Many-body theory of radiative lifetimes of exciton-trion superposition states in doped two-dimensional materials
Farhan Rana, Okan Koksal, Minwoo Jung, Gennady Shvets, Christina Manolatou
MMany-body theory of radiative lifetimes of exciton-trion superposition states in dopedtwo-dimensional materials
Farhan Rana, Okan Koksal, Minwoo Jung, Gennady Shvets, and Christina Manolatou School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853 School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853
Optical absorption and emission spectra of doped two-dimensional (2D) materials exhibit sharppeaks that are often mistakenly identified with pure excitons and pure trions (or charged excitons),but both peaks have been recently attributed to superpositions of 2-body exciton and 4-body trionstates and correspond to the approximate energy eigenstates in doped 2D materials. In this paper, wepresent the radiative lifetimes of these exciton-trion superposition energy eigenstates using a many-body formalism that is appropriate given the many-body nature of the strongly coupled excitonand trion states in doped 2D materials. Whereas the exciton component of these superpositioneigenstates are optically coupled to the material ground state, and can emit a photon and decayinto the material ground state provided the momentum of the eigenstate is within the light cone,the trion component is optically coupled only to the excited states of the material and can emita photon even when the momentum of the eigenstate is outside the light cone. In an electron-doped 2D material, when a 4-body trion state with momentum outside the light cone recombinesradiatively, and a photon is emitted with a momentum inside the light cone, the excess momentumis taken by an electron-hole pair left behind in the conduction band. The radiative lifetimes of theexciton-trion superposition states, with momenta inside the light cone, are found to be in the fewhundred femtoseconds to a few picoseconds range and are strong functions of the doping density.The radiative lifetimes of exciton-trion superposition states, with momenta outside the light cone,are in the few hundred picoseconds to a few nanoseconds range and are again strongly dependenton the doping density. The doping density dependence of the radiative lifetimes of the two peaksin the optical emission spectra follows the doping density dependence of the spectral weights of thesame two peaks observed in the optical absorption spectra as both have their origins in the Coulombcoupling between the excitons and trions in doped 2D materials.
Optical absorption and emission spectra of doped two-dimensional (2D) materials in general, and of transi-tion metal dichalcogenides (TMDs) in particular, exhibitsharp and distinct peaks that are often attributed toneutral and charged excitons (or trions) . Althoughoptical signatures of excitons and trions in doped semi-conductors have been observed for a long time , theirnature, especially of trions, in doped materials had re-mained somewhat of a mystery. For one, it was difficultto understand how a photon, being a boson, could getabsorbed and create a trion, if a trion is taken to befermionic bound state of three particles. Second, it wasnot clear what happened to one of the charged particlesleft behind when a trion emitted a photon. Pauli’s ex-clusion required the left behind charged particle to bedeposited outside the Fermi sea, but the energy and mo-mentum conservation requirements following from Pauli’sexclusion were never observed in the measured photolu-minescence spectra. Third, the variation of the energyseparation of the two peaks observed in the optical ab-sorption spectra, as well as the spectral weight transferbetween these two peaks with doping, did not seem tofollow from the assumption of excitons and trions beingindependent excitations.Several recent works have contributed to resolving thismystery and clarifying the nature of excitons and trionsin doped semiconductors . Recently, the authorshave presented a theoretical model based on two cou-pled Schr¨odinger equations to describe excitons and tri- ons in electron-doped 2D materials . One is a 2-bodySchr¨odinger equation for a conduction band (CB) elec-tron interacting with a valence band (VB) hole, and theother is a 4-body Schr¨odinger equation of two CB elec-trons, one VB hole, and one CB hole interacting witheach other. The CB hole is created when a CB electronis scattered out of the Fermi sea by an exciton. Theeigenstates of the 2-body equation were identified withexcitons and the eigenstates of the 4-body equation wereidentified with trions. A bound trion state is therefore a4-body bosonic state, and not a 3-body fermionic state.The two Schr¨odinger equations are coupled as a resultof Coulomb interactions between the excitons and thetrions in doped materials. The model shows that pureexciton and trion states are not eigenstates of the Hamil-tonian in the presence of doping. However, good approx-imate eigenstates can be constructed from superpositionsof exciton and trion states. This superposition includesboth bound trion states as well as unbound trion states.The latter are exciton-electron scattering states. Thesesuperposition states, first proposed by Suris , resemblethe exciton-polaron variational states proposed by Sidleret al. . The optical conductivity obtained from themodel proposed by the authors explains all the promi-nent features experimentally seen in the optical absorp-tion spectra of doped 2D materials including the obser-vation of two prominent absorption peaks and the vari-ation of their energy splittings and spectral shapes andstrengths with the doping density . Furthermore, the a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec FIG. 1: The nature of couplings involving 2-body exciton and4-body trion states are depicted for an electron-doped mate-rial. The 4-body trion states are coupled to the 2-body ex-citon states via electron-electron and electron-hole Coulombinteractions. Only the exciton states are coupled to the ma-terial ground state via optical coupling. The trion states areoptically coupled to excited states of the material consisting ofa CB electron-hole pair. The trion states include both boundand unbound trion states. peaks observed in the optical absorption spectra of doped2D materials do not correspond to pure exciton or puretrion states. Each peak corresponds to a superpositionof exciton and trion states.While previous papers, including the one by the au-thors, have addressed the problem of light absorption byexcitons and trions , questions related to lightemission and radiative lifetimes of excitons and trions indoped materials remain unanswered. The model devel-oped by the authors , rather interestingly, also showedthat the 4-body trion states have no optical matrix el-ements with the material ground state. The groundstate of, say an electron-doped material, is defined as thestate consisting of a completely full valence band (no VBholes), and a completely full Fermi sea in the conductionband (no CB holes inside and no CB electrons outside theFermi sea). Therefore, the contribution to the materialoptical conductivity from the 4-body trion states resultsalmost entirely from their Coulomb coupling to the 2-body exciton states . The exciton and trion states andthe related couplings are depicted in Fig.1. However, thetrion states, including both bound and unbound trionstates, are optically coupled to the excited states of thematerial consisting of a CB electron-hole pair. In otherwords, a trion state can decay by emitting a photon andleaving behind a CB electron-hole pair. The radiativerate of this process is significant after one has summedover all possible CB electron-hole pairs that can result from the radiative decay of a 4-body trion state.The experimentally relevant radiative lifetimes are notthose of pure exciton and trion states, but of the approx-imate energy eigenstates which, as discussed above, aresuperpositions of exciton and trion states. The goal ofthis paper is to clarify the processes contributing to pho-ton emission from these energy eigenstates in 2D mate-rials and calculate the corresponding radiative lifetimes.Our main results are as follows. The radiative lifetimesof the exciton-trion energy eigenstates, with momenta in-side the light cone, are found to be in the few hundredfemtoseconds to a few picoseconds range and are stronglydependent on the doping density. Within the light cone,the exciton component of these eigenstates provides thedominant contribution to the radiative rates. The radia-tive lifetimes of the exciton-trion superposition states,with momenta outside the light cone, are in the few hun-dred picoseconds to a few nanoseconds range and areagain strong functions of the doping density. Outside thelight cone, only the trion component of these eigenstatescontributes to the radiative rates. The doping densitydependence of the radiative lifetimes of the two peaksin the optical emission spectra follows the doping den-sity dependence of the spectral weights of the same twopeaks observed in the optical absorption spectra as bothhave their origins in the Coulomb coupling between theexcitons and trions in doped 2D materials. I. THEORETICAL MODEL
In this Section we set up the Hamiltonian and derivethe main equations. Although the focus is on electron-doped 2D TMD materials, the arguments are kept gen-eral enough to be applicable to any 2D material.
A. The Hamiltonian
We consider a 2D TMD monolayer located in the z = 0plane inside a uniform medium of dielectric constant (cid:15) .The TMD layer interacts with both TE (electric field inthe z = 0 plane) and TM (magnetic field in the z = 0plane) polarized light modes. The Hamiltonian describ-ing electrons and holes in the TMD layer (near the K and K (cid:48) points in the Brillouin zone) interacting with eachother and with the optical mode in the rotating waveapproximation is , H = (cid:88) (cid:126)k,s E c,s ( (cid:126)k ) c † s ( (cid:126)k ) c s ( (cid:126)k ) + (cid:88) (cid:126)k,s E v,s ( (cid:126)k ) b † s ( (cid:126)k ) b s ( (cid:126)k )+ 1 A (cid:88) (cid:126)q,(cid:126)k,(cid:126)k (cid:48) ,s,s (cid:48) U ( q ) c † s ( (cid:126)k + (cid:126)q ) b † s (cid:48) ( (cid:126)k (cid:48) − (cid:126)q ) b s (cid:48) ( (cid:126)k (cid:48) ) c s ( (cid:126)k )+ 12 A (cid:88) (cid:126)q,(cid:126)k,(cid:126)k (cid:48) ,s,s (cid:48) V ( q ) c † s ( (cid:126)k + (cid:126)q ) c † s (cid:48) ( (cid:126)k (cid:48) − (cid:126)q ) c s (cid:48) ( (cid:126)k (cid:48) ) c s ( (cid:126)k ) z x yx q TMTE
Photon momentum 2D TMD monolayer(z=0 plane)Permittivity = FIG. 2: A 2D TMD monolayer in the z = 0 plane is shown.The two light polarizations are also illustrated. + (cid:88) (cid:126)/q,j ¯ hω ( /q ) a † j ( (cid:126)/q ) a j ( (cid:126)/q )+ 1 √ AL (cid:88) q z , (cid:126)Q,(cid:126)k,j,s (cid:16) g j,s ( (cid:126)/q ) c † s ( (cid:126)k + (cid:126)Q ) b s ( (cid:126)k ) a j ( (cid:126)/q ) + h.c (cid:17) (1)Here, E c,s ( (cid:126)k ) and E v,s ( (cid:126)k ) are the conduction and valenceband energies. s, s (cid:48) represent the spin/valley degrees offreedom in the 2D material, and we assume for simplicitythat the electron and hole effective masses are indepen-dent of the spin/valley. U ( (cid:126)q ) represents Coulomb inter-action between electrons in the conduction and valencebands and V ( (cid:126)q ) represents Coulomb interaction amongthe electrons in the conduction bands. A is the monolayerarea and AL is the volume assumed for field quantization.¯ hω ( (cid:126)/q ) is the energy of a photon with momentum (cid:126)/q , and g j,s ( (cid:126)/q ) is the electron-photon coupling constant for lightwith photon polarization j = TE , TM (see Fig.2). Mostmomentum vectors in the Hamiltonian above are in 2D.Those associated with light are in 3D, carry a slash inthe notation for clarity, and (cid:126)/q = (cid:126)Q + q z ˆ z , where (cid:126)Q is themomentum component in the z = 0 plane. Other thanfor phase factors that are not relevant to the discussionin this paper, g j,s ( (cid:126)/q ) for electron states near the bandedges in 2D TMDs can be given by , g j,s ( (cid:126)/q ) = ev (cid:115) ¯ h (cid:15)ω ( (cid:126)/q ) × (cid:26) q z //q for TM1 for TE (2)where, v is the interband velocity matrix element . B. Exciton States, Trion States, and EnergyEigenstates
As shown by Rana et al. , approximate eigenstates ofthe Hamiltonian in (1) can be written as a superpositionof 2-body exciton and 4-body trion states, | ψ n,s ( (cid:126)Q ) (cid:105) = α n √ A (cid:88) k φ ex ∗ n, (cid:126)Q ( (cid:126)k ) N ex × c † s ( (cid:126)k + λ e (cid:126)Q ) b s ( (cid:126)k − λ h (cid:126)Q ) | GS (cid:105) + (cid:88) m,s (cid:48) β m,s (cid:48) √ A (cid:126)k ,(cid:126)k (cid:54) = (cid:126)p (cid:88) (cid:126)k ,(cid:126)k ,(cid:126)p φ tr ∗ m, (cid:126)Q ( (cid:126)k , s ; (cid:126)k , s (cid:48) ; (cid:126)p, s (cid:48) ) N tr × c † s ( (cid:126)k ) c † s (cid:48) ( (cid:126)k ) b s ( (cid:126)k + (cid:126)k − ( (cid:126)Q + (cid:126)p )) c s (cid:48) ( (cid:126)p ) | GS (cid:105) (3)Here, | GS (cid:105) is the ground state of the electron doped ma-terial. The normalization factors are, N ex = (cid:113) − f c,s ( (cid:126)k + λ e (cid:126)Q ) N tr = (cid:114) (1 + δ s,s (cid:48) ) f c,s (cid:48) ( (cid:126)p ) (cid:104) − f c,s ( (cid:126)k ) (cid:105) (cid:104) − f c,s (cid:48) ( (cid:126)k ) (cid:105) (4)The above energy eigenstate has (in-plane) momentum (cid:126)Q . φ exn, (cid:126)Q ( (cid:126)k + λ h (cid:126)Q ) and φ trm, (cid:126)Q ( (cid:126)k , s ; (cid:126)k , s ; (cid:126)p, s ) are eigen-states of the 2-body exciton and 4-body trion eigenequa-tions, respectively . The corresponding eigenenergiesare, E exn ( (cid:126)Q, s ) and E trm ( (cid:126)Q, s , s ), respectively. λ h =1 − λ e = m h /m ex ( m ex = m e + m h ), where m e ( m h )is the electron (hole) effective mass. m tr = 2 m e + m h , ξ = m e /m tr , and η = m h /m tr . The underlined vector (cid:126)k stands for (cid:126)k + ξ ( (cid:126)Q + (cid:126)p ). The summation over the index m implies summation over all bound and unbound trionstates. Expressions for the coefficients α n and β m,s (cid:48) aregiven later in this paper. The states given above are goodapproximations to the actual eigenstates of the Hamilto-nian in (1) within the purview of single electron-hole pairexcitations and provided one ignores multiple electron-hole pair excitations . In most cases of practical interestinvolving 2D TMDs, only the lowest energy exciton stateneeds to be considered. However, bound trion states aswell as the continuum of unbound trion states need to beincluded since the energy differences involved therein aresmall . This makes the direct calculation of radiativerates using Fermi’s Golden Rule awkward.The optical interaction term in the Hamiltonian in(1) couples the material ground state to only the ex-citon component, and not to the trion components, inthe exciton-trion supersposition energy eigenstates (seeFig.1) . However, excited states of the material contain-ing an electron-hole pair in the CB are optically coupledto the trion components. Given this, two different kindsof radiative transitions are possible and are depicted inFig.3. Fig.3(a) shows photon emission resulting in a de-cay of the energy eigenstate into the material groundstate. The transition rate is determined by | α n | , theweight of the exciton component of the energy eigenstatein (3). This transition is possible only if the momen-tum (cid:126)Q of the energy eigenstate is within the light cone.Fig.3(b) shows photon emission resulting in a decay ofthe energy eigenstate into an excited state of the materialthat has a CB electron-hole pair. The CB electron-holepair is left behind after photon emission from the trioncomponents of the energy eigenstate. Unlike the processin Fig.3(a), the process in Fig.3(b) is possible even if the k Photon
K K’
Exciton-Trion Energy EigenstateHole
E k (a) ElectronHoleElectron k Photon
K K’
Hole
E k (b) ElectronHoleElectronExciton-Trion Energy Eigenstate
FIG. 3: Two different kinds of photon emission processes aredepicted. (a) Photon emission resulting in a decay of theenergy eigenstate into the material ground state. The tran-sition rate is determined by | α n | , the weight of the excitoncomponent of the energy eigenstate in (3). This transitionis possible only if the momentum (cid:126)Q of the energy eigenstateis within the light cone. (b) Photon emission resulting in adecay of the energy eigenstate into an excited state of thematerial that has a CB electron-hole pair. The CB electron-hole pair is left behind after photon emission from the trioncomponents of the energy eigenstate. The transition rate isdetermined by | β m,s (cid:48) | in (3). This transition is possible evenif the momentum (cid:126)Q of the energy eigenstate is outside thelight cone. momentum (cid:126)Q of the energy eigenstate is outside the lightcone. If the emitted photon has an in-plane momentum (cid:126)Q (cid:48) within the light cone, the difference (cid:126)Q − (cid:126)Q (cid:48) is taken bythe electron-hole pair left behind in the CB. The radiativerate for this process is determined by the magnitude ofthe coefficients β m,s (cid:48) of the trion states in the expressionfor the energy eigenstate given in (3).In the Sections that follow, we will calculate separatelythe radiative rates for the two processes in Fig.3. II. RATE FOR RADIATIVE DECAY INTO THEMATERIAL GROUND STATE
We first calculate the rate for the radiative decay ofthe energy eigenstate into the material ground state.This rate is expected to be proportional to the weight ofthe exciton component of the energy eigenstate, and theweight of the exciton component is conveniently given bythe spectral density function which is proportional to theimaginary part of the exciton Green’s function. Thus, weseek an expression for the radiative rate in terms of theexciton Green’s function.
A. Heisenberg Equations
We start from the Heisenberg equation for the photonoperator, (cid:20) ¯ hω ( (cid:126)/q ) + i ¯ h ddt (cid:21) a † j ( (cid:126)/q, t ) = − √ AL (cid:88) (cid:126)k,s g j,s ( (cid:126)/q ) P (cid:126)Q ( (cid:126)k, s ; t )(5)The polarization operator P (cid:126)Q ( (cid:126)k, s ; t ) equals c † s ( (cid:126)k + (cid:126)Q, t ) b s ( (cid:126)k, t ). The Heisenberg equation for the polariza-tion operator is , (cid:20) E c,s ( (cid:126)k + (cid:126)Q ) − E v,s ( (cid:126)k ) + iγ ex + i ¯ h ddt (cid:21) P (cid:126)Q ( (cid:126)k, s ; t ) = − √ AL (cid:88) q z ,j g ∗ j,s ( (cid:126)/q ) a † j ( (cid:126)/q ; t ) (cid:104) − f c,s ( (cid:126)k + (cid:126)Q ) (cid:105) + F (cid:126)Q ( (cid:126)k, s ; t )+ 1 A (cid:88) (cid:126)q U ( (cid:126)q ) P (cid:126)Q ( (cid:126)k + (cid:126)q, s ; t ) (cid:104) − f c,s ( (cid:126)k + (cid:126)Q ) (cid:105) − A (cid:88) (cid:126)q,(cid:126)p,s (cid:48) U ( (cid:126)q ) × T c(cid:126)Q ( (cid:126)k + ( ξ + η ) (cid:126)Q − ξ(cid:126)p, s ; ( ξ + η ) (cid:126)p − ξ (cid:126)Q − (cid:126)q, s (cid:48) ; (cid:126)p, s (cid:48) ; t )+ 1 A (cid:88) (cid:126)q,(cid:126)p,s (cid:48) V ( (cid:126)q ) × T c(cid:126)Q ( (cid:126)k + ( ξ + η ) (cid:126)Q − ξ(cid:126)p + (cid:126)q, s ; ( ξ + η ) (cid:126)p − ξ (cid:126)Q − (cid:126)q, s (cid:48) ; (cid:126)p, s (cid:48) ; t )(6)Here, f c,s ( (cid:126)k ) is the electron occupation probability inthe conduction band (valence band is assumed to becompletely full), γ ex is a phenomenological decoher-ence rate for the polarization that includes dephasingdue to all processes other than exciton-electron scatter-ing. F (cid:126)Q ( (cid:126)k, s ; t ) is a zero-mean delta-correlated quantumLangevin noise source that is introduced by the same pro-cesses that contribute to the decoherence γ ex . The en-ergies E c,s ( (cid:126)k ) include renormalizations due to exchangeat the Hartree-Fock level ( − (1 /A ) (cid:80) (cid:126)q V ( (cid:126)q ) f c,s ( (cid:126)k − (cid:126)q )).Taking the mean value of the operators in (6), ignor-ing the first term and the last two terms on the righthand side (RHS), and Fourier transforming the remain-ing terms results in a 2-body Schr¨odinger equation for theexcitons . The last two terms in (6) on the RHScontain four-body operators T c(cid:126)Q . We define the operator T (cid:126)Q ( (cid:126)k , s ; (cid:126)k , s ; (cid:126)p, s ; t ) as follows, c † s ( (cid:126)k ; t ) c † s ( (cid:126)k ; t ) b s ( (cid:126)k + (cid:126)k − ( (cid:126)Q + (cid:126)p ); t ) c s ( (cid:126)p ; t ) (7)As before, the underlined vector (cid:126)k stands for (cid:126)k + ξ ( (cid:126)Q + (cid:126)p ).The average of the operator T (cid:126)Q describes correlationsarising from Coulomb interactions among four particles:two CB electrons, a VB hole, and a CB hole. (cid:126)Q is thetotal momentum of this 4-body state. We also define theconnected operator T c(cid:126)Q as follows , T (cid:126)Q ( (cid:126)k , s ; (cid:126)k , s ; (cid:126)p, s ; t ) = T c(cid:126)Q ( (cid:126)k , s ; (cid:126)k , s ; (cid:126)p, s ; t ) − f c,s ( (cid:126)p ) P (cid:126)Q ( (cid:126)k − (cid:126)Q, s ; t ) δ (cid:126)k ,(cid:126)p + f c,s ( (cid:126)p ) P (cid:126)Q ( (cid:126)k − (cid:126)Q, s ; t ) δ s ,s δ (cid:126)k ,(cid:126)p (8)The Heisenberg equation for the operator T c(cid:126)Q ( (cid:126)k , s ; (cid:126)k , s ; (cid:126)p, s ) is found to be , (cid:104) E c,s ( (cid:126)k ) + E c,s ( (cid:126)k ) − E v,s ( (cid:126)k + (cid:126)k − ( (cid:126)Q + (cid:126)p )) − E c,s ( (cid:126)p ) + iγ tr + i ¯ h ddt (cid:21) T c(cid:126)Q ( (cid:126)k , s ; (cid:126)k , s ; (cid:126)p, s ; t ) = D (cid:126)Q ( (cid:126)k , s ; (cid:126)k , s ; (cid:126)p, s ; t ) − A (cid:88) (cid:126)q V ( (cid:126)q ) T c(cid:126)Q ( (cid:126)k + (cid:126)q, s ; (cid:126)k − (cid:126)q, s ; (cid:126)p, s ; t ) × (cid:104) − f c,s ( (cid:126)k ) − f c,s ( (cid:126)k ) (cid:105) + 1 A (cid:88) (cid:126)q U ( (cid:126)q ) T c(cid:126)Q ( (cid:126)k + (cid:126)q, s ; (cid:126)k , s ; (cid:126)p, s ; t ) (cid:104) − f c,s ( (cid:126)k ) (cid:105) + 1 A (cid:88) (cid:126)q U ( (cid:126)q ) T c(cid:126)Q ( (cid:126)k , s ; (cid:126)k − (cid:126)q, s ; (cid:126)p, s ; t ) (cid:104) − f c,s ( (cid:126)k ) (cid:105) + 1 A (cid:88) (cid:126)q V ( (cid:126)q ) T c(cid:126)Q ( (cid:126)k + ( ξ + η ) (cid:126)q, s ; (cid:126)k − ξ(cid:126)q, s ; (cid:126)p + (cid:126)q, s ; t ) × (cid:104) f c,s ( (cid:126)p ) − f c,s ( (cid:126)k ) (cid:105) + 1 A (cid:88) (cid:126)q V ( (cid:126)q ) T c(cid:126)Q ( (cid:126)k − ξ(cid:126)q, s ; (cid:126)k + ( ξ + η ) (cid:126)q, s ; (cid:126)p + (cid:126)q, s ; t ) × (cid:104) f c,s ( (cid:126)p ) − f c,s ( (cid:126)k ) (cid:105) − A (cid:88) (cid:126)q U ( (cid:126)q ) T c(cid:126)Q ( (cid:126)k − ξ(cid:126)q, s ; (cid:126)k − ξ(cid:126)q, s ; (cid:126)p + (cid:126)q, s ; t ) f c,s ( (cid:126)p )+ f c,s ( (cid:126)p ) A (cid:88) (cid:126)q V ( (cid:126)q ) (cid:104) − f c,s ( (cid:126)k ) − f c,s ( (cid:126)k ) (cid:105) × (cid:104) P (cid:126)Q ( (cid:126)k − (cid:126)Q + (cid:126)q, s ; t ) δ (cid:126)k − (cid:126)q,(cid:126)p − P (cid:126)Q ( (cid:126)k − (cid:126)Q − (cid:126)q, s ; t ) δ (cid:126)k + (cid:126)q,(cid:126)p δ s ,s (cid:105) − f c,s ( (cid:126)p ) A (cid:88) (cid:126)q U ( (cid:126)q ) (cid:110) P (cid:126)Q ( (cid:126)k − (cid:126)Q, s ; t ) δ (cid:126)k − (cid:126)q,(cid:126)p (cid:104) − f c,s ( (cid:126)k ) (cid:105) − P (cid:126)Q ( (cid:126)k − (cid:126)Q, s ; t ) δ (cid:126)k + (cid:126)q,(cid:126)p δ s ,s (cid:104) − f c,s ( (cid:126)k ) (cid:105)(cid:111) (9)In deriving the above equation, all 6-body operator prod-ucts were reduced to 4-body operator products using therandom phase approximation . By ignoring higherorder correlations we are ignoring the generation of mul-tiple particle-hole pairs in the CB. γ tr is a phenomeno-logical decoherence rate and D (cid:126)Q is the corresponding zero-mean delta-correlated Langevin noise source. If (cid:126)r e , (cid:126)r e , (cid:126)r h , are (cid:126)r h the coordinates of the two electrons,the VB hole, and the CB hole, respectively, then (cid:126)k , (cid:126)k , (cid:126)Q , and (cid:126)p are the momenta associated with the coordi-nates (cid:126)r e − (cid:126)r h , (cid:126)r e − (cid:126)r h , (cid:126)R = ξ ( (cid:126)r e + (cid:126)r e ) + η(cid:126)r h , and (cid:126)R − (cid:126)r h , respectively. Here, (cid:126)R is the center of mass coor-dinate of the two electrons and the VB hole. Taking themean value of the operators in (9), ignoring the last twoterms on the RHS in (9) that involve P (cid:126)Q , and Fouriertransforming the remaining terms will result in a 4-bodySchr¨odinger equation for the trions . Each term on theRHS in the above equation (except the first and the lasttwo) describes Coulomb interaction between two of thefour particles. The last two terms involving P (cid:126)Q describethe generation of four-body correlation from two-bodycorrelations, or the creation of an CB electron-hole pairby an exciton.We should mention here that a classical equation sim-ilar to (9) was obtained by Esser et al. . However, thereare significant differences between (9) and the equationobtained by Esser et al.. In the work of Esser et al.,the connected nature of T c(cid:126)Q was overlooked, the termscontaining interactions with the CB hole were ignored,the phase-space restricting factors were ignored too, and,most importantly, the terms containing the polarization P (cid:126)Q were also missed. Ignoring the coupling to P (cid:126)Q in(9) is equivalent to ignoring exciton-trion coupling viaCoulomb interactions. This coupling is responsible formaking exciton-trion superposition states approximateeigenstates of the interacting system consisting of exci-tons and electrons in a doped material. B. Solution of Heisenberg Equations
The polarization operator P (cid:126)Q ( (cid:126)k, s ; t ) can be decom-posed using the complete set of exciton eigenfunctions φ exn, (cid:126)Q ( (cid:126)k + λ h (cid:126)Q ) as follows, P (cid:126)Q ( (cid:126)k, s ; t ) = (cid:88) n P n, (cid:126)Q ( s ; t ) (cid:113) − f c,s ( (cid:126)k + (cid:126)Q ) φ exn, (cid:126)Q ( (cid:126)k + λ h (cid:126)Q )(10)We assume that at time t , P n, (cid:126)Q ( s ; t ) has a non-zero meanvalue for some particular values of n and s . (cid:104) P n, (cid:126)Q ( s ; t ) (cid:105) can be non-zero if the quantum state is a superposition ofthe material ground state | GS (cid:105) and one of the eigenstatesdescribed in Section I B. Following Milonni , the strat-egy going forward will then be as follows. The Heisen-berg equations will be solved to find how the mean value (cid:104) P n, (cid:126)Q ( s ; t ) (cid:105) decays with time due to radiative transitions,and the lifetime associated with this decay would give theradiative rate. Since we are exclusively interested in ra-diative transitions in this paper, several approximationswill be made in order to keep the focus on the relevantphysics and irrelevant terms will be ignored to keep theanalysis simple.(5) can be be solved by direct integration to give, a † j ( (cid:126)/q, t ) = a † j ( (cid:126)/q, t = 0) e iω ( (cid:126)/q ) t + i √ AL (cid:88) (cid:126)k,s g j,s ( (cid:126)/q )¯ h (cid:90) t e iω ( (cid:126)/q )( t − t (cid:48) ) P (cid:126)Q ( (cid:126)k, s ; t (cid:48) ) dt (cid:48) (11)Next, we find the time dependence of the operator P n, (cid:126)Q ( s ; t ). Using (10) in (6), ignoring the Langevin noisesources on the RHS in (6) and (9) (because these noisesources will not have any effect on the end results soughtin this paper), and using the techniques discussed in aprevious paper by the authors for solving the coupledsystem of equations in (6) and (9), the operator P n, (cid:126)Q ( s ; t )is found to be, P n, (cid:126)Q ( s ; t ) = (cid:90) dω π − i ¯ he iωt P n, (cid:126)Q ( s ; t = 0)¯ hω − E exn ( (cid:126)Q, s ) − iγ ex − Σ ex ∗ n,s ( (cid:126)Q, ω )+ 1 √ AL (cid:88) q z ,j g ∗ j,s ( (cid:126)/q ) (cid:90) d (cid:126)k (2 π ) (cid:113) − f c,s ( (cid:126)k + (cid:126)Q ) φ ex ∗ n, (cid:126)Q ( (cid:126)k + λ h (cid:126)Q ) × (cid:90) dω π (cid:90) t e iω ( t − t (cid:48) ) a † j ( (cid:126)/q ; t (cid:48) )¯ hω − E exn ( (cid:126)Q, s ) − iγ ex − Σ ex ∗ n,s ( (cid:126)Q, ω ) (12)Here, Σ exn,s ( (cid:126)Q, ω ) is the self-energy of the excitons arisingfrom their Coulomb coupling to the trions ,Σ exn,s ( (cid:126)Q, ω ) = (cid:88) m,s (cid:48) (1 + δ s,s (cid:48) ) (cid:12)(cid:12)(cid:12) M m,n ( (cid:126)Q, s, s (cid:48) ) (cid:12)(cid:12)(cid:12) ¯ hω − E trm ( (cid:126)Q, s, s (cid:48) ) + iγ tr (13)The summation over m above implies a summation overall bound and unbound trion states consistent with thevalues of s and s (cid:48) . The expression for the Coulombmatrix elements M m,n ( (cid:126)Q, s, s (cid:48) ) coupling the exciton andtrion states can be found in a previous paper by Ranaet al. . The exciton self-energy thus includes contribu-tion of trion states to the polarization via exciton-trionCoulomb coupling. (12) gives the natural frequencies as-sociated with the material polarization response, givenby the poles of the expression in the denominator, andthese frequencies also correspond to the energy eigen-states of the Hamiltonian . It follows that on fast timescales (of the order of the inverse of the relevant opticalfrequencies), P n, (cid:126)Q ( s ; t ) can be written as, P n, (cid:126)Q ( s ; t (cid:48) ) ≈ P n, (cid:126)Q ( s ; t ) × (cid:90) dω π − i ¯ he − iω ( t − t (cid:48) ) ¯ hω − E exn ( (cid:126)Q, s ) − iγ ex − Σ ex ∗ n,s ( (cid:126)Q, ω ) t (cid:48) > t (cid:90) dω π i ¯ he − iω ( t − t (cid:48) ) ¯ hω − E exn ( (cid:126)Q, s ) + iγ ex − Σ exn,s ( (cid:126)Q, ω ) t (cid:48) < t (14) The above approximation, when used together with (10)in (11), results in an expression for the photon operatorin the standard Markoff approximation , a † j ( (cid:126)/q, t ) = a † j ( (cid:126)/q, t = 0) e iω ( (cid:126)/q ) t − (cid:114) AL (cid:88) n,s g j,s ( (cid:126)/q ) (cid:90) d (cid:126)k (2 π ) (cid:113) − f c,s ( (cid:126)k + (cid:126)Q ) φ exn, (cid:126)Q ( (cid:126)k + λ h (cid:126)Q ) × P n, (cid:126)Q ( s ; t )¯ hω ( (cid:126)/q ) − E exn ( (cid:126)Q, s ) + iγ ex − Σ exn,s ( (cid:126)Q, ω ) (15) C. Radiative Rate
Use of (15) in the first term on the RHS of (6) in-troduces an additional source of damping in the mate-rial polarization which is due to radiative transitions. Toshow this more clearly, we substitute (15) in (6), then usethe decomposition in (10) and project out the equationfor P n, (cid:126)Q ( s ; t ), take the mean value, and retain only thoseterms that are relevant to see this radiative damping toget, d (cid:104) P n, (cid:126)Q ( s ; t ) (cid:105) dt ∼ − R n,s ( (cid:126)Q )2 (cid:104) P n, (cid:126)Q ( s ; t ) (cid:105) (16)where the spontaneous emission rate R n,s ( (cid:126)Q ) is, R n,s ( (cid:126)Q ) = 2 c(cid:15) (cid:90) ∞ Qc dω π (cid:32) ω (cid:112) ω − Q c + (cid:112) ω − Q c ω (cid:33) × Re (cid:104) σ n,s ( (cid:126)Q, ω ) (cid:105) (17)Here, c = 1 / √ (cid:15)µ o is the speed of light in the mediumsurrounding the 2D monolayer. The above result for thespontaneous emission is conveniently expressed in termsof the relevant exciton/trion optical conductivity of the2D TMD monolayer. (17) is the main result of this paper.The optical conductivity of a 2D TMD monolayer, for in-plane light polarization, can be written in terms of theexciton Green’s function , σ ( (cid:126)Q, ω ) = (cid:88) n,s σ n,s ( (cid:126)Q, ω )= i e v ω (cid:88) n,s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d (cid:126)k (2 π ) φ exn, (cid:126)Q ( (cid:126)k + λ h (cid:126)Q ) (cid:113) − f c,s ( (cid:126)k + (cid:126)Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × G exn,s ( (cid:126)Q, ω ) (18)Here, G exn,s ( (cid:126)Q, ω ) is the exciton Green’s function , G exn,s ( (cid:126)Q, ω ) = 1¯ hω − E exn ( (cid:126)Q, s ) + iγ ex − Σ exn,s ( (cid:126)Q, ω ) (19)The energies of the eigenstates in (3) are given by thepoles of the exciton Green’s function. We label theseenergies as E lon,s ( (cid:126)Q ) and E hin,s ( (cid:126)Q ). Earlier, in Section I B,we had remarked that the radiative rate for the energyeigenstate to decay into the ground state is proportionalto the weight of its exciton component given by α n in(3). Assuming, γ tr = γ ex = 0 for simplicity, | α n | foran energy eigenstate equals the residue of the excitonGreen’s function at the energy of the eigenstate, | α n | = (cid:20) − h ∂∂ω ReΣ exn,s ( (cid:126)Q, ω ) (cid:21) − = 11 + (cid:88) m,s (cid:48) (1 + δ s,s (cid:48) ) (cid:12)(cid:12)(cid:12) M m,n ( (cid:126)Q, s, s (cid:48) ) (cid:12)(cid:12)(cid:12) (cid:16) E lo/hin,s ( (cid:126)Q ) − E trm ( (cid:126)Q, s, s (cid:48) ) (cid:17) (20)Before exploring the above results further, it is instruc-tive look at the optical conductivity of 2D materials. Theexciton/trion optical conductivity of electron-doped 2DMoSe was calculated by the authors in a recent paperand the results are reproduced in Fig.4 . The spec-tra shows two prominent absorption peaks which cor-respond to the poles, E lon,s ( (cid:126)Q ) and E hin,s ( (cid:126)Q ), of the exci-ton Green’s function in (19). The spectral weight shiftsfrom the higher energy peak to the lower energy peakas the electron density increases. The energy separationbetween the two peaks also increases nearly linearly withthe electron density . In the literature, the lower en-ergy absorption peak is often identified with the trions(or charged excitons) and the higher energy peak with theexcitons. This identification is true only in the limit ofvery small electron densities. At electron densities largeenough such that the lower energy peak has sufficientspectral weight to be experimentally visible in the ab-sorption spectrum, each peak corresponds to an energyeigenstate that is a superposition of exciton and trionstates, as shown in (3). Furthermore, at large electrondensities, the higher energy peak is broadened due toexciton-electron scattering and acquires a wide pedestal(more visible on its higher energy side) that correspondsto the continuum of unbound trion states (or exciton-electron scattering states). In Fig.4, linewidth broaden-ing due to factors other than exciton-electron scattering,such as phonon scattering, was included by assuming that γ ex = γ tr = 4 meV.The rates, R lon,s ( (cid:126)Q ) = 1 /τ lon,s ( (cid:126)Q ) and R hin,s ( (cid:126)Q ) =1 /τ hin,s ( (cid:126)Q ), corresponding to the lower and higher energypeaks in the absorption spectra, respectively, can be eachobtained by restricting the frequency integral in (17) tothe respective peak. Interestingly, because the integral ofthe optical conductivity in (18) satisfies the sum rule , (cid:90) ∞ ω Re { σ ( (cid:126)Q, ω ) } dω π = e v h (cid:88) s (cid:90) d (cid:126)k (2 π ) (cid:16) − f c,s ( (cid:126)k ) (cid:17) (21) -100 0 100 h - E ex0 (meV) -100 0 100 h - E ex0 (meV) n = 1E12 cm -2 E F = 3.43 meVT = 5Kn = 4E12 cm -2 E F = 13.7 meVT = 5 K n = 2E13 cm -2 E F = 68.6 meVT = 5Kn = 1E13 cm -2 E F = 34.3 meVT = 5Kn = 6E12 cm -2 E F = 20.5 meVT = 5K(a)n = 1E10 cm -2 E F = .034 meVT = 5K (d)(e)(b)(c) (f) on s E Q hin s E Q FIG. 4: Calculated real part of the optical conductivity, σ ,s ( (cid:126)Q = 0 , ω ), for in-plane light polarization is plotted fordifferent electron densities for electron-doped monolayer 2DMoSe . Only the lowest energy exciton state is consideredin the calculations. The spectra are all normalized to thepeak optical conductivity value at zero electron density. T =5K. The frequency axis is offset by the exciton eigenenergy E ex ( (cid:126)Q = 0 , s ) of the two-body Schr¨odinger equation. Twoprominent peaks are seen in the spectra. Each peak corre-sponds to an energy eigenstate state that is a superpositionof exciton and trion states, as shown in (3). Figure is repro-duced from the paper by Rana et al. . one can expect from (17) that the radiative rate for thelower energy absorption peak to increase with the elec-tron density and the radiative rate for the higher en-ergy absorption peak to decrease with the electron den-sity such that the sum rule above is always satisfied. Inaddition, since the area under the two peaks in Fig.4 be-come nearly the same at large electron densities ( 2 × cm − ) (despite the fact that the peak optical conductiv-ity of the lower energy peak is higher), one can expectthe two lifetimes to become comparable at large electrondensities. Numerical simulation results, presented in thenext Section, confirm these findings. D. Numerical Simulations and Results
For simulations, we consider an electron-doped mono-layer of 2D MoSe suspended in air. In monolayerMoSe , spin-splitting of the conduction bands is large( ∼
35 meV ) and the lowest conduction band in each ofthe K and K (cid:48) valleys is optically coupled to the topmostvalence band . We use effective mass values of 0 . m o forboth m e and m h which agree with the recently measuredvalue of 0 . m o for the exciton reduced mass . We use awavevector-dependent dielectric constant (cid:15) ( (cid:126)q ), appropri-ate for 2D materials , to screen the Coulomb potentials.We assume that γ ex = γ tr ∼ . We compute exci-ton and trion eigenfunctions and eigenenergies for differ- Electron Density (10 cm -2 ) on s Q hin s Q T=5K cm -2 )00.20.40.60.81 FIG. 5: The zero-momentum radiative lifetimes, τ lon =0 ,s ( (cid:126)Q =0) and τ hin − ,s ( (cid:126)Q = 0), of the lower and higher energy eigen-states, respectively, of the coupled exciton-trion system (andcorresponding to the lower and higher energy peaks in theoptical absorption spectra in Fig.4) are plotted as a functionof the electron densities for an electron-doped monolayer 2DMoSe suspended in air. T=5 K. The inset shows the samedata on a linear scale. Momentum (10 on s Q hin s Q T=5K
Solid: 1x10 cm -2 Dashed: 6x10 cm -2 FIG. 6: The radiative lifetimes, τ lon =0 ,s ( (cid:126)Q ) and τ hin − ,s ( (cid:126)Q ), ofthe lower and higher energy eigenstates, respectively, of thecoupled exciton-trion system (and corresponding to the lowerand higher energy peaks in the optical absorption spectra inFig.4) are plotted as a function of the in-plane momentum Q for different electron densities (10 cm − and 6 × cm − )for an electron-doped monolayer 2D MoSe suspended in air.T=5 K. ent momenta and electron densities as described by Ranaet al. .Fig.5 shows the zero-momentum radiative lifetimes, τ lon =0 ,s ( (cid:126)Q = 0) and τ hin − ,s ( (cid:126)Q = 0), plotted for differ-ent electron densities. As expected, at very small elec-tron densities the radiative lifetime τ lon =0 ,s ( (cid:126)Q = 0) of the lower energy eigenstate is much longer than the lifetime τ hin =0 ,s ( (cid:126)Q = 0) of the higher energy eigenstate. At verylarge electron densities these two lifetimes become com-parable. At small electron densities, when the entirespectral weight lies with the higher energy absorptionpeak in Fig.4, and the corresponding eigenstate is essen-tially a pure exciton state, the calculated lifetimes forthe higher energy eigenstate agree well with the lifetimespublished previously for excitons in 2D materials .But at larger electron densities (¿10 ), the re-sults in previous work, which treated excitons and trionsas independent excitations, become incorrect.Fig.6 shows the radiative lifetimes, τ lon =0 ,s ( (cid:126)Q ) and τ hin − ,s ( (cid:126)Q ), plotted as a function of the in-plane momen-tum Q (within the light cone) for different electron densi-ties. The light cone momentum is defined as the momen-tum Q for which the energy of the eigenstate, E lon,s ( (cid:126)Q ) or E hin,s ( (cid:126)Q ), equals the photon energy ¯ hQc . The radiativelifetimes are more or less constant for momenta withinthe light cone, decrease rapidly as the momentum ap-proaches the light cone (due to an increase in the densityof photon states), and then diverge for momenta outsidethe light cone (where the excitonic component of the en-ergy eigenstates cannot emit a photon and decay into thematerial ground state). This behavior is well known forpure exciton states in 2D materials , and it carriesover to the coupled exciton-trion energy eigenstates indoped 2D materials. III. RATE FOR RADIATIVE DECAY INTO THEMATERIAL EXCITED STATES
The radiative rates calculated above correspond to theprocess depicted in Fig.3(a) in which the energy eigen-state decays into the material ground state. In this Sec-tion, we calculate the radiative rate for the process inFig.3(b) in which the energy eigenstate decays into an ex-cited state of the material that has an electron-hole pairin the CB. The final state after photon emission consistsof a photon with momentum (cid:126)q (cid:48) = ˆ zq (cid:48) z + (cid:126)Q (cid:48) , a CB holewith momentum (cid:126)p and a CB electron with momentum (cid:126)p + (cid:126)Q − (cid:126)Q (cid:48) . The radiative rate expression must include asummation over all these final states. Furthermore, theradiative rate for the process in Fig.3(b) is expected tobe determined by the magnitude of the coefficients β m,s (cid:48) of the trion states in the expression for the energy eigen-state given in (3). These coefficients are found to be, | β m,s (cid:48) | = (1 + δ s,s (cid:48) ) (cid:12)(cid:12)(cid:12) M m,n ( (cid:126)Q, s, s (cid:48) ) (cid:12)(cid:12)(cid:12) (cid:16) E lo/hin,s ( (cid:126)Q ) − E trm ( (cid:126)Q, s, s (cid:48) ) (cid:17) (cid:88) m (cid:48) ,s (cid:48)(cid:48) (1 + δ s,s (cid:48)(cid:48) ) (cid:12)(cid:12)(cid:12) M m (cid:48) ,n ( (cid:126)Q, s, s (cid:48)(cid:48) ) (cid:12)(cid:12)(cid:12) (cid:16) E lo/hin,s ( (cid:126)Q ) − E trm (cid:48) ( (cid:126)Q, s, s (cid:48)(cid:48) ) (cid:17) (22)The summation over m (cid:48) above implies a summation overall bound and unbound trion states consistent with thevalues of s and s (cid:48)(cid:48) . The expression for the Coulombmatrix elements M m,n ( (cid:126)Q, s, s (cid:48) ) coupling the exciton andtrion states (including bound and unbound trion states)can be found in a previous paper by Rana et al. . A. Radiative Rate
In order to calculate the radiative rates for the pro-cess in Fig.3(b), we avoid truncating the 6-body oper-ator products to 4-body operator products that appearduring the derivation of (9), and then include a Heisen-berg equation for 6-body operator products in our model.The calculations are tedious and not particularly illumi-nating. The final result for the radiative rate R n,s ( (cid:126)Q )can be written in a simple form, R n,s ( (cid:126)Q ) = (cid:88) m,s (cid:48) e v (cid:15) (1 + δ s,s (cid:48) ) (cid:90) dq (cid:48) z π (cid:90) d (cid:126)Q (cid:48) (2 π ) (cid:90) d (cid:126)p (2 π ) × (cid:34) q (cid:48) z Q (cid:48) + q (cid:48) z (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d (cid:126)k (2 π ) × φ trm, (cid:126)Q ( (cid:126)k − ξ ( (cid:126)Q + (cid:126)p ) , s ; ( ξ + η )( (cid:126)Q + (cid:126)p ) − (cid:126)Q (cid:48) , s (cid:48) ; (cid:126)p (cid:48) , s (cid:48) ) × (cid:113) − f c,s ( (cid:126)k ) (cid:12)(cid:12)(cid:12)(cid:12) Re (cid:20) iω S n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω ) | (cid:21) ω = √ q (cid:48) z + Q (cid:48) c (23)The spectral function S n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω ) is, S n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω ) = 1¯ hω − E trm ( (cid:126)Q, s, s (cid:48) ) + ∆ + iγ tr − Σ n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω )(24)Here, ∆ stands for the energy difference E c,s (cid:48) ( (cid:126)p + (cid:126)Q − (cid:126)Q (cid:48) ) − E c,s (cid:48) ( (cid:126)p ), and,Σ n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω ) =(1 + δ s,s (cid:48) ) (cid:12)(cid:12)(cid:12) M m,n ( (cid:126)Q, s, s (cid:48) ) (cid:12)(cid:12)(cid:12) ¯ hω − E exn ( (cid:126)Q, s ) + ∆ + iγ tr − F n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω )(25)where, F n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω ) = (cid:88) m (cid:48) (cid:54) = m,s (cid:48)(cid:48) (cid:54) = s (cid:48) (1 + δ s,s (cid:48)(cid:48) ) (cid:12)(cid:12)(cid:12) M m (cid:48) ,n ( (cid:126)Q, s, s (cid:48)(cid:48) ) (cid:12)(cid:12)(cid:12) ¯ hω − E trm ( (cid:126)Q, s, s (cid:48)(cid:48) ) + ∆ + iγ tr (26)The spectral function S n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω ) has the fol-lowing two important properties: Momentum (10 on s Q hin s Q T=5K
Solid: 1x10 cm -2 Dashed: 6x10 cm -2 FIG. 7: The radiative lifetimes, τ lon =0 ,s ( (cid:126)Q ) and τ hin − ,s ( (cid:126)Q ), ofthe lower and higher energy eigenstates, respectively, of thecoupled exciton-trion system (and corresponding to the lowerand higher energy peaks in the optical absorption spectra inFig.4) are plotted as a function of the in-plane momentum Q for different electron densities (10 cm − and 6 × cm − )for an electron-doped monolayer 2D MoSe suspended in air.The lifetimes shown correspond to the process depicted inFig.3(b) for radiative decay into excited states of the mate-rial. T=5 K. The lifetimes shown are three to four orders ofmagnitude longer than the lifetimes shown earlier in Fig.6 forthe process depicted in Fig.3(a) for radiative decay into thematerial ground state. • Its poles are at the energies of the exciton-trionsuperposition eigenstates shifted by ∆, the energytaken by the electron-hole pair left behind in theCB after photon emission. Therefore, the spec-trum of S n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω ) will have two promi-nent peaks just like the spectrum of optical absorp-tion. Since for Q << k F , the energy shift ∆ will benegligibly small for all p < k F , and the peaks in the S n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω ) spectrum will be more or lessat the same energies as the peaks in the absorptionspectrum. • Assuming γ ex = γ tr = 0, the residue of S n,s,m,s (cid:48) ( (cid:126)Q, (cid:126)p, (cid:126)Q (cid:48) , ω ) at these two poles is exactlyequal to the values of | β m,s (cid:48) | given in (22), whichis satisfying in the light of the discussion above.The radiative rates, R lon,s ( (cid:126)Q ) = 1 /τ lon,s ( (cid:126)Q ) and R hin,s ( (cid:126)Q ) =1 /τ hin,s ( (cid:126)Q ), corresponding to the lower and higher energypeaks in the absorption spectra, respectively, and asso-ciated with the process shown in Fig.3(b), can be eachobtained by restricting the frequency integral in (23) tothe respective spectral peak (the integral over frequencyis implicit in (23) in the q (cid:48) z and (cid:126)Q (cid:48) integrations).0 Electron Density (10 cm -2 )
10 m on s Q
10 m hin s Q T=5K
FIG. 8: The radiative lifetimes, τ lon =0 ,s ( (cid:126)Q ) and τ hin − ,s ( (cid:126)Q ), ofthe lower and higher energy eigenstates, respectively, of thecoupled exciton-trion system (and corresponding to the lowerand higher energy peaks in the optical absorption spectra inFig.4) are plotted as a function of the electron densities for anelectron-doped monolayer 2D MoSe suspended in air. T=5K. The momentum value is chosen to be just outside the lightcone Q ∼ B. Simulation Results
Fig.7 shows the radiative lifetimes, τ lon =0 ,s ( (cid:126)Q ) and τ hin − ,s ( (cid:126)Q ), for radiative decay into the excited states ofthe material, plotted as a function of the in-plane momen-tum Q of the energy eigenstates for two different electrondensities. The radiative lifetimes are finite even outsidethe light cone and have a weak dependence on the mo-mentum Q . More interestingly, the radiative rates shownin Fig.7 are three to four orders of magnitude smallercompared to the radiative rates for decay into the ma-terial ground state shown in Fig.6. This large differencecan be understood as follows. Consider an energy eigen-state of momentum (cid:126)Q , as given in (3), and consider the4-body bound trion state component of the energy eigen-state (the bound trion state has more weight in the eigen-state than all the unbound trion states). The small radiusof the bound trion state ( ∼ − ) means that thephase space occupied by each one of the two CB elec-trons in the bound trion state is fairly large, and is of theorder of a − , where a is the trion radius. When one ofthe two CB electrons in the bound trion state radiativelyrecombines with the VB hole, a CB electron and a CBhole are left behind. Suppose the in-plane momentum ofthe emitted photon is (cid:126)Q (cid:48) , the momentum of the CB elec-tron left behind is (cid:126)p + (cid:126)Q − (cid:126)Q (cid:48) , and the momentum of theCB hole is (cid:126)p . Since (cid:126)Q (cid:48) is restricted to be within the lightcone (the phase space area of which is ∼ ω /c ), only avery small portion of the phase space of the CB electron FIG. 9: Certain processes that have been proposed in theliterature for photon emission involving excitons and trionsin electron-doped materials are depicted. (a) Photon emis-sion process involving a 3-body trion state in which the CBelectron recombines with the VB hole leaving behind anotherCB electron which is deposited outside the Fermi sea .(b) Photon emission process involving an exciton in whichan uncorrelated CB electron from the Fermi sea recombineswith the VB hole, leaving behind an electron-hole pair in theCB . (c) Photon emission process involving a trion in whichan uncorrelated CB electron from the Fermi sea recombineswith the VB hole, leaving behind two electron-hole pairs inthe CB . state prior to the photon emission contributes to pho-ton emission. This phase space fraction is of the orderof ω a /c , which is between 10 − to 10 − . Note that τ hin − ,s ( (cid:126)Q ) > τ lon − ,s ( (cid:126)Q ) in Fig.7, which is the oppositeof the case in Fig.6. This is because the radiative ratesin Fig.7 are proportional to | β m,s (cid:48) | (weight of the trioncomponent in the energy eigenstate), whereas the radia-tive rates in Fig.6 are proportional to | α n | (weight ofthe exciton component in the energy eigenstate). Fig. 8shows the radiative lifetimes, τ lon =0 ,s ( (cid:126)Q ) and τ hin − ,s ( (cid:126)Q ),for momentum Q value just outside the light cone, plot-ted for different electron densities. At very small electrondensities the radiative lifetime τ hin =0 ,s ( (cid:126)Q ) of the higher en-ergy eigenstate is much longer than the lifetime τ lon =0 ,s ( (cid:126)Q )of the lower energy eigenstate, and at very large electrondensities these two lifetimes become comparable. Thefact that τ lon − ,s ( (cid:126)Q ) << τ hin − ,s ( (cid:126)Q ) at very small electrondensities can be understood as follows. At very smallelectron densities, | α n =0 | ∼ | β m =0 ,s (cid:48) | <<
1, andthe higher and lower energy eigenstates are thus nearlypure exciton and pure trion states, respectively, and exci-ton states do not radiatively decay into the excited statesof the material.
IV. CERTAIN OTHER MISCONCEPTIONSREGARDING RADIATIVE RATES
Certain other concepts and processes for radiativetransitions have appeared in the literature in the con-text of excitons and trions in doped 2D materials thatare incorrect in the opinion of the authors. We discussthem briefly here. Fig.9(a) shows a photon emission pro-cess involving a 3-body trion state in which the CB elec-tron recombines with the VB hole leaving behind a CBelectron which is deposited outside the Fermi sea .1This model showed that the energy of the photon emit-ted by a trion state would be red-shifted (with respect tothe photon emitted by an exciton in the same material)by roughly the Fermi energy E F (in addition to the trionbinding energy) which is consumed in promoting the left-behind CB electron to the unoccupied states above theFermi level. The red shift of the photon energy with theFermi energy is in agreement with experiments . How-ever, there are several problems with this photon emis-sion model and with the concept of a 3-body trion stateitself . Recent papers have unambiguously shown thatthe red-shifting of the lower energy eigenstate, linearlywith the Fermi energy, with respect to the higher en-ergy eigenstate is the result of Coulomb interactions .Second, this model incorrectly assumes that the electronsforming the trion state are somehow not a part of the CBelectronic states (as Fig.9(a) depicts) and then concludesthat the electron left-behind after photon emission needsto be deposited back into the CB with enough energy toavoid Pauli blocking. The closest correct model, depictedin Fig.3(b), shows that when a 4-body trion state emitsa photon, the CB electron and the CB hole left-behind(that were a part of the 4-body trion state) remain inthe states they occupied just before the emission of thephoton.Fig.9(b) shows a photon emission process involving anexciton in which an uncorrelated CB electron from theFermi sea recombines with the VB hole, leaving behindan electron-hole pair . A simple calculation using anexciton state as the initial state and a final state consist-ing of a Fermi sea with an electron-hole pair in the CB,and using Fermi’s Golden Rule, will show that the rateof this process, although very small, is roughly propor-tional to the electron density (for small electron densities)which in turn is proportional to the probability of find-ing an uncorrelated electron near the exciton. The catchhere is that the probability of finding an electron of thesame spin/valley near the exciton as that of the electronforming the exciton is not proportional to the electrondensity but is in fact near zero due to Pauli’s principle.Each electron in the conduction band, including the oneforming an exciton, is surrounded by its exchange holeand the size of this exchange hole is much larger thanthe size of the exciton in 2D materials for electron den-sities smaller than ∼ cm − . In our model, when weswitched from the 4-body operator T (cid:126)Q to the connected4-body operator T c(cid:126)Q in (6), we removed terms that con-tributed to the process shown in Fig.9(b), and one of thedifference terms, given in (8), gave the exchange energycontribution, which renormalized the CB energy E c,s ( (cid:126)k )on the LHS in (6). The similar process for trions, shownin Fig.9(c) , would have a negligibly small rate for thesame reason. V. DISCUSSION AND CONCLUSION
The results presented in this paper show that photonscan be emitted by exciton-trion energy egenstates when their momenta (cid:126)Q are inside or outside the light cone. In-side the light cone, radiative rates for transitioning intothe material ground state are nearly four orders of mag-nitude faster than the radiative rates in which the finalstate is an excited state of the material. Outside the lightcone, only radiative decay into an excited state of thematerial is possible. Our results are expected to clarifymany concepts associated with light emission from exci-tons and trions and their superposition states in doped2D materials.It needs to be mentioned here that the radiativelifetimes measured in experiments depend on the typeof measurement performed and therefore some care isneeded in comparing experiments with theory. Radia-tive lifetime measurements are usually performed overexciton/trion ensembles and these ensembles can be pre-pared in experiments in various ways. Ultrafast resonantoptical generation of excitons within the light cone andtheir subsequent probing via 1 s → s excitonic transi-tions using a mid-IR probe pulse have yielded excitonlifetimes in 2D TMDs that match well with theory .Time resolved photoluminescence (PL) measurements onthe other hand rely on the exciton-trion energy eigen-states to relax down to the light cone before they canrecombine radiatively with high efficiency . This relax-ation process is generally bottlenecked by phonon scat-tering times which are usually much slower (around afew picoseconds) than the radiative lifetimes inside thelight cone . In addition, as discussed in this paper,PL collected from both peaks in the emission/absorptionspectra of doped 2D materials are from states that aresuperpositions of exciton and trion states and contributeto PL from both inside and outside the light cone. Al-though the radiative rates outside the light cone are muchsmaller than the rates inside the light cone, the phasespace available outside the light cone for hosting a non-equilibrium exciton-trion population is also much largerand a lot more exciton-trions could be present outsidethe light cone than inside it depending on the nature anddetails of the experiment. An accurate modeling of ra-diative emission from non-equilibrium ensembles requirescomputational approaches well beyond the scope of thiswork . VI. ACKNOWLEDGMENTS
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