A Many-Body Theory of the Optical Conductivity of Excitons and Trions inTwo-Dimensional Materials

Farhan Rana, Okan Koksal, Christina ManolatouSchool of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853∗

The optical spectra of two dimensional (2D) materials exhibit sharp absorption peaks that arecommonly identified with exciton and trions (or charged excitons). In this paper, we show thatexcitons and trions in doped 2D materials can be described by two coupled Schrodinger-like equations- one two-body equation for excitons and another four-body equation for trions. In electron-doped2D materials, a bound trion state is identified with a four-body bound state of an exciton andan excited conduction band electron-hole pair. In doped 2D materials, the exciton and the trionsstates are the not the eigenstates of the full Hamiltonian and their respective Schrodinger equationsare coupled due to Coulomb interactions. The strength of this coupling increases with the dopingdensity. Solutions of these two coupled equations can quantitatively explain all the prominentfeatures experimentally observed in the optical absorption spectra of 2D materials including theobservation of two prominent absorption peaks and the variation of their energy splittings andspectral shapes and strengths with the electron density. The optical conductivity obtained in ourwork satisfies the optical conductivity sum rule exactly. A superposition of exciton and trion statescan be used to construct a solution of the two coupled Schrodinger equations and this solutionresembles the variational exciton-polaron state10, thereby establishing the relationship between ourapproach and Fermi polaron physics27–29.

I. INTRODUCTION

Optical absorption and emission spectra of two-dimensional (2D) materials, most notably transitionmetal dichalcogenides (TMDs), exhibit distinct peaksthat are attributed to neutral and charged excitons (ortrions)1–5. Trions have been discussed extensively in theliterature 2,6–8,10–12. In electron-doped materials, a trionstate has been described in many different ways, i) as abound state of two conduction band (CB) electrons anda valence band (VB) hole, or an electron bound to anexciton3,6,7, ii) as a bound state of two CB electrons anda VB valence band hole, plus a CB hole2,9,14, iii) as anattractive exciton-polaron8, and iv) as a bound molecu-lar state similar to the one that appears in the literatureon Fermi polaron physics10,29. The relationship betweenthese different pictures is not clear. Sidler et al. consid-ered attractive exciton-polarons to be different from tri-ons and claimed to have seen optical signatures of bothin experiments10. Efimkin et al. identified the lower(higher) energy peak appearing in the absorption spec-tra of 2D materials (TMDs in particular) with attractive(repuslive) exciton-polarons. Earlier, Suris had explainedthese two absorption peaks in the context of 2D quantumwell physics as mixed exction-trion states13. The trionstate considered by Combescot et al.6, which consistedof a bound state of two CB electrons and a VB hole andresembled a bound molecular state that appears in theliterature on Fermi Polarons29, where deemed to havenegligible optical matrix element with the ground stateby Sidler et al.10. The variational trion state used previ-ously by the authors2, which consisted of a bound stateof two CB electrons and one VB hole, plus a CB hole,reproduced the measured trion optical absorption spec-tra in 2D transition metal dichalcogenides (TMDs) withfairly good accuracy in the low electron density limit but

it could not explain the splitting of the trion and excitonabsorption peaks as a function of the electron density, norcould it explain the transfer of the spectral weight in theoptical absorption spectra from the exciton to the trionwith the increase in the electron density. Diagrammaticperturbation theory involving summation of ladder dia-grams corresponding to exciton-electron interactions aswell as variational ansatz have been used to describe ex-citons interacting with electrons in electron-doped semi-conductors8–10. The solutions correspond to states thatdescribe screening of the exciton by the electrons, or whatare also called exciton-polarons. Exciton-polaron solu-tions have been successful in capturing the variations ofthe of the energy splittings as well as the spectral weighttransfers observed in the optical absorption spectra as theelectron density is varied. On the other hand, the three-body trion physics has been fairly successful in predict-ing the experimentally observed exciton-trion splittings(or the trion binding energies) in the limit of vanishingelectron density 2,3,11,12. The connection between all theabove pictures of a trion remains unclear.

In this paper, we use the many-body density matrixtechnique to describe describe excitons and trions in elec-tron doped two-dimensional materials 22. Our main re-sults are summarized below.

Excitons and trions in doped materials can be describedby two coupled Schrodinger-like equations. One is thestandard two-body Schrodinger equation of a CB elec-tron interacting with a VB hole (or an exciton)22, andthe other is a four-body Schrodinger equation of twoCB electrons interacting with a VB hole and a CB hole(See Fig. 1). A trimmed version of the latter, obtainedby dropping the CB hole, is the standard three-bodySchrodinger equation that is commonly used to describetrions 2,3,6,7,11,12. The CB hole is created when an elec-tron is scattered out of the Fermi sea. Trions states

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2

+

+

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‐

Trion

+‐

Exciton

VB hole

CB Electron

VB hole

CB hole

CB Electron

CB Electron

FIG. 1: Bound exciton (left) and bound trion (right) statesin an n-doped 2D material are depicted. A bound trion isa 4-body state and consists of two conduction band (CB)electrons, a valence band (VB) hole, and a conduction band(CB) hole. When the background electron density is non-zero, these two states are coupled due to electron-electronand electron-hole Coulomb interactions.

are therefore identified with the eigenstates of this four-body Schrodinger equation. These two Schrodinger equa-tions emerge naturally from the density matrix approach.The most interesting fact is that these two equations arecoupled via Coulomb matrix elements that result fromelectron-electron and electron-hole interactions. There-fore, exciton and trion states are not independent exci-tations in doped materials. The solution of these twocoupled equations provide a simple but exact descrip-tion of an exciton interacting with CB electrons (withinthe purview of an exciton exciting only single electron-hole pair at a time in the conduction band). Solutionsof these two coupled equations can quantitatively explainall the prominent features experimentally observed in theoptical absorption spectra of 2D materials including theobservation of two prominent absorption peaks and thevariation of their energy splittings, spectral strengths,and spectral linewidths with the electron density. Inter-estingly, the exact solution of these two equations resem-bles the variational exciton-polaron solution,9,10 therebyestablishing the relationship between the two approaches.The solutions obtained in this work also explicitly satisfythe optical conductivity sum rule exactly at all carrierdensities.

The eigenspectrum of the four-body Schrodinger equa-tion has bound states, which are bound states of an ex-citon and a CB electron-hole pair, and unbound states,which are exciton-electron scattering states. However,these eigenstates of the four-body Schrodinger equationas well as the eigenstates of the two-body Schrodingerequation are not eigenstates of the full Hamiltonian inthe presence of a non-zero electron density because ofCoulomb interactions. The eigenstates of the full Hamil-tonian are more accurately superpositions of two-bodyexciton and four-body trion states. The two prominentpeaks observed in the optical absorption spectra of doped2D materials correspond to these superpositon eigen-states. The essential physics is as follows. An excitoncan Coulomb excite a CB electron-hole pair and bind itto form a trion. But it is energetically favorable for the

exciton to release this bound CB electron-hole pair andexcite another CB electcron-hole pair and bind it (or scat-ter from it). These virtual transitions can occur at thelowest electron densities, even when the Fermi energy ismuch smaller than the binding energy of the four-bodytrion state. The energy thus gained is responsible forthe observed near-linear increase in the energy splittingof the two absorption peaks with the Fermi energy1,5.The picture described here also captures the essentialphysics associated with the exciton-polarons. The maintwo peaks observed experimentally in the optical absorp-tion spectra result from the coupling between the two-body exciton and four-body trion states and have alsobeen identified as the repulsive and attractive exciton-polarons absorption peaks by Efimkin et al.8. A com-plete and accurate description of the problem requiresusing both the bound and the unbound states of the four-body Schrodinger equation. Exciton-electron scatteringis thereby also included in our analysis.

In this paper, we have focused on single CB electron-hole pair excitation by an exciton and have ignored mul-tiple electron-hole pair excitations that are expected tobecome important when the Fermi energy approachesthe exciton binding energy8. Effects related to Fermiedge singularities15,16, which involve multiple electron-hole pair excitations and are expected to become im-portant if the hole mass or the exciton mass were muchbigger than the electron mass (which is not the case inmost 2D materials including TMDs), are therefore alsoignored. Our work shows that a description based on sin-gle pair excitations can adequately explain the prominentfeatures of the experimentally measured optical absorp-tion spectra of 2D materials.

In the light of the introductory discussion above, thename ”trion” seems like a misnomer since the involvedstates are either two-body or four-body states. But giventhe long history of the use of this terminology, in this pa-per we will use the term “trion” for the eigenstates ofthe four-body Schrodinger equation (Fig. 1). After thiswork had been completed, we became aware of the ear-lier work by Suris14 in which mixed exciton-trion modeshad been introduced to account for the coupling betweenthe excitons and the trions resulting from Coulomb in-teractions. Suris was also the first one to argue that itwould be energetically favorable for the CB hole to bindto the two CB electrons and one VB hole in a boundtrion state as result of trion-exciton coupling. The workpresented here is conceptually along the same lines, butour approach is different in many ways. The use of cou-pled two-body and four-body Schrodinger equations en-ables one to include the effects of the electron density,as well as the effects of exciton-trion coupling, on thewavefunctions and the binding energies of excitons andtrions. Our approach, which does not involve wavevector-independent Coulomb potentials and artificial cut-offs ofmomentum integrals8,10,13, enables us to obtain quanti-tative results for excitons and trions in two dimensionalmaterials. Suris had also ignored direct Coulomb interac-

3

tions between the CB hole and the three other particlesin a bound trion state. Our four-body Schrodinger equa-tion includes these interactions which cannot be ignoredat the moderate to high electron densities at which trionsignatures are experimentally observed in 2D materials.In agreement with the reported experimental results on2D materials, but not in agreement with the conclusionsdrawn by Suris, we do not find any broad absorptionbands in the optical absorption spectra associated withtrions states that do not have a bound CB hole. Incontrast to other previous works8–10, our approach alsoexplicitly takes into account electron’s spin and valleydegrees of freedom and exchange interactions. Finally,our approach also sheds light on the debated question ofwhether or not trions have appreciable optical oscillatorstrength. Our results show that the trion states, definedin this paper as the eigenstates the four-body Schrodingerequation, have no optical oscillator strength and their op-tical activity results only from their Coulomb coupling tothe exciton states.

II. 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. Hamiltonian

We consider a 2D TMD monolayer at z = 0. Light with

a small in-plane momentum ~Q is incident on the layer.The Hamiltonian describing electrons and holes in theTMD layer (near the K and K ′ points in the Brillouinzone) interacting with each other and with the opticalmode in the rotating wave approximation is2,17–19,

H =∑~k,s

Ec,s(~k)c†s(~k)cs(~k) +

∑~k,s

Ev,s(~k)b†s(~k)bs(~k)

+1

A

∑~q,~k,~k′,s,s′

U(q)c†s(~k + ~q)b†s′(

~k′ − ~q)bs′(~k′)cs(~k)

+1

2A

∑~q,~k,~k′,s,s′

V (q)c†s(~k + ~q)c†s′(

~k′ − ~q)cs′(~k′)cs(~k)

+ hω( ~Q)a†( ~Q)a( ~Q)

+1√A

∑~k,s

(gsc†s(~k + ~Q)bs(~k)a( ~Q) + h.c

)(1)

Here, Ec,s(~k) and Ev,s(~k) are the conduction and valenceband energies. s, s′ 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(~q) represents Coulomb inter-action between electrons in the conduction and valence

bands and V (~q) represents Coulomb interaction among

the electrons in the conduction bands. hω( ~Q) is the en-

ergy of a photon with in-plane momentum ~Q, and gs isthe electron-photon coupling constant. Other than forphase factors that are not relevant in the discussion thatfollows, gs for electron states near the band edges in 2DTMDs can be given by18,19,

gs = ev

√h

2〈ε〉ω( ~Q)χ(z = 0) (2)

where, v is the interband velocity matrix element2,17–19,and χ(z) describes the amplitude of the optical mode inthe z-direction.

B. Density Matrix Approach

We use a many body density matrix approachwhich has been fairly successful in modeling excitonphysics22,23, and it has also been previously used for tri-ons in the limit of vanishingly small electron densities24.

We start from the Heisenberg equation for the photonoperator, which after averaging, is22,23,[

hω( ~Q) + ih∂

∂t

]〈a†( ~Q, t)〉 = − 1√

A

∑~k,s

gsP~Q(~k, s; t)

(3)

The polarization P~Q(~k, s; t) equals the equal-time two-

body correlation 〈c†s(~k + ~Q, t)bs(~k, t)〉. Assuming fromnow onwards that in steady state all the relevant equal-time correlation functions have the time dependencee±iωt, the above equation becomes,[hω( ~Q) + iε− hω

]〈a†( ~Q)〉 = − 1√

A

∑~k,s

gsP~Q(~k, s) (4)

where P~Q(~k, s; t) = P~Q(~k, s)eiωt. The equation for

P~Q(~k, s) is,[Ec,s(~k + ~Q)− Ev,s(~k) + iγex − hω

]P~Q(~k, s) =

− 1√Ag∗s 〈a†( ~Q)〉

[1− fc,s(~k + ~Q)

]+

1

A

∑~q

U(~q)P~Q(~k + ~q, s)[1− fc,s(~k + ~Q)

]− 1

A

∑~q,~p,s′

U(~q)

×T c~Q(~k + (ξ + η) ~Q− ξ~p, s; (ξ + η)~p− ξ ~Q− ~q, s′; ~p, s′)

+1

A

∑~q,~p,s′

V (~q)

×T c~Q(~k + (ξ + η) ~Q− ξ~p+ ~q, s; (ξ + η)~p− ξ ~Q− ~q, s′; ~p, s′)

(5)

4

, 'p s

, 'p q s

,k Q q s

,k q s

,k Q s

,k s

, 'p s

, 'p q s

,k Q q s

,k q s

,k Q s

,k s

, 'p s

, 'p q s

,k q s

,k Q q s

,k Q q s

,k s

,k s

,k Q s

, 'p s

, 'p q s

,k Q s

,k Q q s

(a)

(b)

(c)

(d)

Time

FIG. 2: A sampling of diagrams and processes that connecttwo-body and four-body correlations in (5) and (8). Coulombinteractions are depicted by the vertical lines. The horizontallines are electron and hole propagators. Diagrams correspond-ing to exchange processes are not shown.

Here, 〈c†s(~k)cs(~k)〉 = fc,s(~k) is the electron occupationprobability, γex is a phenomenological decoherence ratefor the polarization that includes dephasing due to allprocesses other than exciton-electron scattering. The en-

ergies Ec,s(~k) include renormalizations due to exchange

at the Hartree-Fock level (−(1/A)∑~q V (~q)fc,s(~k − ~q)).

λh = 1 − λe = mh/mex (mex = me + mh), where me

(mh) is the electron (hole) effective mass. (5), withoutthe first term and the last two terms on the RHS, is thestandard eigenvalue equation for excitons22,23. (5) is notHermitian but it can be converted into a Hermitian equa-tion by rescaling the eigenfunctions (see Appendix A).

The Hermitian equation has eigenvalues Eexn ( ~Q, s) and

eigenfunctions φexn, ~Q

(~k + λh ~Q). The eigenfunctions form

a complete set.

The last two terms in (5) on the RHS contain four-body correlations T~Q and correspond to the diagrams

shown in Fig. 2(a,b). Assuming that mtr = 2me + mh,ξ = me/mtr, and η = mh/mtr, we define a four-body

equal-time correlation T~Q(~k1, s1;~k2, s2; ~p, s2; t) as follows,

〈c†s1(~k1, t)c†s2(~k2, t)bs1(~k1 + ~k2 − ( ~Q+ ~p), t)cs2(~p, t)〉 (6)

The underlined vector ~k stands for ~k + ξ( ~Q + ~p). T~Qdescribes the correlations arising from Coulomb interac-tions among four particles: two CB electrons, a VB hole,

and a CB hole. ~Q is the total momentum of this 4-bodystate. We also define a fully connected four-body corre-lation (as defined in the cluster expansion technique23),

T~Q(~k1, s1;~k2, s2; ~p, s2; t) = T c~Q(~k1, s1;~k2, s2; ~p, s2; t)

−fc,s2(~p)P~Q(~k1 − ~Q, s1)δ~k2,~p

+fc,s2(~p)P~Q(~k2 − ~Q)δs1,s2δ~k2,~p (7)

The equation for the connected correlation

T c~Q(~k1, s1;~k2, s2; ~p, s2) is found to be,[Ec,s1(~k1) + Ec,s2(~k2)− Ev,s1(~k1 + ~k2 − ( ~Q+ ~p))

−Ec,s2(~p) + iγtr − hω]T c~Q(~k1, s1;~k2, s2; ~p, s2) =

− 1

A

∑~q

V (~q)T c~Q(~k1 + ~q, s1;~k2 − ~q, s2; ~p, s2)

×[1− fc,s1(~k1)− fc,s2(~k2)

]+

1

A

∑~q

U(~q)T c~Q(~k1 + ~q, s1;~k2, s2; ~p, s2)[1− fc,s1(~k1)

]+

1

A

∑~q

U(~q)T c~Q(~k1, s1;~k2 − ~q, s2; ~p, s2)[1− fc,s2(~k2)

]+

1

A

∑~q

V (~q)T c~Q(~k1 + (ξ + η)~q, s1;~k2 − ξ~q, s2; ~p+ ~q, s2)

×[fc,s2(~p)− fc,s1(~k1)

]+

1

A

∑~q

V (~q)T c~Q(~k1 − ξ~q, s1;~k2 + (ξ + η)~q, s2; ~p+ ~q, s2)

×[fc,s2(~p)− fc,s2(~k2)

]− 1

A

∑~q

U(~q)T c~Q(~k1 − ξ~q, s1;~k2 − ξ~q, s2; ~p+ ~q, s2)fc,s2(~p)

+fc,s2(~p)

A

∑~q

V (~q)[1− fc,s1(~k1)− fc,s2(~k2)

]×[P~Q(~k1 − ~Q+ ~q, s1)δ~k

2−~q,~p

−P~Q(~k2 − ~Q− ~q, s2)δ~k1+~q,~pδs1,s2

]−fc,s2(~p)

A

∑~q

U(~q){P~Q(~k1 − ~Q, s1)δ~k2−~q,~p

[1− fc,s2(~k2)

]− P~Q(~k2 − ~Q, s2)δ~k1+~q,~pδs1,s2

[1− fc,s1(~k1)

]}(8)

In deriving the above equation, all six-body correlationsare reduced to four-body correlations using the clusterexpansion23. By ignoring higher order correlations weare ignoring the generation of multiple particle-hole pairsin the CB. Here, γtr is a phenomenological decoherencerate. If ~re1, ~re2, ~rh1, are ~rh2 the coordinates of the twoelectrons, the VB hole, and the CB hole, respectively,

then ~k1, ~k2, ~Q, and ~p are the momenta associated with

the coordinates ~re1−~rh1, ~re2−~rh1, ~R = ξ(~re1+~re2)+η~rh1,

and ~R − ~rh2, respectively. Here, ~R is the center of masscoordinate of the two electrons and the VB hole. Ig-noring the last two terms on the RHS in (8) that in-volve P~Q, Fourier transform of the remaining terms will

result in a four-body Schrodinger equation. Each termon the RHS in the above equation (except the last two)describes Coulomb interaction between two of the fourparticles.The last two terms involving P~Q capture the

5

generation of four-body correlation from two-body cor-relation, or the creation of an CB electron-hole pair byan exciton, and and correspond to the diagrams shownin Fig. 2(c,d). We should mention here that an equa-tion similar to (8) was obtained by Esser et al.24, but inthat work the connected nature of T c~Q was overlooked,

the terms containing interactions with the CB hole wereignored, the phase-space restricting factors were ignoredtoo, and, most importantly, the terms containing the po-larization P~Q were also missed. Ignoring the coupling to

P~Q in (8) is equivalent to ignoring exciton-trion coupling

via Coulomb interactions.(5) and (8) are a closed system of coupled Schrodinger

equations for two-body and four-body systems.

C. Trion States

The trion states are defined here as the eigenstatesof the four-body Schrodinger equation given in (8). (8)is not Hermitian but it can be converted into a Her-mitian equation (see Appendix B), with a few suit-able approximations, and the eigenfunctions thereforeform a complete set. The eigenfunctions are written as

φtrm, ~Q

(~k1, s1;~k2, s2; ~p, s2). The corresponding eigenener-

gies are Etrm( ~Q, s1, s2). The eigenstates include boundfour-body states, unbound exciton-electron scatteringstates, and completely unbound four-body states. Thelatter have high energies and may be ignored here. The

eigenfunctions φtrm, ~Q

(~k1, s1;~k2, s2; ~p, s2) are either sym-

metric or antisymmetric in ~k1 and ~k2 depending on thevalues of s1 and s2 and on the spin state of the two elec-trons (singlet or triplet).

D. The Fate of the Conduction Band Hole

The 4th, 5th, and 6th terms on the RHS in the four-body Schrodinger equation include interactions involvingthe CB hole that is generated when an electron is scat-tered out of the Fermi sea by the exciton to form a trion.The CB hole wavefunction can have a radius no smallerthan ∼ 1/kF in real space and therefore it can be muchlarger than the exciton and the three-body trion radii atsmall electron densities. For this reason, interactions in-volving the CB hole have been ignored in previous works8–10,14. Here we argue that the CB hole needs to be takeninto account in bound trion states. Signatures of boundtrion states are observed only at moderate to high elec-tron densities in 2D materials (n > 1012 cm−2) at whichFermi energy can be appreciable. The LHS of (8) has theenergies of the photoexcited CB electron, the VB hole,and the initial and final energies of the CB electron scat-tered out of the Fermi sea. All energies include renormal-ization due to exchange. An electron within the Fermi seahas a larger energy renormalization than an electron well

outside the Fermi sea22. Consequently, when an electronis scattered out of the Fermi sea to bind to an excitonand form a tightly bound trion state, the difference in itsinitial and final exchange energies, as given by the termson the LHS of (8), can be pretty large - so much so thata bound trion state may not even be energetically possi-ble except at very small electron densities. The inclusionof the terms on the RHS of (8), which include Coulombinteractions involving the CB hole, make up for this en-ergy difference and make bound trion states possible andenergetically favorable provided the CB hole, along withthe CB electron, gets bounded to the exciton to make afour-body bound state depicted in Fig.(1). In additionto direct Coulomb interactions involving the CB hole,and as shown in the earlier work of Suris14, exciton-trioncoupling, and the energy gained therewith, also favorsthe binding of the CB hole bound to the exciton and theCB electron in a four-body bound state.

E. Exciton Self-Energy and Optical Conductivity

A formal solution of (8) can be written in terms of itseigenfunctions as,

T c~Q(~k1, s1;~k2, s2; ~p, s2) = −(1 + δs1,s2)

×∑m

√fc,s2(~p)

[1− fc,s1(~k1)

] [1− fc,s2(~k2)

]

×φtrm, ~Q

(~k1, s1;~k2, s2; ~p, s2)

hω − Etrm( ~Q, s1, s2)− iγtr1

A4

∑~k′1,~p

′,~q

φtr∗m,~Q

(~k′1, s1; (ξ + η)~p′ − ξ ~Q+ ~q, s2; ~p′, s2)

×√fc,s2(~p′) [1− fc,s2(~p′ + ~q)]

√

1− fc,s1(~k′1)

× V (~q)P~Q(~k′1 − ~Q+ ~q, s1)− U(~q)

P~Q(~k′1 − ~Q, s1)√

1− fc,s1(~k′1)

(9)

The summation over m above implies a summation overall bound and unbound trion states consistent with thevalues of s1 and s2. The above solution can be used in(5) to obtain the polarization,

P~Q(~k, s) =g∗s√A〈a†( ~Q)〉

∑n

√1− fc,s(~k + ~Q)

×φexn, ~Q

(~k + λh ~Q)

hω − Eexn ( ~Q, s)− iγex − Σex∗n,s ( ~Q, ω)∫d2~k′

(2π)2φex∗n, ~Q

(~k′ + λh ~Q)

√1− fc,s(~k′ + ~Q)

(10)

6

Here, the summation over n implies a summation over allbound and unbound exciton states. The expression for

the exciton self energy Σexn,s( ~Q, ω) is given below. Theabove result can be used in (4) to obtain an expression

for the optical conductivity σ( ~Q, ω) of the 2D material,

σ( ~Q, ω) = ie2v2

ω

×∑n,s

∣∣∣∣∣∫

d2~k′

(2π)2φexn, ~Q

(~k′ + λh ~Q)

√1− fc,s(~k′ + ~Q)

∣∣∣∣∣2

hω − Eexn ( ~Q, s) + iγex − Σexn,s( ~Q, ω)

(11)

The exciton self-energy can be expressed as,

Σexn,s( ~Q, ω) =∑m,s′

(1 + δs,s′)∣∣∣Mm,n( ~Q, s, s′)

∣∣∣2hω − Etrm( ~Q, s, s′) + iγtr

(12)

The summation over m above implies a summation overall bound and unbound trion states consistent with thevalues of s and s′. Mm,n( ~Q, s, s′) equals,

1

A3

∑~k,~p,~q

φtr∗m,~Q

(~k − ξ(~p+ ~Q), s; (ξ + η)~p− ξ ~Q+ ~q, s′; ~p, s′)

×√fc,s′(~p) [1− fc,s′(~p+ ~q)]

×

{√[1− fc,s(~k)

] [1− fc,s(~k + ~q)

]V (~q)

× φexn, ~Q

(~k − λe ~Q+ ~q)− U(~q)φexn, ~Q

(~k − λe ~Q)}

(13)

It is evident from the above expression that the coupling

term Mm,n( ~Q, s, s′) increases with the electron density.The self-energy expression in (12) assumes that an exci-ton state does not couple to a different exciton state dueto Coulomb interactions via an intermediate trion state.Given that the energy separation between the lowest en-ergy exciton state and the higher energy exciton statesin 2D materials can be in the hundreds of milli electronvolts range4, the approximation made in this assumptionis expected to be very good for the lowest energy excitonstate.

The expressions in (11) and (12) constitute the mainresults of this work. The optical conductivity in (11) cor-responds to the diagrams and processes shown in Fig. 3.Only fully connected diagrams contribute to the opticalconductivity. Diagrams with multiple particle-hole exci-tations at the same time are not included in this work.

Q

Q

(a) (c)

(d) (e)

(g) (i)

(f)

Photon

(h)

,k s

,k Q s (b)

, 'p s

FIG. 3: A sampling of diagrams and processes that contributeto the optical conductivity. Curved horizontal lines representelectron and hole propagators. The double horizontal line isthe photon propagator. Coulomb interactions are depictedby the vertical lines. The small circles represent electron-holepairs excited in the conduction band by the photogeneratedexciton. Only fully connected diagrams contribute to the op-tical conductivity. Diagrams with multiple electron-hole pairexcitations in the conduction band are not included in thiswork.

F. Unbound Trion States: Exciton-ElectronScattering

The expression for the exciton self-energy in (12) in-cludes coupling with all bound trion states as well as un-bound trion states. The latter are just exciton-electronscattering states and need to be treated carefully. Thesymmetric/anti-symmetric four-body wavefunction, with

center of mass momentum ~Q, of a state consisting of an

exciton scattered from initial momentum ~Q to ~Q − ~qo,and an electron scattered from initial momentum ~po in-side the Fermi sea to momentum ~po+~qo outside the Fermisea, can be approximated as,

φtrm, ~Q

(~k1, s1;~k2, s2; ~p, s2) ≈[√A

2φexn, ~Q−~qo

(~k1 + (ξ − λe) ~Q+ ξ~po + λe~qo)

×δ~k2,(ξ+η)~po−ξ ~Q+~qo±{~k1 ↔ ~k2

}]×√Afc,s2(~po)δ~po,~p

(14)

The summation over the index m in (12) would now in-volve a summation over all momenta ~po inside the Fermisea and all transferred momenta ~qo, as well as all symmet-ric and anti-symmetric states consistent with the valuesof s and s′. The energy of the above state, for small

values of ~Q and ~qo, is,

Etrm( ~Q, s1, s2)→ Eexn ( ~Q = 0, s1, s2) +h2q2

o

2mT

−h2

(~Q

mex− ~pome

)· ~qo

7

(15)

Here, Eexn ( ~Q = 0, s1, s2) = 0.5(Eexn ( ~Q = 0, s1)+Eexn ( ~Q =0, s2)). mT = mexme/(mex+me), is the reduced mass ofthe exciton and the electron. Use of the state in (14) toevaluate exciton-electron scattering contributions to theself-energy in (12) is equivalent to the use of the Born ap-proximation in exciton-electron scattering 25. The contri-bution of exciton-electron scattering to the exciton self-energy is found to be,

Σexn,s( ~Q, ω)|ex−e =∑u,s′

∫d2~qo(2π)2

∫d2~po(2π)2

fc,s′(~po)

×(1 + δs,s′)

∣∣∣hn,u( ~Q, ~po, ~qo, s, s′)∣∣∣2

hω −[Eexn ( ~Q = 0, s, s′) +

h2q2o2mT

− h2(

~Qmex− ~po

me

)+ iγtr

](16)

The summation over the variable u = ±1 involves a sum-mation over all symmetric and antisymmetric unboundtrion states (or exciton-electron scattering states) con-sistent with the values of s and s′. The expression for

hn,u( ~Q, ~po, ~qo, s, s′) is given in the Appendix.

G. Optical Conductivity Sum Rule

This optical conductivity sum rule for 2D TMDs canbe derived from the restricted Thomas-Reiche-Kuhn op-tical conductivity sum rule26 and, assuming a full valenceband, can be expressed as26,∫ ∞

0

ωRe{σ~Q(ω)} dω2π

=e2v2

2h

∑s

∫d2~k

(2π)2

(1− fc,s(~k)

)(17)

Here, v is the interband velocity matrix element be-tween the valence and conduction band Bloch states (seeSec. II A). Band filling is incorporated into the abovesum rule. The completeness of the exciton eigenfunctions

φexn, ~Q

(~k) can be used to see that the derived optically con-

ductivity in (11) satisfies the above sum rule exactly.

III. VARIATIONAL EIGENSTATES ANDCONNECTION WITH EXCITON-POLARON

STATES

Variational states for exciton-polarons have been con-structed in previous works 9,10. These variational statesresemble the Fermi polaron states of an impurity atomin a cold Fermi gas 27–29. Here we show that variationalstates can be constructed using the eigenstates of the two-body and four-body Schrodinger equations in (5) and (8),respectively, and which give results for the eigenenergiesin exact agreement with the exciton self-energy given in(12). The eigenenergies can be obtained from the poles

of the exciton Green’s function and these energies are theroots of the equation,

hω − Eexn ( ~Q, s) + iγex − Σexn,s(~Q, ω) = 0 (18)

where the exciton self-energy is as given in (12). Since thetwo-body and four-body Schrodinger equations are cou-pled via the Coulomb matrix elements, one can constructapproximate eigenstates of the Hamiltonian (within thepurview of single CB electron-hole pair excitations) by asimple superposition as follows,

|ψn,s( ~Q)〉 =αn√A

∑k

φex∗n, ~Q

(~k)

Nex

×c†s(~k + λe ~Q)bs(~k − λh ~Q)|GS〉

+∑m,s′

βm√A3

~k1,~k

26=~p∑

~k1,~k2,~p

φtr∗m,~Q

(~k1, s;~k2, s′; ~p, s′)

Ntr

× c†s(~k1)c†s′(~k2)bs(~k1 + ~k2 − ( ~Q+ ~p))cs′(~p)|GS〉

(19)

where |GS〉 is the ground state of the electron doped ma-terial. The above state resembles a Fermi-polaron varia-tional state10,27–29. The normalization terms are,

Nex =

√1− fc,s(~k + λe ~Q)

Ntr =

√(1 + δs,s′)fc,s′(~p)

[1− fc,s(~k1)

] [1− fc,s′(~k2)

](20)

The underlined vectors, ~k1 and ~k2, are defined as ear-lier in Section II B. The states in the superposition areproperly normalized and are orthogonal. The restric-

tions ~k1,~k2 6= ~p in the summations in the second term

follow from the fact that the states of the four-bodySchrodinger equation correspond to fully connected di-agrams and should have no direct optical matrix elementwith the ground state |GS〉. This restriction also keepsthe superposed trion states in the variational state or-thogonal to the exciton states. When the variationalstate given above is used to minimize energy with respectto the Hamiltonian given earlier in (1), the eigenfunc-

tions φtrm, ~Q

(~k1, s;~k2, s′; ~p, s′) and φex

n, ~Q(~k), as expected,

are found to obey the coupled two-body and four-bodySchrodinger equations, the trion states are found tobe coupled to the exciton states via the Coulomb ma-

trix elements Mm,n( ~Q, s, s′) given earlier (see (13)), andthe energy eigenvalues hω are found to obey (18) pro-vided γex and γtr are set to zero. Therefore, the samephysics is captured by our coupled two-body and four-body Schrodinger equations and the exciton-polaron for-malism. Furthermore, the optical conductivity calcu-lated using the above variational state also matches theone found earlier in (11). The formalism presented hereshows that the variational polaron state can be written

8

FIG. 4: Calculated energies, Etrm=0( ~Q = 0, s1, s2) and

Eexn=0( ~Q = 0, s1), of the lowest energy bound trion and ex-

citon states, respectively, referenced to the material bandgap,are plotted as a function of the electron density (and Fermienergy) for monolayer 2D MoSe2 on SiO2. Trion binding en-ergy Etr

b is also plotted. T = 5K.

in terms of the eigenstates of the two-body and the four-body Schrodinger equations (i.e. in terms of the excitonand trion eigenstates) and that all bound and unboundtrion states must be included in the variational polaronstate. Signatures of the resulting quantum coherence be-tween the exciton and trion states have been observedexperimentally32.

IV. NUMERICAL SIMULATION RESULTS ANDDISCUSSION

For simulations, we consider a monolayer of 2D MoSe2

on a SiO2 substrate. In monolayer MoSe2, spin-splittingof the conduction bands is large (∼35 meV30) and thelowest conduction band in each of the K and K ′ valleysis optical coupled to the topmost valence band21. We useeffective mass values of 0.7mo for both me and mh whichagree with the recently measured value of 0.35mo for theexciton reduced mass31. We use a wavevector-dependentdielectric constant ε(~q) for the Coulomb potentials ap-propriate for 2D materials, as described in our earlierwork2, to screen the Coulomb potentials. We shouldemphasize here that besides γex and γtr, and unlike inprevious works8,10, there are no other free parameters inour theoretical model and no artificial upper cut-offs ofmomenta integrals to avoid divergences.

A. Trion and Exciton Radii and Energies

As discussed in previous works2,3,8–10, exchange corre-lations favor only singlet trion bound states in MoSe2, inwhich the exciton belongs to one valley and the bound

1010 1011 1012 1013

Electron Density (cm-2)

0.5

1

1.5

2

0.034 0.34 34.0Fermi Energy (meV)

3.4

atr

aex

btr

FIG. 5: Calculated trion radii, atr and btr, for the variationaltrion wavefunction, and the exciton radius aex for the varia-tional exciton wavefunction are plotted as a function of theelectron density (and Fermi energy) for monolayer 2D MoSe2on SiO2. T = 5K.

electron-hole pair belongs to the other valley. We writea product variational wavefunction for the lowest energy

four-body bound singlet trion eigenstate with ~Q = 0, asfollows,

φtrm=0, ~Q=0

(~k1, s1;~k2, s2; ~p, s2) = χ(~k1, s1;~k2, s2)ϕ(~p, s2)

(21)We assume the following symmetric variational wave-function for χ(~r1, s1;~r2, s2) (assuming s1 6= s2)2,3,12,

χ(~r1, s1;~r2, s2) ∝[e−|~r1|/atr−|~r2|/btr + {~r1 ↔ ~r2}

](22)

The trion radii, atr and btr, are variational parameters.The form of the CB hole wavefunction ϕ(~p, s2) is cho-sen so as to minimize Coulomb energy and maximizecoupling between the trion and exciton states, as dis-cussed in Sec. II D. Since the trion radii are expectedto be much smaller than the size of the CB hole, whichcannot be smaller than ∼ 1/kF , the simplest and theeasiest way to get the smallest CB hole is to assumethat ϕ(~p, s2) =

√fc,s2(~p)/ns2 , where proper wavefunc-

tion normalization requires ns2 = A−1∑

~p fc,s2(~p). For

the lowest energy ~Q = 0 bound exciton state we use thevariational wavefunction2,3,12,

φexn=0, ~Q=0

(~r) ∝ e−|~r|/aex (23)

Using the radii, atr, btr, and aex, as variational param-

eters, we find the eigenenergies, Etrm=0( ~Q = 0, s1, s2)

and Eexn=0( ~Q = 0, s1), as a function of the electrondensity. The results are shown in Fig. 4 which plotsthese energies with respect to the material bandgap Eg.The corresponding trion and exciton radii are plotted

9

in Fig. 5. The trion binding energy Etrb , defined as

Eexn=0( ~Q = 0, s1) − Etrm=0( ~Q = 0, s1, s2), is also plotted.The exciton binding energy decreases with the electrondensity due to phase space filling2. The trion bindingenergy first increases with the electron density and thenit decreases. The initial increase is due to two reasons,

• Suppose an exciton with center of mass momen-

tum ~Q = 0 grabs an electron with momentum ~pwithin the Fermi sea to form a four-body bound

state with momentum ~Q = 0. The center of masskinetic energy of the four-body state would beh2p2/(2mt)−h2p2/(2me). The first term is the cen-ter of mass kinetic energy of the two CB electronsand one VB hole in the four-body bound state. Thesecond term represents the kinetic energy of the CBhole in the four-body bound state. These energiesare included in the terms on the LHS of (8). Av-eraging this energy with respect to the CB holewavefunction ϕ(~p, s) contributes a factor EF /3 tothe trion binding energy.

• At small electron densities, phase space filling re-stricts electron-electron Coulomb repulsion morethan electron-hole Coulomb attraction.

As the electron density increases further, the reducedphase space diminishes electron-hole Coulomb attractionas well and the binding energy of the trion decreasesrapidly. It remains an open question if the trion bind-ing energy eventually goes to zero or not at high enoughelectron densities. The approximations made in this workdo not permit us to generate reliable results for electrondensities higher than 2 × 1013 cm−2. Interestingly, theFermi energy remains smaller than the trion binding en-ergy for electron densities smaller than ∼ 1013 1/cm3.Note that the binding energies of the exact energy eigen-functions are expected to be larger than our variationalsolutions. In this work, no bound trions states with ananti-symmetric wavefunction were found even for van-ishingly small electron densities. In addition, no boundtrion states were found in which the CB hole was notbound to the two CB electrons and the VB hole.

Fig. 5 shows that the exciton and the larger trion radiidecrease with the electron density because phase spaceblocking inside the Fermi surface causes the wavefunc-tions to spread out more in the momentum space2. Forelectron densities higher than ∼ 5× 1012 cm−2, the twotrion radii are almost identical and approximately equalto 0.9 nm.

B. Optical Conductivity Spectra

Fig. 6 shows the calculated real part of the optical con-ductivity σ~Q=0(ω) plotted for different electron densities

for monolayer 2D MoSe2 on SiO2. The expression in (11)has been used to generate the plots in Fig. 6. In our opti-cal conductivity calculations, we have included only the

FIG. 6: Calculated real part of the optical conductivity,σ~Q=0(ω), is plotted for different electron densities for mono-layer 2D MoSe2 on SiO2. The spectra are all normalized topeak optical conductivity value at zero electron density. T =5K. The frequency axis is offset by the exciton eigenenergyEex

0 ( ~Q, s) of the two-body Schrodinger equation.

T = 5K

FIG. 7: Calculated energy difference between the two domi-nant peaks in the optical absorption spectra in Fig. 6 is plot-ted as a function of the Fermi energy EF for monolayer 2DMoSe2 on SiO2. The dashed line has unit slope and showsthat the calculated energy difference varies as EF at highelectron densities.

lowest energy bound exciton and bound trion states, aswell as the unbound trion states that describe exciton-electron scattering for the lowest energy bound excitonstate in the Born approximation, as discussed in Sec. II F.The values of γex and γtr were both chosen to be 4 meV.

The spectra shows two distinct peaks. At small elec-tron densities, the higher energy peak dominates and car-ries all the spectral weight. As the electron density in-creases, the spectral weight shifts from the higher energypeak to the lower energy peak. The higher and lowerenergy peaks have also been called repulsive and attrac-tive exciton-polaron peaks, respectively, by Efimkin et

10

T = 5K

FIG. 8: The maximum optical conductivity (real part) valuesof the two absorption peaks in Fig. 6, normalized to the max-imum optical conductivity value at zero electron density, areplotted as a function of the electron density for monolayer 2DMoSe2 on SiO2.

al.8. Very often in the literature they are just referred toas the exciton peaks and the trion peak, respectively1,5.These two peaks arise from the Coulomb coupling of theexcitons and the trions, as discussed earlier in this paper.For very small electron densities, their energies coincidewith those of excitons and trions as obtained from thetwo-body and the four-body Schrodinger equations. Thecoupling between the excitons and the trions, described

by the matrix elements Mm,n( ~Q, s, s′) in (13), increaseswith the electron density and, therefore, the energy dif-ference between the two peaks in the absorption spectraalso increases with the electron density. Fig. 7 plots thisenergy difference as a function of the Fermi energy EFfor monolayer 2D MoSe2 on SiO2. The dashed line hasunit slope and shows that the calculated energy differ-ence varies approximately as EF at high electron densi-ties. Fig. 6 shows that as the electron density increases,the lower energy peak shifts down to lower energies muchmore than the upward motion of the higher energy peak.This happens because the continuum of exciton-electronscattering states lies just above the higher energy peakand prevent the higher energy peak from moving upwardstoo much. Fig. 8 plots the peak optical conductivity(real part) of the two absorption peaks, normalized tothe peak optical conductivity at zero electron density,as a function of the electron density. The lower energypeak begins to dominate when the electron density ex-ceeds 6 × 1012 cm−2. As the electron density increases,the higher energy peak also loses spectral weight to thebroad continuum of single electron-hole pair excitationsfrom exciton-electron scattering. This results not onlyin the broadening of the higher energy peak but also inthe appearance of a broad pedestal around the base ofthe peak that is more prominent on its higher energyside. The lower energy peak, on the other hand, does notbroaden as the electron density increases. Fig.9 shows the

0 5 10 15 20Electron Density (1012 cm-2)

0

5

10

15

20

25

30

FW

HM

Lin

ewid

th -

2ex

(m

eV)

FIG. 9: The (FWHM) linewidth (with 2γex subtracted) of thehigher energy absorption peak in Fig. 6 is plotted as a functionof the electron density for monolayer 2D MoSe2 on SiO2. Theplotted linewidth with 2γex subtracted displays the linewidthresulting from exciton-electron interactions. The linewidth ofthe lower energy absorption peak in Fig. 6 does not changewith the electron density.

(FWHM) linewidth (with 2γex subtracted) of the higherenergy absorption peak plotted as a function of the elec-tron density for monolayer 2D MoSe2 on SiO2. The plot-ted linewidth with 2γex subtracted displays the linewidthresulting from just exciton-electron interactions. Thecontribution to the linewidth from exciton-electron inter-actions increases almost linearly with the electron density(and the Fermi energy).

The integrated area under the plotted conductivityspectra in Fig. 6 is almost conserved (but not exactlyconserved) as the electron density increases, in agreementwith the sum rule in (17). These observations are all ingood agreement with the experimental results reportedfor two dimensional materials1,5, and with previous the-oretical works8,9,13.

V. CONCLUSION

In this paper, we presented a theoretical model thatexplains the behavior of trions and excitons in doped 2Dmaterials. Coulomb scattering couples the exciton andtrion states in doped materials. This coupling is welldescribed by two coupled Schrodinger equations for ex-citons and trions that we derived using the many bodydensity matrix technique. The calculated optical con-ductivity was shown to explain the prominent features ofthe experimentally measured optical absorption spectraand also satisy the optical conductivity sum rule exactly.The eigensolution of the coupled Schrodinger equations,constructed using a superposition of exciton and trionstates, had the same form as a Fermi polaron state and re-vealed the connection between our approach and polaronphysics. The work presented here will help to clarify thephysics associated with excitons and trions in doped 2D

11

materials.There are still several questions that remain open in

this area. For example, it is not clear if the trioneigenequation has bound states at very high electron den-sities (much larger than ∼ 1013 cm−2). If not, then howdoes the conductivity spectra evolve at very high elec-tron densities? At high enough electron densities, mul-tiple electron-hole pair excitations, ignored in this work,are also expected to become important. Their inclusionis expected to broaden the lower energy absorption peakas well and steal spectral weight from it. Indeed, multi-ple pair excitations have already been shown to play animportant role in Fermi polaron physics in atomic sys-tems34. Finally, the role of Fermi edge affects, whichinvolves multiple electron-hole pair excitations, in thiscontext is not clear but they are also expected to be-come increasingly important at high electron densities.Exploring answers to these questions will be the subjectof future work.

VI. ACKNOWLEDGMENTS

The authors would like to acknowledge helpful dis-cussions with Nick Vamivakas, Minwoo Jung, FrancescoMonticone, Jacob Khurgin, and Gennady Shvets, andsupport from CCMR under NSF grant number DMR-1120296, ONR under grant number N00014-12-1-0072,and NSF EFRI-NewLaw under grant number 1741694.

Appendix A: Appendix: Hermitian Two-BodyExciton Schrodinger Equation

Equation (5), without the first term and the last twoterms on the RHS, is the standard eigenvalue equationfor excitons22,23. However, the equation is not Hermi-tian. It can be converted into a Hermitian equation. We

define P ~Q(~k, s) as P~Q(~k, s)/

√1− fc,s(~k + ~Q). We also

include[1− fc,s(~k + ~Q)

]in the last term on the RHS

in (5) containg the Coulomb potential V (~q). This addedfactor does not show up at this level in the density matrixtechnique but its inclusion ensures the Hermiticity of theset of coupled two-body and four-body Schrodinger equa-tions. Physically, it restricts the phase space for electronscattering just like the first and the second terms on theRHS. With these definitions and changes, we obtain,[

Ec,s(~k + ~Q)−Ev,s(~k) + iγex − hω]P ~Q(~k, s) =

− 1√Ag∗s 〈a†( ~Q)〉

√1− fc,s(~k + ~Q)

+

√1− fc,s(~k + ~Q)

A

∑~q

U(~q)

×P ~Q(~k + ~q, s)

√1− fc,s(~k + ~Q+ ~q)

− 1

A

∑~q,~p,s′

U(~q)

×T c~Q(~k + (ξ + η) ~Q− ξ~p, s; (ξ + η)~p− ξ ~Q− ~q, s′; ~p, s′)√

1− fc,s(~k + ~Q)

+1

A

∑~q,~p,s′

V (~q)

√1− fc,s(~k + ~Q)

×T c~Q(~k + (ξ + η) ~Q− ξ~p+ ~q, s; (ξ + η)~p− ξ ~Q− ~q, s′; ~p, s′)

(A1)

The homogeneous part of the above equation is now aHermitian eigenvalue equation. It has a complete set of

orthonormal eigenfunctions φexn, ~Q

(~k+λh ~Q). In the limit of

very low electron density, when phase space filling effectscan be ignored, and assuming Q << kF , the eigenener-gies of the bound exciton states can be expressed as,

Eexn ( ~Q, s) = Eexn ( ~Q = 0, s) +h2Q2

2mex(A2)

Appendix B: Appendix: Hermitian Four-Body TrionSchrodinger Equation

Equation (8), without the last two terms on the RHS,is a four-body eigenvalue equation for trions. The equa-tion is not Hermitian. It can be converted into a Her-mitian eigenvalue equation with a few approximations.

The term[1− fc,s1(~k1)− fc,s2(~k2)

]on the RHS can be

replaced by[1− fc,s1(~k1)

] [1− fc,s2(~k2)

]. The differ-

ence between the two, −fc,s1(~k1)fc,s2(~k2), stems fromthe fact that the four-body correlation function T c~Q can

be non-zero if the correlations are between electrons out-side the Fermi sea or if they are between holes insidethe Fermi sea. In this work, correlation between holesin the Fermi sea may be ignored since the trion radiiare smaller than the inverse Fermi momentum for elec-tron densities smaller than 2 × 1013 cm−2. Similarly,

one can replace the terms[fc,s2(~p)− fc,s1/2(~k1/2)

]on

the RHS by fc,s2(~p)[1− fc,s1/2(~k1/2)

]. We then define

Tc~Q(~k1, s1;~k2, s2; ~p, s2) as,

Tc~Q(~k1, s1;~k2, s2; ~p, s2) =

T c~Q(~k1, s1;~k2, s2; ~p, s2)√fc,s2(~p)

[1− fc,s1(~k1)

] [1− fc,s2(~k2)

] (B1)

With the above approximations and definitions, we ob-tain,[

Ec,s1(~k1) + Ec,s2(~k2)−Ev,s1(~k1 + ~k2 − ( ~Q+ ~p))

12

−Ec,s2(~p) + iγtr − hω]Tc~Q(~k1, s1;~k2, s2; ~p, s2) =

−

√[1− fc,s1(~k1)

] [1− fc,s2(~k2)

]A

∑~q

V (~q)×

Tc~Q(~k1 + ~q, s1;~k2 − ~q, s2; ~p, s2)×√[

1− fc,s1(~k1 + ~q)] [

1− fc,s2(~k2 − ~q)]

+

√1− fc,s1(~k1)

A

∑~q

U(~q)Tc~Q(~k1 + ~q, s1;~k2, s2; ~p, s2)×

√1− fc,s1(~k1 + ~q)

+

√1− fc,s2(~k2)

A

∑~q

U(~q)Tc~Q(~k1, s1;~k2 − ~q, s2; ~p, s2)×

√1− fc,s2(~k2 − ~q)

+

√fc,s2(~p)

[1− fc,s1(~k1)

]A

×∑~q

V (~q)Tc~Q(~k1 + (ξ + η)~q, s1;~k2 − ξ~q, s2; ~p+ ~q, s2)×

√fc,s2(~p+ ~q)

[1− fc,s1(~k1 + ~q)

]

+

√fc,s2(~p)

[1− fc,s2(~k2)

]A

×∑~q

V (~q)Tc~Q(~k1 + ξ~q, s1;~k2 − (ξ + η)~q, s2; ~p− ~q, s2)×

√fc,s2(~p− ~q)

[1− fc,s2(~k2 − ~q)

]−√fc,s2(~p)

A×∑

~q

U(~q)Tc~Q(~k1 − ξ~q, s1;~k2 − ξ~q, s2; ~p+ ~q, s2)×

√fc,s2(~p+ ~q)

+

√fc,s2(~p)

[1− fc,s1(~k1)

] [1− fc,s2(~k2)

]A

∑~q

V (~q)×

[P~Q(~k1 − ~Q+ ~q, s1)δ~k2−~q,~p

−P~Q(~k2 − ~Q− ~q, s2)δ~k1+~q,~pδs1,s2

]−√fc,s2(~p)

A

∑~q

U(~q)×P~Q(~k1 − ~Q, s1)√1− fc,s1(~k1)

δ~k2−~q,~p

√1− fc,s2(~k2)

−P~Q(~k2 − ~Q, s2)√

1− fc,s2(~k2)δ~k

1+~q,~pδs1,s2

√1− fc,s1(~k1)

(B2)

The homogeneous part of the above equation is now aHermitian eigenvalue equation. It has a complete set of

orthonormal eigenfunctions φtrm, ~Q

(~k1, s1;~k2, s2; ~p, s2). In

the limit of very low electron density, when phase spacefilling effects can be ignored, and assuming Q << kF , theeigenenergies of the bound trion states can be expressedas,

Etrm( ~Q, s1, s2) = Etrm( ~Q = 0, s1, s2) +h2Q2

2mt(B3)

We should emphasize here that if the eigenstate in theform given in (19) is used as a variational state withthe Hamiltonian given in (1) then the resulting coupledequations for the two-body and four-body correlationswould be identical to the equations (A1) and (B2) givenin the Appendices.

Appendix C: Appendix: Expression forhn,u( ~Q, ~p, ~q, s, s′) in Equation (16)

The expression for hn,u( ~Q, ~p, ~q, s, s′) appearing in (16)is given below,

√2hn,u( ~Q, ~p, ~q, s, s′) =

∫d2~k

(2π)2V (~q)

√1− fc,s′(~p+ ~q)

×√

1− fc,s(~k + λe ~Q)

√1− fc,s(~k + λe ~Q− ~q)

×φex∗n, ~Q−~q(

~k − λh~q) φexn, ~Q(~k)

+u

∫d2~k

(2π)2V (~p+ ~q − λe ~Q− ~k)

√1− fc,s(~p+ ~q)

×√

1− fc,s(~k + λe ~Q)

√1− fc,s′(~k + λe ~Q− ~q)

×φex∗n, ~Q−~q(

~k − λh~q) φexn, ~Q(~k)

−∫

d2~k

(2π)2U(~q)

√1− fc,s′(~p+ ~q)

×φex∗n, ~Q−~q(

~k + λe~q) φexn, ~Q

(~k)

−u∫

d2~k

(2π)2U(~p− λe ~Q− ~k)

√1− fc,s′(~k + λe ~Q)

×φex∗n, ~Q−~q(

~k + λe~q) φexn, ~Q

(~p+ ~q − λe ~Q)

(C1)

Here, u = ±1 depending on whether the exciton-electronscattering state (or the unbound trion state), as given in(14), is symmetric or antisymmetric (consistent with thevalues of s and s′).

13

Appendix D: Appendix: The Importance ofIncluding Exciton-Electron Scattering States

(Unbound Trion States)

An important point that needs to emphasized is thatwithout including exciton-electron scattering, that takesspectral weight away from the higher energy peak asshown in Fig. 6, the peak optical conductivity of thelower energy peak can never exceed the peak conduc-tivity of the higher energy peak. This follows from thebasic physics of two coupled systems and can be seen as

follows. Assuming γex = γtr = γ for simplicity, ~Q = 0,and ignoring exciton-electron scattering states, the polesof the exciton’s Green’s function will be at energies givenby (18) and are found to be,

hω =Eex0 + Etr0

2− iγ±

√(Eex0 − Etr0

2)2 + |M0,0|2 (D1)

The corresponding spectral weights for the lower and thehigher energy peaks in the optical absorption spectrawould be proportional to W and 1 − W , respectively,where,

W =

√(Eex0 − Etr0

2

)2

+ |M0,0|2 −(Eex0 − Etr0

2

)

2

√(Eex0 − Etr0

2

)2

+ |M0,0|2

(D2)

When the electron density is zero, M0,0 is zero, and thespectral weight all lies in the higher energy exciton peakin the optical absorption spectrum. As the electron den-sity increases, the spectral weight begins to shift to thelower energy peak. But even when the electron den-sity, and therefore M0,0, are very large, the value of Wnever exceeds 1/2. Including the contribution of exciton-electron scattering states (or unbound trion states) istherefore necessary in producing the results shown inFig. 6 (where the peak optical conductivity of the lowerenergy peak is shown to become much larger than thepeak optical conductivity of the higher energy peak athigh electron densities) and accurately reproducing theexperimental observations1,5.

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