The trion as an exciton interacting with a carrier

M. Combescot*, O. Betbeder-Matibet

GPS, Universite Denis Diderot and Universite Pierre et Marie Curie, CNRS, Tour 23, 2 place Jussieu, 75251, Paris Cedex 05, France

Received 27 November 2002; accepted 27 January 2003 by B. Jusserand

Abstract

The X2 trion is essentially an electron bound to an exciton. However, due to the composite nature of the exciton, there

is no way to write an exciton–electron interaction potential. We can overcome this difficulty by using a commutation

technique similar to the one we introduced for excitons interacting with excitons, which allows to take exactly into account the

close-to-boson character of the excitons. From it, we can obtain the X2 trion creation operator in terms of excitons and

electrons. We can also derive the X2 trion ladder diagram between an exciton and an electron. These are the basic tools for

future works on many-body effects involving trions.

q 2003 Elsevier Science Ltd. All rights reserved.

PACS: 71.35. 2 y

Keywords: D. Excitons and related phenomena

The stability of semiconductor trions has been predicted

long ago [1,2]. However, their binding energies being

extremely small in bulk materials, clear experimental

evidences [3–5] of these exciton–electron bound states

have been achieved recently only, due to the development of

good semiconductor quantum wells, the reduction of

dimensionality enhancing all binding energies.

It is now possible to study these exciton–electron bound

states as well as their interactions [6–8] with other carriers.

However, many-body effects with trions are even more

subtle than many-body effects with excitons: Being made of

indistinguishable carriers, the interchange of these fermions

with other carriers is quite tricky to handle properly.

We have recently developed a novel procedure [9] to

treat many-body effects between excitons without the help

of any bosonization procedure. It allows to take exactly into

account the possible exchanges between their components

responsible for their close-to-boson character. In this

communication, we first derive a similar procedure for an

exciton interacting with electrons. It generates an exciton–

electron coupling induced by direct Coulomb processes and

an exciton–electron coupling induced by the possible

exchange of the exciton electron with the other electrons,

the interplay between the two giving rise to very many

subtle exchange Coulomb processes.

We then use this commutation technique to get the trion

creation operators in terms of (exact) exciton and electron

and we show how it is possible to hide all tricky exchange

couplings in the prefactors of the trion operators, provided

that we force them to have a very specific invariance.

In a last part, we derive the trion ladder diagram between

an exciton and an electron. Although it ends by being

conceptually similar to the exciton ladder diagrams [10]

between an electron and a hole, its derivation faces at first a

major difficulty due to the composite nature of the exciton.

Indeed, the electrons being indistinguishable, there is no

way to identify the exciton–electron potential responsible

for the trion binding, similar to the Coulomb attracting

potential which exists between electron and hole. In

addition, the spins are known to be unimportant for

excitons—if we neglect electron–hole exchange—while

they are crucial for trions, the singlet and triplet states

having different energy. Since the bare Coulomb scattering

is spin independent, it is not clear at first how the total

electronic spin affects these trion ladder diagrams.

Although we here speak in terms of exciton and electron,

the extension of the present work to Xþ trions with electron

replaced by hole is formally straightforward, although quite

0038-1098/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0038-1098(03)00103-0

Solid State Communications 126 (2003) 687–691

www.elsevier.com/locate/ssc

* Corresponding author. Fax: þ33-1-43-54-28-78.

E-mail address: combescot@gps.jussieu.fr (M. Combescot).

heavy due to the valence band degeneracy—and its resulting

non-diagonal heavy hole–light hole Coulomb interaction.

We wish to stress that the purpose of this work is not to

find a new clever way to estimate the trion binding energies.

The wide literature on H2 or H2þ in atomic physics [11]

already provides much more accuracy than needed in Solid

State Physics. What we want to do here, is to put the trion

into the general framework of an (exact) exciton interacting

with indistinguishable carriers, in order to provide con-

venient tools for further works on many-body effects

involving trions.

1. Commutation technique for an exciton interacting

with electrons

Following the commutation technique we developed for

excitons interacting with excitons, in the case of excitons

interacting with electrons, we are led to introduce the two

parameters Jdirn0k0nk and Ln0k0nk defined as

½V†n;sn

; a†k;s� ¼

Xn0;k0

Jdirn0k0nkB†

n0 ;sna†

k0 ;s; ð1Þ

½Dn0;sn0 ;n;sn; a†

k;s� ¼Xk0

dsn0 ;sLn0k0nka†

k0 ;sn: ð2Þ

The ‘creation potential’ V†n;sn

and the deviation-from-boson

operator Dn0 ;sn0 ;n;snare the ones defined as for excitons

interacting with excitons [9], namely ½H;B†n;sn

� ¼ EnB†n;sn

þ

V†n;sn

and ½Bn0 ;sn0;B†

n;sn� ¼ dn;n0dsn ;sn0

2 Dn0 ;sn0 ;n;sn: H is the

exact semiconductor Hamiltonian and B†n;sn

is the creation

operator of the exciton n ¼ ðnn;QnÞ; with energy En ¼

ennþ "2Q2

n=2ðme þ mhÞ and electron spin sn: (The hole

‘spin’ being unimportant for trions, we drop it to simplify

the notations). B†n;sn

is linked to the electron and hole

creation operators a†k;s and b†

k by

B†n;sn

¼X

p

kplwnnlb†

2pþahQna†

pþaeQn ;sn; ð3Þ

b†kh

a†ke ;s

¼Xn

kwnlahke 2 aekhlB†n;keþkh ;s

; ð4Þ

where ae ¼ 1 2 ah ¼ me=ðme þ mhÞ and kklwnl is the

exciton relative motion wave function in k space. By

using the explicit expression of V†n;sn

deduced from its

definition [9], Eq. (1) leads to

Jdirn0k0nk ¼ dQn0 þk0 ;QnþkWnn0 nn

ðQn0 2 QnÞ; ð5Þ

Wn0nðqÞ ¼ Vqkwn0 leiahq·r 2 e2iaeq·rlwnl; ð6Þ

Vq being the Fourier transform of the Coulomb potential.

Wn0nðqÞ characterizes the scattering of a n exciton into a n0

state under a q Coulomb excitation. For physical under-

standing, let us mention that Jdirn0k0nk also reads

Jdirn0k0nk ¼

ðdre dre0 drhf

pn0 ðre; rhÞf

pk0 ðre0 Þ

�e2

lre0 2 rel2

e2

lre0 2 rhl

" #fnðre; rhÞfkðre0 Þ; ð7Þ

where fkðrÞ ¼ krlkl ¼ eik·r=ffiffiffiV

pis the free-particle wave

function while fnðre; rhÞ ¼ krehlwnnlkRehlQnl; with reh ¼

re 2 rh and Reh ¼ ðmere þ mhrhÞ=ðme þ mhÞ; is the total

wave function of the n exciton: Jdirn0k0nk thus corresponds to

the direct scattering of a ðn; kÞ exciton–electron state into a

ðn0;k0Þ state induced by the Coulomb interactions between

the exciton and the electron, when the n and n0 excitons are

made with the same electron–hole pair ðe; hÞ:

If we turn to Ln0k0nk; Eqs. (2) and (3) and the explicit

expression of Dn0;s0;n;s deduced from its definition [9] lead to

Ln0k0nk ¼ dK0 ;KLnn0 ;p0;nn ;p

; ð8Þ

Ln0p0np ¼ kwn0 lp þ aep0lkp0 þ aeplwnl; ð9Þ

with k ¼ p þ beK;Qn ¼ 2p 1 bxK and be ¼ 1 2 bx ¼

me=ð2me þ mhÞ: We can mention that Ln0k0nk also reads

Ln0k0nk ¼ð

dre dre0 drhfpn0 ðre; rhÞf

pk0 ðre0 Þfnðre0 ; rhÞfkðreÞ; ð10Þ

which clearly shows that the ðn; kÞ and ðn0; k0Þ states are

coupled by Ln0k0nk due to their exchange of electrons,

independently from any Coulomb process. This possible

exchange of electrons also leads to

B†n;sa†

k;s ¼ 2Xn0;k0

Ln0k0nkB†n0;s

a†k0;s

; ð11Þ

while two exchanges reduce to identity:Xn00 ;k00

Ln0k0n00k00Ln00k00nk ¼ dnn0dkk0 : ð12Þ

2. Trion creation operators

The X2 trions being made of two electrons and one hole,

their creation operators a priori write in terms of b†kh

a†ke

a†ke0:

According to Eq. (4), they can also be written in terms of

B†na†

k; with Qn þ k being equal to the trion total momentum

Ki:

Let us look for these trion creation operators as

T†i;S;Sz

¼Xn;p

f ðhi;SÞn;p T†

n;p;Ki;S;Sz; ð13Þ

where i stands for ðhi;KiÞ; and the T†s are the

M. Combescot, O. Betbeder-Matibet / Solid State Communications 126 (2003) 687–691688

exciton–electron creation operators defined by

T†n;p;K;1;^1 ¼ B†

n;2pþbxK;^a†pþbeK;^;

T†n;p;K;S;0 ¼ ðB†

n;2pþbxK;þa†pþbeK;2

2 ð21ÞSB†n;2pþbxK;2a†

pþbeK;þÞ=ffiffi2

p;

ð14Þ

with S ¼ ð0; 1Þ: Due to Eqs. (8) and (11), these operators

are such that

T†n;p;K;S;Sz

¼ ð21ÞSXn0 ;p0

Ln0p0npT†n0 ;p0 ;K;S;Sz

; ð15Þ

while, due to the indistinguishability of the electrons,

the corresponding exciton–electron states do not form

an orthogonal basis, since

kvlTn0 ;p0 ;K0;S0 ;S0zT†

n;p;K;S;Szlvl

¼ dK0 ;KdS0 ;SdS0z ;Szðdn0;ndp0 ;p þ ð21ÞSLn0p0npÞ; ð16Þ

We can use Eq. (15) to replace the f prefactors in Eq.

(13) by F defined as

Fðhi;SÞn;p ¼

1

2f ðhi;SÞn;p þ ð21ÞS

Xn0 ;p0

Lnpn0p0 fðhi;SÞ

n0;p0

0@

1A; ð17Þ

so that, due to Eq. (12), these prefactors now verify

Fðhi;SÞn;p ¼ ð21ÞS

Xn0 ;p0

Lnpn0p0Fðhi;SÞ

n0;p0 ; ð18Þ

which just states that they stay invariant under the

possible electron exchange corresponding to Eq. (11).

From Eq. (13), with f replaced by F; and Eqs. (16) and

(18), one can easily check that

kvlTn;p;K;S;SzT†hi ;Ki;Si ;Siz

lvl ¼ 2dS;SidSz ;Siz

dK;KiF ðhi;SÞ

n;p : ð19Þ

If we now enforce T†i;S;Sz

lvl to be a trion, i.e. an

eigenstate of H; we must have

kvlTn;p;Ki;S;SzðH 2 1i;SÞT

†i;S;Sz

lvl ¼ 0: ð20Þ

By using Eqs. (1), (5), (14) and (19), we then derive the

‘Schrodinger equation’ fulfilled by the trion prefactors

en þ"2p2

2mt

2 Ehi;S

!Fðhi;SÞn;p

þXn0 ;p0

Wnn0 ð2p þ p0ÞFðhi;SÞ

n0 ;p0 ¼ 0;

ð21Þ

where we have set Ei;S ¼ 1hi;Sþ "2K2

i =2ð2me þ mhÞ; mt

being the exciton–electron relative motion mass, m21t ¼

m21e þ ðme þ mhÞ

21:

It can be surprising to have only direct Coulomb

scatterings, through Wnn0 ð2p þ p0Þ; appearing in the trion

Schrodinger Eq. (21). Exchange processes between elec-

trons, through Lnpn0p0 ; seem absent. They are actually hidden

in the Fss, more precisely in their invariance relation (18).

It is possible to relate the prefactors F of the trion

operator to the trion orbital wave function in a quite easy

way: Indeed Eqs. (13–14) lead to write this wave function as

Ci;Sðre; re0 ; rhÞ ¼ kRee0hlKil½cðhi;SÞðreh; ue0;ehÞ

þð21ÞSðe $ e0Þ�=2

; kRee0hlKilcðhi;SÞkreh; ue0;ehl;

ð22Þ

where we have set

cðn;sÞðr; uÞ ¼ffiffi2

p Xv;p

Fðh;sÞn;p krlwnlkulpl ð23Þ

Ree0h ¼ ðmere þ me0re0 þ mhrhÞ=ð2me þ mhÞ is the trion

center of mass position, and ue0Fh ¼ re0 2 Reh is the distance

between the electron e0 and the center of mass of the exciton

made with ðe; hÞ: The two terms of the first line of Eq. (22) are

indeed equal due to Eqs. (9) and (18). Consequently, the

prefactors of the trion expansion on the exciton–electron

operators are given by

ffiffi2

pFðh;sÞn;p ¼ cðh;sÞ

n;p ¼ð

dr dukwnlrlkplulcðn;sÞðr;uÞ ð24Þ

cðh;sÞn;p is thus the generalized Fourier transform “in the exciton

sense” of the relative motion wave function cðn;sÞðr; uÞ. In the

standard Fourier transform, kwnlrlwould be replaced by kp0lrl.

3. Trion ladder diagram

It is widely known [10] that excitons correspond to

‘ladder diagrams’ between one electron and one hole

propagators. By writing the semiconductor Hamiltonian as

H ¼ H0 þ V ; these diagrams simply result from the

iteration of ða 2 HÞ21 ¼ ða 2 H0Þ21 þ ða 2 HÞ21Vða 2

H0Þ21 acting on one free electron–hole pair.

For trions, the problem is much more tricky. The most

naıve idea is to look for the diagrammatic expansion of the

trion in terms of two electron and one hole propagators, with

all possible electron–electron and electron–hole inter-

actions between them. In one of our previous works on

interacting excitons, namely an exciton interacting with a

distant metal [10,12,13], we have, however, shown that

these usual free electron and hole diagrams turn out to be

extremely complicated when compared to the ‘exciton

diagrams’, in which appear exciton propagators instead of

free electron and free hole propagators. This exciton

diagram procedure, however, faces a major difficulty at

first, since, due to the composite nature of the exciton, there

is no way to write the exact semiconductor Hamiltonian H

as H00 þ V 0; with V 0 being an exciton-electron potential, so

that there is no simple iteration of ða 2 HÞ21 in terms of a V 0

interaction.

It is, however, possible to overcome this difficulty by

using our ‘commutation technique’. From the definition of

M. Combescot, O. Betbeder-Matibet / Solid State Communications 126 (2003) 687–691 689

the creation potential V†n;sn

; it is easy to show that

1

a 2 HB†

n;sn¼ B†

n;sn

1

a 2 H 2 En

þ1

a 2 HV†

n;sn

1

a 2 H 2 En

: ð25Þ

With a ¼ Vþ ih; this equation, along with Eqs. (1) and (5),

gives ðV2 H þ ihÞ21 acting on one exciton–electron pair

as

1

V2 H þ ihT†

n;p;K;S;Szlvl

¼1

V2 En;p;K þ ih

"T†

n;p;K;S;Szlvl

þXn0 ;p0

Wn0nð2p0 þ pÞ1

V2 H þ ihT†

n0;p0 ;K;S;Szlvl#;

ð26Þ

where En;p;K ¼ en þ "2p2=2mt þ "2K2=2ð2me þ mhÞ is the

energy of the free exciton–electron pair.

The iteration of the above equation leads to

1

V2 H þ ihT†

n;p;K;S;Szlvl

¼Xn0 ;p0

An0p0npðV;KÞT†n0 ;p0;K;S;Sz

lvl; ð27Þ

where An0p0npðV;KÞ given by

An0p0npðV;KÞ ¼

"dn0 ;ndp0;p

þ1

V2 En0 ;p0;K þ ih

(Wn0nð2p0 þ pÞ

þXn1 ;p1

Wn0n1ð2p0 þ p1ÞWn1n

ð2p1 þ pÞ

V2 En1 ;p1 ;Kþ ih

þ · · ·

)#

£1

V2 En;p;K þ ih; ð28Þ

can be formally rewritten as

An0p0npðV;KÞ

¼ dn0 ;ndp0 ;p þ~Wn0p0npðV;KÞ

V2 En0 ;p0 ;K þ ih

" #1

V2 En;p;K þ ih: ð29Þ

~Wn0p0npðV;KÞ appears as a ‘renormalized exciton–electron

interaction’. It verifies the integral equation

~Wn0p0npðV;KÞ ¼ Wn0nð2p0 þ pÞ þXn1 ;p1

�~Wn0p0n1p1

ðV;KÞWn1nð2p1 þ pÞ

V2 En1 ;p1 ;Kþ ih

: ð30Þ

Its iteration is shown in Fig. 1b. Before going further, let us

note that

1

V2 En1 ;p1 ;Kþ ih

¼ð idv1

2pgeðVþ v1; p1 þ beKÞgxð2v1; n1;2p1 þ bxKÞ

¼ GxeðV; n1; p1;KÞ; ð31Þ

with geðv; kÞ ¼ ðv2 "2k2=2me þ ihÞ21 being the usual free

electron Green’s function for an empty Fermi sea, while

gxðv; nÞ ¼ ðv2 En þ ihÞ21 is the free boson – exciton

Green’s function, as if the excitons were non-interacting

bosons, i.e. if all the J ’s and L’s were set equal to zero. Gxe

thus appears as the propagator of an exciton–electron pair.

It is quite similar to the electron–hole pair propagator Geh

appearing in exciton diagrams (see for example Eq. 2.10 of

Ref. [10]).

The simplest way to obtain An0p0npðV;KÞ is to insert the

trion closure relation between the two operators of the LHS

of Eq. (27) and to project this equation over kvlTn00 ;p00 ;K;S;Sz:

By using Eqs. (16,19,24), we get

Xhi

4cðhi;SÞ

n00 ;p00 ðcðhi;SÞn;p Þp

V2 Ehi ;K;S þ ih

¼ An00p00npðV;KÞ þ ð21ÞSXn0;p0

Ln00p00n0p0An0p0npðV;KÞ: ð32Þ

An00p00npðV;KÞ is then obtained by adding the above

Fig. 1. (a) Direct Coulomb scattering between one ‘free’ exciton and

one free electron. (b) Renormalized free exciton–electron inter-

action as given by the integral equation (30). It corresponds to the

sums of one, two,…ladder rungs between one ‘free’ exciton and one

free electron.

M. Combescot, O. Betbeder-Matibet / Solid State Communications 126 (2003) 687–691690

equations for S ¼ 0 and S ¼ 1: Using Eq. (29), we thus get

the renormalized exciton–electron interaction, i.e. the sum

of the exciton–electron ladder ‘rungs’, as

j ~Wn0p0npðV;KÞ ¼

"2 dn0;ndp0 ;p þ

1

GxeðV; n0; p0;KÞ

Xhi ;S

£cðhi;SÞ

n0 ;p0 ðcðhi;SÞn;p Þp

V2 Ehi ;K;S þ ih

#1

GxeðV; n; p;KÞ: ð33Þ

This result is quite similar to the ‘renormalized electron–

hole Coulomb interaction’ appearing in electron–hole

ladder diagrams, as given for example in Eq. (2.18) of

Ref. [10].

4. Conclusion

This work relies on a new commutation technique for

excitons interacting with electrons which takes exactly into

account the possible exchange between carriers, i.e. the

close-to-boson character of the excitons. Using it, we have

generated the trion creation operator in terms of excitons

and electrons. We have also generated the exciton–electron

ladder diagram associated to these trions, in terms of exciton

propagators and electron propagators.

Just as the exciton creation operator in terms of electrons

and holes or the exciton ladder diagram between one

electron and one hole propagators, do not help to solve the

Schrodinger equation for one exciton, but allow to deal with

many-body effects involving excitons, the trion creation

operator and the trion ladder diagram derived here are of no

use to solve the Schrodinger equation for one trion, but

constitute the appropriate tools for further works on many-

body effects with trions.

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