PHYSICAL REVIEW X 7, 041040 (2017)

Marrying Excitons and Plasmons in Monolayer Transition-Metal Dichalcogenides

† Dinh Van Tuan,1,* Benedikt Scharf,2,3,4 Igor Žutić,2 and Hanan Dery1,5, 1Department of Electrical and Computer Engineering, University of Rochester, Rochester, New York 14627, USA 2Department of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA 3Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany 4Institute for Theoretical Physics and Astrophysics, University of Würzburg, Am Hubland, 97074 Würzburg, Germany 5Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA (Received 14 June 2017; revised manuscript received 7 September 2017; published 17 November 2017) Just as photons are the quanta of light, plasmons are the quanta of orchestrated charge-density oscillations in conducting media. Plasmon phenomena in normal metals, superconductors, and doped semiconductors are often driven by long-wavelength Coulomb interactions. However, in crystals whose Fermi surface is comprised of disconnected pockets in the Brillouin zone, collective electron excitations can also attain a shortwave component when electrons transition between these pockets. In this work, we show that the band structure of monolayer transition-metal dichalcogenides gives rise to an intriguing mechanism through which shortwave plasmons are paired up with excitons. The coupling elucidates the origin for the optical sideband that is observed repeatedly in monolayers of WSe2 and WS2 but not understood. The theory makes it clear why exciton-plasmon coupling has the right conditions to manifest itself distinctly only in the optical spectra of electron-doped tungsten-based monolayers.

DOI: 10.1103/PhysRevX.7.041040 Subject Areas: Condensed Matter Physics

I. INTRODUCTION AND MOTIVATION Understanding its microscopic origin has been an outstand- ing open question ever since its first observation in the Transition-metal dichalcogenides become direct band- photoluminescence (PL) measurements of ML-WSe2 by gap semiconductors when thinned to a single atomic Jones et al. [10]. So far, this optical transition has been monolayer [1,2]. Stacking such two-dimensional crystals attributed to the fine structure of negatively charged excitons, via van der Waals forces alleviates the need to fabricate biexcitons, or defects in various experiments [9–17].We logic and optoelectronic devices with lattice-matched show that this optical transition actually corresponds to an crystals [3–5]. Furthermore, the lack of space inversion exciton-plasmon quasiparticle due to intervalley Coulomb symmetry in these monolayers lifts the spin degeneracy at excitations, where other identifications are not consistent the edges of the conduction and valence bands [6,7].Asa with its spectral position, dependence on electron density, result, time-reversal partners the spin and valley degrees of and the fact that it does not emerge in hole-doped conditions freedom (d.o.f.) such that spin-up charge carriers populate or in molybdenum-based MLs. To explain the behavior of valleys (low-energy pockets) on one side of the Brillouin this optical sideband, we use the empirical data of differential zone, while spin-down ones populate valleys on the opposite reflectance spectroscopy of ML-WSe2 at 4 K, as shown side [6]. The intense recent research on the spin-valley in Fig. 1 where this sideband is indicated by X−0.The partnership has led to understanding of optical selection monolayer was embedded between thin layers of hexagonal rules and transport phenomena in monolayer transition-metal boron nitride in a dual-gated structure [15]. The bias voltage dichalcogenides (ML-TMDs) [8,9]. was the same in the top and bottom gate electrodes; thus, only In this work, we focus on a unique optical sideband the charge density in the monolayer is controlled, while no that emerges in emission or absorption-type experiments out-of-plane electrical field is introduced. The relation – of electron-doped ML-WSe2 and ML-WS2 [9 17]. between gate voltage and charge density is such that 1 V corresponds to about 1012 cm−2 [15]. *[email protected] Figure 1(a) shows the spectrum for low and intermediate † [email protected] electron densities. The neutral-exciton spectral line, denoted by X0, dominates the spectrum, and its intensity Published by the American Physical Society under the terms of decays with increasing the electron density. Also noticeable the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to are three smaller spectral lines in the low-energy side of the the author(s) and the published article’s title, journal citation, spectrum (as highlighted in the enlarged box). These and DOI. features cannot be completely quenched because of

2160-3308=17=7(4)=041040(19) 041040-1 Published by the American Physical Society VAN TUAN, SCHARF, ŽUTIĆ, and DERY PHYS. REV. X 7, 041040 (2017)

(a)− (b) Figures 1(c) and 1(d) show the absorption spectrum in hole-doped conditions. It includes the neutral and charged- exciton spectral lines (X0 and Xþ), where the latter behaves A similarly to X− in terms of its decay, blueshift, and broadening at elevated densities. There is no optical side-

− band. The reflectivity spectrum of ML-MoSe2 is similar (not shown) [15]. It only includes two spectral lines, both in

(c) − (d) − the electron and hole-doping regimes, which behave − − similarly to X0 and X in Figs. 1(c) and 1(d). They share − − þ − − no common properties with the optical sideband of tung- sten-based MLs. It is emphasized that in this work we neither model nor focus on the microscopic origin of the charged exciton. Sidler et al. recently suggested that it originates from Fermi polarons, wherein the exciton is attracted to the surrounding Fermi sea via weak van der Waals forces [27,28]. Jadczak et al., on the other hand, FIG. 1. Optical reflection spectroscopy of ML-WSe2 at 4 K. attributed the charged exciton to trions and further eluci- The gate voltage controls the charge density in the ML. (a,b) dated its strong coupling to phonons due to the resonance Optical spectra for low and high electron densities, respectively. between the trion binding energy and that of the homopolar The optical sideband is indicated by X−0. (c,d) The respective phonon mode [29]. results for hole densities. The sample is n type, so a gate voltage −1 The observed behavior of the optical sideband in of about V is required to deplete the ML from resident tungsten-based compounds is largely identical in reflec- electrons. These results were kindly provided to us by Wang, Mak, and Shan. Reference [15] includes details of the experiment tance and PL experiments, including the unique feature that with a similar device. the sideband redshifts in both cases [10,11,15].One noticeable difference, however, is the relative amplitude of the signal. Specifically, reflectance measurements show disordered regions that prevent complete charge depletion that the largest oscillator strength is that of the neutral across the entire sample [18]. The spectral lines that are exciton at low densities [15,16,20,30,31]. The oscillator A B denoted by X− and X− were recently attributed to two types strength of the optical sideband, on the other hand, is of negatively charged excitons in tungsten-based mono- evidently weaker as one can notice by comparing the y-axis layers [14,19,20]. The lowest-energy spectral line, denoted scales in Figs. 1(a) and 1(b). The scenario is opposite in PL by X−0, is the optical sideband. In addition to the fact that measurements, where one finds that the strongest light A B 0 X− and X− are clearly resolved from that of X− even at low emission comes from the optical sideband at elevated densities, as highlighted in the enlarged box of Fig. 1(a), electron densities, while that of the neutral exciton at their behavior is strikingly different, rendering their iden- low densities is evidently weaker [10,11,15]. This apparent A tification relatively easy. For example, X− has the known contradiction is settled in PL experiments for the following properties of trions in semiconductor quantum wells reasons: (i) Neutral excitons in tungsten-based monolayers [21–26]: It is amplified without enhanced broadening when experience intervalley scattering after photoexcitation, the electron population increases in the low-density regime. rendering them less optically active in the light emission In other words, in the regime where the Fermi energy is process [32–35]. (ii) The quantum efficiency of light emis- measurably smaller than the binding energy of the charged sion can be improved at elevated electron densities because A exciton (energy difference between X− and X0, which is of screening of charged defects that function as nonradiative A about 30–35 meV). Figure 1(b) shows that X− starts to recombination centers for excitons. (iii) We show that several blueshift, decay, and broaden at intermediate and elevated photon-emission mechanisms contribute to the optical side- densities, in which the Fermi energy is no longer negligible band in the PL case. compared with the charged-exciton binding energy. In this The main contribution of this work is in elucidating the density regime, the trion cannot avoid many-body inter- microscopic origin for the optical sideband, revealing an actions with the Fermi sea. The optical sideband, on the intriguing pairing phenomenon between excitons and other hand, shares none of these features. Figure 1(b) shows plasmons, which has no parallel in other known two- that when the electron density continues to increase, X−0 dimensional (2D) systems. We explain why the optical redshifts and possibly drains part of the oscillator strength sideband is manifested in the spectra of electron-doped of the decaying charged exciton spectral lines. Its spectral ML-WSe2 and ML-WS2 while being absent in the hole- line shows no evident broadening, and its saturated doped case or in ML-MoSe2 and ML-MoS2. The theoreti- intensity at elevated densities hardly decays (whereas the cal framework in this work studies the oscillator strength of other spectral lines strongly decay, blueshift, and broaden). excitons through the electron-hole pair function in the

041040-2 MARRYING EXCITONS AND PLASMONS IN MONOLAYER … PHYS. REV. X 7, 041040 (2017) presence of dynamical screening. The theory extends Fig. 2, indirect excitons have lower energy than direct ones beyond the quasistatic random-phase approximation or in ML-WX2 and vice versa in ML-MoX2 [32]. the Shindo approximation for the dynamical effects [36]. Next, we briefly discuss the two plasmon species in It therefore allows one to model the decay of the neutral ML-TMDs. The first type of plasmon is characterized by its exciton at elevated charge densities without introducing long wavelength [41–43]. These plasmons originate from phenomenological screening parameters or ansatzes the long-range part of the Coulomb potential through which regarding the energy dependence of the pair function. In electrons cross the Fermi surface with small crystal addition, the theory allows one to model the emergence of momentum transfer (i.e., an intravalley process). They dynamical optical sidebands, where we focus on the one are common to all conducting media and are largely that arises from the exciton-plasmon coupling at large independent of the band structure since their wavelength electron densities. is much longer than the lattice constant [44]. In the context of optical transitions in semiconductors, plasmons screen the electron-hole attraction, and they shrink the band-gap II. BASIC PROPERTIES OF EXCITONS energy by assisting electrons (or holes) of similar spins to AND PLASMONS IN ML-TMDs avoid each other [36]. The energy dispersion of long- Before we can explain the coupling between plasmons wavelength plasmons in 2D semiconductors follows, and excitons, we briefly summarize the relevant properties sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 of each of these quasiparticles separately. Excitons in 2e EFk ElðkÞ¼ ; ð1Þ ML-TMDs are classified based on the spin and valley ϵðkÞ configuration of the electron-hole pair [32–34]. Figure 2 shows the valleys pertinent to the formation of low-energy where e is the elementary charge, k is the wave number of 2 2 2 excitons in ML-WX and ML-MoX , where X is the the plasmon, and EF ¼ πℏ n=me is the Fermi energy chalcogen atom. Excitons are bright or dark (optically related to the charge density and effective mass of the active or inactive) if the spins of the electron in the electron (n and me). The static dielectric function ϵðkÞ takes conduction band and the missing electron in the valence into account the experimental sample geometry, as depicted band are parallel or antiparallel, respectively. Scattering in Fig. 3. Its form follows [45], between dark and bright excitons necessitates a spin-flip of the electron or hole, which is typically a much slower 1 1 1 þðp þ p Þe−kd þ p p e−2kd · b t b t ; 2 – ϵ ¼ ϵ −2kd ð Þ process than the lifetime of bright excitons [37 40]. ðkÞ ML 1 − pbpte Therefore, we neglect dark excitons and show that the dynamical relation between two types of bright excitons is where d denotes the thickness of the semiconductor, ϵ sufficient to explain the optical measurements in Fig. 1. The including the van der Waals gap, and ML is the dielectric first type is the direct exciton in which the electron and hole constant of the ML. Here, pb (pt) is the polarization due to have the same valley index, giving rise to direct-gap optical the dielectric contrast between the ML and the bottom (top) transitions. The second type is the indirect exciton in which layer, the electron and hole reside in opposite valleys. Optical ϵ − ϵ ML bðtÞ transitions of indirect excitons in perfect crystals are p ¼ ; ð3Þ bðtÞ ϵ þ ϵ mediated by external agents such as shortwave phonons, ML bðtÞ which are needed to make up for the large momentum where ϵb (ϵt) is the dielectric constant of the bottom (top) mismatch of indirect excitons and photons. As shown in layer. In the long-wavelength regime, one can readily show that the effective dielectric constant is largely set by the

(a) (b)

FIG. 2. Direct and indirect excitons in the time-reversed K-point valleys. Panels (a) and (b) show the involved electronic FIG. 3. The geometry studied in this work is that of a ML-TMD states in tungsten- and molybdenum-based monolayers, respec- sandwiched between two other materials. The thickness of the tively. The indirect exciton has lower energy in the former case. ML, including the van der Waals gap, is d. The dielectric Δ ϵ ϵ Spin of the bands is color coded, and is the conduction-band constants of the ML, bottom, and top layers are ML, b, and splitting energy. ϵt, respectively.

041040-3 VAN TUAN, SCHARF, ŽUTIĆ, and DERY PHYS. REV. X 7, 041040 (2017) average dielectric constants of the top and bottom layers amplitude is similar to the value of the intervalley 2π 2 ϵ [45,46], Coulomb potential, e = MLK0, that was used to derive   Eq. (5), indicating that α ∼ 0.1 remains a good estimate 2 2 ≪1 ϵ ϵ ϵ ϵ ϵ kd⟶ t þ b ϵ − t þ b kd even when microscopic corrections are taken into account. ðkÞ 2 þ ML 2ϵ 2 : ð4Þ ML Note that the existence of short-wave plasmons only depends on the fact that the short-range Coulomb potential is not negligible, regardless of its exact value. The energy The second type of plasmon originates from the short- dispersion in Eq. (5) is derived by assuming that the short- range Coulomb potential through which electrons transition range Coulomb potential hardly changes when kF ≪ K0, between valleys [35,47–50]. Because of the relatively large where kF is the Fermi wave number. Indeed, kF is more spin splitting in the valence band of ML-TMDs [6], such than 1 order of magnitude smaller than K0 even when the transitions are mostly relevant in the conduction band density is of the order of 1013 cm−2. The region of free- wherein the much smaller spin splitting can be comparable plasmon propagation is then limited to K0 − ðα=3ÞkF < to the Fermi energy, as shown in Fig. 4(a). Collective q 1 if the thickness of the ML is comparable to the IN THE PRESENCE OF PLASMONS lattice constant [46]. Courtade et al. have used empirical values of the fine structure of charged excitons and To study how excitons interact with plasmons, we use the estimated that the amplitude of the short-range Coulomb finite-temperature Green’s functions formalism [51] and potential is of the order of a few eV · Å2 [20]. This quantify the optical signature of bright excitons when surrounded by collective charge excitations [36]. The (a) connection to the absorption spectrum is then established via the imaginary part of the electron-hole pair function, X −1 AðωÞ ∝ β ImfGpðq → 0; k;z;Ω → Eω þ iδÞg: ð6Þ k;z

Here, ℏω is the photon energy and β−1 ¼ k T is the (b) B thermal energy. Note that k and q are 2D crystal wave vectors, where the electron in the pair takes on k þ q and the hole − k. In the limit where q → 0, a direct optical transition is rendered because of the small momentum of photons. Throughout the analysis below, pair functions of direct and indirect excitons are evaluated by assigning q ¼ 0 and q ¼ K0, respectively. Dynamical effects due to plasma oscillations are embodied by the dependence of G FIG. 4. The intervalley Coulomb interaction in ML-TMDs. p Ω (a) Spin-conserving charge excitations from the −K to the K on z and , which are odd (fermion) and even (boson) valleys. Electrons need to overcome the spin-splitting energy gap imaginary Matsubara energies, z ¼ð2l þ 1Þπi=β and in the conduction band. (b) The resulting shortwave charge Ω ¼ 2lπi=β, where l is an integer [36]. The sums over fluctuations in the monolayer. k and z in Eq. (6) integrate out the fermion d.o.f. In the last

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FIG. 5. Feynman diagram representation of the two-particle Dyson equation under the screened ladder approximation. The top and bottom propagators denote the Green’s functions of electrons and holes, respectively. The wiggly line denotes the dynamically screened attractive potential. This diagram sums over all processes through which the electron-hole pair is repeatedly scattered by the attractive potential. step, Pad´e analytical continuation is performed in order to picture considers the interaction of excitons with collective evaluate the pair function at real photon energies [52]: electron excitations, it neglects exciton-exciton inter- þ Ω → Eω þ iδ, where δ → 0 and Eω ¼ ℏω − μe − μh is actions. Accordingly, the screened ladder approximation expressed in terms of photon energy and quasichemical can be used to explain experiments in which the ML is not potentials in the conduction and valence bands. subjected to intense photoexcitation. Figure 5 shows the To find the pair function in Eq. (6), we make use of the Feynman diagram of the two-particle Green’s function so-called screened ladder approximation. It describes before contraction (i.e., before integrating out any of its repeated interaction of the electron-hole pair with the quantum numbers). This diagram is formally written as dynamically screened potential [36]. While this physical

1 X q k k Ω q k k Ω q k k Ω k − k − 0 q k k 0 Ω Gpð ; i; f;z; Þ¼Gp;0ð ; i; f;z; Þþβ Gp;0ð ; i; 1;z; ÞVsð 1 2;z z ÞGpð ; 2; f;z; Þ; ð7Þ 0 k1;k2;z

0 0 where Vsðk − k ;z− z Þ is the dynamically screened contracted pair function Gpðq; k;z;ΩÞ, from which we can potential to be discussed later. Here, Gpðq; ki; kf;z;ΩÞ estimate the absorption according to Eq. (6). We note that and Gp;0ðq; ki; kf;z;ΩÞ are the interacting and noninter- while this step seems trivial, the approach so far was to acting pair functions, respectively. The noninteracting case contract over the odd Matsubara energy (z) and then use the is given by the product of the electron and hole Green’s so-called Shindo approximation to describe dynamical functions screening [36,53]. However, dynamical screening can be better described upon contraction in momentum rather than q k k Ω k q Ω − −k δ energy space. This contraction keeps both dynamical Gp;0ð ; i; f;z; Þ¼Geð i þ ; zÞGhð i;zÞ ki;kf ; parameters (even and odd Matsubara energies), while ð8Þ eliminating kf does not compromise any further accuracy in the solution of the pair function if the starting point is the where here and below we assume the area of the monolayer screened ladder approximation. After contraction, we is 1. The Kronecker delta function allows us to integrate out simplify the form of Eq. (7), the final wave vector kf in Eq. (7) and to calculate the

 X  −1 0 0 0 0 Gpðq; k;z;ΩÞ¼Geðk þ q; Ω − zÞGhð−k;zÞ 1 þ β Vsðk − k ;z− z ÞGpðq; k ;z; ΩÞ : ð9Þ k0;z0

This equation can be solved by matrix inversion tech- between ML-TMDs and conventional semiconductor quan- niques, as explained in the Appendix. tum wells [54]. A central part of the theory is the inclusion of the A. Effect of long-wavelength plasmons on excitons dynamically screened potential in Eq. (9). We invoke the We begin by evaluating the pair functions of direct single-plasmon pole approximation (SPP), which provides excitons when Vs, Ge, and Gh in Eq. (9) are affected only a relatively compact form for the dynamical screening. by long-wavelength plasmons. Neglecting shortwave plas- Using the more rigorous random-phase approximation does mons in the first step allows us to find common attributes not lead to qualitative changes or to increased

041040-5 VAN TUAN, SCHARF, ŽUTIĆ, and DERY PHYS. REV. X 7, 041040 (2017) computational complexity when solving Eq. (9). When X 2π 2 k − q Σ k − e fið Þ x;ið Þ¼ ϵ ; ð14Þ including long-wavelength plasma excitations, the dynami- q ðqÞ q cally screened Coulomb potential under the SPP approxi- mation reads [36] where fi is the Fermi-Dirac distribution. The exchange   contribution from the electron-electron interaction can only 2 2 2πe 1 E ðkÞ shift the filled valleys (i.e., fi ≠ 0). The correlation term, on V ðk;z− z0Þ¼ 1 þ l ; ð10Þ s ϵðkÞ k ðz − z0Þ2 − E2 the other hand, affects all bands, and it comes from k emission or absorption of long-wavelength plasmons where (intravalley processes), rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2π 2 1 2 Σ k e ElðqÞ 1 alk c;ið ;zÞ¼ ϵ 2 Ek ¼ ElðkÞ þ ð11Þ q ðqÞ q Eq 4

041040-6 MARRYING EXCITONS AND PLASMONS IN MONOLAYER … PHYS. REV. X 7, 041040 (2017) the exciton decay when the background charge density thereby reinforcing the important role of BGR and dynami- increases. The effective mass for both electrons and holes is cal screening. It is noted that, unlike the experimental 0.36m0 [56], where m0 is the free-electron mass (see findings of Fig. 1, the simulated exciton peak in Fig. 6 is Appendix A1). The dielectric environment is modeled not completely quenched at elevated densities. In addition 1 ϵ 7 25 by sandwiching the ML (d ¼ nm, ML ¼ . ) between to the fact that we do not consider charged excitons, which thick layers of hexagonal boron nitride (ϵb ¼ ϵt ¼ 2.7) drain part of the oscillator strength of neutral excitons [58], [15,57]. The arrows in Fig. 6 indicate the continuum the model tends to overestimate the Coulomb attraction at redshift due to BGR. Above this energy, the excitons are elevated densities. The reason is that broadening effects are no longer bound. Note that the 2s bound state merges into not introduced in the electron and hole Green’s functions of the continuum already at relatively small densities because Eq. (9); they are introduced in the analytical continuation of its smaller binding energy. The BGR is calculated from phase after the pair function is solved (in the Appendix). Σ k ε − μ0 k 0 the value of ið ; k;i i Þ at ¼ , by assuming a rigid An additional important result in Fig. 6, which is shift of the bands. The cutoff energies we have used for the supported by Figs. 1(a) and 1(c), is that the neutral exciton ε spectral position is hardly affected by the background bottom and top spin-split valleys in Eq. (15) are qc ¼ 60 ε 45 charge density because of the mutual offset between meV and qc ¼ meV, respectively. These cutoff energies are the same for all densities. Using larger BGR and the screening-induced reduction in the binding (smaller) cutoff energies, the peak position will gradually energy of excitons; the long-wavelength plasmons are redshift (blueshift) as one increases the charge density. The responsible for both effects [54]. One can therefore realize inset of Fig. 6 shows that the BGR is strongest when the that the binding energy of excitons in ML-TMDs can be of charge density is ramped up from zero to about 1012 cm−2. the order of a few hundred meV only at vanishing charge The continuum redshift is much slower between about 1012 densities. All in all, long-wavelength plasmons lead to 13 −2 qualitatively similar effects in ML-TMDs and conventional and 10 cm , during which excitons become unbound semiconductor quantum wells [55]. The only change is a and merge into the continuum. quantitatively larger effect in ML-TMDs due to reduced Two effects cause the decay of the neutral-exciton dielectric screening and larger electron mass [59,60]. spectral line with the increase in electron density. The dominant one is the broadening effect when the exciton energy approaches the continuum [see arrows in Fig. 6 and B. Effect of shortwave plasmons on excitons Eq. (A2) in the Appendix]. The decay that is caused by In the next step, the unique features of ML-TMDs are reduction in the oscillator strength plays a secondary role revealed by turning on the coupling between shortwave due to the mitigated effect of screening in 2D. This plasmons and excitons. In the long-wavelength case, behavior is reminiscent of excitonic enhancement of free the exciton-plasmon coupling is implicit: It is embodied electron-hole transitions in semiconductor quantum wells in the self-energies of the electron and hole as well as in the [55], in which the overlap between the electron and hole attractive screened potential (i.e., the coupling to plasmons wave functions remains sizable when excitons enter the is “concealed” in the dressed propagator and Coulomb lines continuum. The simulated decay of the neutral exciton in in Fig. 5). An explicit exciton-plasmon interaction is Fig. 6 is corroborated in the differential-reflectivity mea- inhibited if the extension of the exciton is smaller than surements, which resolve the oscillator strength and broad- the wavelength of the plasmon because of the charge ening of the spectral lines [15,16]. Figures 1(a) and 1(c) neutrality of the former. This physical picture changes show that when the charge density is ramped up by gate for shortwave plasmons, which can be effectively paired up voltage, the measured decay of neutral excitons is not with an exciton in spite of its charge neutrality. Here, the compensated by an equivalent increase of charged excitons, short-range nature of the charge fluctuations, as can be seen

(a) (b)

FIG. 7. Coupling between excitons and shortwave plasmons. (a) Cartoon of the exciton and shortwave charge fluctuations. The plasmon wavelength is shorter than the exciton extent, which is further blown up because of screening. (b) Feynman diagram representation of the renormalized direct-exciton pair function due to shortwave plasmons. The self-energy term takes into account the coupling between direct and indirect excitons (D and I) due to the electron-plasmon interaction. The hole (bottom propagator) acts as a spectator, while the electron interacts with the plasmon. The double wiggly lines in this diagram denote the plasmon propagator, and the vertices denote the electron- plasmon interaction. The diagram is an infinite sum of the virtual processes D þ DID þ DIDID þ.

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1 − in Fig. 7(a), allows the hole or electron to act as a spectator 5 1011 cm 2 0.08 while the plasmon interacts with the opposite charge. 1 1012 12 The energy required to excite a shortwave plasmon at 0.8 0.06 2 10 low temperatures is greater than that available thermally 4 1012 (Δ ≫ k T). In addition, we recall that free-propagating 0.04 B 12 plasmons are independent d.o.f., which do not interact with 0.6 6 10 individual charged particles of the electronic system. As a 0.02 result, plasmon signatures in the optical spectrum can be 0 12 0.4 −0.4 −0.35 −0.3 8 10 resolved by supplying energy from outside the electron system in amounts greater than the plasmon energy. This is Absorption (a.u.) the case when we shake up the electronic system by 0.2 exciting electron-hole pairs. Figure 7(b) shows the Feynman diagram for the explicit coupling between an 0 exciton and a shortwave plasmon when the hole acts as the −0.5 −0.4 −0.3 −0.2 −0.1 spectator. The plasmon propagator is the double wiggly e line. This Feynman diagram corresponds to renormaliza- tion of the pair function of direct excitons according to FIG. 8. The absorption spectrum of direct excitons in electron- doped ML-WX2 after renormalization due to interaction with G ðq → 0; k; ΩÞ shortwave plasmons. A dynamical sideband emerges in the low- ~ q → 0 k Ω p energy side of the absorption spectrum, as highlighted in the Gpð ; ; Þ¼1 − Σ k Ω q → 0 k Ω ; ð16Þ sð ; ÞGpð ; ; Þ enlarged box. The sideband initially redshifts and intensifies when the electron density increases, showing saturated behavior where the pair Green’s functions were contracted by at elevated densities. integrating out the fermion Matsubara energies. The mixing with indirect excitons comes from the self-energy term, X the optical transition, and in this respect, its amplitude Σ k Ω β−1 2 Ω − Ω0 q0 q0 k Ω0 sð ; Þ¼ Mq0 Dð ; ÞGpð ; ; Þ; ð17Þ should be compared with that seen in differential reflec- q0;Ω0 tivity rather than photoluminescence experiments (we further address this issue later). Comparing the results in where Mq0 is the interaction between the plasmon and the Figs. 1(b) and 8, however, we still find two quantitative electron component of the exciton, and DðΩ; qÞ is the free- disagreements. The first one is that the main peak does not plasmon propagator. The sum in Eq. (17) is restricted to completely decay at elevated densities. As explained 0 intervalley transitions, q → K0, so that the pair function on before, we attribute this disagreement to the fact that the right-hand side is that of indirect excitons. The broadening effects are introduced only after solving the interaction and propagator terms follow [44], two-particle Dyson equation (in the Appendix). The second pffiffiffiffiffiffi disagreement is that the relative amplitude of the simulated 2 0 3πℏ jK0 − q j dynamical sideband is about 3 times weaker than the Mq0 ¼ ; ð18Þ me oscillator strength of the optical sideband. We attribute this mismatch to the difficulty to simulate a stronger 2E ðqÞ dynamical sideband through an increase of the integration D Ω; q s : ð Þ¼Ω2 − 2 ð19Þ Es ðqÞ cutoff in Eq. (17) outside the small window of free-plasmon 0 propagation (e.g., when jK0 − q j ≳ αkF). The reason for Figure 8 shows the renormalized absorption spectrum of this is that the high-order Pad´e polynomials used to perform direct excitons in electron-doped ML-WX2 at 10 K. A analytical continuation, Ω → Eω þ iδ, are notoriously sen- dynamical sideband emerges in the low-energy side of the sitive to minute numerical errors [62]. spectrum. We have considered tungsten-based settings such We now focus on the redshift and initial amplification of that indirect excitons have lower energy than direct ones the dynamical sideband, which are common to both [Fig. 2(a)]. The spin-orbit coupling splits the conduction- experiment [as shown in Figs. 1(b)] and our simulations band valleys by 26 meV [61]. All other parameters are the (highlighted in the enlarged box of Fig. 8). The redshift is same as in Fig. 6. The behavior of the simulated dynamical caused by the electron-electron exchange interaction when sideband is similar to the empirical behavior of the optical the lower spin-split valleys are being filled. This interaction sideband in electron-doped ML-WX2 [9–16], as shown in leads to a stronger energy redshift of the populated lower Fig. 1(b). Both the theoretical and the experimental results valleys compared with the unpopulated upper ones (inset of show a redshift and amplification of the optical sideband Fig. 6), thereby increasing the plasmon energy via Δ in when increasing the electron density until they saturate. We Eq. (5). The initial amplification of the dynamical sideband note that the pair function reflects the oscillator strength of is commensurate with the increase in the number of free-

041040-8 MARRYING EXCITONS AND PLASMONS IN MONOLAYER … PHYS. REV. X 7, 041040 (2017)

(a) (b) doped case. Figure 9(b) shows the renormalized absorption spectra of hole- and electron-doped ML-WX2. The − − dynamical sideband is practically absent in the hole-doped

− case, which is similar to the experimental findings in Figs. 1(c) and 1(d). We can understand this behavior by noting that for electron doping, the splitting of the con- duction-band valleys corresponds to both plasmon energy −−−−− −−−− e e and the energy difference between direct and indirect excitons (the role of electron-hole exchange is relatively FIG. 9. The renormalized absorption spectrum of direct ex- small as we explain in Appendix A4). For hole doping, on citons in ML-MoX2 and in the case of hole doping. (a) The case the other hand, the plasmon energy is governed by the large of electron-doped ML-MoX2 before (solid lines) and after splitting of the valence-band valleys (about 400 meV in (dashed lines) renormalization. The dynamical sideband is not ML-WX2), while the energy difference between direct and seen. (b) The case of electron-doped (solid line) and hole-doped indirect excitons is still governed by the much smaller (dotted line) ML-WX2. The dynamical sideband is negligible for splitting in the conduction band. The large mismatch hole doping, as highlighted in the magnified box. It emerges about 400 meV below the exciton line because of the large between the two suppresses the emergence of a visible valence-band splitting. dynamical sideband in hole-doped ML-TMDs.

C. Stokes vs anti-Stokes optical sidebands α 3 propagating plasmon modes (larger value of kF= ). Figure 8 shows that the exciton-plasmon resonance Saturation at high densities occurs when the energy differ- emerges about 2Δ below the direct-exciton spectral line δ ence between direct and indirect excitons, ED−I, deviates in electron-doped ML-WX2. Figure 9(a) shows that it from the energy of shortwave plasmons, about Δ. coincides with the spectral line of the direct exciton in Specifically, indirect excitons experience a blueshift due electron-doped ML-MoX2. These calculations did not to electron filling of the lower valleys, while direct ones are reveal the emergence of an optical sideband above the not subjected to this effect since they originate from direct-exciton spectral line because of the small population electronic states in the empty upper valleys. Continuing of shortwave plasmons at the simulated low temperature 10 ( ) → 0 ∼ Δ ≫ to increase the electron density eventually leads to a (T ¼ K), g EsðqÞ when EsðqÞ kBT. slow decay of the dynamical sideband due to increased Specifically, the self-energy of direct excitons includes “ ” δ Δ off-resonance conditions between ED−I and both plasmon emission and absorption processes, corre- (Appendix A5). sponding to “Stokes” and “anti-Stokes” sidebands below Our model shows that the dynamical sideband is evident and above the energy of the indirect exciton, respectively only in electron-doped ML-WX2, providing strong evi- (see Appendix A2). The amplitude of the Stokes sideband − ( − ) 1 ( ) dence that it corresponds to the measured optical sideband is commensurate with g EsðqÞ ¼ þ g EsðqÞ ,while [9–16]. Figure 9(a) shows that the dynamical sideband is that of the anti-Stokes sideband is commensurate with ( ) absent in electron-doped ML-MoX2, simulated by revers- g EsðqÞ . The vanishing value of the latter at low temper- ing the order of the conduction-band valleys such that atures explains why Figs. 8 and 9 do not include signatures direct excitons have lower energy [Fig. 2(b)]. The dynami- of high-energy optical sidebands due to plasmon absorption. cal sideband is not seen because of “wrong conditions.” At high temperatures, on the other hand, it may be Namely, the sideband is positioned at about one plasmon possible to observe a distinct high-energy optical sideband energy below the energy of the indirect exciton, which in in the absorption spectrum of electron-doped ML-MoSe2. the case of ML-MoX2, coincides with the direct-exciton The value of Δ in this material is non-negligible [32,56], spectral line. This behavior makes it hard to resolve the resulting in a well-resolved anti-Stokes sideband whose optical signature of exciton-plasmon quasiparticles in energy is about 2Δ above the spectral line of the direct ML-MoX2. Figure 9(a) indeed shows that the renormali- exciton. This high-energy optical sideband is expected to zation only affects the spectral region of the exciton peak blueshift when the electron density increases because of the without the emergence of dynamical sidebands in other increasing value of Δ. Its amplitude, however, should be spectral regions. In electron-doped ML-WX2, on the other measurably weaker than that of the Stokes sideband in ( ) ( − ) hand, the dynamical sideband can be distinctly recognized ML-WX2 since the ratio jg EsðqÞ =g EsðqÞ j remains since the indirect exciton has lower energy; thus, overall, evidently smaller than 1 even close to room temperature the sideband emerges about 2Δ below the direct-exciton (see Appendix A2). In addition, the observation of a high- spectral line. energy optical sideband in high-temperature ML-MoSe2 Further evidence that the revealed exciton-plasmon may be elusive if it resides close to or within the continuum. phenomenon corresponds to the optical sideband in elec- In other words, the exciton-plasmon coupling is manifested tron-doped ML-WX2 is provided by simulating the hole- at elevated electron densities in which the exciton energy

041040-9 VAN TUAN, SCHARF, ŽUTIĆ, and DERY PHYS. REV. X 7, 041040 (2017) approaches the continuum because of the strong BGR. The A. Plasmon-assisted optical transitions emergence of an exciton-plasmon resonance above the We have attributed the optical sideband in the reflectance direct-exciton spectral line is hampered by broadening spectra to the creation of an exciton-plasmon quasiparticle. effects, similar to the behavior of neutral excitons when It was found by renormalization of the pair function of their energy approaches the continuum (Fig. 6). The direct excitons, through the sum of infinite plasmon- plasmon emission process, on the other hand, is relatively induced virtual transitions to intermediate indirect-exciton Δ robust since the increased value of with electron density states [Fig. 7(b)]. This physical picture should not be causes the low-energy optical sideband to redshift, thereby confused with that of a plasmon-assisted optical absorption, keeping it protected from merging into the redshifting in which an indirect exciton is created through two virtual continuum. Accordingly, broadening effects of the Stokes transitions into and from a direct exciton. To better under- sideband are mitigated, and ultrafast dissociation of the stand this difference and its implication in the context of the exciton-plasmon quasiparticle to the continuum of elec- experimental data (Fig. 1), we assume low temperatures tron-hole pairs is suppressed. Appendix A2 includes where only emission of a shortwave plasmon is feasible further details on the Stokes and anti-Stokes optical during these processes (Δ ≫ kBT). sidebands. 1. Plasmon-assisted photon absorption IV. DISCUSSION (reflectivity spectrum) When comparing the results in this work with exper- In photon absorption at low temperatures, the plasmon- imental findings, one should recall that the theory only assisted process creates an indirect exciton and a shortwave evaluates the pair function and its renormalization due to plasmon in the final state. The overall energy conservation of the absorption process is such that the photon energy is coupling with plasmons. Accordingly, the theory does not the sum of the indirect exciton and plasmon energies, capture optical transitions that are associated with three- body complexes [21,22,63–70], Fermi polarons [27,28], ℏω ¼ E ð−q ÞþE ðq Þ; ð20Þ and exciton optical transitions next to localization centers I f s f [71]. Nonetheless, the theory in this work covers two where the crystal momenta of the plasmon and indirect important regimes. The first one is the low-density regime, ≲ 1012 −2 exciton are opposite in order to guarantee momentum n cm , in which the neutral exciton dominates the conservation (the photon carries negligible momentum). absorption spectrum, and its decay is not compensated by Here, the photon is absorbed into a direct-exciton virtual an equivalent increase of the trion peak, as shown by the state, followed by plasmon emission that transfers the empirical data in Fig. 1(a). The dynamical picture we have exciton to its final state. Crystal momentum must be developed quantifies the exciton decay due to long- conserved during virtual transitions to and from intermedi- wavelength plasmons (Fig. 6), where we find good agree- ate states because of translation symmetry. As a second- ment with reflectivity experiments [15]. The second case is order process, the absorption amplitude is weak unless the ≳ 5 1012 −2 the high-doping regime, n × cm , in which both energy of the intermediate virtual state resonates with the reflectivity and PL experiments show that only one peak real energy level of the direct exciton. Recalling the energy survives in the spectrum of electron-doped ML-WX2 diagram in Fig. 2, Eq. (20) implies that this condition is met [10,15], as shown by Fig. 1(b). Trions cease to exist at in electron-doped ML-WX2 but not in ML-MoX2. In other very large densities since their effective radius is larger than words, the plasmon-assisted photon absorption is a second- the average distance between electronsffiffiffi of the Fermi sea. order resonance process in electron-doped ML-WX2, 1 p The latter is proportional to = n, while the trion extension meaning that the neutral-exciton spectral line (X0) has is of the order of 3–5nminML-TMDs[69,70]. The main two contributions. The first one is from the first-order focus of this work is on the high-doping regime, and we absorption process, which creates direct excitons. The attribute the observed optical transition to a new type of second contribution is from the second-order resonance quasiparticle where the exciton is coupled to a shortwave process, which creates indirect excitons via emission of charge fluctuation. We provided evidence that the behavior shortwave plasmons. of the optical sideband when increasing the electron density Therefore, an important resulting question is how one matches that of the exciton-plasmon quasiparticle, and we can resolve the relative contribution of the plasmon-assisted further showed that it emerges in electron-doped ML-WX2 optical transitions to the oscillator strength of X0 in but not in the hole-doped or ML-MoX2 cases (Figs. 8 electron-doped ML-WX2. Clearly, the contribution has and 9). Any alternative explanation that aims at deciphering negligible weight at vanishing electron densities because the origin for the optical sideband should be consistent with of the minute range of free-propagating plasmon modes. this observation. Below, we discuss several key aspects that When the charge density increases, however, we can one may find helpful when interpreting experimental identify this contribution by looking for differences findings and when comparing them with our theory. between the reflectance data in the electron-doped and

041040-10 MARRYING EXCITONS AND PLASMONS IN MONOLAYER … PHYS. REV. X 7, 041040 (2017) hole-doped regimes. Indeed, Figs. 1(a) and 1(c) reveal a spectral region of the direct exciton (X0). Unlike phonon- clear difference in the measured behavior of X0, whose assisted optical transitions [32,72–74], the amplitude of the blueshift and broadening are evidently stronger in the plasmon-assisted ones increases with the electron density electron-doped case. As we suggest below, the plasmon- because of the increased range of free-propagating assisted optical transitions can give rise to this behavior. plasmon modes. Without the contribution of shortwave plasmons, one would expect the blueshift of X0 to be stronger in the hole- 2. Plasmon-assisted photon emission (PL spectrum) doped case. Specifically, the blueshift of neutral excitons in In photon emission at low temperatures, the plasmon- the absorption spectrum of semiconductors is induced by assisted process creates a shortwave plasmon and a filling one or both of the bands that are involved in the photon in the final state through annihilation (radiative optical transition [54]. This condition is not readily met in recombination) of an indirect exciton. The overall energy electron-doped ML-WX2 since only the bottom conduction conservation of the emission process is such that valleys are populated, whereas the optical transition involves the top valleys. Both experiment and theory show ℏω −q − q ¼ EIð fÞ Esð fÞ: ð21Þ that electron population in the top valleys of ML-WX2 begins at densities that are comparable to 1013 cm−2 Here, the indirect exciton in the initial state is transitioned [15,32]. The situation is different for the hole-doped case into a direct-exciton virtual state by plasmon emission, at which hole filling in the top of the valence band is followed by the photon emission. Recalling the energy measurable at smaller densities. Thus, while such behavior diagram in Fig. 2, Eq. (21) implies that the resonance should lead to weaker or at most comparable blueshift of condition is now met in electron-doped ML-MoX2,in the direct exciton in the electron-doped case, the experi- which the energy of the intermediate virtual state resonates ment shows the opposite. Furthermore, the simulated decay with the real energy level of the direct exciton. However, of the pair function in Fig. 6 better resembles the observed this second-order resonance process has no real value at decay of X0 in the hole-doped case. This fact suggests a missing component in the theory such that dynamical low-temperature PL since the population of indirect exci- screening and BGR are not enough to fully explain the tons in ML-MoX2 is negligible at low temperatures: Most excitons in these compounds remain direct after photo- behavior of X0 in electron-doped ML-WX2. Plasmon-assisted optical transitions can complete the excitation and energy relaxation [Fig. 2(b)]. Therefore, we puzzle and explain the stronger blueshift and broadening of do not expect plasmon-assisted optical transitions to have a measurable signature in the PL spectrum of ML-MoX2 at X0 in electron-doped ML-WX2. There are two comple- menting facts that support this claim. First, plasmon- low temperatures. assisted optical absorption stems from a resonance process The case of electron-doped ML-WX2 is unique if we consider the renormalization of direct excitons that led to in electron-doped ML-WX2, suggesting a non-negligible effect. Second, Eq. (20) can explain the enhanced blueshift the exciton-plasmon quasiparticle. In the reflectance spec- in the experiment since the plasmon energy increases at trum, we have concluded that the contributions from the elevated densities (through the increased value of Δ). creation of indirect excitons through the plasmon-assisted Finally, the enhanced broadening of X0 in ML-WX2 when photon absorption and the creation of exciton-plasmon the electron density increases can be explained by the quasiparticles are spectrally resolved: The latter corre- 0 closer proximity of the photon energy to the redshifting sponds to the optical sideband (X− ), and the former can continuum (or possibly merging with the continuum). We only affect the spectral region of the bare neutral exciton will provide a complete description of these phenomena (X0). In the PL spectrum, on the other hand, Eq. (21) alongside second-order perturbation theory to describe the suggests that the photon energy should coincide with that of plasmon-assisted optical transitions in a future work. the optical sideband. In other words, both the annihilation All in all, the optical sideband in the reflectance experi- of the exciton-plasmon quasiparticle and the annihilation of 0 ment of electron-doped ML-WX2 (X− ) is attributed to the indirect excitons through plasmon-assisted photon emis- exciton-plasmon quasiparticle, originating from renormali- sion contribute to X−0 in the PL spectrum. The latter zation of the self-energy of the direct exciton. Its energy is contribution may not be negligible because of a large one (two) plasmon(s) below that of the indirect (direct) population of indirect excitons after energy thermalization exciton. Photon absorption due to plasmon-assisted optical in ML-WX2 [Fig. 2(a)]. In addition, while the energy of the transitions, on the other hand, is a second-order process that intermediate virtual state does not resonate with the real converts photons into indirect excitons through plasmon energy level of the bare direct exciton, it matches the low emission at low temperatures. It becomes important in energy of the dressed one (i.e., the exciton-plasmon electron-doped ML-WX2, in which the intermediate virtual quasiparticle). Therefore, the recombination of indirect exciton state resonates with the real energy level of the direct excitons due to plasmon-assisted photon emission may exciton. The signature of this process in the reflectance have a measurable signature on the spectral weight of X−0, experiment of electron-doped ML-WX2 is limited to the explaining in part why its magnitude is much stronger in PL

041040-11 VAN TUAN, SCHARF, ŽUTIĆ, and DERY PHYS. REV. X 7, 041040 (2017) compared with reflectance spectra (see related discussion in spectroscopy: The generated wave is amplified when the Sec. I) [10,15]. We note that modeling of the density- detuning between the input beams approaches 2Δ. dependent PL spectrum is beyond the scope of this work, partly because of the lack of knowledge of the nonradiative B. Three-body complexes versus recombination processes. For example, charged impurities exciton-plasmon quasiparticles that function as nonradiative recombination centers are It is beyond the scope of this work to provide a complete screened at elevated densities, resulting in improved quan- three-body dynamical function since it is impossible to tum efficiency for radiative recombination. Conversely, numerically solve its corresponding Dyson equation with- nonradiative Auger processes are amplified at elevated out simplifying approximations. The dynamical model we – densities [75 77], thereby reducing the quantum efficiency. have developed describes the behavior of the pair function and its renormalization due to coupling with shortwave 3. High temperatures plasmons. The optical sideband, which we have associated So far, we have discussed plasmon-assisted optical with the exciton-plasmon quasiparticle, has nearly opposite transitions at low temperatures, in which only emission behavior from that of charged excitons (see discussion of of shortwave plasmons is possible during the photon Fig. 1 in Sec. I). Accordingly, the identification of charged absorption or emission. At high temperatures, on the other excitons complexes and exciton-plasmon quasiparticles hand, we should also consider the absorption of shortwave can be readily achieved. Studying how the dynamically plasmons during these optical processes. Repeating the screened potential influences the oscillator strength and previous analyses, one can see that the conditions for broadening of charged excitons is a work in progress. Here, second-order resonance processes are switched between we mention some of the known theoretical approaches to ML-MoX2 and ML-WX2. Detecting the signatures of these study their behavior. The most computationally convenient processes in the high-temperature reflectance or PL spectra way to deal with trions is to dispense with the dynamical may be elusive for the same reasons we have mentioned terms altogether [20,63–68], but then one can only con- regarding the anti-Stokes optical sideband in Sec. III C. clude about their binding energy. Such static models cannot describe the density-dependent rise, decay, energy shift, 4. Other optical spectroscopy tools to and broadening of trions in the spectrum. Alternatively, one probe the plasmon-exciton coupling can neglect dynamical terms in the exciton-electron inter- In addition to differential reflectivity and photolumines- action [27] or a priori assume the dynamical form of the ’ cence experiments, it may be possible to probe the coupling Green s function without solving the corresponding between excitons and shortwave plasmons in electron- dynamical Dyson equation [21,28]. In the latter case, doped samples using Raman-type experiments or nonlinear one can describe the behavior of charged excitons by optical spectroscopy. For example, the energy of scattered invoking an exciton-electron scattering function, where photons in the Stokes sideband of a Raman experiment is dynamical terms in the scattering potential are neglected. expected to be 2Δ below that of the incident photon. It One open question in the context of our work deals with involves a virtual process through which two counter- possible coupling between trions and shortwave plasmons. propagating shortwave plasmons are emitted, so the sum of Figures 1(a) and 1(b) do not show optical sidebands whose Δ 2Δ their momenta becomes negligible and matches the minute energy is or below that of the charged exciton spectral Δ momentum of photons. In the first step of this process, the line; the value of is about 26 meV at low densities [61], incident photon induces a virtual transition to a direct and it increases at elevated ones [35]. The absence or exciton intermediate level, which in the second step is weakness of trion-plasmon coupling is attributed to the scattered to the indirect exciton intermediate level because relatively small oscillator strength of the charged exciton of plasmon emission. In the next step, the indirect exciton is compared with that of the neutral exciton (Fig. 1). The scattered back to the direct exciton intermediate level by oscillator strength is inversely proportional to the square of emission of a counterpropagating plasmon. Finally, the the complex radius, where the radius is about 3 times larger system emits a photon at the Stokes energy (the energy of for trions [69,78]. This fact is also reflected in the much the incident photon minus the energy of two plasmons). smaller binding energy of the charged exciton. Here, we The overall process is expected to be weak since two reemphasize that the decay of the neutral-exciton spectral plasmon-mediated virtual transitions are involved. The line with the increase in electron density is mostly caused amplitude of the scattered light can increase by using by broadening effects when the exciton energy approaches near-resonance conditions. The amplitude of the scattered the redshifting continuum (see discussion of Fig. 6). The light and its energy can be controlled by employing a gate quasiparticle made by a neutral exciton coupled to a voltage that tunes the value of Δ through electron filling of shortwave plasmon is resilient to this broadening since the lower spin-split valleys. In other words, the gate voltage its energy is kept away from the redshifting continuum due provides an additional control parameter in the experiment. to the increasing value of 2Δ when the density is ramped A similar approach can be used in nonlinear wave-mixing up. For that reason, the oscillator strength of the optical

041040-12 MARRYING EXCITONS AND PLASMONS IN MONOLAYER … PHYS. REV. X 7, 041040 (2017) sideband neither decays nor broadens at elevated densities attraction between two neutral excitons. If the latter was [Fig. 1(b) and Fig. 8]. Quantitatively, the oscillator strength the microscopic origin for the observed biexciton spectral amplitude is manifested in a relatively large value of the line, then it should be insensitive to whether the two pair function at even Matsubara energies, compared with excitons are direct, indirect, or a mixture of the two. As the respective dynamical Green’s function that would a result, one should be able to observe the biexciton spectral describe charged excitons. Finally, we mention that the line in the PL of both ML-MoX2 and ML-WX2 under optical sideband is weak but noticeable even when EF is comparable pumping intensities. The fact that this scenario significantly smaller than the binding energy of the charged is not observed in experiments precludes such microscopic exciton [enlarged boxes of Figs. 1(a) and 8]. In this regime, origins. Furthermore, previous theoretical calculations have the negatively charged exciton manages to stay away from shown that the binding energy of the biexciton is about electrons and plasma excitations of the Fermi sea, as 20 meVeither in ML-MoX2 or ML-WX2, by using a variety evident from the absence of broadening and blueshift of of four-body computational models with static 2D A B – X− and X− in Fig. 1(a). Neutral electron-hole pairs, on the Coulomb (Keldysh) potentials [63,65,69,78,80 82].PL other hand, are not repelled by the surrounding Fermi sea experimental reports, on the other hand, are available only and therefore can be coupled to shortwave plasmons at all for ML-WX2 in which the biexciton emerges about – finite densities. 50 60 meV below the neutral-exciton spectral line [11,12,79]. Our claim regarding the coupling between C. Strong photoexcitation conditions direct and indirect excitons through shortwave charge fluctuations is indeed self-consistent with the fact that Recently, You et al. observed the emergence of a strong the biexciton spectral line has not been reported to date in spectral line in the PL of ML-WSe2 when subjected to strong PL experiments of ML-MoX2. The low-temperature pop- excitation [79]. This peak showed quadratic dependence on ulation of indirect excitons in these compounds is negli- the intensity of the pump signal and did not shift appreciably gible because of their higher energy, as shown in Fig. 2(b). when increasing the pumping intensity. This behavior Moreover, the coupling between indirect and direct exci- suggests that the emerged spectral line is due to exciton- tons has the same wrong conditions that were discussed in exciton interactions [79]. A similar behavior was also the context of Fig. 9(a). Namely, the renormalization of observed in the PL of ML-WS2 [11,12]. To date, however, direct excitons does not lead to a distinct spectral line, this behavior was not reported in the PL of ML-MoX2. which can be resolved from that of the neutral exciton. We argue that these observations can be settled by con- In addition, our explanation is consistent with the fact sidering the shortwave Coulomb excitations. Specifically, that the indirect-direct biexciton spectral line in ML-WX2 the strong pump intensity generates a relatively large density emerges at the same spectral position of the optical side- of direct excitons, the majority of which relax in energy and band in electron-doped ML-WX2 (about 2Δ below the become indirect in ML-WX2 [Fig. 2(a)] [32,33]. The band direct-exciton spectral line, which is of the order of about structure of indirect excitons is similar to the one shown in 50–60 meV in these materials) [10,11,15,16]. One differ- Fig. 4(a) [7], where their electron component generates the ence between the two cases, however, is that the energy same type of shortwave Coulomb excitations. A direct shift of the direct-indirect biexciton is weak when the exciton can then interact with these excitations in the same photoexcitation intensity increases [79], compared with way as before, leading to renormalization of its self-energy. the redshift of the optical sideband when increasing the The observed biexciton spectral line emerges through this electron density [10]. The reason for this is that, unlike gate- renormalization. The main difference from the optical side- induced free electrons, bound and neutral excitons do not band, which emerges under conditions of weak photoexci- lead to an appreciable change of Δ. The value of Δ can tation in electron-doped ML-WX2, is that now the pumping change because of the long-wavelength exchange interac- is responsible not only for the exciton population but also for tion, which is effective in the case of free electrons [35].Asa the generation of shortwave Coulomb excitations through the result, the indirect-direct biexciton shows a relatively weak electron components of indirect excitons. These two respon- energy shift when increasing the photoexcitation intensity (as sibilities explain the quadratic dependence of the observed long as excitons remain bound). optical transition on the pumping intensity. In this view, the measured phenomena can still be identified as an exciton- V. CONCLUSION AND PERSPECTIVES exciton interaction but with the notion that the coupling is between direct and indirect excitons due to shortwave charge The identified pairing of shortwave plasmons and fluctuations. excitons explains the puzzling observations of the emerging The importance of our theory is therefore in suggesting a low-energy optical sideband in the spectra of electron- microscopic origin for the exciton-exciton optical transition doped WSe2 and WS2, while being conspicuously absent in the PL of strongly photoexcited ML-WX2. We claim that with hole doping, MoSe2 and MoS2 [9–16]. Elucidating the it originates from the intervalley Coulomb excitations and exciton-plasmon mechanism and the importance of not from other sources such as van der Waals-type dynamical screening paves the way to harness unexplored

041040-13 VAN TUAN, SCHARF, ŽUTIĆ, and DERY PHYS. REV. X 7, 041040 (2017) manifestations of spin-valley coupling and the realization APPENDIX: COMPUTATIONAL DETAILS of other dynamical sidebands, such as those arising because Equation (9) is solved with matrix inversion. Each row of the exciton-phonon coupling or the coupling between (or column) in the matrix is indexed via fki;zng. The zone-edge phonons and intervalley plasmons. 0 angular dependence of k − k2 in V ðk − k2;z− z Þ is The findings of this work reveal the unique nature of i s i averaged out, and then all the 2D wave vectors in Eq. (9) shortwave (intervalley) plasmons in monolayer transition- are treated as scalars. We use a uniform grid with a metal dichalcogenides. Recalling that freely propagating spacing of Δk ¼ k =N , where k is chosen such that shortwave plasmons are independent d.o.f. in the electronic max k max ε ¼ 1 eV. For the fermion Matsubara energies, z ¼ system, these quasiparticles can be directly detected in kmax n iπð2nz þ 1ÞkBT, nz runs between −Nz to þNz. The boson electron-rich monolayers (i.e., not through their coupling to Ω excitons). Using THz spectroscopy, for example, one Matsubara energy in Eq. (9) is treated as a parameter in the matrix. In other words, we solve this equation for each should expect to see a resonance in the absorption spectrum Ω 2π ∈ − when the photon energy is about twice the spin splitting in n ¼ i nΩkBT with nΩ ½ NΩ;NΩ. For numerical 50 160 the conduction band (≲0.1 eV). Such far-infrared photons calculations, we have used Nk ¼ , Nz ¼ , 90 can couple to two counterpropagating shortwave plasmons, NΩ ¼ , and the code was written in FORTRAN and thereby conserving both energy and momentum. As before, parallelized with OpenMP. For every density, the calcu- 0 Ω Ω the plasmon energy and amplitude can be tuned by a gate lation of Gpðq ¼ ;k;z; Þ or Gpðq ¼ K0;k;z; Þ takes voltage. In addition, we expect dynamical screening to be about 7 hours when using 44 CPUs (2 GHz) with 1 TB important in the emerging area of magnetic proximity shared DRAM. ’ effects in monolayer transition-metal dichalcogenides [83]. After the calculation of the pair Green s function, we use It has been experimentally shown that magnetic substrates Eq. (6) to calculate the absorption spectrum. Specifically, can strongly alter the optical response of excitons, induced we first calculate the function by time-reversal symmetry breaking between the K and −K X Z  valleys [84–86]. Equivalently, magnetic proximity effects −1 AðΩÞ¼β Im dkkGpðq ¼ 0;k;z;ΩÞ ; ðA1Þ or strong magnetic fields should lead to symmetry breaking z between counterpropagating shortwave plasmons. Beyond monolayer transition-metal dichalcogenides, and then use Pad´e polynomials to extract the analytical virtual exchange of shortwave plasmons may give rise to form of AðΩ → Eω þ iδÞ in the upper complex plane from pairing mechanisms in superconductors whose Fermi knowledge of its values at discrete even Matsubara energies surfaces comprise distinct pockets in the Brillouin zone. [52]. The broadening term is taken as [54] Similar to our formalism, plasmon-mediated Cooper pair- ing can be modeled when a dynamical screening descrip- δ2 δ ≡ δðEωÞ¼δ1 þ ; ðA2Þ tion is employed [87–90], and it was argued to facilitate 1 þ exp½ðEc − EωÞ=δ3 superconductivity in cuprate metal oxides [91] and other transition-metal systems through plasmon-phonon hybridi- where δj (j ¼f1; 2; 3g) are three parameters with the zation [92,93], as well as in semiconductors and electron- following meaning: δ1 reflects the inhomogeneous broad- hole liquids [94–97]. ening due to disorder, adatoms, and charge puddles; δ2 and δ3 reflect the enhanced homogeneous broadening when the ACKNOWLEDGMENTS transition energy belongs to the continuum (Eω >Ec). Far below the continuum, we get δðEωÞ → δ1. Far above the The authors are indebted to Zefang Wang, Kin Fai Mak, continuum edge, we get δðEωÞ → δ1 þ δ2. See Chapter 13 and Jie Shan for providing the reflectivity measurements in in Ref. [54] for more details. In this work, we have used Fig. 1 and for many useful discussions. We also thank δ1 ¼ 10 meV, δ2 ¼ 70 meV, and δ3 ¼ 20 meV. Xiaodong Xu for discussing and sharing valuable exper- The results shown in Figs. 8 and 9 were obtained using imental results prior to their publication. This work is mainly the renormalization in Eq. (16). The self-energy term, supported by the Department of Energy, Basic Energy whose expression is provided in Eq. (17), is calculated Sciences (Grant No. DE-SC0014349). The computational by summing over even Matsubara energies and by inte- work at Rochester was also supported by the National grating over q0. The integration interval for the latter is Science Foundation (Grant No. DMR-1503601) and the 0 jq − K0j ≲ γðα=3ÞkF, where γ ¼ 2. In other words, we Defense Threat Reduction Agency (Grant No. HDTRA1-13- have assumed that plasmons are not immediately damped 1-0013). The work at Buffalo was supported by the outside the region of free-plasmon propagation. Department of Energy, Basic Energy Sciences (Grant No. DE-SC0004890). The work at Regensburg and Würzburg was supported by the DFG (Grants No. SCHA 1. Effective mass model 1899/2-1 and No. SFB 1170 “ToCoTronics”), and by the We emphasize that the goal of this work is not to ENB Graduate School on Topological Insulators. reproduce the exact binding energy of excitons in

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ML-TMDs. Rather, we are interested in the effects of inspecting the Bose-Einstein distribution factors that are dynamical screening and coupling between excitons and associated with the distinct anti-Stokes and Stokes side- shortwave plasmons. We use the effective mass approxi- bands, jgðΔMoÞ=gð−ΔWÞj, which emerge in ML-MoSe2 mation since it allows us to simplify the calculation while and ML-WX2, respectively. For example, assuming that keeping the main dynamical effects intact. In the calcu- these optical sidebands persist at 200 K and assigning Δ ∼ lation of the pair functions of direct and indirect excitons, 40 meV for both materials at some comparable elevated we have assumed similar effective masses for electrons in densities [35], the amplitude of the Stokes sideband the lower and upper valleys of the conduction band. We becomes about 20 times stronger than that of the anti- ignore the (small) change in the values of the effective Stokes sideband. masses at the edges of the lower and upper valleys since the binding energy of excitons in ML-TMDs is much larger 3. Type B excitons than the conduction-valley splitting energy. In other words, Type B excitons correspond to optical transitions from the matrix we use to solve the pair function includes the lower spin-split valence band. We expect the coupling electronic states whose energies are hundreds of meVabove between type B excitons in ML-TMDs and shortwave the edge of the band. For such states, the exact value of the plasmons to be weak because of the ultrashort lifetime of effective mass in the bottom of the band is meaningless. We type B excitons: They experience ultrafast intervalley spin- therefore choose the same value for both lower and upper conserving energy relaxation of holes to the lower spin- valleys, which represents the effective mass of the split valleys. Furthermore, at elevated charge densities in “extended” conduction band. which the plasma excitations become relevant, the band- gap renormalization is strong enough so that type B optical 2. Emission and absorption of plasmons transitions are within the continuum. In other words, when The self-energy of the exciton due to the interaction with the densities of resident holes or electrons are larger than 12 −2 shortwave plasmons, given in Eq. (17), includes both about 10 cm , type B excitons can experience ultrafast emission and absorption terms. Formally, these terms are dissociation to free electrons and holes. The coupling to derived from the identity [36,51] shortwave plasmons is less effective in this case. X −1 0 0 0 0 β DðΩ − Ω ; q ÞGpðq ; k; Ω Þ 4. Electron-hole exchange Ω0 I The relatively large binding energy of excitons in ML- 1 1 Ω0 Ω − Ω0 q0 q0 k Ω0 TMDs can lead to evident energy splitting between bright ¼ d 0 Dð ; ÞGpð ; ; Þ; 2πi C expðβΩ Þ − 1 and dark excitons [98–100]. The theory presented in our ðA3Þ work, however, involves only bright excitons (singlet spin configuration of the electron and hole). Therefore, both where the sum runs over even Matsubara energies, and the direct and indirect bright excitons should experience integration contour C in the upper plane encircles the poles similar effects due to the electron-hole exchange interac- in the positive sense. Using Cauchy’s residue theorem, the tion. As a result, the energy splitting between direct and Bose-Einstein distribution factors for absorption and emis- indirect excitons remains equivalent to the spin splitting of ( ) the conduction band. Furthermore, the emergence of the sion, g EsðqÞ , result from the poles in the plasmon Ω0 Ω dynamical sideband is relevant at elevated charge densities propagator, ¼ EsðqÞ [see Eq. (19)], where expðβΩÞ¼expð2πinÞ¼1. Furthermore, by approximat- in which the wave functions of excitons are blown up ing the pair function as a simple pole expression, because of screening (smaller binding energy). The elec- 0 −1 tron-hole exchange is a weaker effect at such conditions. Gp ∝ ðΩ − E0Þ , where E0 is the exciton energy, we can analytically derive the Stokes and anti-Stokes side- bands. The former stems from the emission term 5. More simulation results ∝ ( − ) Ω − Ω → ℏω δ g EsðqÞ =½ þ EsðqÞ E0, where þ i It is important to realize that the effects of dynamical − → 1 and gð EsðqÞÞ at low temperatures. Similarly, the screening and interaction with plasmons are largely inde- ∝ ( ) Ω − − absorption term stems from g EsðqÞ =½ EsðqÞ E0. pendent of the exact parameters that we choose for the Section III C describes some of the difficulties that may effective mass and dielectric constant. For example, in the arise when one tries to observe the anti-Stokes sideband in main text, we have assumed that the monolayer is the absorption spectrum of high-temperature ML-MoSe2. embedded in hexagonal boron nitride, for which there is Other difficulties that impede its detection may be caused discrepancy in the reported values of the dielectric constant. by plasmon damping due to coupling with phonons and by Recent experiments use values of ϵ ¼ 2.7 [15,16], and we the relatively small population of shortwave plasmons have used this parameter in Figs. 6, 8, and 9. However, ∼ Δ due to the increased value of EsðqÞ at elevated previous studies of hexagonal boron nitride reported values electron densities. This behavior can be understood by that are nearly twice as large, ϵ ¼ 5.06 [101]. We have used

041040-15 VAN TUAN, SCHARF, ŽUTIĆ, and DERY PHYS. REV. X 7, 041040 (2017)

2 enable stronger pairing between excitons and shortwave 8 −2 1 10 cm plasmons, as can be seen from the shoulder and enhanced 11 0.1 5 10 blueshift of the main peak at elevated densities. At the same 1.5 1 1012 time, the redshift of the dynamical sideband is also some- what enhanced, and it shows slight decay at elevated 2 1012 0.05 12 densities. Finally, Fig. 11 shows the absorption spectrum 4 10 when using γ ¼ 2 and the effective isotropic model for the 1 6 1012 0 values of the thickness and dielectric constant of the ML −0.26 −0.24 −0.22 −0.2 0 6 ϵ 16 3 [102]: d ¼ . Å and ML ¼ . . The behavior of the

Absorption (a.u.) optical sideband is essentially the same, providing further 0.5 evidence that the exciton-plasmon is a dynamical effect with no qualitative dependence on the exact parameter values that one uses in the static dielectric function. 0 −0.3 −0.25 −0.2 −0.15 −0.1 e [1] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, FIG. 10. Renormalized absorption spectrum in electron-doped Atomically Thin MoS2: A New Direct-Gap Semiconductor, ML-WX2 embedded in layers with a dielectric constant, Phys. Rev. Lett. 105, 136805 (2010). ϵ ϵ 5 06 b ¼ t ¼ . . This figure shows the same qualitative behavior [2] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, ϵ ϵ 2 7 as in Fig. 8 in which we have used b ¼ t ¼ . . G. Galli, and F. Wang, Emerging Photoluminescence in Monolayer MoS2, Nano Lett. 10, 1271 (2010). [3] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, Electronics and Optoelectronics of 2 Two-Dimensional Transition Metal Dichalcogenides, − 1 108 cm 2 Nat. Nanotechnol. 7, 699 (2012). 11 [4] A. K. Geim and I. V. Grigorieva, Van der Waals 0.1 5 10 Heterostructures, Nature (London) 499, 419 (2013). 1.5 12 1 10 [5] K. F. Mak and J. Shan, Photonics and Optoelectronics of 2 1012 0.05 2D Semiconductor Transition Metal Dichalcogenides, 4 1012 Nat. Photonics 10, 216 (2016). [6] D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Coupled 1 6 1012 0 Spin and Valley Physics in Monolayers of MoS2 and −0.3 −0.28 −0.26 −0.24 Other Group-VI Dichalcogenides, Phys. Rev. Lett. 108,

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