, Russia(Received 11 May 2018; revised manuscript received 16 July 2018; published 18 September 2018)Transitions metal dichalcogenides (TMDs) are direct gap semiconductors in the monolayer (ML) limitwith fascinating optical and spin-valley properties. The strong optical absorption of up to 20% for a singleML is governed by excitons, electron-hole pairs bound by Coulomb attraction. Excited exciton states inMoSe2 and MoTe2 monolayers have so far been elusive because of their low oscillator strength and stronginhomogeneous broadening. Here, we show that encapsulation in hexagonal boron nitride results in anemission line width of the A∶1s exciton below 1.5 meV and 3 meV in our MoSe2 and MoTe2 monolayersamples, respectively. This allows us to investigate the excited exciton states by photoluminescenceupconversion spectroscopy for both monolayer materials. The excitation laser is tuned into resonance withthe A∶1s transition, and we observe emission of excited exciton states up to 200 meV above the laserenergy. We demonstrate bias control of the efficiency of this nonlinear optical process. We discuss theorigin of the upconversion effect. Our model calculations suggest an exciton-exciton (Auger) scatteringmechanism specific to TMD MLs involving an excited conduction band, thus generating high-energyexcitons with small wave vectors. The optical transitions are further investigated by white light reflectivity,photoluminescence excitation, and resonant Raman scattering, confirming their origin as excited excitonicstates in monolayer thin semiconductors.DOI: 10.1103/PhysRevX.8.031073 Subject Areas: Optoelectronics, Semiconductor PhysicsI. INTRODUCTIONTransition metal dichalcogenides such as MoS2, WS2,WSe2, MoSe2, and MoTe2 are direct band-gap semicon-ductors when thinned down to one monolayer [1–6]. Theirband gap is situated in the visible to near infrared of theoptical spectrum. Since the Coulomb interaction is strong inthis ultimate two-dimensional (2D) limit, the optical proper-ties are dominated by excitons—bound electron-hole pairs[7–14]. Recently, encapsulation in hexagonal boron nitride(hBN) of TMD monolayers (MLs) has resulted in consid-erable narrowing of the exciton transition linewidth down to1 meV [15–24]. This result gives us access to fine features ofthe exciton spectra that dominate the linear and nonlinearoptical properties of monolayer semiconductors.Optical excitation of a semiconductor at the bandgap typically results in luminescence at lower energydue to energy relaxation of charge carriers and excitons.In Figs. 1(a)–1(d), we show that resonant laser excitationof the lowest-energy exciton resonance A∶1s results inpronounced photoluminescence (PL) emission at higherenergy than the excitation laser for four different TMDMLmaterials. This effect is generally termed upconversion andhas been observed for different semiconductor structuressuch as InP=InAs heterojunctions, CdTe quantum wells,and InAs quantum dots [25–29], albeit based on differentmicroscopic mechanisms. Upconversion has previouslybeen reported for WSe2 [16,30] and MoS2 [31] MLs.These experiments allow detailed insight into the light-matterinteraction physics of excitons in TMD monolayers: First,clarifying the origin of the upconversion signal is in itself acrucial problem, as the origin of excess energy needs to beidentified and the role of exciton-exciton scattering mecha-nisms is revealed. Second, upconversion spectroscopy givesus access to the excited exciton states that govern absorptionand emission, which are so far not well understood in MLMoSe2 and MoTe2. The relative motion of the electron andhole in the exciton, in analogy to the hydrogen atom andpositronium, is characterized by a principle quantum numbern ¼ 1; 2; 3…, where typical photoluminescence emissionstems from the n ¼ 1 exciton of the A-exciton series, labeledA∶1s. The optical absorption in energy above the A∶1s*glazov@coherent.ioffe.ru†urbaszek@insa-toulouse.frPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 8, 031073 (2018)2160-3308=18=8(3)=031073(17) 031073-1 Published by the American Physical Societyhttps://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.8.031073&domain=pdf&date_stamp=2018-09-18https://doi.org/10.1103/PhysRevX.8.031073https://doi.org/10.1103/PhysRevX.8.031073https://doi.org/10.1103/PhysRevX.8.031073https://doi.org/10.1103/PhysRevX.8.031073https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/optical transition will be determined by the excited statesA∶2s, 3s, etc. and the B-exciton series, separated fromthe A-exciton mainly by the spin-orbit splitting in the200–400-meV range [32]. Upconversion allows us toaccess excited exciton states for MoSe2 and MoTe2, whichis not possible in samples that are not encapsulated inhBN as the excited A-excitons spectrally overlap with theB-exciton series. We demonstrate bias control of theupconversion process. We provide an in-depth study ofexciton states in MoTe2, comparing upconversion withphotoluminescence excitation spectroscopy (PLE) andwhite light reflectivity. In the last part of the paper, weprovide a theoretical model and discuss the origin ofupconversion in TMD monolayers. Our model calculationssuggest that Auger-type exciton-exciton scattering couldbe very efficient in TMD MLs as compared to othersemiconductor nanostructures due to (i) the strongCoulomb interaction, which makes it possible to relaxthe single-electron momentum conservation [33,34], and(ii) the possibility of a resonant processes involving excitontransfer to an excited energy band.II. EXCITED-STATE SPECTROSCOPYIN ML MoSe2MonolayerMoSe2 is a very versatileTMDmaterial ideallysuited to explore coupling to optical cavities [35,36], inves-tigating voltage control of monolayer mirrors [37,38], andthe interplay between charged and neutral excitons [39].Most of these experiments are based on the optical responseof the lowest-energy exciton state A∶1s, but very little isknown about excited exciton states that govern opticalabsorption at higher energies and energy relaxation pathwaysfor PL emission.The experimental results for the high-quality MoSe2samples encapsulated in hBN [40] are summarized inFig. 2; details of the experimental setup can be foundin Appendix A. In differential white light reflectivity atT ¼ 4 K, we clearly observe the A- and B-exciton 1s states[41–43], the A∶1s resonance has a full width athalf maximum (FWHM) of the order of 2 meV [44]. InFig. 2(a), we show an intriguing result: Excitation of thesample with a low-power, continuous-wave (cw), narrow-linewidth (<1 μeV) laser at the A∶1s energy results inemission of the B∶1s transition at higher energy. As we scanthe laser across the A∶1s resonance, the PL intensity (blackgraph) has a clear maximum in intensity when the laser isexactly at the A∶1s resonance; the blue data points representthe integrated upconversion intensity for different laserenergies. In addition to B∶1s, another transition appearsabout 150 meVabove the A∶1s in upconversion PL, whichwe tentatively assign to the excited A-exciton A∶2s state.This transition is also visible in reflectivity in Fig. 2(b). Forsamples directly exfoliated onto SiO2, the excited A-excitonstates were not directly accessible due to their overlap withthe B-exciton 1s state. A fingerprint of the A∶2p state wasreported in two-photon PLE experiments [43], where theB∶1s state absorption is strongly reduced [45].Now, we investigate the origin of the upconversion PLin ML MoSe2 shown in Figs. 1(b) and 2(a). We comparethe evolution of standard and upconversion PL intensityas a function of laser power in Fig. 2(c). The slope ofthe upconversion intensity versus laser power (1.64)integrated over the combined A∶2s and B∶1s emissionis roughly twice as high as for standard PL (0.88),consistent with a two-photon (two-exciton) process beingat the origin of this nonlinear optical effect. We notethat the exact power dependence determination forupconversion PL depends slightly on the spectral rangeused for signal integration. For example, the slope for anarrow spectral range at the A∶2s (B∶1s) resonance is(a)(b)(c)(d)FIG. 1. Upconversion spectroscopy in TMD monolayers. Wepresent, for four different monolayer materials, resonant excita-tion experiments of the A∶1s exciton at T ¼ 4 K, which result inPL emission at higher energy. The laser energy—equal to theA∶1s transition energy—is marked by a vertical arrow. Theupconversion emission peaks are labeled A∶2s and B∶1s, wherethe origin of these peaks is confirmed in complementary experi-ments such as reflectivity and PLE. The results for WSe2 arereproduced from Ref. [16], and the MoS2 results from Ref. [31].The inset in panel (a) shows a scheme of the sample. The inset inpanel (b) shows the exciton-exciton Auger process, where oneexciton annihilates and another one acquires total momentum andenergy of the two particles.B. HAN et al. PHYS. REV. X 8, 031073 (2018)031073-2roughly 2 (1.8). In Fig. 2(d), we plot the upconversionemission as the laser is scanned across the A∶1s reso-nance. Upconversion is only detectable over a 2-meVrange of laser energy when the laser is in resonance withthe A∶1s state. This indicates that upconversion is aresonant process, as observed for WSe2 monolayers [16].This conclusion gets additional support from upconver-sion experiments in a charge tunable device [Fig. 2(e)],presented in Fig. 2(f). At a bias of þ10 V, the excitationlaser is tuned into resonance with the neutral A∶1s state.As the applied voltage is lowered to þ1 V, electrons areadded to the monolayer, decreasing absorption strength atthe neutral exciton resonance. The upconversion signal isnot detectable anymore for a bias of −10 V, as absorptionat the exciton resonance is inefficient; this is because thecharged exciton absorption, at a different energy, domi-nates [37,38,41]. Differential reflectivity of our deviceshows a strong neutral exciton resonance at þ10 V,whereas for −10 V, the charged exciton transitiondominates; see reflectivity measurements on the devicein Appendix B. These experiments confirm that upcon-version PL emission has its origin in resonant neutralexciton generation and can be controlled electrically incharge tunable structures. Note that for a quantitativeanalysis of the bias tuning of the upconversion efficiency,the slight shift of the central A∶1s transition energy(neutral exciton) with bias also needs to be taken intoaccount. However, in our sample, this shift in energy forthe A∶1s regime is considerably smaller than the tran-sition linewidth.(a)(b)(c)(d)(e) (f)FIG. 2. Control of upconversion in ML MoSe2 for T ¼ 4 K. Sample No. 1: (a) Scanning a Ti-Sa laser across the A∶1s transition,which results in upconversion emission of the A:2s and B∶1s transitions (black curve for excitation exactly at resonance). Blue symbolsgive the integrated upconversion intensity as a function of laser energy. (b) Reflection contrast for the same sample spot, confirming theenergy positions of A∶1s, A∶2s, and B∶1s transitions. (c) The power dependence of the upconversion signal shows an increase with aslope of roughly 1.64 (black symbols), as compared to the standard A∶1s exciton emission with a slope of roughly half (0.88—redsymbols) using a HeNe laser for excitation. (d) Contour plot (blue—below 50 counts; red—greater than 2000 counts) of upconversionPL intensity as the excitation laser is swept across the A∶1s resonance. Sample No. 2: (e) Schematics of the charge tunable device.(f) Voltage control of upconversion. The signal is maximal in the neutral regime and gets weaker as the n-type regime favors trion andnot neutral exciton absorption; the B∶1s and A∶2s emission are marked; a third emission peak of yet-to-be-determined origin appears atlower energy.EXCITON STATES IN MONOLAYER MoSe2 AND MoTe2 … PHYS. REV. X 8, 031073 (2018)031073-3III. EXCITED-STATE SPECTROSCOPYIN ML MoTe2MoTe2 is a very interesting layered material [46–48],which provides the fascinating opportunity to switch betweensemiconducting 2H and metallic phases by tuning strain orcarrier concentration [49,50]. This material allows us toworktowards devices based on bias-controlled phase changes inmonolayer MoTe2 [51,52]. The first studies of 2H −MoTe2flakes exfoliated onSiO2 have identifiedmonolayers as directband gap semiconductors [53,54]; interestingly, the natureof the gap of the bilayer is still under discussion [55,56]. So,in practice, the monolayers have to be distinguished frombilayers in Raman scattering experiments; see Fig. 7a.MoTe2MLs have an optical band gap at T ¼ 4 K at 1.17 eV,corresponding to an emission wavelength of 1050 nm.Therefore, its alloying with other TMD materials, such asMoS2 and MoSe2, allows one, in principle, to cover the fullspectral range from 630 to 1050 nm for optoelectronicsapplications. As optical absorption is not only strong at theexcitonic band gap (A∶1s) but also for higher-lying excitonstates [7–14], better knowledge of the excited excitonspectrum is needed. This knowledge also allows us, inprinciple, to get an estimation of the exciton binding energyby comparing the data with model calculations of excitonstates in a screened 2D potential [10,31,57,58].Here, we show the striking impact of hBN encapsulationon the optical properties of monolayer MoTe2. The PLspectrum in Fig. 3(b) shows very narrow emission lines(FWHM linewidth of 3 meV) for the neutral exciton atthe A∶1s state at 1.17 eV, approaching the optical qualityreported for hBN encapsulated MoS2 and WSe2 mono-layers [15–21]. We confirm the high sample quality inreflectivity experiments in Fig. 3(a), which show thistransition basically at the same energy as in PL, indicatingnegligible neutral exciton localization. In reflectivity, wealso see a broader transition about 250 meV above theA∶1s, which we ascribe to the B∶1s state, followingcomparison with the data from the literature [53,54,59].We also observe in reflectivity a transition 120 meV abovethe A∶1s state, not reported previously, which we ascribeto the A:2s state. Strikingly, when exciting with a laser(a)(b)(c)(d)FIG. 3. Exciton spectroscopy in MoTe2 monolayers encapsulated in hBN for T ¼ 4 K. (a) Differential reflectivity spectrum. Theenergy positions of the exciton transitions A∶1s, A∶2s, and B∶1s are marked. (b) Excitation with a HeNe laser at 1.96 eV, resulting inhot PL of the A∶2s state and PL for the A∶1s state. The low-energy peak labeled “T” might be related to the trion or phonon replica.(c) Photoluminescence excitation measurements detecting the emission from the A∶1s exciton. Peaks related to the resonant excitationof the A∶2s and B∶1s are marked. (d) Upconversion PL. The laser tuned into resonance with the A∶1s state results in emission of about120 meV higher energy, same as in Fig. 1(d).B. HAN et al. PHYS. REV. X 8, 031073 (2018)031073-4energy of 1.96 eV, we also see hot PL emission of this A∶2stransition in Fig. 3(b).To further investigate the nature of these excited excitonstates, we carry out PLE experiments. We monitor the PLemission of the A∶1s state [as in Fig. 3(b)] as a function ofthe laser excitation power. PLE probes absorption, whichgives information on the higher-lying electronic transitions,and subsequent relaxation to the A∶1s state, usually byemitting phonons. In our experiments, we observe clearindications of both processes: absorption by excited excitonstates and phonon-assisted energy relaxation. In Fig. 3(c),we see clear resonances in PLE exactly at the same energiesas the reflectivity spectrum for the A∶2s and B∶1s states.The PL emission is enhanced by orders of magnitude wherethe laser is resonant with these excited exciton states,indicating efficient absorption and energy relaxation. Moredetails on phonon-assisted relaxation and associated Ramanscattering on this sample are described in Appendix B.As discussed in the previous section for MoSe2 MLs,a powerful technique for investigating exciton states isphotoluminescence upconversion. Here, a cw laser excitesthe MoTe2 monolayer at the A∶1s resonance, and emissionat higher energies is monitored. In Fig. 3(d), we indeedobserve emission 120 meV above the A∶1s state; thisemission is exactly at the same energy as the transitionascribed to the A∶2s state with the three other spectroscopytechniques: reflectivity, hot PL, and PLE, which are allcompared in Figs. 3(a)–3(d). We note that the upconversionPL of MoTe2 consists of the A∶2s emission superimposedon a broad background. The emission of this globalupconversion signal increases linearly with laser powerand not nearly quadratically as expected. The detection oflight emission above the laser excitation energy is clearly anonlinear process due to the required energy conservation,and we further investigate the reasons for this unexpectedpower dependence.IV. ENERGY SEPARATION BETWEENA∶1s AND A∶2s STATESThe values of the energy separation between the A∶1sand A∶2s states,Δ12 ≡ EA∶2s − EA∶1s; ð1Þobtained by the upconversion spectroscopy and otheroptical techniques in our experiments are summarized inTable I, together with other excitonic parameters calculatedand taken from the literature. As an important result ofour measurements, we find that the values of Δ12 in hBN-encapsulatedMoSe2 andMoTe2 are comparable to the onesin WSe2 [22,60,61]. The fact that the effective masses ofelectrons and holes in MoSe2 and MoTe2 are calculated tobe about a factor of 2 larger than those in WSe2 and WS2[32] (see also Table I for the summary of values) implies, inprinciple, values of Δ12 in Mo-based monolayers largerthan those measured here. However, in two-dimensionalsemiconductors, the electron-hole interaction law stronglydeviates from the 1=r dependence due to dielectric screen-ing effects [7,57,58] and can be described by the expressionVðrÞ ¼ πe22ϰr0�H0�rr0�− N0�rr0��; ð2Þwhere r is the in-plane electron-hole separation, ϰ is theeffective dielectric constants of the surroundingmedia (takento be the same for all the structures under study because it ismainly determined by the hBN environment in our encapsu-lated samples [31]), r0 is the screening radius, and H0 andN0are the Struve and Neumann functions, respectively. Notethat in some references [31,60,62], ρ0 ¼ ϰr0 is used toprovide a direct link with the monolayer polarizability.Accounting for this effect by means of a simple variationalapproach (see Appendix C for details of calculations), weobtain good agreement for the A∶1s − A∶2s separation; seeTable I. Here, the only free parameter is r0, in reasonableagreementwith data in the literature.Note that bothmeasuredand calculated separations are in agreement with previousexperimental and theoretical results available for WSe2 andMoTe2; see, e.g., Refs. [31,60,61].A more detailed understanding of the energy positionsof the excited states and more sophisticated modelingmay also require going beyond the simple variationalscheme, applying various extensions of the screenedpotential [57,58,60], and accounting for the bands non-parabolicity [63]. The exact values of the effective massare also discussed in the literature, with interesting newinsights from transport measurements in gated samples[64–66], which provide larger values of effective masses.TABLE I. Exciton parameters obtained in experiments andcalculated values. The effective masses of the electron, me, andhole, mh, are taken from Ref. [32] (averaged over differentcalculation methods and subbands), m0 is the free electron mass,and μ ¼ memh=ðme þmhÞ is the reduced mass of the electron-hole pair; the effective dielectric constant of the surroundingmedia was chosen as in Ref. [31], ϰ ¼ 4.5; r0 was chosen toobtain values similar to experimental values of Δ12, Eq. (1). Wealso present ρ0 ¼ ϰr0, which is used in some references; seeAppendix C for details.MoS2 MoSe2 WSe2 MoTe2Δexper12 (meV) 175 150 130 120me=m0 0.45 0.53 0.34 0.57mh=m0 0.54 0.6 0.36 0.64μ=m0 0.25 0.28 0.17 0.3r0 (Å) 6.67 10 8.2 14.4ρ0 (Å) 30 45 37 65EtheorB (meV) 214 186 162 156Δtheor12 (meV) 174 148 133 121EXCITON STATES IN MONOLAYER MoSe2 AND MoTe2 … PHYS. REV. X 8, 031073 (2018)031073-5V. THEORY OF EXCITON UPCONVERSIONTo briefly summarize the main experimental results, wehave demonstrated that TMDMLs such as MoTe2, MoSe2,MoS2, and WSe2 exhibit strong photoluminescence upcon-version. At resonant laser excitation of the A∶1s groundexcitonic state, luminescence from excited states, such asA∶2s and B∶1s, is detected. The effect vanishes fornonresonant excitation or if the oscillator strength of theA∶1s exciton is reduced by the gate doping in chargetunable samples. Furthermore, the analysis of the upcon-version PL intensity as a function of excitation powerdemonstrates that this effect is nonlinear. In this section, weprovide a theoretical model of the upconversion effectobserved in TMD MLs. In this process, the emitted photonenergy is larger than that of the absorbed photon. That iswhy, in order to fulfill energy conservation, a third body, anexciton or a phonon, should be involved. However, at atemperature of 4 K, the thermal phonon energies are lessthan 0.3 meV; thus, lattice vibrations cannot provideeffective transfer of excitons up to several 100 meV abovethe excitation energy. Additionally, the experimental datashown in Fig. 2(c) (see also Ref. [16]) clearly demonstratesthe presence of an optical nonlinearity: The upconversionintensity scales quadratically with the number of photo-excited excitons.A first discussion of possible origins of the upconversionsignal can be found in Ref. [16]. The main ingredients areas follows: the absorption of multiple m ≥ 2 photons andthe consequent energy relaxation of photogenerated elec-tron-hole pairs. Further insight into the exact mechanism isprovided by the dependence of the upconversion photo-luminescence intensity Iup on the incident radiation inten-sity I. The data presented in Fig. 2(c) for the MoSe2 sample[see also published data in Fig. 2(b) of Ref. [16] on WSe2]show that this dependence is superlinear and roughly scalesas Iup ∝ I2α, where 0.5 < α < 1. Importantly, the emissionfrom the A∶1s state under quasiresonant excitation withthe same power scales as IA∶1s ∝ Iα, i.e., with a powertwice as small. Since the emission intensity is proportionalto the occupancy of the corresponding excitonic states,the experimental results demonstrate that the formation ofthe upconversion signal requires two excitons in the groundstate. Hence, the most plausible scenario is related to theexciton-exciton interaction, i.e., the Auger process. Thiseffect is discussed in detail below; we briefly address theother scenarios of upconversion involving direct two-photon absorption and also phonon-assisted processes.Thus, in order to describe the upconversion theoretically,we need to take into account exciton-exciton interactionprocesses where one of the excitons is annihilated whilethe second exciton acquires large extra energy, as depictedin the inset in Fig. 1(b) [67,68]. Subsequently, this excitonrelaxes toward the radiative states (particularly, A∶2s andB∶1s), and hot luminescence from these states is observedsince the radiative recombination time is competitive (i.e.,short enough) compared to the energy relaxation time.This mechanism of generating highly excited excitonscan be viewed as Auger-like exciton-exciton annihilation.At first glance, it seems to be quite weak because, in orderto satisfy the energy and momentum conservation laws,the initial kinetic energy of the involved particles should bevery large [69,70]. As we show here, this effect is veryefficient in TMD MLs due to (i) strong Coulomb inter-action, which makes it possible to relax the single-electronmomentum conservation [33,34], and (ii) the possibility ofresonant processes involving exciton transfer to an excitedenergy band [16]. In other words, we include the particularenergy spacing between different conduction bands in thetheory of this four-particle interaction.The schematics of exciton-exciton Auger processes ispresented in Fig. 4. Panel (a) shows an example of astandard Auger process, which is possible in any semi-conductor: Because of the Coulomb interaction, oneelectron recombines with a hole, while another carrier istransferred to a highly excited state. Figure 4(b) illustrates avery different process, which is possible in the studiedTMD MLs due to their specific band structure: It turns outthat there is an excited conduction band (denoted as c0)whose distance to the conduction band, E0g, approximatelysatisfies the conditionE0g ≲ Eg − EB; ð3Þwhere EB ≈ 0.3…0.5 eV [7] is the exciton binding energyfor measurements directly on SiO2 (see Table II),with lower values for EB of the order of 200 meV inFIG. 4. Model of exciton upconversion—single-particle pic-ture. Filled circles denote electrons; open circles denote unoccu-pied states in the valence band. Dashed arrows show realelectronic transitions. (a) Intraband process enabled by excitoniceffects. The resulting high-energy exciton involves carriers inbands c and v; the final momentum Kf ¼ K1 þ K2 is large.(b) Resonant Auger process resulting in high-energy, low-wave-vector Kf excitons involving the excited conduction band c0.B. HAN et al. PHYS. REV. X 8, 031073 (2018)031073-6hBN-encapsulated samples (see Table I and Ref. [61]).Thus, in the course of exciton-exciton annihilation, theelectron can be promoted to the c0 band with a relativelysmall wave vector rather than be scattered to a large-wave-vector state within the same band.The rate of the Auger processes is described by aparameter RA such that the generation rate of highlyenergetic excitons is given bydn0Xdt¼ RAn2X: ð4ÞHere, n0X is the density of highly energetic excitons, and nXis the density of photoexcited excitons in the A∶1s state.Recombination and energy relaxation processes are disre-garded in Eq. (4). The same rate RAn2X describes the decayrate of A∶1s excitons due to the nonradiative exciton-excitonannihilation: dnX=dt ¼ −RAn2X [33,56,67–69,72–75]. Therate RA can be expressed by means of the Fermi golden rulein the formRA ¼ 2πn2XℏXK1;K2;νjMXXj2fðK1ÞfðK2Þ× δ½Eg − 2EB − E0g þ EðK1Þ þ EðK2Þ − EνðKfÞ�:ð5ÞHere, MXX is the matrix element of the exciton-excitoninteraction,K1 andK2 are the center-of-masswave vectors ofthe two interacting excitons,fðKÞ is the distribution functionof photoexcited A∶1s excitons, and EðKÞ ¼ ℏ2K2=2M isthe exciton dispersion, withM being its effective mass. Thesubscript ν denotes the quantum numbers of the final excitonstate, which include the electron band index [c or c0 for theprocesses shown, respectively, in Figs. 4(a) or 4(b)], as wellas of the internalmotion (2s; 2p;… including the continuumstates), andEνðKÞ is the dispersion of the exciton in the finalstate, which accounts for its binding energy. The excitonwave vector in the final state, Kf, is found from themomentum conservation law: Kf ¼ K1 þ K2. For the der-ivation of Eq. (5), we assume that the occupation of the finalstates is negligible and omit the corresponding occupationfactor; we also disregard the anisotropy and nonparabolicityof exciton dispersion.The analytical results are obtained in Appendix D underthe model assumptions EB ≪ Eg; E0g, which allow us tomake use of k · p-perturbation theory for calculating theexcitonic states and transition rates [45]. The analysisdemonstrates that the dominating contribution to RA isgiven by the resonant processes described in Fig. 4(b),where the electron in the exciton is promoted to the excitedband c0. The Coulomb interaction between the electronscan result in the process shown in Fig. 4(b), where one pairrecombines while the remaining electron occurs in the c0band. So, one c0v pair is present in the end. Since in thisprocess, both electrons change their quantum states, theresonant Auger scattering is due to the electron-electroninteraction only, while the electron-hole interaction doesnot play a role, in contrast to the conventional exciton-exciton scattering where all charge carriers remain in thesame bands [76,77]. Moreover, our analysis shows that theexchange contribution to the matrix element dominates; seeAppendix D.Our next goal is to evaluate the scattering rate for excitonsunder the resonant condition Eg ¼ E0g þ 2EB − EB;n, wherethe energy released at the nonradiative recombination ofthe A∶1s exciton with K ¼ 0 is equal to the energy of theexcited c0-band exciton in the ns state. To that end, thedistribution function fðKÞ of excitons should be determined.It is controlled by the interplay of the generation process andall types of relaxation processes, including exciton-excitonand exciton-phonon scattering as well as the radiative decayfor the excitonswith small wavevectorKwithin the radiativecone and the redistribution of excitons between the brightand dark (spin-forbidden) states. The generation rate of theexcitons under the resonant excitation used in our experi-ments can be recast in the resonant form as [cf. Eq. (E2) andRefs. [16,44]]dnXdt����gen∝1ðℏω − EA∶1sÞ2 þ ℏ2Γ2AIℏω; ð6Þwhere EA∶1s ¼ Eg − EB is the resonant energy of the A∶1sexciton, and ΓA is the damping rate of this exciton. Here, weabstain from presenting and solving the full kinetic equationmodel for exciton distribution and analyze the timescalesof themost important processes. The calculations [44,78–80]show that, at cryogenic temperatures of 2;…; 4 K, theexciton-acoustic phonon scattering time is in the picosecondtimescale, i.e., on the same order of magnitude as theradiative decay rate. Thus, even at low temperatures, photo-created excitons can efficiently leave the light cone to thenonradiative states. Additionally, as recently shown [44,80],the pronounced high-energy tails in the exciton absorptionare caused by the exciton-phonon interaction, which can beaccounted for by the ω-dependent ΓA in Eq. (6). These tailsare also quite visible in the upconversion excitation spectra,TABLE II. Band gap energies. Data from DFT calculationssummarized in Ref. [71] for S and Se-based MLs and fromRef. [55] for MoTe2 MLs. Here, c0 corresponds to the cþ 2 bandin the notation of Refs. [32,71]. The direct comparison of thevalues with experimental data in Fig. 1 is not possible due to thedifferent levels of DFT approximations used.Energy (eV) MoS2 MoSe2 WSe2 MoTe2Eg 1.8 1.6 1.7 1.7E0g 1.1…1.2 1 1.4 1.3EXCITON STATES IN MONOLAYER MoSe2 AND MoTe2 … PHYS. REV. X 8, 031073 (2018)031073-7Fig. 2(d); such processes can efficiently generate the excitonswith largewave vectors and kinetic energies [44]. In contrast,the rate of exciton-exciton Auger annihilation is about anorder of magnitude smaller; see Appendix D and below.Thus, excitons are likely to thermalize [81].Finally, neglecting the difference of exciton masses inthe initial and final states and assuming, on the basis of thearguments above, that excitons are thermalized with thetemperature T, we haveRA ¼ πℏkBT���� 2πe2γ3γ6ϰa2BEgE0g����2jAnj2: ð7ÞHere, kB is the Boltzmann constant, aB is the exciton Bohrradius, γ3 and γ6 are the interband momentum matrixelements (in units of m0=ℏ, m0 being the free electronmass) for electron transition from c to, respectively, v andc0 bands, and ϰ is the effective high-frequency dielectricconstant. In Eq. (7), An is the dimensionless overlapintegral, which depends on the screening parameter r0 inthe interaction potential (2); see Appendix D for details.The values of jAnj2 for several excitonic states are shown inFig. 5. For n ¼ 1 and reasonable material parameters [71],the quantity RA at T ¼ 4 K in Eq. (7) can be estimated to be1;…; 10 cm2=s. This quantity can be reduced by a factor10;…; 100 if the resonant condition is not fulfilled. Therate of intraband transitions, where the electron remains inthe same band, Fig. 4(a), can be estimated by replacing thefactor γ6=ðaBE0gÞ by ðEB=EgÞ3=2, which produces a para-metrically smaller, ∼EB=Eg contribution; see Appendix D.In the nonresonant case the transition at K1 ¼ K2 ¼ 0 isnot possible, and the detuning Δ ¼ E0g − Eg þ 2EB − EB;nis present for the intraband process. This intraband processis only possible when excitons with the kinetic energy onthe order of Δ are present. The rate equation (7) thereforeacquires an exponential factor exp ð−jΔj=kBTÞ < 1.The experimentally observed Auger rates in the literatureare 1–2 orders of magnitude smaller than the resonantcontribution to RA investigated here [67,68,72–75,82]. Theexact values of RA will also vary with sample temperatureand environment, demonstrating that the exact resonanceconditions are not fulfilled in the studied structures.Furthermore, depending on the material, e.g., for W-basedmonolayers, a considerable part of the excitonic populationcan be in the spin-dark states, which also affects the Augerrates because, for the Auger process, at least one excitonshould be spin allowed. Our experiments are carried out at4 K, whereas many exciton-exciton scattering studies arecarried out at elevated temperatures. The presence ofdisorder in the sample, especially without hBN encapsu-lation, may enable us to simultaneously fulfill the energyand momentum conservation in TMD MLs, making addi-tional scenarios possible.Let us briefly address other possible origins of theupconversion effect. The straightforward one is due to thehot photoluminescence excited by two-photon absorption.The details of the process are given in Appendix E.The estimates show that this process is about 3 ordersof magnitude weaker than the resonant Auger processdescribed above. Moreover, strictly speaking, it is charac-terized by the Iup ∝ I2 dependence, in contrast to theexperimental data. However, the present analysis cannotfully rule out such a process or more sophisticated processes,where the exciton-phonon interaction is involved at one ofthe steps of the two-photon absorption. The detailed calcu-lation of the rates of these processes can be superficialbecause of the imprecise knowledge of the band-structureparameters and, therefore, the impossibility to check towhich extent all needed resonance conditions are fulfilled.We believe that this analysis will stimulate further exper-imental and theoretical studies aimed, in particular, atspecifying band parameters of two-dimensional materials.To summarize, our analysis suggests that excitonupconversion photoluminescence in TMD MLs is pos-sibly due to a specific nonlinear process: Two excitonsgenerated by the resonant laser collide; as a result, one ofthose excitons recombines nonradiatively, while the otheris promoted to a highly excited state (most likely relatedto an excited conduction subband according to the band-structure calculation). Subsequently, the excited excitonloses its energy, and a hot PL from the radiative A∶2sand B∶1s states is observed. This scenario describes themain experimental findings: (i) upconversion PL from thestates A∶2s and B∶1s, which are visible in the hot PL[Figs. 1, 2(a), 2(b), 2(d), and 3], (ii) quadratic dependenceof the upconversion intensity on the number of excitonsin the ground state, Fig. 2(c), (iii) resonant characterof the process as a function of excitation laser energy,Fig. 2(d), and (iv) absence of the upconversion in thepresence of doping, where the exciton resonance vanishes,Fig. 2(f).FIG. 5. Dependence of the coefficient jAnj2 on n for r0 ¼ 3aB.The red line shows the approximation jAnj2 ¼ 2.47 × 10−7=n3.The inset shows dependences jA1j2 (red), jA2j2 (blue), and jA3j2(green) on the screening radius r0. Dashed lines are fits jA1j2 ¼0.015ðaB=r0Þ2, jA2j2 ¼ 3.46 × 10−5ðaB=r0Þ4, and jA3j2¼6.63×10−6ðaB=r0Þ4.B. HAN et al. PHYS. REV. X 8, 031073 (2018)031073-8VI. CONCLUSIONWe identify excited exciton states in high-quality MoSe2and MoTe2 monolayer samples, which govern absorptionand emission above the A∶1s exciton resonance. We alsoidentify the A∶2s state 150 meV (120 meV) above theA∶1s state in ML MoSe2 (MoTe2). We show that excitedexciton states can be studied in photoluminescence upcon-version experiments. In addition to being a highly selectivespectroscopic tool applicable to several TMD materials[16,31], this nonlinear optical effect also gives insights intoexciton-exciton interactions—relevant physical processesthat are also used for studying population inversion andother density-dependent phenomena [75,83–85]. We dis-cuss the possibility that, in TMD monolayers, the gener-ation of high-energy excitons with Auger-like scatteringprocesses is efficient due to the strong Coulomb interactionand resonant excitation of higher-lying conduction bands.ACKNOWLEDGMENTSWeacknowledge funding fromANR2D-vdW-Spin, ANRVallEx, Labex NEXT projects VWspin and MILO, ITNSpin-NANO Marie Sklodowska-Curie Grant AgreementNo. 676108, and ITN 4PHOTON No. 721394. X.M. alsoacknowledges the Institut Universitaire de France. Growthof hexagonal boron nitride crystals was supported by theElemental Strategy Initiative conducted by theMEXT, Japanand the CREST (JPMJCR15F3), JST. L. E. G. and M.M. G.acknowledge partial support from LIA ILNACS, RFBRProjects No. 17-02-00383 and No. 17-52-16020, RussianFederation President Grant No. MD-1555.2017.2, and theBASIS Foundation.APPENDIX A: EXPERIMENTAL METHODSThe samples are fabricated by mechanical exfoliation ofbulk MoSe2 and MoTe2 (commercially available from 2Dsemiconductors) and very-high-quality hexagonal boronnitride (hBN) crystals [40] on 83-nm SiO2 on a Si substrate.The experiments are carried out at T ¼ 4 K in a confocalmicroscope built in a vibration-free, closed-cycle cryostat.The excitation/detection spot diameter is ∼1 μm. Themonolayer (ML) is excited by a continuous-wave Ti-Salaser (700–1020 nm) or a HeNe laser (633 nm). The PLsignal is dispersed in a spectrometer and detected with aSi-CCD camera (λ < 1 μm) or InGaAs detector(λ > 1 μm). The typical excitation power is 3 μW.APPENDIX B: ADDITIONAL DATACharge tuning in ML MoSe2.—Figure 6 demonstratesthe reflectivity spectrum of the charge tunable MoSe2device clearly showing the redistribution of the oscillatorstrength between the neutral and charged excitons, which isstudied in the context of upconversion in Figs. 2(e) and 2(f).Resonant and nonresonant Raman scattering in MLMoTe2.—Scattering with phonons in Raman processesallows us to distinguish monolayers from multilayers.This is especially useful for MoTe2, where the bilayeralso shows clear and narrow PL emission. In Fig. 7a, wecompare results for a monolayer and a bilayer, the absenceof the low-energy A1g peak allows us to identify monolayersamples [86,87]. The experiments in Fig. 7a are essentiallybased on nonresonant Raman scattering; i.e., neither thelaser energy nor the emitted light after phonon scattering isresonant with a particular electronic state. This is differentin Fig. 7b: Here, we tune the laser to an excess energy ofabout 20 meVabove the A∶1s resonance. In addition to PLemission (orange peak), we see a spectrally sharper feature(shaded blue) superimposed on the PL, which shifts withexcitation laser energy. This peak corresponds to Ramanscattering with the A01 phonon, which is particularlyefficient as the final state after scattering corresponds to areal electronic state [17,86,88–91] in these single-resonantRaman scattering experiments.In Fig. 7c, we report double-resonant Raman experi-ments [22,89]. As the laser energy is scanned across theA∶2s state, we see that the PL of the A∶1s exciton isenhanced; see intensity plotted in Fig. 3(c) as a function oflaser energy. In addition, we observe in Fig. 7c that aRaman feature is crossing the PL line, exactly 120 meVbelow the respective laser energy. When the laser is at theA∶2s energy, the Raman process is double resonant [86,89]as the initial state (A∶2s) and the final state (A∶1s) arereal electronic states. This case has already been observedbetween exciton states in ML WSe2 on SiO2 [89]. Pleasenote that very different electronic states and phonons arediscussed in the double-resonant Raman experiments inRef. [86]. Detailed MoTe2 Raman studies are also reportedfor monolayers and multilayers in Ref. [48]. In experimentsin hBN-encapsulatedMLWSe2 samples, similar experimentsEnergy (eV) DifferentialReflectivity (arb. u.) A:1s TbiasFIG. 6. Charge tunable device. In differential reflectivity on thecharge tunable device of Figs. 2(e) and 2(f), we identify the A∶1sstate and, at more negative bias, the charged exciton state (markedT for trion).EXCITON STATES IN MONOLAYER MoSe2 AND MoTe2 … PHYS. REV. X 8, 031073 (2018)031073-9have been interpreted as being due to phonon-related proc-esses only [15]. This interpretation seems unlikely in view offollow-up studies of ML WSe2 in magnetic fields, whichclearly showed that the excited state is an excitonic transitionand not just a phonon replica [61]. For the case of MoTe2,we have a strong case for the transition at 120 meV abovethe A∶1s to be attributed to a real electronic transition, as thistransition is confirmed in Figs. 3(a)–3(d) by four comple-mentary spectroscopy techniques and in Fig. 7c by double-resonant Raman scattering.APPENDIX C: BINDING ENERGIES OFEXCITONIC STATES IN VARIATIONALAPPROACHIn order to calculate the energy separation betweenthe ground and excited excitonic states, we apply a simplevariational approach. The wave functions of the groundand first excited states are sought in the two-dimensionalhydrogenic formΦ1sðrÞ ¼ffiffiffiffiffiffiffiffi2πa2re−r=a; ðC1aÞΦ2sðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffi227πa2r �1 −2r3a�e−r=3a; ðC1bÞwith the only trial parameter a, which is the same for bothfunctions. Such an approach has the advantage of auto-matically providing orthogonal ground and excited states atthe cost of simplicity. To test the calculation, we reproduce,within an error of≤2%, the results of Ref. [31], where moresophisticated numerical approaches were used, for A∶1sand A∶2s excitons in the monolayer of MoS2 with the sameset of band structure and screening parameters.APPENDIX D: CALCULATION OF AUGERRATES1. Resonant interband processWe consider three bands, c, v, c0, schematically illustratedin Fig. 4 with two excited electron-hole pairs (excitons) with(a)(b) (c)FIG. 7. Raman spectroscopy in ML MoTe2, T ¼ 4 K. (a) Nonresonant Raman scattering using a HeNe laser. (b) Single-resonantRaman scattering as the excitation laser energy is one phonon energy A01 above the A∶1s state. (c) Double-resonant Raman experiments,as a phonon multiple ensures efficient relaxation from the optically excited A∶2s state to the emitting A∶1s state.B. HAN et al. PHYS. REV. X 8, 031073 (2018)031073-10an electron occupying the lowest conduction band c andwithan empty state in the valence band v. Here, for simplicity, wedisregard the spin degree of freedom of change carriers,assuming that the spin is conserved in the course of exciton-exciton interaction. Furthermore, we focus on the states inthe vicinity of one of the band extrema (Kþ or K− valley)and disregard the processes involving intervalley transfer ofelectron-hole pairs studied in Refs. [92]. The Coulombinteraction between the electrons can result in the processshown in Fig. 4(b), where one pair recombines while theremaining electron occurs in the c0 band. So, one c0v pair ispresent in the end. Since in this process both electrons changetheir quantum states, the resonant Auger scattering is due tothe electron-electron interaction only, while, e.g., electron-hole interaction does not play a role.In the free-particle picture, we have two electrons in thec band, which occur in the c0 and v bands after Coulombscattering. The two-electron wave functions of the consid-ered system in the initial and final states can be presented asjii ¼ 1ffiffiffi2p ½Ψckcðr1ÞΨck̃cðr2Þ − Ψckcðr2ÞΨck̃cðr1Þ�;jfi ¼ 1ffiffiffi2p ½Ψvkvðr1ÞΨc0kc0 ðr2Þ −Ψvkvðr2ÞΨc0kc0 ðr1Þ�: ðD1ÞHere, kc, kv are the electron and unoccupied-state wavevectors in one of the excitons, and k̃c, k̃v are the electronand unoccupied-state wave vectors in another exciton.The wave function in each band n ¼ c, c0, v is a productof the Bloch amplitude and the plane wave:ΨnkðrÞ ¼ eik·runkðrÞ; ðD2Þand the normalization area is set to unity. In the k · pmodel,the Bloch amplitudes have the formuckc ¼ uc þℏm0kc · pvcEc − Evuv þℏm0kc · pc0cEc − Ec0uc0 ; ðD3Þuc0kc0 ¼ uc0 þℏm0kc0 · pvc0Ec0 − Evuv þℏm0kc0 · pcc0Ec0 − Ecuc; ðD4Þuvkv ¼ uv þℏm0kv · pcvEv − Ecuc þℏm0kv · pc0vEv − Ec0uc0 ; ðD5Þwhere un denotes the Bloch amplitude at the extremumpoint, m0 is the free electron mass, and pnn0 are themomentum matrix elements between the states in the bandsn and n0 (n, n0 ¼ c, c0, v).The wave functions jii, jfi in Eq. (D1) are antisymme-trized with respect to the permutations of electrons, givingrise to the direct and exchange contributions. The matrixelement of the direct interaction, where the electron fromthe state with the wave vector kc recombines with the holefrom the same exciton and transfers to the state kv, can beconveniently presented in the formMdir ¼ huvkv juckcihuc0kc0 juck̃ci× VCðK1ÞδK1;kc−kvδK1;kc0−k̃c; ðD6Þwhere K1 ¼ kc − kv is the exciton center-of-mass momen-tum (note that the hole state corresponds to the time-reversed counterpart of the unoccupied state) andVCðqÞ ¼2πe2ϰqð1þ qr0ÞðD7Þis the 2D Fourier image of the Coulomb potential, with ϰbeing the background average constant of the surroundingstructure and r0 being the dielectric screening parameter[57,58,93]. Note that this parameter should be taken in thehigh-frequency limit because the energy transferred in thecourse of the exciton-exciton interaction is on the orderof the band gap Eg.Taking into account thatℏm0k · pvc ¼ γ�3kþ;ℏm0k · pc0c ¼ γ6k−; ðD8Þwhere γ3 and γ6 are the band-structure parameters intro-duced in Refs. [32,71] and k� ¼ kx � iky, we obtainhuvkv juckci ¼γ�3KþEc − Ev; huc0kc0 juck̃ci ¼γ6K−Ec0 − Ec:ðD9ÞFinally, the direct interaction matrix element takes a simpleformM1ðK1Þ≡ VCðK1Þγ�3γ6K21EgE0gδK1;kc−kvδK1;kc0−k̃c: ðD10ÞTaking into account the excitonic effect, Eq. (D10) shouldbe averaged over the exciton wave function [45,92,94].Furthermore, we need to note that, in the initial state, thereare two unoccupied states in the valence band. As a result,we have (K1, K2 ≪ a−1B )MdirðK1;K2;Kf; nÞ ¼ δν;1sδKf;K1þK2Φ1sð0Þ×12½VCðK1ÞK21 þ VCðK2ÞK22�γ�3γ6EgE0g:ðD11ÞWe recall that K1, K2 are the wave vectors of excitons in theinitial state, Kf ¼ K1 þ K2 is the wave vector of the excitonin the final state, and the subscript ν enumerates the relativemotion states of the remaining electron-hole pair. In thederivation of Eq. (D11), we neglected the difference of theelectron effective masses in c and c0 bands and assumedthat, initially, both excitons occupy 1s state; Φ1sðρÞ is theEXCITON STATES IN MONOLAYER MoSe2 AND MoTe2 … PHYS. REV. X 8, 031073 (2018)031073-11envelope function of the relative motion. Correspondingly,the final-state relative motion envelope function remainsthe same.The typical center of mass wave vectors involved inexciton-exciton scattering are on the order of the thermalwave vector KT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MkBT=ℏ2pand are much smallerthan the screening wave vector r−10 ; therefore, the directexciton-exciton scattering matrix element Mdir is propor-tional to the first powers of the exciton wave vectors:Mdir ∝ KT .Next, we consider an exchange process, where theelectron occupies the empty state in the valence bandrelated to the hole in the other exciton; i.e., the electron withthe wave vector kc transfers to the valence band state withthe wave vector k̃v. As a result, for uncorrelated electron-hole pairs, we have−M1ðjkc − k̃vjÞ; ðD12Þfor the exchange contribution, where M1 is defined inEq. (D10). In order to transform Eq. (D12) to the formconvenient for averaging over the exciton wave functions,we introduce the relative motion wave vectors for twoinitial and final exciton states in accordance withk1 ¼kc þ kv2; k2 ¼k̃c þ k̃v2;kf ¼ kc0 þ kv2¼ k1 þK22: ðD13ÞHere, we assumed that the effective masses of the electronand hole are the same, in agreement with microscopiccalculations [32,71], see also Table II. Taking into accountthat, as before, the center-of-mass wave vectors K1, K2 ∼KT are small compared with the inverse Bohr radius a−1B ofthe exciton, we omit Kf in kc − k̃v ¼ k1 − k2 þ Kf=2 andin k1 þ Kf=2 and arrive atMexchðK1;K2;Kf; νÞ≈ −δKf;K1þK2×Xk1;k2M1ðjk1 − k2jÞC�νðk1ÞC1sðk2ÞC1sðk1Þ: ðD14ÞHere, CνðkÞ are the Fourier transforms of the relativemotion exciton functions ΦνðρÞ:CνðkÞ ¼Zdρeik·ρΦνðρÞ:It follows fromEq. (D14) that only s-shell states contribute tothe matrix element. As compared with its direct counterpart,the transferred momentum here is jk1 − k2j ∼ a−1B . SinceM1ðqÞ ∝ q for qr0 ≪ 0, the direct contribution is by a factorKTaB smaller than the exchange one. Thus, in what follows,we consider the exchange contribution only.In order to analyze the exchange process in more detail,we first consider a limit where the screening is very strong,i.e., where jk1 − k2jr0 ≫ 1. In this case, we can approxi-mate M1ðqÞ by a constant and arrive atMexchðK1;K2;Kf; νÞ ≈ −δν;1sδKf;K1þK22πe2ϰr0γ�3γ6EgE0gΦ1sð0Þ:ðD15ÞFor arbitrary screening, we evaluate the sum in Eq. (D14),making use of the two-dimensional hydrogenic functions.For the bound states ν ¼ ns, we have [95]CnsðkÞ ¼ 2ffiffiffiffiffiffi2πpaB�2n − 11þ κ2n�3=2Pn−1�κ2n − 1κ2n þ 1�; ðD16Þwith κn ¼ ð2n − 1ÞkaB and PnðxÞ being the Legendrepolynomial. As a result,MexchðK1;K2;Kf; nsÞ ¼Anðr0Þa2B2πe2γ�3γ6ϰEgE0gδKf;K1þK2:ðD17ÞDependence of jAnj2 on n for r0 ¼ 3aB is shown in Fig. 5.We see that the squared matrix element decreases rapidlywithn. It follows fromEq. (D16) thatCn0 ∝ n−3=2 atn → ∞;therefore, jMbj2 ∼ n−3. Figure 5 shows that this asymptotic isvalid already at n ≥ 3.The inset to Fig. 5 shows the dependences of thescattering probability on the screening radius r0. Finalstates with n ¼ 1, 2, 3 are considered. As mentioned above,scattering into the ns state for the short-range interaction ispossible at n ¼ 1 only. The probability of this processdecreases as 1=r20, while for n ≥ 2, it drops as 1=r40.The corresponding asymptotes are shown by dashed linesin Fig. 5.2. Intraband Auger processLet us now briefly address the intraband process depictedin Fig. 4(a), where the charge carriers remain in the samebands after the scattering. For free carriers, the process hasa high threshold, requiring the initial and final wave vectorsto be on the order offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMEg=ℏ2q; otherwise, the energy andmomentum conservation laws cannot be satisfied simulta-neously. Taking into account the excitonic effect, the processbecomes allowed because one can find, in the relative motionwavefunction Fourier image [Eq. (D16)], sufficiently largewave vectors due to the Coulomb interaction. In other words,the electron-hole Coulomb interaction either in the initial orin the final state may relax the momentum conservation inthe course of the Auger scattering. However, the two-bandB. HAN et al. PHYS. REV. X 8, 031073 (2018)031073-12approximation is insufficient to give a correct result since(i) the model approximations for the band dispersions are, asa rule, invalid at the kinetic energies ∼Eg due to the k · pinteraction with remote bands [32,69] and (ii) the asymptoticform of CνðkÞ at large wave vectors can strongly differ froma simplified hydrogenic model (D16) [7,12,94]. Thus, wepresent only analytical estimations based on the parabolicapproximations for the band dispersions, assuming that theratio Eg=EB is very large, which allows us to take intoaccount the Coulomb effects perturbatively.We start from the direct process. Instead of Eq. (D10), wehave, for the free carrier scattering,VCðK1Þγ�3K1;þEgδK1;kc−kvδK1;kc0−k̃c: ðD18ÞAs compared with Eq. (D10), the factor ∝ γ6=E0g is absentdue to the fact that only one charge carrier changes theband. Making use of the following notations,kc0 ¼ kf þK1 þ K22¼ kf þKf2;k̃c ¼ kf þK2 − K12; ðD19Þwe obtain, for the exciton Auger scattering matrix element,the following expression:M0dirðK1;K2;Kf; kfÞ ¼ δKf;K1þK2Φ1sð0ÞVCðK1Þγ�3K1;þEg×ZdreiK1·r=2Φ�kf;lðrÞΦ1sðrÞþ K1 ↔ K2: ðD20ÞHere, we take into account that only excitons with positiveenergies of relative motion can be in the final state; i.e.,kf corresponds to the continuum electron-hole pair statemodified by the Coulomb interaction. The final-state wavevector can be estimated from the energy conservationcondition with the result kf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEg − 2EBÞM=ð2ℏ2Þq. Inthis estimate, we neglect thermal energy of excitons ascompared with Eg − 2EB and, as before, take the sameeffective masses for an electron and a hole.Since K1 ≪ a−1B , kf, the integral in Eq. (D20) can beevaluated by decomposing the exponent in the series. AtK1 ¼ 0, the integral in Eq. (D20) equals zero due toorthogonality of the functions of discrete and continuousspectra. Therefore, we take into account the K1-linear term:ZdreiK1·r=2Φ�kf;lðrÞΦ1sðrÞ ≈ aBK1DðkfÞ; ðD21ÞwhereDðkfÞ ¼iaB2Zdr r cosφΦ�kf;1ðrÞΦ1sðrÞ ≪ 1: ðD22ÞWithin the hydrogenic model, which is used hereafter forcrude estimations, DðkfÞ ∝ ðkfaBÞ−3 and the ratio of theinterband contribution (D11) and the intraband contribution(D20) can be estimated as����MdirM0dir���� ∼ γ6aBE0g�EgEB�3=2∼EgEB≫ 1: ðD23ÞThus, the resonant interband process is dominant. A similarestimate holds for the exchange contributions.3. Auger recombination rateThe exciton Auger recombination rate at resonant inter-band scattering of two 1s excitons into the ns exciton stateis given by [cf. Eq. (5) of the main text]RAn2X ¼ 2πℏXK1;K2jMexchðK1;K2;Kf; nsÞj2fðK1ÞfðK2Þ× δ½Eg − 2EB − E0g þEðK1Þ þEðK2Þ− EnsðKfÞ�:ðD24ÞIn order to calculate the rate of transitions, we take intoaccount that the matrix element MexchðK1;K2;Kf; nsÞdepends on the principal quantum number n of the finalstate and is independent of the initial wave vectors ofexcitons.Let us first assume that there is exact resonance; i.e., for acertain value of n at K1 ¼ K2 ¼ Kf, we haveEg ¼ E0g þ 2EB − EB;n; ðD25Þwhere EB;n is the binding energy of the ns state. Removingthe energy conservation δ function and assuming thatfðKÞ ¼ N exp�−ℏ2K22MkBT�; ðD26Þi.e., that the excitons are distributed according to theBoltzmann law at the temperature T, and N is thenormalization constant determined by the conditionnX ¼ gXKfðKÞ;where the factor g accounts for the spin and valleydegeneracy, we haveRA ¼ Rn; Rn ¼πℏkBT���� 2πe2Anγ�3γ6ϰa2BEgE0g����2: ðD27ÞNote that the Auger process is active for collisions of bright(spin-allowed) excitons with bright or dark ones, while forEXCITON STATES IN MONOLAYER MoSe2 AND MoTe2 … PHYS. REV. X 8, 031073 (2018)031073-13the dark-dark scattering, the process is strongly suppressed.In the latter case, the k · p admixture with the valence bandis minor, and the recombination via the discussed channelis not effective. The Auger decay rate can be recast in thealternative formRAn2X ≡ nXτA; ðD28Þwhere we introduced the Auger recombination time τAðnXÞ.Since e2=aB ∼ EB, we have an estimate in the case of theresonance with the 1s state1τA∼nXℏkBT�Ega20aB�2; ðD29Þwhere a0 is the lattice constant. At T ¼ 4 K, a0 ¼ 3 Å,aB ¼ 1 nm, EB ¼ 0.5 eV, and the exciton densitynX ¼ 109 cm−2, this estimate yields τA ∼ 25 ps.Let us now take into account the detuningΔ ¼ E0g − Eg − EB;n þ 2EB: ðD30ÞAccordingly, we have the sum over K1;2 in the followingform:XK1;K2fðK1ÞfðK2Þδ�ℏ2ðK21 þ K22 − jK1 þ K2j2Þ2M− Δ�¼ n2X2kBTe−jΔj=kBT: ðD31ÞWe see that the difference with the case of zero detuning isthe exponent. The Auger recombination rate is given byRA ¼ Rne−jΔj=kBT: ðD32ÞAPPENDIX E: TWO-PHOTON ABSORPTIONVIA EXCITON STATEIn order to analyze the two-photon absorption via theexciton state, we present a simple three-state model describ-ing the dynamics of the ground state of the crystal j0i(no excitons), jA∶1si (A∶1s exciton), and excited state jfi(the exciton formed by the electron in the c0 band and the holein the valence band v). Expanding the wavefunction of thesystem over these states and introducing the decompositioncoefficients, respectively,C0,CA∶1s, andCf, we obtain, fromsecond-order time-dependent perturbation theory,C0 ¼ 1; ðE1aÞCA∶1s ¼VA0ℏω − EA∶1s þ iℏΓA; ðE1bÞCf ¼ VfAVA0ð2ℏω − Ef þ iℏΓfÞðℏω − EA∶1s þ iℏΓAÞ: ðE1cÞHere, ω is the energy of the incident radiation, EA∶1s ¼Eg − EB is the excitation energy of the A exciton, Ef is theenergy of the exciton in the final state jfi, and ΓA and Γfare the dampings of these states. In Eqs. (E1), the matrixelements VA0 and VAf are the matrix elements of excitonexcitation for the processes v → c and c → c0, respectively;see below for explicit expressions. In this simplifiedapproach, the steady-state populations of the A∶1s stateand of the f state readNA∶1s ¼ jCA∶1sj2 ¼jVA0j2ðℏω − EAÞ2 þ ℏ2Γ2A; ðE2aÞNf ¼ jCfj2 ¼jVfAj2ð2ℏω − EfÞ2 þ ℏ2Γ2f× NA∶1s: ðE2bÞNote that Eq. (E2a) gives the spectral shape of one-photonabsorption in the vicinity of A∶1s exciton resonance,while Eq. 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