Marrying Excitons and Plasmons in Monolayer Transition ... · The theory makes it clear why...

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, USA2Department of Physics, University at Buffalo,

State University of New York, Buffalo, New York 14260, USA3Institute 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-densityoscillations in conducting media. Plasmon phenomena in normal metals, superconductors, and dopedsemiconductors are often driven by long-wavelength Coulomb interactions. However, in crystals whoseFermi surface is comprised of disconnected pockets in the Brillouin zone, collective electron excitationscan also attain a shortwave component when electrons transition between these pockets. In this work, weshow that the band structure of monolayer transition-metal dichalcogenides gives rise to an intriguingmechanism through which shortwave plasmons are paired up with excitons. The coupling elucidates theorigin for the optical sideband that is observed repeatedly in monolayers of WSe2 and WS2 but notunderstood. The theory makes it clear why exciton-plasmon coupling has the right conditions to manifestitself 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

Transition-metal dichalcogenides become direct band-gap semiconductors when thinned to a single atomicmonolayer [1,2]. Stacking such two-dimensional crystalsvia van der Waals forces alleviates the need to fabricatelogic and optoelectronic devices with lattice-matchedcrystals [3–5]. Furthermore, the lack of space inversionsymmetry in these monolayers lifts the spin degeneracy atthe edges of the conduction and valence bands [6,7]. As aresult, time-reversal partners the spin and valley degrees offreedom (d.o.f.) such that spin-up charge carriers populatevalleys (low-energy pockets) on one side of the Brillouinzone, while spin-down ones populate valleys on the oppositeside [6]. The intense recent research on the spin-valleypartnership has led to understanding of optical selectionrules and transport phenomena inmonolayer transition-metaldichalcogenides (ML-TMDs) [8,9].In this work, we focus on a unique optical sideband

that emerges in emission or absorption-type experimentsof electron-doped ML-WSe2 and ML-WS2 [9–17].

Understanding its microscopic origin has been an outstand-ing open question ever since its first observation in thephotoluminescence (PL) measurements of ML-WSe2 byJones et al. [10]. So far, this optical transition has beenattributed to the fine structure of negatively charged excitons,biexcitons, or defects in various experiments [9–17]. Weshow that this optical transition actually corresponds to anexciton-plasmon quasiparticle due to intervalley Coulombexcitations, where other identifications are not consistentwith its spectral position, dependence on electron density,and the fact that it does not emerge in hole-doped conditionsor in molybdenum-based MLs. To explain the behavior ofthis optical sideband, we use the empirical data of differentialreflectance spectroscopy of ML-WSe2 at 4 K, as shownin Fig. 1 where this sideband is indicated by X−0. Themonolayer was embedded between thin layers of hexagonalboron nitride in a dual-gated structure [15]. The bias voltagewas the same in the top and bottomgate electrodes; thus, onlythe charge density in the monolayer is controlled, while noout-of-plane electrical field is introduced. The relationbetween gate voltage and charge density is such that 1 Vcorresponds to about 1012 cm−2 [15].Figure 1(a) shows the spectrum for low and intermediate

electron densities. The neutral-exciton spectral line,denoted by X0, dominates the spectrum, and its intensitydecays with increasing the electron density. Also noticeableare three smaller spectral lines in the low-energy side of thespectrum (as highlighted in the enlarged box). Thesefeatures cannot be completely quenched because of

*[email protected]†[email protected]

Published 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 7, 041040 (2017)

2160-3308=17=7(4)=041040(19) 041040-1 Published by the American Physical Society

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disordered regions that prevent complete charge depletionacross the entire sample [18]. The spectral lines that aredenoted by XA− and XB− were recently attributed to two typesof negatively charged excitons in tungsten-based mono-layers [14,19,20]. The lowest-energy spectral line, denotedby X−0, is the optical sideband. In addition to the fact thatXA− and XB− are clearly resolved from that of X−0 even at lowdensities, as highlighted in the enlarged box of Fig. 1(a),their behavior is strikingly different, rendering their iden-tification relatively easy. For example, XA− has the knownproperties of trions in semiconductor quantum wells[21–26]: It is amplified without enhanced broadening whenthe electron population increases in the low-density regime.In other words, in the regime where the Fermi energy ismeasurably smaller than the binding energy of the chargedexciton (energy difference between XA− and X0, which isabout 30–35 meV). Figure 1(b) shows that XA− starts toblueshift, decay, and broaden at intermediate and elevateddensities, in which the Fermi energy is no longer negligiblecompared with the charged-exciton binding energy. In thisdensity regime, the trion cannot avoid many-body inter-actions with the Fermi sea. The optical sideband, on theother hand, shares none of these features. Figure 1(b) showsthat when the electron density continues to increase, X−0redshifts and possibly drains part of the oscillator strengthof the decaying charged exciton spectral lines. Its spectralline shows no evident broadening, and its saturatedintensity at elevated densities hardly decays (whereas theother spectral lines strongly decay, blueshift, and broaden).

Figures 1(c) and 1(d) show the absorption spectrum inhole-doped conditions. It includes the neutral and charged-exciton spectral lines (X0 and Xþ), where the latter behavessimilarly to XA− in terms of its decay, blueshift, andbroadening 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 inthe electron and hole-doping regimes, which behavesimilarly to X0 and Xþ in Figs. 1(c) and 1(d). They shareno common properties with the optical sideband of tung-sten-based MLs. It is emphasized that in this work weneither model nor focus on the microscopic origin of thecharged exciton. Sidler et al. recently suggested that itoriginates from Fermi polarons, wherein the exciton isattracted to the surrounding Fermi sea via weak van derWaals forces [27,28]. Jadczak et al., on the other hand,attributed the charged exciton to trions and further eluci-dated its strong coupling to phonons due to the resonancebetween the trion binding energy and that of the homopolarphonon mode [29].The observed behavior of the optical sideband in

tungsten-based compounds is largely identical in reflec-tance and PL experiments, including the unique feature thatthe sideband redshifts in both cases [10,11,15]. Onenoticeable difference, however, is the relative amplitudeof the signal. Specifically, reflectance measurements showthat the largest oscillator strength is that of the neutralexciton at low densities [15,16,20,30,31]. The oscillatorstrength of the optical sideband, on the other hand, isevidently weaker as one can notice by comparing the y-axisscales in Figs. 1(a) and 1(b). The scenario is opposite in PLmeasurements, where one finds that the strongest lightemission comes from the optical sideband at elevatedelectron densities, while that of the neutral exciton atlow densities is evidently weaker [10,11,15]. This apparentcontradiction is settled in PL experiments for the followingreasons: (i) Neutral excitons in tungsten-based monolayersexperience intervalley scattering after photoexcitation,rendering them less optically active in the light emissionprocess [32–35]. (ii) The quantum efficiency of light emis-sion can be improved at elevated electron densities becauseof screening of charged defects that function as nonradiativerecombination centers for excitons. (iii)We show that severalphoton-emission mechanisms contribute to the optical side-band in the PL case.The main contribution of this work is in elucidating the

microscopic origin for the optical sideband, revealing anintriguing pairing phenomenon between excitons andplasmons, which has no parallel in other known two-dimensional (2D) systems. We explain why the opticalsideband is manifested in the spectra of electron-dopedML-WSe2 and ML-WS2 while being absent in the hole-doped case or in ML-MoSe2 and ML-MoS2. The theoreti-cal framework in this work studies the oscillator strength ofexcitons through the electron-hole pair function in the

(a) (b)

(c) −−−−

−−−

−

−

−

(d)

FIG. 1. Optical reflection spectroscopy of ML-WSe2 at 4 K.The gate voltage controls the charge density in the ML. (a,b)Optical spectra for low and high electron densities, respectively.The optical sideband is indicated by X−0. (c,d) The respectiveresults for hole densities. The sample is n type, so a gate voltageof about −1 V is required to deplete the ML from residentelectrons. These results were kindly provided to us by Wang,Mak, and Shan. Reference [15] includes details of the experimentwith a similar device.

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

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presence of dynamical screening. The theory extendsbeyond the quasistatic random-phase approximation orthe Shindo approximation for the dynamical effects [36].It therefore allows one to model the decay of the neutralexciton at elevated charge densities without introducingphenomenological screening parameters or ansatzesregarding the energy dependence of the pair function. Inaddition, the theory allows one to model the emergence ofdynamical optical sidebands, where we focus on the onethat arises from the exciton-plasmon coupling at largeelectron densities.

II. BASIC PROPERTIES OF EXCITONSAND PLASMONS IN ML-TMDs

Before we can explain the coupling between plasmonsand excitons, we briefly summarize the relevant propertiesof each of these quasiparticles separately. Excitons inML-TMDs are classified based on the spin and valleyconfiguration of the electron-hole pair [32–34]. Figure 2shows the valleys pertinent to the formation of low-energyexcitons in ML-WX2 and ML-MoX2, where X is thechalcogen atom. Excitons are bright or dark (opticallyactive or inactive) if the spins of the electron in theconduction band and the missing electron in the valenceband are parallel or antiparallel, respectively. Scatteringbetween dark and bright excitons necessitates a spin-flip ofthe electron or hole, which is typically a much slowerprocess than the lifetime of bright excitons [37–40].Therefore, we neglect dark excitons and show that thedynamical relation between two types of bright excitons issufficient to explain the optical measurements in Fig. 1. Thefirst type is the direct exciton in which the electron and holehave the same valley index, giving rise to direct-gap opticaltransitions. The second type is the indirect exciton in whichthe electron and hole reside in opposite valleys. Opticaltransitions of indirect excitons in perfect crystals aremediated by external agents such as shortwave phonons,which are needed to make up for the large momentummismatch of indirect excitons and photons. As shown in

Fig. 2, indirect excitons have lower energy than direct onesin ML-WX2 and vice versa in ML-MoX2 [32].Next, we briefly discuss the two plasmon species in

ML-TMDs. The first type of plasmon is characterized by itslong wavelength [41–43]. These plasmons originate fromthe long-range part of the Coulomb potential through whichelectrons cross the Fermi surface with small crystalmomentum transfer (i.e., an intravalley process). Theyare common to all conducting media and are largelyindependent of the band structure since their wavelengthis much longer than the lattice constant [44]. In the contextof optical transitions in semiconductors, plasmons screenthe electron-hole attraction, and they shrink the band-gapenergy by assisting electrons (or holes) of similar spins toavoid each other [36]. The energy dispersion of long-wavelength plasmons in 2D semiconductors follows,

ElðkÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2e2EFkϵðkÞ

s; ð1Þ

where e is the elementary charge, k is the wave number ofthe plasmon, and EF ¼ πℏ2n=me is the Fermi energyrelated to the charge density and effective mass of theelectron (n andme). The static dielectric function ϵðkÞ takesinto account the experimental sample geometry, as depictedin Fig. 3. Its form follows [45],

1

ϵðkÞ ¼1

ϵML·1þ ðpb þ ptÞe−kd þ pbpte−2kd

1 − pbpte−2kd; ð2Þ

where d denotes the thickness of the semiconductor,including the van der Waals gap, and ϵML is the dielectricconstant of the ML. Here, pb (pt) is the polarization due tothe dielectric contrast between the ML and the bottom (top)layer,

pbðtÞ ¼ϵML − ϵbðtÞϵML þ ϵbðtÞ

; ð3Þ

where ϵb (ϵt) is the dielectric constant of the bottom (top)layer. In the long-wavelength regime, one can readily showthat the effective dielectric constant is largely set by the

(a) (b)

FIG. 2. Direct and indirect excitons in the time-reversedK-point valleys. Panels (a) and (b) show the involved electronicstates in tungsten- and molybdenum-based monolayers, respec-tively. The indirect exciton has lower energy in the former case.Spin of the bands is color coded, and Δ is the conduction-bandsplitting energy.

FIG. 3. The geometry studied in this work is that of a ML-TMDsandwiched between two other materials. The thickness of theML, including the van der Waals gap, is d. The dielectricconstants of the ML, bottom, and top layers are ϵML, ϵb, andϵt, respectively.

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

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average dielectric constants of the top and bottom layers[45,46],

ϵðkÞ ⟶kd≪1 ϵt þ ϵb2

þ�ϵML −

ϵ2t þ ϵ2b2ϵML

�kd2: ð4Þ

The second type of plasmon originates from the short-range Coulomb potential through which electrons transitionbetween valleys [35,47–50]. Because of the relatively largespin splitting in the valence band of ML-TMDs [6], suchtransitions are mostly relevant in the conduction bandwherein the much smaller spin splitting can be comparableto the Fermi energy, as shown in Fig. 4(a). Collectiveintervalley excitations result in shortwave plasmons, whosecharge-fluctuations profile is illustrated in Fig. 4(b), andwhose energy dispersion follows [35]

EsðqÞ ¼ Δþ�1þ 3

α

�~εq þ

α

3EF: ð5Þ

Note that Δ is the spin-splitting energy in the conductionband, α ¼ ðK0asÞ−1 ∼ 0.1, and ~εq ¼ ℏ2ðK0 − qÞ2=2me.Here, K0 ¼ 4π=3a is the wave number that connects thetime-reversed valleys (a ∼ 3.2 Å is the lattice constant),and as ¼ ℏ2ϵML=e2me is the effective Bohr radius at theseshort wavelengths. Microscopic corrections due to theshort-range Coulomb potential can be incorporated inEq. (5) via a change in the value of α. The reason forthis is that the dielectric constant loses its meaning whenkd > 1 if the thickness of the ML is comparable to thelattice constant [46]. Courtade et al. have used empiricalvalues of the fine structure of charged excitons andestimated that the amplitude of the short-range Coulombpotential is of the order of a few eV · Å2 [20]. This

amplitude is similar to the value of the intervalleyCoulomb potential, 2πe2=ϵMLK0, that was used to deriveEq. (5), indicating that α ∼ 0.1 remains a good estimateeven when microscopic corrections are taken into account.Note that the existence of short-wave plasmons onlydepends on the fact that the short-range Coulomb potentialis not negligible, regardless of its exact value. The energydispersion in Eq. (5) is derived by assuming that the short-range Coulomb potential hardly changes when kF ≪ K0,where kF is the Fermi wave number. Indeed, kF is morethan 1 order of magnitude smaller than K0 even when thedensity is of the order of 1013 cm−2. The region of free-plasmon propagation is then limited to K0 − ðα=3ÞkF <q < K0 þ ðα=3ÞkF [35]. In other words, when the wavenumber of a shortwave plasmon belongs to a small regionof nearly perfect time-reversal intervalley conditions(q ∼ K0), it does not interact with individual electronsand should be treated as an independent degree of freedomin the electronic system [44].In addition to their different length scales, intravalley and

intervalley plasmons in ML-TMDs differ in their energyspectrum. The spin splitting in the conduction bandproduces a gapped dispersion for shortwave plasmonscompared with the gapless dispersion in the long-wavelength case: EsðK0Þ¼ΔþαEF=3 while Elð0Þ ¼ 0.The value of Δ is governed by the spin-orbit couplingat vanishing electron densities. Its value increases atelevated densities because of the electron-electronexchange interaction [35].

III. PAIR FUNCTION OF EXCITONSIN THE PRESENCE OF PLASMONS

To study how excitons interact with plasmons, we use thefinite-temperature Green’s functions formalism [51] andquantify the optical signature of bright excitons whensurrounded by collective charge excitations [36]. Theconnection to the absorption spectrum is then establishedvia the imaginary part of the electron-hole pair function,

AðωÞ ∝ β−1Xk;z

ImfGpðq → 0;k; z;Ω → Eω þ iδÞg: ð6Þ

Here, ℏω is the photon energy and β−1 ¼ kBT is thethermal energy. Note that k and q are 2D crystal wavevectors, where the electron in the pair takes on kþ q andthe hole − k. In the limit where q → 0, a direct opticaltransition is rendered because of the small momentum ofphotons. Throughout the analysis below, pair functionsof direct and indirect excitons are evaluated by assigningq ¼ 0 and q ¼ K0, respectively. Dynamical effects due toplasma oscillations are embodied by the dependence of Gpon z and Ω, which are odd (fermion) and even (boson)imaginary Matsubara energies, z ¼ ð2lþ 1Þπi=β andΩ ¼ 2lπi=β, where l is an integer [36]. The sums overk and z in Eq. (6) integrate out the fermion d.o.f. In the last

(a)

(b)

FIG. 4. The intervalley Coulomb interaction in ML-TMDs.(a) Spin-conserving charge excitations from the −K to the Kvalleys. Electrons need to overcome the spin-splitting energy gapin the conduction band. (b) The resulting shortwave chargefluctuations in the monolayer.


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step, Padé analytical continuation is performed in order toevaluate the pair function at real photon energies [52]:Ω → Eω þ iδ, where δ → 0þ and Eω ¼ ℏω − μe − μh isexpressed in terms of photon energy and quasichemicalpotentials in the conduction and valence bands.To find the pair function in Eq. (6), we make use of the

so-called screened ladder approximation. It describesrepeated interaction of the electron-hole pair with thedynamically screened potential [36]. While this physical

picture considers the interaction of excitons with collectiveelectron excitations, it neglects exciton-exciton inter-actions. Accordingly, the screened ladder approximationcan be used to explain experiments in which the ML is notsubjected to intense photoexcitation. Figure 5 shows theFeynman diagram of the two-particle Green’s functionbefore contraction (i.e., before integrating out any of itsquantum numbers). This diagram is formally written as

Gpðq;ki;kf; z;ΩÞ ¼ Gp;0ðq;ki;kf; z;ΩÞ þ1

β

Xk1;k2;z0

Gp;0ðq;ki;k1; z;ΩÞVsðk1 − k2; z − z0ÞGpðq;k2;kf; z0;ΩÞ; ð7Þ

where Vsðk − k0; z − z0Þ is the dynamically screenedpotential to be discussed later. Here, Gpðq;ki;kf; z;ΩÞand Gp;0ðq;ki;kf; z;ΩÞ are the interacting and noninter-acting pair functions, respectively. The noninteracting caseis given by the product of the electron and hole Green’sfunctions

Gp;0ðq;ki;kf; z;ΩÞ ¼ Geðki þ q;Ω− zÞGhð−ki; zÞδki;kf ;ð8Þ

where here and below we assume the area of the monolayeris 1. The Kronecker delta function allows us to integrate outthe final wave vector kf in Eq. (7) and to calculate the

contracted pair function Gpðq;k; z;ΩÞ, from which we canestimate the absorption according to Eq. (6). We note thatwhile this step seems trivial, the approach so far was tocontract over the odd Matsubara energy (z) and then use theso-called Shindo approximation to describe dynamicalscreening [36,53]. However, dynamical screening can bebetter described upon contraction in momentum rather thanenergy space. This contraction keeps both dynamicalparameters (even and odd Matsubara energies), whileeliminating kf does not compromise any further accuracyin the solution of the pair function if the starting point is thescreened ladder approximation. After contraction, wesimplify the form of Eq. (7),

Gpðq;k; z;ΩÞ ¼ Geðkþ q;Ω − zÞGhð−k; zÞ�1þ β−1

Xk0;z0

Vsðk − k0; z − z0ÞGpðq;k0; z0;ΩÞ�: ð9Þ

This equation can be solved by matrix inversion tech-niques, as explained in the Appendix.

A. Effect of long-wavelength plasmons on excitons

We begin by evaluating the pair functions of directexcitons when Vs, Ge, and Gh in Eq. (9) are affected onlyby long-wavelength plasmons. Neglecting shortwave plas-mons in the first step allows us to find common attributes

between ML-TMDs and conventional semiconductor quan-tum wells [54].A central part of the theory is the inclusion of the

dynamically screened potential in Eq. (9). We invoke thesingle-plasmon pole approximation (SPP), which providesa relatively compact form for the dynamical screening.Using the more rigorous random-phase approximation doesnot lead to qualitative changes or to increased

FIG. 5. Feynman diagram representation of the two-particle Dyson equation under the screened ladder approximation. The top andbottom propagators denote the Green’s functions of electrons and holes, respectively. The wiggly line denotes the dynamically screenedattractive potential. This diagram sums over all processes through which the electron-hole pair is repeatedly scattered by the attractivepotential.


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computational complexity when solving Eq. (9). Whenincluding long-wavelength plasma excitations, the dynami-cally screened Coulomb potential under the SPP approxi-mation reads [36]

Vsðk; z − z0Þ ¼2πe2

ϵðkÞ1

k

�1þ E

2lðkÞ

ðz − z0Þ2 − E2k

�; ð10Þ

where

Ek ¼ ElðkÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ alk

4

rð11Þ

is the pole energy, and al ¼ ℏ2ðϵb þ ϵtÞ=2e2me is theeffective Bohr radius at long wavelengths. The solution ofEq. (9) is greatly simplified when using the static approxi-mation Vsðk; 0Þ or when studying the zero density caseElðkÞ ¼ 0. In these cases, one first calculates the sumP

zGeðkþ q;Ω − zÞGhð−k; zÞ and then directly calcu-lates the absorption spectrum from the contracted pairfunction Gpðq;k;ΩÞ. The static approximation, however,tends to overestimate the role of screening, and one has tointroduce phenomenological terms in the static potential tomitigate the screening effect [54,55]. A better descriptionfor the screening is provided by keeping the dynamicalterms in the potential, Vsðk; z − z0Þ. This approach, how-ever, renders the solution of Eq. (9) evidently morecomputationally demanding compared with the static case.The second signature of long-wavelength plasmons is

introducing the band-gap renormalization (BGR) throughthe self-energy terms in the electron and hole Green’sfunctions,

Giðk; zÞ ¼1

z − εk;i − Σiðk; zÞ þ μi; ð12Þ

where i ¼ fe; hg, and εk;i ¼ ℏ2k2=2mi is the noninteract-ing electron (or hole) energy. The self-energy term Σiðk; zÞstems from the interaction of the electron (or hole) withthe dynamically screened potential, and μi ¼ μi;0 þΣiðkF; EF − μi;0Þ is the chemical potential. Here, μi;0 isthe temperature- and density-dependent chemical potentialof the noninteracting electron gas. Using the GW approxi-mation, the self-energy term is written as [36]

Σiðk; zÞ ¼ −β−1Xq;Ω

Vsðq;ΩÞGi;0ðk − q; z − ΩÞ

¼ Σx;iðkÞ þ Σc;iðk; zÞ; ð13Þ

where Gi;0 is the Green’s function of the noninteractingparticle (Σ ¼ 0). The self-energy is separated into exchangeand correlation parts. The former follows

Σx;iðkÞ ¼ −Xq

2πe2

ϵðqÞfiðk − qÞ

q; ð14Þ

where fi is the Fermi-Dirac distribution. The exchangecontribution from the electron-electron interaction can onlyshift the filled valleys (i.e., fi ≠ 0). The correlation term, onthe other hand, affects all bands, and it comes fromemission or absorption of long-wavelength plasmons(intravalley processes),

Σc;iðk; zÞ ¼Xq

the exciton decay when the background charge densityincreases. The effective mass for both electrons and holes is0.36m0 [56], where m0 is the free-electron mass (seeAppendix A 1). The dielectric environment is modeledby sandwiching the ML (d ¼ 1 nm, ϵML ¼ 7.25) betweenthick layers of hexagonal boron nitride (ϵb ¼ ϵt ¼ 2.7)[15,57]. The arrows in Fig. 6 indicate the continuumredshift due to BGR. Above this energy, the excitons areno longer bound. Note that the 2s bound state merges intothe continuum already at relatively small densities becauseof its smaller binding energy. The BGR is calculated fromthe value of Σiðk; εk;i − μ0i Þ at k ¼ 0, by assuming a rigidshift of the bands. The cutoff energies we have used for thebottom and top spin-split valleys in Eq. (15) are εqc ¼60 meV and εqc ¼ 45 meV, respectively. These cutoffenergies are the same for all densities. Using larger(smaller) cutoff energies, the peak position will graduallyredshift (blueshift) as one increases the charge density. Theinset of Fig. 6 shows that the BGR is strongest when thecharge density is ramped up from zero to about 1012 cm−2.The continuum redshift is much slower between about 1012

and 1013 cm−2, during which excitons become unboundand merge into the continuum.Two effects cause the decay of the neutral-exciton

spectral line with the increase in electron density. Thedominant one is the broadening effect when the excitonenergy approaches the continuum [see arrows in Fig. 6 andEq. (A2) in the Appendix]. The decay that is caused byreduction in the oscillator strength plays a secondary roledue to the mitigated effect of screening in 2D. Thisbehavior is reminiscent of excitonic enhancement of freeelectron-hole transitions in semiconductor quantum wells[55], in which the overlap between the electron and holewave functions remains sizable when excitons enter thecontinuum. The simulated decay of the neutral exciton inFig. 6 is corroborated in the differential-reflectivity mea-surements, which resolve the oscillator strength and broad-ening of the spectral lines [15,16]. Figures 1(a) and 1(c)show that when the charge density is ramped up by gatevoltage, the measured decay of neutral excitons is notcompensated by an equivalent increase of charged excitons,

thereby reinforcing the important role of BGR and dynami-cal screening. It is noted that, unlike the experimentalfindings of Fig. 1, the simulated exciton peak in Fig. 6 isnot completely quenched at elevated densities. In additionto the fact that we do not consider charged excitons, whichdrain part of the oscillator strength of neutral excitons [58],the model tends to overestimate the Coulomb attraction atelevated densities. The reason is that broadening effects arenot introduced in the electron and hole Green’s functions ofEq. (9); they are introduced in the analytical continuationphase after the pair function is solved (in the Appendix).An additional important result in Fig. 6, which is

supported by Figs. 1(a) and 1(c), is that the neutral excitonspectral position is hardly affected by the backgroundcharge density because of the mutual offset betweenBGR and the screening-induced reduction in the bindingenergy of excitons; the long-wavelength plasmons areresponsible for both effects [54]. One can therefore realizethat the binding energy of excitons in ML-TMDs can be ofthe order of a few hundred meV only at vanishing chargedensities. All in all, long-wavelength plasmons lead toqualitatively similar effects in ML-TMDs and conventionalsemiconductor quantum wells [55]. The only change is aquantitatively larger effect in ML-TMDs due to reduceddielectric screening and larger electron mass [59,60].

B. Effect of shortwave plasmons on excitons

In the next step, the unique features of ML-TMDs arerevealed by turning on the coupling between shortwaveplasmons and excitons. In the long-wavelength case,the exciton-plasmon coupling is implicit: It is embodiedin the self-energies of the electron and hole as well as in theattractive screened potential (i.e., the coupling to plasmonsis “concealed” in the dressed propagator and Coulomb linesin Fig. 5). An explicit exciton-plasmon interaction isinhibited if the extension of the exciton is smaller thanthe wavelength of the plasmon because of the chargeneutrality of the former. This physical picture changesfor shortwave plasmons, which can be effectively paired upwith an exciton in spite of its charge neutrality. Here, theshort-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 plasmonwavelength is shorter than the exciton extent, which is further blown up because of screening. (b) Feynman diagram representation of therenormalized direct-exciton pair function due to shortwave plasmons. The self-energy term takes into account the coupling between directand indirect excitons (D and I) due to the electron-plasmon interaction. The hole (bottom propagator) acts as a spectator, while the electroninteracts 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 DþDIDþDIDIDþ � � �.


041040-7

in Fig. 7(a), allows the hole or electron to act as a spectatorwhile the plasmon interacts with the opposite charge.The energy required to excite a shortwave plasmon at

low temperatures is greater than that available thermally(Δ ≫ kBT). In addition, we recall that free-propagatingplasmons are independent d.o.f., which do not interact withindividual charged particles of the electronic system. As aresult, plasmon signatures in the optical spectrum can beresolved by supplying energy from outside the electronsystem in amounts greater than the plasmon energy. This isthe case when we shake up the electronic system byexciting electron-hole pairs. Figure 7(b) shows theFeynman diagram for the explicit coupling between anexciton and a shortwave plasmon when the hole acts as thespectator. The plasmon propagator is the double wigglyline. This Feynman diagram corresponds to renormaliza-tion of the pair function of direct excitons according to

~Gpðq → 0;k;ΩÞ ¼Gpðq → 0;k;ΩÞ

1 − Σsðk;ΩÞGpðq → 0;k;ΩÞ; ð16Þ

where the pair Green’s functions were contracted byintegrating out the fermion Matsubara energies. The mixingwith indirect excitons comes from the self-energy term,

Σsðk;ΩÞ ¼ β−1Xq0;Ω0

M2q0DðΩ −Ω0;q0ÞGpðq0;k;Ω0Þ; ð17Þ

where Mq0 is the interaction between the plasmon and theelectron component of the exciton, and DðΩ;qÞ is the free-plasmon propagator. The sum in Eq. (17) is restricted tointervalley transitions, q0 → K0, so that the pair function onthe right-hand side is that of indirect excitons. Theinteraction and propagator terms follow [44],

Mq0 ¼ffiffiffiffiffiffi3π

pℏ2jK0 − q0jme

; ð18Þ

DðΩ;qÞ ¼ 2EsðqÞΩ2 − E2s ðqÞ

: ð19Þ

Figure 8 shows the renormalized absorption spectrum ofdirect excitons in electron-doped ML-WX2 at 10 K. Adynamical sideband emerges in the low-energy side of thespectrum. We have considered tungsten-based settings suchthat indirect excitons have lower energy than direct ones[Fig. 2(a)]. The spin-orbit coupling splits the conduction-band valleys by 26 meV [61]. All other parameters are thesame as in Fig. 6. The behavior of the simulated dynamicalsideband is similar to the empirical behavior of the opticalsideband in electron-doped ML-WX2 [9–16], as shown inFig. 1(b). Both the theoretical and the experimental resultsshow a redshift and amplification of the optical sidebandwhen increasing the electron density until they saturate. Wenote that the pair function reflects the oscillator strength of

the optical transition, and in this respect, its amplitudeshould be compared with that seen in differential reflec-tivity rather than photoluminescence experiments (wefurther address this issue later). Comparing the results inFigs. 1(b) and 8, however, we still find two quantitativedisagreements. The first one is that the main peak does notcompletely decay at elevated densities. As explainedbefore, we attribute this disagreement to the fact thatbroadening effects are introduced only after solving thetwo-particle Dyson equation (in the Appendix). The seconddisagreement is that the relative amplitude of the simulateddynamical sideband is about 3 times weaker than theoscillator strength of the optical sideband. We attributethis mismatch to the difficulty to simulate a strongerdynamical sideband through an increase of the integrationcutoff in Eq. (17) outside the small window of free-plasmonpropagation (e.g., when jK0 − q0j≳ αkF). The reason forthis is that the high-order Padé polynomials used to performanalytical continuation, Ω → Eω þ iδ, are notoriously sen-sitive to minute numerical errors [62].We now focus on the redshift and initial amplification of

the dynamical sideband, which are common to bothexperiment [as shown in Figs. 1(b)] and our simulations(highlighted in the enlarged box of Fig. 8). The redshift iscaused by the electron-electron exchange interaction whenthe lower spin-split valleys are being filled. This interactionleads to a stronger energy redshift of the populated lowervalleys compared with the unpopulated upper ones (inset ofFig. 6), thereby increasing the plasmon energy via Δ inEq. (5). The initial amplification of the dynamical sidebandis commensurate with the increase in the number of free-

−0.5 −0.4 −0.3e

−0.2 −0.10

0.2

0.4

0.6

0.8

1

Abs

orpt

ion

(a.u

.)

5 1011 cm−2

1 1012

2 1012

4 1012

6 1012

8 1012−0.4 −0.35 −0.30

0.02

0.04

0.06

0.08

FIG. 8. The absorption spectrum of direct excitons in electron-doped ML-WX2 after renormalization due to interaction withshortwave plasmons. A dynamical sideband emerges in the low-energy side of the absorption spectrum, as highlighted in theenlarged box. The sideband initially redshifts and intensifieswhen the electron density increases, showing saturated behaviorat elevated densities.


041040-8

propagating plasmon modes (larger value of αkF=3).Saturation at high densities occurs when the energy differ-ence between direct and indirect excitons, δED−I, deviatesfrom the energy of shortwave plasmons, about Δ.Specifically, indirect excitons experience a blueshift dueto electron filling of the lower valleys, while direct ones arenot subjected to this effect since they originate fromelectronic states in the empty upper valleys. Continuingto increase the electron density eventually leads to aslow decay of the dynamical sideband due to increased“off-resonance” conditions between δED−I and Δ(Appendix A 5).Our model shows that the dynamical sideband is evident

only in electron-doped ML-WX2, providing strong evi-dence that it corresponds to the measured optical sideband[9–16]. Figure 9(a) shows that the dynamical sideband isabsent in electron-doped ML-MoX2, simulated by revers-ing the order of the conduction-band valleys such thatdirect excitons have lower energy [Fig. 2(b)]. The dynami-cal sideband is not seen because of “wrong conditions.”Namely, the sideband is positioned at about one plasmonenergy below the energy of the indirect exciton, which inthe case of ML-MoX2, coincides with the direct-excitonspectral line. This behavior makes it hard to resolve theoptical signature of exciton-plasmon quasiparticles inML-MoX2. Figure 9(a) indeed shows that the renormali-zation only affects the spectral region of the exciton peakwithout the emergence of dynamical sidebands in otherspectral regions. In electron-doped ML-WX2, on the otherhand, the dynamical sideband can be distinctly recognizedsince the indirect exciton has lower energy; thus, overall,the sideband emerges about 2Δ below the direct-excitonspectral line.Further evidence that the revealed exciton-plasmon

phenomenon corresponds to the optical sideband in elec-tron-doped ML-WX2 is provided by simulating the hole-

doped case. Figure 9(b) shows the renormalized absorptionspectra of hole- and electron-doped ML-WX2. Thedynamical sideband is practically absent in the hole-dopedcase, which is similar to the experimental findings inFigs. 1(c) and 1(d). We can understand this behavior bynoting that for electron doping, the splitting of the con-duction-band valleys corresponds to both plasmon energyand the energy difference between direct and indirectexcitons (the role of electron-hole exchange is relativelysmall as we explain in Appendix A 4). For hole doping, onthe other hand, the plasmon energy is governed by the largesplitting of the valence-band valleys (about 400 meV inML-WX2), while the energy difference between direct andindirect excitons is still governed by the much smallersplitting in the conduction band. The large mismatchbetween the two suppresses the emergence of a visibledynamical sideband in hole-doped ML-TMDs.

C. Stokes vs anti-Stokes optical sidebands

Figure 8 shows that the exciton-plasmon resonanceemerges about 2Δ below the direct-exciton spectral linein electron-doped ML-WX2. Figure 9(a) shows that itcoincides with the spectral line of the direct exciton inelectron-doped ML-MoX2. These calculations did notreveal the emergence of an optical sideband above thedirect-exciton spectral line because of the small populationof shortwave plasmons at the simulated low temperature(T ¼ 10 K), g(EsðqÞ) → 0 when EsðqÞ ∼ Δ ≫ kBT.Specifically, the self-energy of direct excitons includesboth plasmon emission and absorption processes, corre-sponding to “Stokes” and “anti-Stokes” sidebands belowand above the energy of the indirect exciton, respectively(see Appendix A 2). The amplitude of the Stokes sidebandis commensurate with −g( − EsðqÞ) ¼ 1þ g(EsðqÞ), whilethat of the anti-Stokes sideband is commensurate withg(EsðqÞ). The vanishing value of the latter at low temper-atures explains why Figs. 8 and 9 do not include signaturesof high-energy optical sidebands due to plasmon absorption.At high temperatures, on the other hand, it may be

possible to observe a distinct high-energy optical sidebandin the absorption spectrum of electron-doped ML-MoSe2.The value of Δ in this material is non-negligible [32,56],resulting in a well-resolved anti-Stokes sideband whoseenergy is about 2Δ above the spectral line of the directexciton. This high-energy optical sideband is expected toblueshift when the electron density increases because of theincreasing value of Δ. Its amplitude, however, should bemeasurably weaker than that of the Stokes sideband inML-WX2 since the ratio jg(EsðqÞ)=g( − EsðqÞ)j remainsevidently smaller than 1 even close to room temperature(see Appendix A 2). In addition, the observation of a high-energy optical sideband in high-temperature ML-MoSe2may be elusive if it resides close to or within the continuum.In other words, the exciton-plasmon coupling is manifestedat elevated electron densities in which the exciton energy

(a)

e e

(b)

− − − −

−−

−

− − − − −

FIG. 9. The renormalized absorption spectrum of direct ex-citons in ML-MoX2 and in the case of hole doping. (a) The caseof electron-doped ML-MoX2 before (solid lines) and after(dashed lines) renormalization. The dynamical sideband is notseen. (b) The case of electron-doped (solid line) and hole-doped(dotted line) ML-WX2. The dynamical sideband is negligible forhole doping, as highlighted in the magnified box. It emergesabout 400 meV below the exciton line because of the largevalence-band splitting.


041040-9

approaches the continuum because of the strong BGR. Theemergence of an exciton-plasmon resonance above thedirect-exciton spectral line is hampered by broadeningeffects, similar to the behavior of neutral excitons whentheir energy approaches the continuum (Fig. 6). Theplasmon emission process, on the other hand, is relativelyrobust since the increased value of Δ with electron densitycauses the low-energy optical sideband to redshift, therebykeeping it protected from merging into the redshiftingcontinuum. Accordingly, broadening effects of the Stokessideband are mitigated, and ultrafast dissociation of theexciton-plasmon quasiparticle to the continuum of elec-tron-hole pairs is suppressed. Appendix A 2 includesfurther details on the Stokes and anti-Stokes opticalsidebands.

IV. DISCUSSION

When comparing the results in this work with exper-imental findings, one should recall that the theory onlyevaluates the pair function and its renormalization due tocoupling with plasmons. Accordingly, the theory does notcapture optical transitions that are associated with three-body complexes [21,22,63–70], Fermi polarons [27,28],and exciton optical transitions next to localization centers[71]. Nonetheless, the theory in this work covers twoimportant regimes. The first one is the low-density regime,n≲ 1012 cm−2, in which the neutral exciton dominates theabsorption spectrum, and its decay is not compensated byan equivalent increase of the trion peak, as shown by theempirical data in Fig. 1(a). The dynamical picture we havedeveloped quantifies the exciton decay due to long-wavelength plasmons (Fig. 6), where we find good agree-ment with reflectivity experiments [15]. The second case isthe high-doping regime, n≳ 5 × 1012 cm−2, in which bothreflectivity and PL experiments show that only one peaksurvives in the spectrum of electron-doped ML-WX2[10,15], as shown by Fig. 1(b). Trions cease to exist atvery large densities since their effective radius is larger thanthe average distance between electrons of the Fermi sea.The latter is proportional to 1=

ffiffiffin

p, while the trion extension

is of the order of 3–5 nm in ML-TMDs [69,70]. The mainfocus of this work is on the high-doping regime, and weattribute the observed optical transition to a new type ofquasiparticle where the exciton is coupled to a shortwavecharge fluctuation. We provided evidence that the behaviorof the optical sideband when increasing the electron densitymatches that of the exciton-plasmon quasiparticle, and wefurther showed that it emerges in electron-doped ML-WX2but not in the hole-doped or ML-MoX2 cases (Figs. 8and 9). Any alternative explanation that aims at decipheringthe origin for the optical sideband should be consistent withthis observation. Below, we discuss several key aspects thatone may find helpful when interpreting experimentalfindings and when comparing them with our theory.

A. Plasmon-assisted optical transitions

We have attributed the optical sideband in the reflectancespectra to the creation of an exciton-plasmon quasiparticle.It was found by renormalization of the pair function ofdirect excitons, through the sum of infinite plasmon-induced virtual transitions to intermediate indirect-excitonstates [Fig. 7(b)]. This physical picture should not beconfused with that of a plasmon-assisted optical absorption,in which an indirect exciton is created through two virtualtransitions into and from a direct exciton. To better under-stand this difference and its implication in the context of theexperimental data (Fig. 1), we assume low temperatureswhere only emission of a shortwave plasmon is feasibleduring these processes (Δ ≫ kBT).

1. Plasmon-assisted photon absorption(reflectivity spectrum)

In photon absorption at low temperatures, the plasmon-assisted process creates an indirect exciton and a shortwaveplasmon in the final state. The overall energy conservationof the absorption process is such that the photon energy isthe sum of the indirect exciton and plasmon energies,

ℏω ¼ EIð−qfÞ þ EsðqfÞ; ð20Þ

where the crystal momenta of the plasmon and indirectexciton are opposite in order to guarantee momentumconservation (the photon carries negligible momentum).Here, the photon is absorbed into a direct-exciton virtualstate, followed by plasmon emission that transfers theexciton to its final state. Crystal momentum must beconserved during virtual transitions to and from intermedi-ate states because of translation symmetry. As a second-order process, the absorption amplitude is weak unless theenergy of the intermediate virtual state resonates with thereal energy level of the direct exciton. Recalling the energydiagram in Fig. 2, Eq. (20) implies that this condition is metin electron-doped ML-WX2 but not in ML-MoX2. In otherwords, the plasmon-assisted photon absorption is a second-order resonance process in electron-doped ML-WX2,meaning that the neutral-exciton spectral line (X0) hastwo contributions. The first one is from the first-orderabsorption process, which creates direct excitons. Thesecond contribution is from the second-order resonanceprocess, which creates indirect excitons via emission ofshortwave plasmons.Therefore, an important resulting question is how one

can resolve the relative contribution of the plasmon-assistedoptical transitions to the oscillator strength of X0 inelectron-doped ML-WX2. Clearly, the contribution hasnegligible weight at vanishing electron densities becauseof the minute range of free-propagating plasmon modes.When the charge density increases, however, we canidentify this contribution by looking for differencesbetween the reflectance data in the electron-doped and


041040-10

hole-doped regimes. Indeed, Figs. 1(a) and 1(c) reveal aclear difference in the measured behavior of X0, whoseblueshift and broadening are evidently stronger in theelectron-doped case. As we suggest below, the plasmon-assisted optical transitions can give rise to this behavior.Without the contribution of shortwave plasmons, one

would expect the blueshift of X0 to be stronger in the hole-doped case. Specifically, the blueshift of neutral excitons inthe absorption spectrum of semiconductors is induced byfilling one or both of the bands that are involved in theoptical transition [54]. This condition is not readily met inelectron-doped ML-WX2 since only the bottom conductionvalleys are populated, whereas the optical transitioninvolves the top valleys. Both experiment and theory showthat electron population in the top valleys of ML-WX2begins at densities that are comparable to 1013 cm−2

[15,32]. The situation is different for the hole-doped caseat which hole filling in the top of the valence band ismeasurable at smaller densities. Thus, while such behaviorshould lead to weaker or at most comparable blueshift ofthe direct exciton in the electron-doped case, the experi-ment shows the opposite. Furthermore, the simulated decayof the pair function in Fig. 6 better resembles the observeddecay of X0 in the hole-doped case. This fact suggests amissing component in the theory such that dynamicalscreening and BGR are not enough to fully explain thebehavior of X0 in electron-doped ML-WX2.Plasmon-assisted optical transitions can complete the

puzzle and explain the stronger blueshift and broadening ofX0 in electron-doped ML-WX2. There are two comple-menting facts that support this claim. First, plasmon-assisted optical absorption stems from a resonance processin electron-doped ML-WX2, suggesting a non-negligibleeffect. Second, Eq. (20) can explain the enhanced blueshiftin the experiment since the plasmon energy increases atelevated densities (through the increased value of Δ).Finally, the enhanced broadening of X0 in ML-WX2 whenthe electron density increases can be explained by thecloser proximity of the photon energy to the redshiftingcontinuum (or possibly merging with the continuum). Wewill provide a complete description of these phenomenaalongside second-order perturbation theory to describe theplasmon-assisted optical transitions in a future work.All in all, the optical sideband in the reflectance experi-

ment of electron-doped ML-WX2 (X−0) is attributed to theexciton-plasmon quasiparticle, originating from renormali-zation of the self-energy of the direct exciton. Its energy isone (two) plasmon(s) below that of the indirect (direct)exciton. Photon absorption due to plasmon-assisted opticaltransitions, on the other hand, is a second-order process thatconverts photons into indirect excitons through plasmonemission at low temperatures. It becomes important inelectron-doped ML-WX2, in which the intermediate virtualexciton state resonates with the real energy level of the directexciton. The signature of this process in the reflectanceexperiment of electron-doped ML-WX2 is limited to the

spectral region of the direct exciton (X0). Unlike phonon-assisted optical transitions [32,72–74], the amplitude of theplasmon-assisted ones increases with the electron densitybecause of the increased range of free-propagatingplasmon modes.

2. Plasmon-assisted photon emission (PL spectrum)

In photon emission at low temperatures, the plasmon-assisted process creates a shortwave plasmon and aphoton in the final state through annihilation (radiativerecombination) of an indirect exciton. The overall energyconservation of the emission process is such that

ℏω ¼ EIð−qfÞ − EsðqfÞ: ð21Þ

Here, the indirect exciton in the initial state is transitionedinto a direct-exciton virtual state by plasmon emission,followed by the photon emission. Recalling the energydiagram in Fig. 2, Eq. (21) implies that the resonancecondition is now met in electron-doped ML-MoX2, inwhich the energy of the intermediate virtual state resonateswith the real energy level of the direct exciton. However,this second-order resonance process has no real value atlow-temperature PL since the population of indirect exci-tons in ML-MoX2 is negligible at low temperatures: Mostexcitons in these compounds remain direct after photo-excitation and energy relaxation [Fig. 2(b)]. Therefore, wedo not expect plasmon-assisted optical transitions to have ameasurable signature in the PL spectrum of ML-MoX2 atlow temperatures.The case of electron-doped ML-WX2 is unique if we

consider the renormalization of direct excitons that led tothe exciton-plasmon quasiparticle. In the reflectance spec-trum, we have concluded that the contributions from thecreation of indirect excitons through the plasmon-assistedphoton absorption and the creation of exciton-plasmonquasiparticles are spectrally resolved: The latter corre-sponds to the optical sideband (X−0), and the former canonly affect the spectral region of the bare neutral exciton(X0). In the PL spectrum, on the other hand, Eq. (21)suggests that the photon energy should coincide with that ofthe optical sideband. In other words, both the annihilationof the exciton-plasmon quasiparticle and the annihilation ofindirect excitons through plasmon-assisted photon emis-sion contribute to X−0 in the PL spectrum. The lattercontribution may not be negligible because of a largepopulation of indirect excitons after energy thermalizationin ML-WX2 [Fig. 2(a)]. In addition, while the energy of theintermediate virtual state does not resonate with the realenergy level of the bare direct exciton, it matches the lowenergy of the dressed one (i.e., the exciton-plasmonquasiparticle). Therefore, the recombination of indirectexcitons due to plasmon-assisted photon emission mayhave a measurable signature on the spectral weight of X−0,explaining in part why its magnitude is much stronger in PL


041040-11

compared with reflectance spectra (see related discussion inSec. I) [10,15]. We note that modeling of the density-dependent PL spectrum is beyond the scope of this work,partly because of the lack of knowledge of the nonradiativerecombination processes. For example, charged impuritiesthat function as nonradiative recombination centers arescreened at elevated densities, resulting in improved quan-tum efficiency for radiative recombination. Conversely,nonradiative Auger processes are amplified at elevateddensities [75–77], thereby reducing the quantum efficiency.

3. High temperatures

So far, we have discussed plasmon-assisted opticaltransitions at low temperatures, in which only emissionof shortwave plasmons is possible during the photonabsorption or emission. At high temperatures, on the otherhand, we should also consider the absorption of shortwaveplasmons during these optical processes. Repeating theprevious analyses, one can see that the conditions forsecond-order resonance processes are switched betweenML-MoX2 and ML-WX2. Detecting the signatures of theseprocesses in the high-temperature reflectance or PL spectramay be elusive for the same reasons we have mentionedregarding the anti-Stokes optical sideband in Sec. III C.

4. Other optical spectroscopy tools toprobe the plasmon-exciton coupling

In addition to differential reflectivity and photolumines-cence experiments, it may be possible to probe the couplingbetween excitons and shortwave plasmons in electron-doped samples using Raman-type experiments or nonlinearoptical spectroscopy. For example, the energy of scatteredphotons in the Stokes sideband of a Raman experiment isexpected to be 2Δ below that of the incident photon. Itinvolves a virtual process through which two counter-propagating shortwave plasmons are emitted, so the sum oftheir momenta becomes negligible and matches the minutemomentum of photons. In the first step of this process, theincident photon induces a virtual transition to a directexciton intermediate level, which in the second step isscattered to the indirect exciton intermediate level becauseof plasmon emission. In the next step, the indirect exciton isscattered back to the direct exciton intermediate level byemission of a counterpropagating plasmon. Finally, thesystem emits a photon at the Stokes energy (the energy ofthe incident photon minus the energy of two plasmons).The overall process is expected to be weak since twoplasmon-mediated virtual transitions are involved. Theamplitude of the scattered light can increase by usingnear-resonance conditions. The amplitude of the scatteredlight and its energy can be controlled by employing a gatevoltage that tunes the value of Δ through electron filling ofthe lower spin-split valleys. In other words, the gate voltageprovides an additional control parameter in the experiment.A similar approach can be used in nonlinear wave-mixing

spectroscopy: The generated wave is amplified when thedetuning between the input beams approaches 2Δ.

B. Three-body complexes versusexciton-plasmon quasiparticles

It is beyond the scope of this work to provide a completethree-body dynamical function since it is impossible tonumerically solve its corresponding Dyson equation with-out simplifying approximations. The dynamical model wehave developed describes the behavior of the pair functionand its renormalization due to coupling with shortwaveplasmons. The optical sideband, which we have associatedwith the exciton-plasmon quasiparticle, has nearly oppositebehavior from that of charged excitons (see discussion ofFig. 1 in Sec. I). Accordingly, the identification of chargedexcitons complexes and exciton-plasmon quasiparticlescan be readily achieved. Studying how the dynamicallyscreened potential influences the oscillator strength andbroadening of charged excitons is a work in progress. Here,we mention some of the known theoretical approaches tostudy their behavior. The most computationally convenientway to deal with trions is to dispense with the dynamicalterms altogether [20,63–68], but then one can only con-clude about their binding energy. Such static models cannotdescribe the density-dependent rise, decay, energy shift,and broadening of trions in the spectrum. Alternatively, onecan neglect dynamical terms in the exciton-electron inter-action [27] or a priori assume the dynamical form of theGreen’s function without solving the correspondingdynamical Dyson equation [21,28]. In the latter case,one can describe the behavior of charged excitons byinvoking an exciton-electron scattering function, wheredynamical terms in the scattering potential are neglected.One open question in the context of our work deals with

possible coupling between trions and shortwave plasmons.Figures 1(a) and 1(b) do not show optical sidebands whoseenergy isΔ or 2Δ below that of the charged exciton spectralline; the value of Δ is about 26 meV at low densities [61],and it increases at elevated ones [35]. The absence orweakness of trion-plasmon coupling is attributed to therelatively small oscillator strength of the charged excitoncompared with that of the neutral exciton (Fig. 1). Theoscillator strength is inversely proportional to the square ofthe complex radius, where the radius is about 3 times largerfor trions [69,78]. This fact is also reflected in the muchsmaller binding energy of the charged exciton. Here, wereemphasize that the decay of the neutral-exciton spectralline with the increase in electron density is mostly causedby broadening effects when the exciton energy approachesthe redshifting continuum (see discussion of Fig. 6). Thequasiparticle made by a neutral exciton coupled to ashortwave plasmon is resilient to this broadening sinceits energy is kept away from the redshifting continuum dueto the increasing value of 2Δ when the density is rampedup. For that reason, the oscillator strength of the optical


041040-12

sideband neither decays nor broadens at elevated densities[Fig. 1(b) and Fig. 8]. Quantitatively, the oscillator strengthamplitude is manifested in a relatively large value of thepair function at even Matsubara energies, compared withthe respective dynamical Green’s function that woulddescribe charged excitons. Finally, we mention that theoptical sideband is weak but noticeable even when EF issignificantly smaller than the binding energy of the chargedexciton [enlarged boxes of Figs. 1(a) and 8]. In this regime,the negatively charged exciton manages to stay away fromelectrons and plasma excitations of the Fermi sea, asevident from the absence of broadening and blueshift ofXA− and XB− in Fig. 1(a). Neutral electron-hole pairs, on theother hand, are not repelled by the surrounding Fermi seaand therefore can be coupled to shortwave plasmons at allfinite densities.

C. Strong photoexcitation conditions

Recently, You et al. observed the emergence of a strongspectral line in the PL ofML-WSe2 when subjected to strongexcitation [79]. This peak showed quadratic dependence onthe intensity of the pump signal and did not shift appreciablywhen increasing the pumping intensity. This behaviorsuggests that the emerged spectral line is due to exciton-exciton interactions [79]. A similar behavior was alsoobserved in the PL of ML-WS2 [11,12]. To date, however,this behavior was not reported in the PL of ML-MoX2.We argue that these observations can be settled by con-

sidering the shortwave Coulomb excitations. Specifically,the strong pump intensity generates a relatively large densityof direct excitons, the majority of which relax in energy andbecome indirect in ML-WX2 [Fig. 2(a)] [32,33]. The bandstructure of indirect excitons is similar to the one shown inFig. 4(a) [7], where their electron component generates thesame type of shortwave Coulomb excitations. A directexciton can then interact with these excitations in the sameway as before, leading to renormalization of its self-energy.The observed biexciton spectral line emerges through thisrenormalization. The main difference from the optical side-band, which emerges under conditions of weak photoexci-tation in electron-doped ML-WX2, is that now the pumpingis responsible not only for the exciton population but also forthe generation of shortwaveCoulomb excitations through theelectron components of indirect excitons. These two respon-sibilities explain the quadratic dependence of the observedoptical transition on the pumping intensity. In this view, themeasured phenomena can still be identified as an exciton-exciton interaction but with the notion that the coupling isbetween direct and indirect excitons due to shortwave chargefluctuations.The importance of our theory is therefore in suggesting a

microscopic origin for the exciton-exciton optical transitionin the PL of strongly photoexcited ML-WX2. We claim thatit originates from the intervalley Coulomb excitations andnot from other sources such as van der Waals-type

attraction between two neutral excitons. If the latter wasthe microscopic origin for the observed biexciton spectralline, then it should be insensitive to whether the twoexcitons are direct, indirect, or a mixture of the two. Asa result, one should be able to observe the biexciton spectralline in the PL of both ML-MoX2 and ML-WX2 undercomparable pumping intensities. The fact that this scenariois not observed in experiments precludes such microscopicorigins. Furthermore, previous theoretical calculations haveshown that the binding energy of the biexciton is about20 meVeither in ML-MoX2 or ML-WX2, by using a varietyof four-body computational models with static 2DCoulomb (Keldysh) potentials [63,65,69,78,80–82]. PLexperimental reports, on the other hand, are available onlyfor ML-WX2 in which the biexciton emerges about50–60 meV below the neutral-exciton spectral line[11,12,79]. Our claim regarding the coupling betweendirect and indirect excitons through shortwave chargefluctuations is indeed self-consistent with the fact thatthe biexciton spectral line has not been reported to date inPL experiments of ML-MoX2. The low-temperature pop-ulation of indirect excitons in these compounds is negli-gible because of their higher energy, as shown in Fig. 2(b).Moreover, the coupling between indirect and direct exci-tons has the same wrong conditions that were discussed inthe context of Fig. 9(a). Namely, the renormalization ofdirect excitons does not lead to a distinct spectral line,which can be resolved from that of the neutral exciton.In addition, our explanation is consistent with the fact

that the indirect-direct biexciton spectral line in ML-WX2emerges at the same spectral position of the optical side-band in electron-doped ML-WX2 (about 2Δ below thedirect-exciton spectral line, which is of the order of about50–60 meV in these materials) [10,11,15,16]. One differ-ence between the two cases, however, is that the energyshift of the direct-indirect biexciton is weak when thephotoexcitation intensity increases [79], compared withthe redshift of the optical sideband when increasing theelectron density [10]. The reason for this is that, unlike gate-induced free electrons, bound and neutral excitons do notlead to an appreciable change of Δ. The value of Δ canchange because of the long-wavelength exchange interac-tion, which is effective in the case of free electrons [35]. As aresult, the indirect-direct biexciton shows a relatively weakenergy shiftwhen increasing thephotoexcitation intensity (aslong as excitons remain bound).

V. CONCLUSION AND PERSPECTIVES

The identified pairing of shortwave plasmons andexcitons explains the puzzling observations of the emerginglow-energy optical sideband in the spectra of electron-doped WSe2 and WS2, while being conspicuously absentwith hole doping, MoSe2 andMoS2 [9–16]. Elucidating theexciton-plasmon mechanism and the importance ofdynamical screening paves the way to harness unexplored


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manifestations of spin-valley coupling and the realizationof other dynamical sidebands, such as those arising becauseof the exciton-phonon coupling or the coupling betweenzone-edge phonons and intervalley plasmons.The findings of this work reveal the unique nature of

shortwave (intervalley) plasmons in monolayer transition-metal dichalcogenides. Recalling that freely propagatingshortwave plasmons are independent d.o.f. in the electronicsystem, these quasiparticles can be directly detected inelectron-rich monolayers (i.e., not through their coupling toexcitons). Using THz spectroscopy, for example, oneshould expect to see a resonance in the absorption spectrumwhen the photon energy is about twice the spin splitting inthe conduction band (≲0.1 eV). Such far-infrared photonscan couple to two counterpropagating shortwave plasmons,thereby conserving both energy and momentum. As before,the plasmon energy and amplitude can be tuned by a gatevoltage. In addition, we expect dynamical screening to beimportant in the emerging area of magnetic proximityeffects in monolayer transition-metal dichalcogenides [83].It has been experimentally shown that magnetic substratescan strongly alter the optical response of excitons, inducedby time-reversal symmetry breaking between theK and −Kvalleys [84–86]. Equivalently, magnetic proximity effectsor strong magnetic fields should lead to symmetry breakingbetween counterpropagating shortwave plasmons.Beyond monolayer transition-metal dichalcogenides,

virtual exchange of shortwave plasmons may give rise topairing mechanisms in superconductors whose Fermisurfaces comprise distinct pockets in the Brillouin zone.Similar to our formalism, plasmon-mediated Cooper pair-ing can be modeled when a dynamical screening descrip-tion is employed [87–90], and it was argued to facilitatesuperconductivity in cuprate metal oxides [91] and othertransition-metal systems through plasmon-phonon hybridi-zation [92,93], as well as in semiconductors and electron-hole liquids [94–97].

ACKNOWLEDGMENTS

The authors are indebted to Zefang Wang, Kin Fai Mak,and Jie Shan for providing the reflectivity measurements inFig. 1 and for many useful discussions. We also thankXiaodong Xu for discussing and sharing valuable exper-imental results prior to their publication. This work is mainlysupported by the Department of Energy, Basic EnergySciences (Grant No. DE-SC0014349). The computationalwork at Rochester was also supported by the NationalScience Foundation (Grant No. DMR-1503601) and theDefenseThreat ReductionAgency (Grant No. HDTRA1-13-1-0013). The work at Buffalo was supported by theDepartment of Energy, Basic Energy Sciences (GrantNo. DE-SC0004890). The work at Regensburg andWürzburg was supported by the DFG (Grants No. SCHA1899/2-1 and No. SFB 1170 “ToCoTronics”), and by theENB Graduate School on Topological Insulators.

APPENDIX: COMPUTATIONAL DETAILS

Equation (9) is solved with matrix inversion. Each row(or column) in the matrix is indexed via fki; zng. Theangular dependence of ki − k2 in Vsðki − k2; z − z0Þ isaveraged out, and then all the 2D wave vectors in Eq. (9)are treated as scalars. We use a uniform grid with aspacing of Δk ¼ kmax=Nk, where kmax is chosen such thatεkmax ¼ 1 eV. For the fermion Matsubara energies, zn ¼iπð2nz þ 1ÞkBT, nz runs between −Nz to þNz. The bosonMatsubara energy Ω in Eq. (9) is treated as a parameter inthe matrix. In other words, we solve this equation for eachΩn ¼ i2πnΩkBT with nΩ ∈ ½−NΩ; NΩ�. For numericalcalculations, we have used Nk ¼ 50, Nz ¼ 160,NΩ ¼ 90, and the code was written in FORTRAN andparallelized with OpenMP. For every density, the calcu-lation of Gpðq ¼ 0; k; z;ΩÞ or Gpðq ¼ K0; k; z;ΩÞ takesabout 7 hours when using 44 CPUs (2 GHz) with 1 TBshared DRAM.After the calculation of the pair Green’s function, we use

Eq. (6) to calculate the absorption spectrum. Specifically,we first calculate the function

AðΩÞ ¼ β−1Xz

Im

�ZdkkGpðq ¼ 0; k; z;ΩÞ

�; ðA1Þ

and then use Padé polynomials to extract the analyticalform of AðΩ → Eω þ iδÞ in the upper complex plane fromknowledge of its values at discrete even Matsubara energies[52]. The broadening term is taken as [54]

δ≡ δðEωÞ ¼ δ1 þ δ21þ exp½ðEc − EωÞ=δ3�

; ðA2Þ

where δj (j ¼ f1; 2; 3g) are three parameters with thefollowing meaning: δ1 reflects the inhomogeneous broad-ening due to disorder, adatoms, and charge puddles; δ2 andδ3 reflect the enhanced homogeneous broadening when thetransition energy belongs to the continuum (Eω > Ec). Farbelow the continuum, we get δðEωÞ → δ1. Far above thecontinuum edge, we get δðEωÞ → δ1 þ δ2. See Chapter 13in Ref. [54] for more details. In this work, we have usedδ1 ¼ 10 meV, δ2 ¼ 70 meV, and δ3 ¼ 20 meV.The results shown in Figs. 8 and 9 were obtained using

the renormalization in Eq. (16). The self-energy term,whose expression is provided in Eq. (17), is calculatedby summing over even Matsubara energies and by inte-grating over q0. The integration interval for the latter isjq0 −K0j ≲ γðα=3ÞkF, where γ ¼ 2. In other words, wehave assumed that plasmons are not immediately dampedoutside the region of free-plasmon propagation.

1. Effective mass model

We emphasize that the goal of this work is not toreproduce the exact binding energy of excitons in


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ML-TMDs. Rather, we are interested in the effects ofdynamical screening and coupling between excitons andshortwave plasmons. We use the effective mass approxi-mation since it allows us to simplify the calculation whilekeeping the main dynamical effects intact. In the calcu-lation of the pair functions of direct and indirect excitons,we have assumed similar effective masses for electrons inthe lower and upper valleys of the conduction band. Weignore the (small) change in the values of the effectivemasses at the edges of the lower and upper valleys since thebinding energy of excitons in ML-TMDs is much largerthan the conduction-valley splitting energy. In other words,the matrix we use to solve the pair function includeselectronic states whose energies are hundreds of meVabovethe edge of the band. For such states, the exact value of theeffective mass in the bottom of the band is meaningless. Wetherefore choose the same value for both lower and uppervalleys, which represents the effective mass of the“extended” conduction band.

2. Emission and absorption of plasmons

The self-energy of the exciton due to the interaction withshortwave plasmons, given in Eq. (17), includes bothemission and absorption terms. Formally, these terms arederived from the identity [36,51]

β−1XΩ0

DðΩ − Ω0;q0ÞGpðq0;k;Ω0Þ

¼ 12πi

ICdΩ0

1

expðβΩ0Þ − 1DðΩ − Ω0;q0ÞGpðq0;k;Ω0Þ;

ðA3Þ

where the sum runs over even Matsubara energies, and theintegration contour C in the upper plane encircles the polesin the positive sense. Using Cauchy’s residue theorem, theBose-Einstein distribution factors for absorption and emis-sion, g(� EsðqÞ), result from the poles in the plasmonpropagator, Ω0 ¼ Ω� EsðqÞ [see Eq. (19)], whereexpðβΩÞ ¼ expð2πinÞ ¼ 1. Furthermore, by approximat-ing the pair function as a simple pole expression,Gp ∝ ðΩ0 − E0Þ−1, where E0 is the exciton energy, wecan analytically derive the Stokes and anti-Stokes side-bands. The former stems from the emission term∝ g( − EsðqÞ)=½Ωþ EsðqÞ − E0�, where Ω → ℏωþ iδand gð−EsðqÞÞ → 1 at low temperatures. Similarly, theabsorption term stems from ∝ g(EsðqÞ)=½Ω − EsðqÞ − E0�.Section III C describes some of the difficulties that may

arise when one tries to observe the anti-Stokes sideband inthe absorption spectrum of high-temperature ML-MoSe2.Other difficulties that impede its detection may be causedby plasmon damping due to coupling with phonons and bythe relatively small population of shortwave plasmonsdue to the increased value of EsðqÞ ∼ Δ at elevatedelectron densities. This behavior can be understood by

inspecting the Bose-Einstein distribution factors that areassociated with the distinct anti-Stokes and Stokes side-bands, jgðΔMoÞ=gð−ΔWÞj, which emerge in ML-MoSe2and ML-WX2, respectively. For example, assuming thatthese optical sidebands persist at 200 K and assigning Δ ∼40 meV for both materials at some comparable elevateddensities [35], the amplitude of the Stokes sidebandbecomes about 20 times stronger than that of the anti-Stokes sideband.

3. Type B excitons

Type B excitons correspond to optical transitions fromthe lower spin-split valence band. We expect the couplingbetween type B excitons in ML-TMDs and shortwaveplasmons to be weak because of the ultrashort lifetime oftype B excitons: They experience ultrafast intervalley spin-conserving energy relaxation of holes to the lower spin-split valleys. Furthermore, at elevated charge densities inwhich the plasma excitations become relevant, the band-gap renormalization is strong enough so that type B opticaltransitions are within the continuum. In other words, whenthe densities of resident holes or electrons are larger thanabout 1012 cm−2, type B excitons can experience ultrafastdissociation to free electrons and holes. The coupling toshortwave plasmons is less effective in this case.

4. Electron-hole exchange

The relatively large binding energy of excitons in ML-TMDs can lead to evident energy splitting between brightand dark excitons [98–100]. The theory presented in ourwork, however, involves only bright excitons (singlet spinconfiguration of the electron and hole). Therefore, bothdirect and indirect bright excitons should experiencesimilar effects due to the electron-hole exchange interac-tion. As a result, the energy splitting between direct andindirect excitons remains equivalent to the spin splitting ofthe conduction band. Furthermore, the emergence of thedynamical sideband is relevant at elevated charge densitiesin which the wave functions of excitons are blown upbecause of screening (smaller binding energy). The elec-tron-hole exchange is a weaker effect at such conditions.

5. More simulation results

It is important to realize that the effects of dynamicalscreening and interaction with plasmons are largely inde-pendent of the exact parameters that we choose for theeffective mass and dielectric constant. For example, in themain text, we have assumed that the monolayer isembedded in hexagonal boron nitride, for which there isdiscrepancy in the reported values of the dielectric constant.Recent experiments use values of ϵ ¼ 2.7 [15,16], and wehave used this parameter in Figs. 6, 8, and 9. However,previous studies of hexagonal boron nitride reported valuesthat are nearly twice as large, ϵ ¼ 5.06 [101]. We have used


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this value in Fig. 10, illustrating that such changes do notaffect the findings of this work. The only change, as seenfrom the values of the x axis, is a smaller binding energydue to the larger dielectric screening. Yet, the roles ofdynamic screening and the emergence of the dynamicalsidebands are the same. Here, we have also used a slightlylarger cutoff, jq0 −K0j≲ γðα=3ÞkF, where γ ¼ 3 instead ofγ ¼ 2, which we used in Figs. 8 and 9. We use this value to

enable stronger pairing between excitons and shortwaveplasmons, as can be seen from the shoulder and enhancedblueshift of the main peak at elevated densities. At the sametime, the redshift of the dynamical sideband is also some-what enhanced, and it shows slight decay at elevateddensities. Finally, Fig. 11 shows the absorption spectrumwhen using γ ¼ 2 and the effective isotropic model for thevalues of the thickness and dielectric constant of the ML[102]: d ¼ 0.6 Å and ϵML ¼ 16.3. The behavior of theoptical sideband is essentially the same, providing furtherevidence that the exciton-plasmon is a dynamical effectwith no qualitative dependence on the exact parametervalues that one uses in the static dielectric function.

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−0.3 −0.25 −0.2 −0.15 −0.10

0.5

1

1.5

2A

bsor

ptio

n (a

.u.)

1 108 cm−2

5 1011

1 1012

2 1012

4 1012

6 1012

−0.26 −0.24 −0.22 −0.20

0.05

0.1

e

FIG. 10. Renormalized absorption spectrum in electron-dopedML-WX2 embedded in layers with a dielectric constant,ϵb ¼ ϵt ¼ 5.06. This figure shows the same qualitative behavioras in Fig. 8 in which we have used ϵb ¼ ϵt ¼ 2.7.

−0.35 −0.3 −0.25 −0.2e

−0.15 −0.1 −0.050

0.5

1

1.5

2

Abs

orpt

ion

(a.u

.)

1 108 cm−2

5 1011

1 1012

2 1012

4 1012

6 1012

−0.3 −0.28 −0.26 −0.240

0.05

0.1

FIG. 11. Renormalized absorption spectrum in electron-dopedML-WX2 embedded in layers with a dielectric constant,ϵb ¼ ϵt ¼ 3.4. This figure shows the same qualitative behavioras in Figs. 8 and 10. Here, the effective thickness and dielectricconstant of the monolayer are d ¼ 0.6 Å and ϵML ¼ 16.3,respectively.


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