Electronic structure and absorption spectrum of biexciton obtained by using exciton basis

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by  Shiue-Yuan Shiau

Electronic structure and absorption spectrum of biexciton obtained by using exciton
basis
Shiue-Yuan Shiau1,2, Monique Combescot3 and Yia-Chung Chang1∗
1Research Center for Applied Sciences, Academia Sinica, Taipei, 115 Taiwan
2 Department of Physics and National Center for Theoretical Sciences,
National Cheng Kung University, Tainan, 701 Taiwan and
3Institut des NanoSciences de Paris, Universit´e Pierre et Marie Curie, CNRS, 4 place Jussieu, 75005 Paris
(Dated: January 31, 2013)
We approach the biexciton Schr¨odinger equation not through the free-carrier basis as usually
3 done,butthroughthefree-excitonbasis,exciton-excitoninteractionsbeingtreatedaccordingtothe
1 recentlydevelopedcompositebosonmany-bodyformalismwhichallowsanexacthandlingofcarrier
0 exchange between excitons, as induced by the Pauli exclusion principle. We numerically solve the
2 resulting biexciton Schr¨odinger equation with the exciton levels restricted to the ground state and
wederivethebiexcitongroundstateaswellastheboundandunboundexcitedstatesasafunction
n
of hole-to-electron mass ratio. The biexciton ground-state energy we find, agrees reasonably well
a
J with variational results. Next, we use the obtained biexciton wave functions to calculate optical
absorption in the presence of a dilute exciton gas in quantum well. We find a small asymmetric
0
peak with a characteristic low-energy tail, identified with the biexciton ground state, and a set of
3
large peaksassociated with biexciton unboundstates, i.e., exciton-exciton scattering states.

]
l
e
I. INTRODUCTION aresmallerthanusualexcitonline width; so,trionpeaks
-
r fall on the side of exciton lines. This small trion binding
t
s energy can be physically understood by seeing the trion
The existence of composite particles in semiconduc-
.
at tors has been predicted long ago1. Bound states made as an electron or a hole bound to an exciton. The effec-
tiveattractionthenisdipole-likewhichmakesinteraction
m of conduction band electrons and valence band holes re-
between exciton and free carrier much weaker than be-
sultfromthe Coulombattractionbetweenthese carriers.
-
tweenelementarycharges. Aclearsignatureoftrionshas
d To name the simplest ones, these bound states are exci-
n tons(X)madeofoneelectronplusonehole,trions(X∓) been obtained recently only in semiconductor quantum
o wells4–7, the reduction of dimensionality increasing all
madeoftwoelectronsplusonehole,ortwoholesplusone
c binding energies as seen from the exciton energy which
electron,andbiexcitons(XX)madeoftwoelectronsplus
[ twoholes. Moreexoticcompositeobjectsmadeofalarge goes from R(3D) in 3D to R(2D) = 4R(3D) in 2D, and to
X X X
1 number of correlatedfermionpairs, called“electron-hole infinity in 1D.
v
droplets”,alsoexist,withacarrierdensityfarlargerthan What makes bound trions hard to observe has also to
6
the one in which excitons are formed. be traced back to their oscillator strength which is one
6
2 The simplest composite particle, the exciton, made trionvolumedividedby onesamplevolumesmallerthan
7 of one conduction electron and one valence hole, is the exciton oscillator strength8. This drastic reduction
. very similar to an Hydrogen atom, if we neglect in- factorcanbephysicallyunderstoodastheprobabilityfor
1
terband Coulomb processes2. Exciton has bound and a photocreatedexciton to localize over a trion volume, a
0
3 unbound (scattering) states which can be analytically free carrier initially spread over the whole sample. As
1 determined3. Exciton bound states appear as large nar- a result, the trion peak commonly observed in heavily
: row peaks in the photon absorption spectrum, the peak doped samples in which a large electron density exists,
v
i intensity depending on the so-called “exciton oscillator should not be interpreted as a signature of elementary
X strength”. The reason bound exciton peaks are easy to trion, but rather as an exciton interacting in a coherent
r observe is twofold: first, when a plane-wavephoton with way with all the electrons present in the sample. Such
a
momentum Q transforms into a bound exciton, the cou- a many-body effect is singular and leads to a broad ab-
pling is quite good because the center-of-mass motion of sorption line, as experimentally shown9.
the bound exciton also is a plane wave with same mo- Mathematically, the derivation of trion eigenstates
mentum Q. Second, excitons, made of an even number amounts to solving a three-body problem which has no
of fermions, have a bosonic nature; so, they can be piled known analytical solution. Attempts to tackle such a
up all at the same energy, the absorption peak intensity problem inevitably rely on some truncation scheme in
increasing linearly with the density of excitons already addition to heavy numerics in order to possibly obtain
present in the sample. satisfactory results. Recently, we showed how, using a
Observation of composite particles like trions is more physicallyrelevantviewpoint,we canapproachone trion
complex. It has been hampered for quite a long time, as an exciton interacting with an electron10. We have
partly due to the smallbinding energies that trions have constructed a Schr¨odinger equation for trion using the
in bulk samples, at best one order of magnitude smaller electron-exciton basis and solved it by restricting this
than the exciton binding energy. Such binding energies basis to the first few low-lying exciton levels,—which is
2
reasonablesincetheenergyscaleforexcitationoftheex- indistinguisability of the fermionic components of these
citon internal motion, of the order of one Rydberg, is composite particles. By restricting the exciton levels to
much larger than the other energy scales. The resulting the ground state only, it becomes possible to numeri-
trionbindingenergieswefindin2Dand3Dagreereason- cally solve the biexciton Schr¨odinger equation quite eas-
ablywellwiththemostaccuratevariationalresults. One ily. The values we obtain for the biexciton ground state
important advantage of this approach is to allow reach- energiesin2Dand3Dareingoodagreementwith varia-
ing the triongroundand excitedstates onequal footing, tionalresults. Oneimportantadvantageoftheprocedure
these excited states being out of reach from standard is that the biexciton Schr¨odinger equation can be cast
variational methods. into a generalized eigenvalue problem; so, we can reach
Following Lampert’s prediction1, an even more com- bound and unbound excitedstates atonce, with a single
plexcompositeparticle,thebiexciton,hasbeenobserved matrix diagonalization.
inbulk materialssuchasCuCl [Ref.11], Cu O[Ref. 12], Inasecondstep,weusetheobtainedbiexcitonrelative
2
andAgBr[Ref.13,14]. Morerecently,thebiexcitonbind- motion wave functions to calculate the photon absorp-
ing energy has been measured in GaAs quantum well15 tionspectruminquantumwells,assumedtobeexact2D
and found to be one order of magnitude larger than in systems. Instead of considering biexciton as generated
bulk samples, a result supported by calculations done through two-photon absorption23–26, we here study one
one year later16. Since then, other aspects of biexcitons photocreated exciton interacting with a dilute exciton
in confined structures, such as optical enhancement in gas. Usingsimilarargumentsasthoseweusedforbound
biexciton formation17,18 and the influence of dimension- trion,wefindthatthebiexcitonoscillatorstrengthisone
ality on the biexciton binding15,19,20, have been studied. biexciton volume divided by one sample volume smaller
Biexcitons in quantum wires have also been reported21. than the exciton oscillator strength. This would make
observingthebiexcitonlineverydifficult. However,biex-
In view of the successful application of the compos-
ite boson many-body formalism to trion10, we, in this citons, like excitons, are boson-like particles: They can
thus be packed up all at the same energy level. As a re-
work,go on along the same line to tackle biexciton. The
biexcitonproblema priori isanevenmorecomplexfour- sult, the biexcitonabsorptionline increaseslinearlywith
excitondensityprovidedthatthedensityislowenoughto
body problem, with two electrons (e ,e ) and two holes
1 2
possiblyneglectmany-bodyeffectsbetweenthephotocre-
(h ,h ) involved. The idea is to start with two electron-
1 2
ated exciton and the free excitons present in the sam-
holepairsboundintotwoexcitonsbythestrongelectron-
ple. Thecalculatedphotonabsorptionspectrumshowsa
holeCoulombattraction. Theexciton-excitonattraction,
smallpeak,withacharacteristiclow-energytail,originat-
although quite weak since it essentially is dipole-like, al-
ing from the biexciton molecular state, and large peaks
lows two free excitons to form a molecule with a binding
centered on the exciton ground levels, which are asso-
energysubstantiallysmallerthantheexcitonbindingen-
ciated with exciton-exciton scattering states. Both, the
ergy. To approach a system made of two electrons plus
bound and unbound biexciton peak intensities decrease
two holes, the exciton basis is physically quite appeal-
when the temperature increases. It can also be shown
ing because the strong exciton binding energy is then
that the intensity of the absorption line for one biexci-
included into the problem at the zeroth order. We are
ton made from a photocreated exciton and anexciton of
left with solving a Schr¨odinger equation for the weaker
the exciton gas, increases linearly with photon number
biexcitonbindingenergy. Inthisapproach,thefour-body
and exciton density. This is in contrast to the biexciton
system is pictured as two interacting excitons: one exci-
absorption line associated with two-photon absorption
tonismadeofthe(e ,h )pair,whiletheotherismadeof
1 1
which increases quadratically with photon number and
(e ,h )pair,thesetwoexcitonshoweverexchangingtheir
2 2
thus becomes dominant at high laser intensity.
carrierstobepossiblymadeof(e ,h )and(e ,h ). Such
1 2 2 1
a two-exciton picture could be thought, at first sight, to The present paper is organized as follows:
lead to an easy problem because of the weak exciton- In Sec. II, we briefly discuss the relation which exists
exciton attraction compared with the strong electron- between biexciton written in the free-carrier basis, and
hole attraction. However,this weakattractionisthe one biexciton written in the exciton basis. We also intro-
responsiblefortwoexcitonstobeboundintoamolecule. duce the four commutators necessaryto properly handle
So,inordertoreachboundstatesandfindtheassociated many-body effects involving composite excitons.
poles, this exciton-exciton interaction has to be treated In Sec. III, we study triplet biexciton states made of
in an exact way. same-spin electrons and same-spin holes and we derive
With this goal in mind, we here construct a biexciton the corresponding Schr¨odinger equation.
Schr¨odinger equation in terms of the exciton basis us- InSec.IV,westudysingletandtripletbiexcitonstates
ing the recently developed composite boson many-body made of opposite-spin electrons and opposite-spin holes.
theory22. In much the same spirit as Feynman diagrams These biexciton states are first constructed in terms of
for elementary particles, this theory takes advantage of two free electrons plus two free holes, and then in terms
“shiva diagrams” to visually identify many-body effects of two free excitons, in order to reveal important par-
involvedamongcompositeparticles. Itmoreoverenables ity relations. We then concentrate on singlet biexciton
treating exactly carrier exchange which results from the states with center-of-mass momentum equal to zero and
3
p′ p p′−αQ′ p−αQ
n e e j n=(ν ,−Q′) e e j=(ν ,−Q)
n j
p′ p −p′−αQ′ −p−αQ
h h h h
k′ k −k′+αQ′ −k+αQ
h h h h
m i m=(ν ,Q′) i =(ν,Q)
m i
k′ k k′+αQ′ k+αQ
e e e e
(a) FIG. 2: Pauli scattering λ(cid:0)((ννnm,−,QQ′′))((ννji,,−QQ))(cid:1) for carrier ex-
k′ p change between an exciton i = (νi,Q) and an exciton j =
n h h j (νj,−Q)(seeEq.(B.4)). Theexciton(νi,Q)isalinearcom-
bination of electron-hole pair (k+αeQ,−k+αhQ) where
ke′ pe αe =1−αh =me/(me+mh) (see Eq. (A.7)).
band. Theusualbasisforsuchabiexcitonsystemisthen
p′ k
e e made of states with two free electrons and twofree holes
m i
p′ k a† a† b† b† v . (1)
h h ke1,s1 ke2,s2 kh1,m1 kh2,m2| i
Totransformthis free-carrierbasisinto anexcitonbasis,
(b)
we make use of the relations which exist between free
electron-holepaircreationoperatorsandexcitoncreation
FIG. 1: (a) λh(cid:0)mn ji(cid:1) = λ(cid:0)mn ji(cid:1) for hole exchange, the exci- operators,namely,
tons m and i having the same electron. (b) Pauli scattering
λe(cid:0)mn ji(cid:1)=λ(cid:0)mn ij(cid:1)forelectron exchange,theexcitonsmand Bi†;simi = a†ke,sib†kh,mihkhke|ii, (2)
i havingthe same hole. kXekh
a† b† = B† ik k , (3)
ke,si kh,mi i;simih | e hi
we restrict the exciton levels to the ground state. This Xi
nicely reduces the biexciton Schr¨odinger equation to a
where i denotes the i exciton state. Using Eq. (3),
1D integral equation. | i
we can rewrite the two-free-electron-hole pair states of
In Sec. V, we numerically solve this 1D integral equa-
Eq. (1) in terms of exciton states as
tion to obtain the biexciton binding energies for the
ground and excited states as a function of hole-to- B† B† v , (4)
electron mass ratio. We also show the biexciton relative i;simi j;sjmj| i
motionwavefunctionsfortheboundstateaswellasfora with (s ,s ) = (s ,s ) and (m ,m ) = (m ,m ).
i j 1 2 i j 1 2
fewunboundstates. Finally,weusethesewavefunctions Note that the basis made of [(s ,m );(s ,m )] and
1 1 2 2
to calculate the photonabsorptionspectrumin the pres- [(s ,m );(s ,m )]areequallyvalid. Thismeansthat,for
1 2 2 1
enceofadilute excitongasforvariouslowtemperatures. s =s andm =m ,thebasiscanbemadeeitheroftwo
1 2 1 2
In the last section, we conclude. bri6ght excitons,6 ( 1/2,3/2) and (1/2, 3/2), or of two
− −
dark excitons, (1/2,3/2) and ( 1/2, 3/2), the bright
− −
exciton basis however being more convenient for prob-
II. BIEXCITON ON THE EXCITON BASIS lems dealing with photons. Note that bright and dark
excitons are degenerateif we neglect interbandCoulomb
We consider a system made of two electrons and two processes.
holes in a semiconductor: the electrons carry a spin While the great advantage of the exciton basis is to
s= 1/2whilethe holescarryanangularmomentumm containpartoftheelectron-holeinteraction,actuallythe
±
that we will also call spin. In bulk semiconductors, the strongpartleadingtoexcitonboundstates,itsmaindis-
holeangularmomentumcanbem=( 3/2, 1/2),while advantageistobeovercomplete;asadirectconsequence,
± ±
in narrowquantumwells,it reduces to m= 3/2due to thisbasisisnotorthogonal. Itispossibletoovercomethe
±
the heavy-lighthole energy splitting induced by the well difficulties induced by the non-orthogonality of the ex-
confinement. For simplicity, here we shall neglect the citon basis through the commutation technique recently
role of light holes and consider heavy holes only. Fur- developedforcompositebosonmany-bodyeffects22. The
thermore, we neglect the warping of the semiconductor keys of this formalism rely on just four commutators be-
valencebandandapproximateitbyaspherical,parabolic tween exciton operators B†: Fermion exchanges follow
i
4
fromtwocommutatorswhichread,intheabsenceofspin p′ p p p
degrees of freedom, as n e e j n e e j
p p p′ p
h h h h
q q
hBm,Bi†i = δmi−Dmi, (5) m kh kh i m kh′ kh i
D ,B† = λ n j +λ n j B†. (6) k′ k k k
mi j h m i e m i i e e e e
h i Xn (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) n pe′ pe j n pe pe j
D is called “deviation-from-boson” operator because,
mi p p p′ p
without it, Bi† would reduce to an elementary boson op- h q h h q h
erator. The Pauli scattering λh mn ji corresponds to a kh′ kh kh kh
holeexchangebetweenexcitons(i(cid:0),j),t(cid:1)heexcitonsmand m i m i
ihavingthe sameelectron,asdefinedinEq.(B.1)ofthe ke ke ke′ ke
appendix and shown in the diagram of Fig. 1(a). In the
same way, the Pauli scattering λ n j corresponds to
e m i
an electron exchange, the excitons(cid:0)m a(cid:1)nd i having the FIG. 3: Direct Coulomb scattering ξdir(cid:0)n j(cid:1) between exci-
mi
same hole, as defined in Eq. (B.2) and shown in the dia- ton i and exciton j. The “out” exciton m is made with the
gram of Fig. 1(b). λ n j , which is equal to λ mj , same electron-hole pair as the i exciton. Similarly for exci-
h m i e n i
is often written as λ (cid:0)n j (cid:1)for simplicity. (cid:0) (cid:1) tonsnand j. This exciton-excitonscattering consists of four
m i terms: two repulsive interactions between electrons and be-
The other two co(cid:0)mmu(cid:1)tators that handle fermion-
tweenholes,andtwoattractiveinteractionsbetweenelectron
fermion interactions, are
and hole.
H,B† = E B†+V†, (7)
i i i i
h i III. BIEXCITON MADE OF ELECTRON-HOLE
V†,B† = ξdir n j B† B†. (8) PAIRS WITH SAME SPIN s1 =s2,m1=m2
i j m i m n
h i Xmn (cid:0) (cid:1)
Let us start with triplet biexcitons made of same-spin
The “creation potential” V† generates, through (8), electrons and same-spin holes and drop the spin indices
i
the direct Coulomb scattering ξdir n j of excitons i tomakenotationsofthis sectionlighter. We lookforthe
m i
and j. It consists of four Coulomb(cid:0) pro(cid:1)cesses between biexciton eigenstates
the fermionic components of these two excitons: one
electron-electron repulsion, one hole-hole repulsion, and (H η)Ψ(η) =0 (9)
−E | i
twoelectron-holeattractions,asshowninthediagramof
in the two-free-excitonbasis ij =B†B† v , namely,
Fig. 3. The precise expression of the hole-hole part of | i i j| i
this scattering is given in Eq. (B.5) of the appendix.
These four commutators are used to get the biexciton |Ψ(η)i= φ(ijη)|iji= φ(ijη)Bi†Bj†|vi. (10)
Schr¨odinger equations derived in the next sections. To Xij Xij
include the electron and hole degrees of freedom, let us
Since B†B† =B†B†, we can replace the above prefactor
focus on the two relevant cases: i j j i
(a) Carriers with same spins, s = s and m = m . by (φ(η) +φ(η))/2. The biexciton state then appears as
1 2 1 2 ij ji
This corresponds to triplet states for the electron part (η) (η)
inEq.(10)butwiththesymmetryconditionφ =φ .
(S = 1,Sz = 1) and triplet-like states for the hole ij ji
paert (S =e 3,S±z = 3). The associated orbital wave Equations(7)and(8)allowustorewritethebiexciton
h h ± Schr¨odinger equation (9) as
functions then have to be odd with respect to exchange
ofthetwoelectronsorthetwoholesinordertofulfillthe 0 = φ(η) (E )ij + ξdir sj rs
Pauli exclusion principle. Xij ij h ij −Eη | i Xrs (cid:0)r i(cid:1)| ii
(b) Carriers with opposite spins, s = s and m =
1 2 1
m . The resulting spin configuration−then depends = (E )φ(η)+ ξdir sj φ(η) rs(11)
−on t2he way electrons and holes are linearly combined: Xrs h rs−Eη rs Xij (cid:0)r i(cid:1) ij i| i
We can either have a triplet state for the electron part
(S =1,Sz =0) and a triplet-like state for the hole part with Ers =Er+Es. In the standardcase,i.e., when the
e e
(S = 3,Sz = 0), or a singlet state for the electron part basis is made of orthogonal states, the above equation
(Sh=0,Szh=0)anda singlet-likestate forthe hole part forcesthebrackettobezero. Thesituationismoresubtle
e e
(S =0,Sz =0). This case thus requires a more careful with the exciton basis because, due to Eqs. (5) and (6),
h h the scalar product of two-exciton states reads as
derivationsincetheorbitalwavefunctionfortripletstate
must be odd as in the case (a), but even for the singlet
v B B B†B† v = δ δ λ n j + m n .
cseotnfiogfusirnagtiloetn.stTahtees.biexcitongroundstatebelongstothe h | m n i j| i h mi nj − (cid:0)m i(cid:1)i h ←→(1i2)
5
p p p′ p
n e e j n e e j
n=(ν ,−Q′) p′−αeQ′ p−αeQ j=(ν ,−Q) kh′ ph kh ph
n j q q
−p′−αQ′ −p−αQ
h h p′ k p k
q m h h i m h h i
k k k′ k
e e e e
−k′+αQ′ −k+αQ
m=(νm,Q′) h h i =(νi,Q) n pe pe j n pe′ pe j
k′+αQ′ k+αQ k p k′ p
e e h h h h
q q
p′ k p k
FIG. 4: Part of the direct Coulomb scattering m h h i m h h i
ξdir(cid:0)((ννnm,−,QQ′′))((ννji,,−QQ))(cid:1) between a i = (νi,Q) exciton and ke′ ke ke ke
a j = (νj,−Q) exciton coming from hole-hole repulsion
(second diagram of Fig. 3).
FIG.5: “In”exchange-Coulomb scattering ξin(cid:0)n j(cid:1)between
mi
exciton i and exciton j. The exciton pair exchanges their
By projecting Eq. (11) onto mn, we then find, since holes after Coulomb interactions, while the excitons m and i
h |
φ(η) =φ(η), keep thesame electron.
ij ji
0=(E )φ(η) + ξˆ(η) n j φ(η), (13)
mn−Eη mn m i ij
Xij (cid:0) (cid:1)
p′−αQ′ p−αQ
e e
where ξˆ(η) n j is defined as n=(ν ,−Q′) j=(ν ,−Q)
m i n −p′−αQ′ −p−αQ j
(cid:0) (cid:1) h h
ξˆ(η) n j =ξdir n j ξin n j λ n j (E ). q
m i m i − m i − m i ij −Eη
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (14)
with ξin n j = λ(n s)ξdir sj being “in” −k′+αQ′ −k+αQ
exchange-(cid:0)Cmouil(cid:1)ombscaPttresringmwirthCou(cid:0)lromi(cid:1)binteractions m=(νm,Q′) k′+αhQ′ k+αhQ i =(νi,Q)
taking place betweenthe “in” excitonpair (i,j), i.e., be- e e
fore hole exchange (see the diagram of Fig. 5).
ξˆ(η) n j canappearasaneffective Coulombscatter-
m i
ingbet(cid:0)ween(cid:1)twoexcitonsalthoughitalsodependsonthe FIG. 6: Part of the “in” exchange-Coulomb scattering
ebffieexcctiitvoensceantetregryinEgηc.orTrhesepfiornsdtttowothceosnttarnibduatridoncsomtobitnhais- ξin(cid:0)((ννnm,−,QQ′′))((ννji,,−QQ))(cid:1) between a i = (νi,Q) exciton and a
j = (νj,−Q) exciton, coming from hole-hole repulsion (first
tionofCoulombprocessesappearing,forexample,inthe diagram of figure 5). The exciton pair exchanges their holes
time evolution of two excitons. The third term is more after Coulomb interactions (see Eq. (B.15)), the excitons m
interestingbecauseitonlycomesfromthePauliexclusion and i keepingthe same electron.
principle. Since the associated Pauli scattering λ n j
m i
is dimensionless,it must goalongwith anenergyt(cid:0)o pro(cid:1)-
duceanenergy-likescattering,thisenergyactuallybeing
an energy difference in order to be gap independent. as
We now introduce a similar “out” exchange-
Coulomb scattering ξout n j , defined as ξout n j =
Pbetrwsξedeinr(tmnhers)“λou(cid:0)trs”ji(cid:1)e,xCciot(cid:0)oumnlomip(cid:1)abirint(emra,cnt)ioonnscteakei(cid:0)nxmcghpial(cid:1)nagcee 0=(Emn−Eη)φ(mηn) +Xij ξˆs(yηm) (cid:0)mn ji(cid:1)φi(jη), (16)
has occurred. It is easy to check that ξin n j =
m i
ξout j n ∗. Moreover, the “in” and “out” e(cid:0)xcha(cid:1)nge- the effective Coulomb scattering, defined as
i m
(cid:2)Coulo(cid:0)mb s(cid:1)c(cid:3)atterings are not independent: their differ-
ence is related to Pauli scattering via
1
ξin(cid:0)mn ji(cid:1)−ξout(cid:0)mn ji(cid:1)=(Emn−Eij)λ(cid:0)mn ji(cid:1), (15) ξˆs(yηm) (cid:0)mn ji(cid:1) = ξdir(cid:0)mn ji(cid:1)− 2hξin(cid:0)mn ji(cid:1)+ξout(cid:0)mn ji(cid:1)
as easy to recover from calculating the matrix element +λ n j (E +E 2 ) , (17)
v B B HB†B† v with H acting either on the right or (cid:0)m i(cid:1) mn ij − Eη i
h | m n i j| i
on the left.
ByinsertingEq.(15)intoEq.(14),wecansymmetrize now being symmetrical with respect to the “in” and
the biexcitonSchr¨odingerequation(13) fortriplet states “out” exciton pairs (i,j) and (m,n).
6
IV. BIEXCITON MADE OF ELECTRON-HOLE The biexciton ground state belongs to the set of singlet
PAIRS WITH OPPOSITE SPINS states (S =S =0).
e h
s =−s , m =−m
1 2 1 2
When the electron spins and hole spins are opposite, B. Exciton basis
the derivationofbiexciton singletandtripletstates with
Sz = Sz = 0 requires a more careful analysis of the To write the biexciton on the exciton basis, we use
e h
parityconditioninducedbythefree-carrierfermionicna- Eq. (3) to rewrite the electron-hole state of Eq. (21) in
ture. Tothisend,wefirstconstructbiexcitoneigenstates terms of bright exciton operators. We find
in the free-carrier basis and derive the parity condition
imposed by the Pauli exclusion principle for singlet and Ψ(η) = φ(η,Se,Sh)B† B† v , (23)
| Se,Shi ij i,−1 j,1| i
triplet wave functions. We then use Eq. (3) to rewrite Xij
the biexciton eigenstates in terms of exciton operators
and rederive the parity condition in the exciton basis. where the prefactor φ(η,Se,Sh), defined as
ij
Finally, we write down the biexciton Schr¨odinger equa-
tion. φ(η,Se,Sh) = ik k′ j k′k φ(η,Se,Sh) ,
ij kXe,k′hkX′e,khh | e hih | e hi ke,k′e,kh,k′h
(24)
A. Free-carrier basis
is the biexciton wave function “in the exciton basis”.
Using k k p p = δ δ and the exciton clo-
The biexciton eigenstates for opposite electron spins sure relathiohn,e| eihii = Ik,epiet ikshepahsy to show that from
and opposite hole spins can be written on free-carrier Eqs.(24),(B.1P),ia|nidh(|B.2),theparityconditionsforelec-
states as, tron exchange or hole exchange read as
Ψ(η) = ψ(η,Se,Sh) (18)
| Se,Shi kXe,k′ekXh,k′h ke,k′e,kh,k′h φm(ηn,Se,Sh) = (−1)SeXij λe(cid:0)mn ji(cid:1)φi(jη,Se,Sh)
×ha†ke,+1/2a†k′e,−1/2−(−1)Sea†ke,−1/2a†k′e,+1/2i = (−1)Sh λh mn ji φi(jη,Se,Sh). (25)
b† b† ( 1)Shb† b† v . Xij (cid:0) (cid:1)
×h kh,+3/2 k′h,−3/2− − kh,−3/2 k′h,+3/2i| i
Bynotingthatλ n j =λ mj ,thesetwoequations
This writing covers singlet state, (S = S = 0), as well h m i e n i
e h give the parity con(cid:0)ditio(cid:1)n for e(cid:0)xcito(cid:1)n exchange as
as “triplet” state, (S = 1,S = 3), for which we have a
e h
sum instead of a difference of pair states—the biexciton φ(η,Se,Sh) =( 1)(Se+Sh)φ(η,Se,Sh). (26)
triplet state Sz =0 being degeneratewith respectto the ij − ji
ones constructed on (s =s , m =m ) in Sec. III.
1 2 1 2
Since a†ke,−1/2a†k′e,+1/2 = −a†k′e,+1/2a†ke,−1/2, it is pos- We now turn to the biexciton Schr¨odinger equation in
sible to rewrite Eq. (18) as the exciton basis. We find, from the commutators (7)
and (8),
Ψ(η) = ψˆ(η,Se,Sh) a† a† (19)
| Se,Shi kXe,k′ekXh,k′h ke,k′e,kh,k′h ke,+1/2 k′e,−1/2 0 = (H −Eη(Se,Sh))|Ψ(Sηe),Shi (27)
×hb†kh,+3/2b†k′h,−3/2−(−1)Shb†kh,−3/2b†k′h,+3/2i|vi, = Xrs n(cid:16)Ers−Eη(Se,Sh)(cid:17)φr(ηs,Se,Sh)
where ψˆk(ηe,,Ske′e,,Skhh),k′h = ψk(ηe,,Ske′e,,Skhh),k′h + (−1)Seψk(η′e,,Skee,,Skhh),k′h + ξdir sj φ(η,Se,Sh) B† B† v .
follows the parity condition Xij (cid:0)r i(cid:1) ij o r,−1 s,1| i
ψˆ(η,Se,Sh) =( 1)Seψˆ(η,Se,Sh) . (20)
ke,k′e,kh,k′h − k′e,ke,kh,k′h As v B B B† B† v reduces to δ δ for ex-
If we do the same for the hole part, we end up with citohns|mna,d1eomf,−ca1rrire,−rs1wist,1h|diifferentspins,mth,repnr,sojection
|ΨS(ηe),Shi=kXe,k′ekXh,k′hφk(ηe,,Ske′e,,Skhh),k′h 0of=th(iEs equatio(Sne,oSnht)o)φh(vη|,BSen,S,1hB)+m,−1 sξimdirplyngjiveφs(η,Se,Sh).
×a†ke,+1/2a†k′e,−1/2b†kh,+3/2b†k′h,−3/2|vi, (21) mn−Eη mn Xij (cid:0)m i(cid:1) ij (28)
hwahserteheφek(xηe,p,Skee′ec,,Sktheh)d,k′hpa=rityψˆck(ηoe,,nSkde′e,,iSkthiho),kn′h, n+am(−el1y)Shψˆk(ηe,,Ske′e,,Skh′h),kh Nenodteutphawti,thifwtheeinsasmteeadeqpuroatjeiocntibtuotntwoithhv|mBma,n1Bdnn,−i1n,tewre-
changed. So,theresultingSchr¨odingerequation(28)has
φk(ηe,,Ske′e,,Skhh),k′h =(−1)Seφk(η′e,,Skee,,Skhh),k′h =(−1)Shφk(ηe,,Ske′e,,Skh′h),kh. to be solved self-consistently with the parity conditions
(22) (25),—which is not convenient numerically.
7
Moreover, the Schr¨odinger equation (28) for spin molecular state. This can be understood by considering
triplet states (S =1,S =3) does not readily reduce to energiesclosetotheenergy2ε oftwogroundstateexci-
e h ν0
Eq. (13), whereas the Schr¨odinger equation must be the tonslabeledby(ν ,Q=0),whichistheexpectedbiexci-
0
same for all triplet states since they are degenerate. To tonenergyfortemperaturemuchsmallerthantheenergy
possibly relate these two Schr¨odinger equations and also for exciton internal motion excitation. So, the biexciton
avoid handling the parity conditions (25), we can intro- binding energy δ =2ε (Se,Sh) is expected to be far
ducetwonewsetsoffunctionsϕ(c;η,Se,Sh) withc=(e,h) smallerthantheηdifferenν0ce−bEeηtweenexcitonbindingener-
ij
and rewrite φi(jη,Se,Sh) as gteiersinεgνiξ−diεrν00.0Weistheeqnuanlottoetzhearot;thsoe,dξirdeirctesCsoeuntloiamllbysvcaant--
00
ishes for sm(cid:0)all(cid:1)momentum transfer (see Eq. (B.10)). By
φ(η,Se,Sh) =ϕ(c;η,Se,Sh)+( 1)Sc λ j n ϕ(c;η,Se,Sh),
ij ij − c i m mn contrast, the “in” exchange-Coulomb scattering is given
Xmn (cid:0) (cid:1) by27,28
(29)
so that the parity condition (25) is automatically ful- 315π3 a 2
filled whatever ϕi(jc;η,Se,Sh). By inserting the aboveequa- ξin 00 = −(cid:18)8π− 512 (cid:19)(cid:16) LX(cid:17) RX(3D) in 2D ,
tion into Eq. (28), we find a Schr¨odinger equation for (cid:0)00(cid:1)  26π aX 3R(3D) in 3D
ϕ(c;η,Se,Sh). It reads − 3 (cid:16) L (cid:17) X
ij  (33)
with ξin 00 equal to ξout 00 according to Eq. (15).
0 = (cid:16)Emn−Eη(Se,Sh)(cid:17)ϕm(c;nη,Se,Sh) ACosualomrebs(cid:0)u0lstc0,a(cid:1)tttheerinsugms, ξoifnth0(cid:0)e00“0i+n(cid:1)”ξoauntd0“0ou,t”reenxdcehrasntghee-
+ ξdir n j +( 1)Scξout n j (30) 00 00
Xij h (cid:0)m i(cid:1) − (cid:0)m i(cid:1) effectivescatteringξˆs(yc;mη)o(cid:0)vera(cid:1)lllargely(cid:0)neg(cid:1)ativeforsmall
excitonmomenta,whilethethirdtermofEq.(32),ofthe
+( 1)Sc E (Se,Sh) λ n j ϕ(c;η,Se,Sh). orderofthebiexcitonbindingenergy,issmall. Thislarge
− (cid:16) mn−Eη (cid:17) c(cid:0)m i(cid:1)i ij negativeeffective scatteringallowsbound-statesolutions
If we now use Eq. (15) to rewrite ξout n j in terms of to the Schr¨odinger equation (31). It is then crucial to
m i treat the exchange-Coulomb interactions adequately if
ξin n j , the above equation also read(cid:0)s (cid:1)
m i one aims at getting reliable results for the ground and
(cid:0) (cid:1)
excited states. Detailed discussions about the depen-
0=(E (Se,Sh))ϕ(c;η,Se,Sh)+ ξˆ(c;η) n j ϕ(c;η,Se,Sh),
mn−Eη mn sym m i ij dence of Coulomb scatterings on electron-to-hole mass
Xij (cid:0) (cid:1) ratio and on relative motion momentum of the exciton
(31)
pair can be found in Refs. 29 and 30. For completeness,
wheretheeffectivescatteringnowhasasymmetricalform
inAppendix II,wehaverederivedthevariousscatterings
with respect to “in” and “out” states
appearing in the biexciton Schr¨odinger equations.
Equation(23), for i=(ν ,k+K/2)andj =(ν , k+
ξˆ(c;η) n j = ξdir n j + (−1)Sc ξin n j +ξout n j K/2), K being the biexcitoin center-of-mass momje−ntum
sym (cid:0)m i(cid:1) (cid:0)m i(cid:1) 2 h (cid:0)m i(cid:1) (cid:0)m i(cid:1) and k the relative motion momentum between the ex-
+λ n j E +E 2 (Se,Sh) .(32) citon pair, allows us to rewrite the biexciton operator
c(cid:0)m i(cid:1)(cid:16) mn ij − Eη (cid:17)i in terms of two bright excitons as (see also Eq. (29) of
It is then easy to see that, for spin triplet state, S = 3 Ref. 31)
h
and c = h, the Schr¨odinger equations (31) and (13) are
B† = φ(η,Se,Sh)(ν ,ν )B† B† .
indeed identical. ηK k i j νi,k+K/2;−1 νj,−k+K/2;1
Equations (13) and (28), in the absence of Coulomb νXiνj;k
scatterings ξdir n j and Pauli scatterings λ n j , (34)
m i m i Sinceallλandξscatteringsdonotdependonthecenter-
wouldleadtoφ(η(cid:0)) =(cid:1)1forE = andzeroothe(cid:0)rwis(cid:1)e.
mn mn Eη of-mass momentum K, we can, without any loss of gen-
The biexciton would then reduce to two free excitons as
erality, set K=0 from now on. Such a biexciton is then
expected. Interactions between excitons, through both
made of two excitons with opposite momenta.
Coulomb and Pauli scatterings, produce a difference be-
Since the parity condition (25) is difficult to numer-
tween the biexciton energy and the energy of two free
ically implement in the Schr¨odinger equation (28) for
excitons.
φ(η,Se,Sh)(ν ,ν ),weinsteadsolveasomewhatmorecom-
BycomparingtheSchr¨odingerequations(16)and(17) k i j
plicatedequation(31)in whichthe parity condition(29)
for same-spin carriers with the Schr¨odinger equations
(31)and(32)foropposite-spincarriers,weseethatthere is already enforced, the function ϕk(c;η,Se,Sh)(νi,νj) in
is a sign change in front of the exchange part of the ef- Eq. (29) being a free parameter.
fective scattering (32), depending on singlet and triplet Let us from now on focus on the biexciton singlet
states. This sign change which results from the Pauli state (Se = Sh = 0) and consider c = h without any
exclusion principle, is the reason why two excitons in a loss of generality. By restricting the exciton level to the
singlet state (S = S = 0) can bind together into a ground state ν and by setting ϕ(h;η,0,0)(ν ,ν ) = ϕ(η),
e h 0 k 0 0 k
8
the Schr¨odinger equation (31) for the singlet state then
0.5
reduces to
0.45
y 2D
−δηhϕk(η)+Xk′ λ(cid:16)(ν(ν00,−,kk))(ν(ν00,−,kk′)′)(cid:17)ϕ(kη′)i g Energ 0.03.54 GS
≃ Mk2Xϕ(kη)+Xk′ ξ˜(cid:16)(ν(ν00,−,kk))(ν(ν00,−,kk′)′)(cid:17)ϕ(kη′), (35) n Bindin 0.002..523 1st2nd
o
where M =m +m and cit 0.15 3rd
X e h x
e 0.1
Bi
ξ˜ (ν0,−k)(ν0,−k′) =ξdir (ν0,−k)(ν0,−k′) 0.05
(cid:16) (ν0,k) (ν0,k′) (cid:17) (cid:16) (ν0,k) (ν0,k′) (cid:17) 00 0.5 1 1.5 2 2.5 3
+12hξin(cid:16)(ν(ν00,−,kk))(ν(ν00,−,kk′)′)(cid:17)+ξout(cid:16)((νν00,−,kk))(ν(ν00,−,kk′)′)(cid:17) log10(mh/me)
+λ (ν0,−k)(ν0,−k′) (k′2+k2) . (36)
(cid:16) (ν0,k) (ν0,k′) (cid:17) MX i FIG. 7: (Color online) Binding energies of the 2D biexciton
ground state(GS) and the first three bound excited states in
R(2D) = 4R(3D) unit, as a function of the hole-to-electron
X X
The scatterings λ and ξ˜ depend on (k,k′) through mass ratio mh/me. The ground state binding energy has a
k, k′ andthe angleθkk′ betweenkandk′, the explicit minimum for me=mh.
| | | |
values of these scatterings being given in Appendix II.
Since the biexciton singlet state has a zero angular mo-
mentum, the ϕ(η) function depends on k = k only. So, 0.25
k | |
wecanfirstaveragethevariousscatteringsovertheθk,k′. gy 3D
Letuscallλ(k,k′)andξ˜(k,k′)theseaveragedquantities. er 0.2
n
Wethenendupwitha1Dintegralequationforthefunc- E GS
tion ϕ(η). It reads as ng 0.15
k di
Bin 1st
δ ϕ(η)+ λ(k,k′)ϕ(η) = k2 ϕ(η)+ ξ˜(k,k′)ϕ(η). on 0.1 2nd
− ηh k Xk′ k′ i MX k Xk′ k′ xcit 3rd
e 0.05
(37) Bi
This equation is numerically solved to get the binding 00 0.5 1 1.5 2 2.5 3
energies of the biexciton ground and excited states for log (m /m )
10 h e
varioushole-to-electronmass ratios. We canalso getthe
ϕ(η) functions, which are related to the biexciton wave
p
functions φ(η) with proper symmetry via Eq. (29). The FIG.8: (Coloronline)SameasFig.7for3D;theenergyunit
p
excited states are expected to mainly come from vibra- now is RX(3D). The curves are qualitatively similar to the 2D
tional modes. To reach rotational modes, it is necessary curves,though thebindingenergies are significantly smaller.
to include p and d exciton levels. As a consequence, the
λ and ξ scatterings, as well as the biexciton wave func-
tionsφ(η),wouldgetanangulardependence. Theprecise ground-state wave functions hp|ν0i in 2D and 3D
k
treatmentofthe angulardependence inthesescatterings
a √2π
is rather complex and definitely beyond the scope of the p1s (2D) = X , (38)
present work. A relatively simple, yet nontrivial, biexci- h | i (cid:16) L (cid:17)(1+a2Xp2/4)3/2
ton state follows from just considering two ground-state a 3/2 8√π
p1s (3D) = X . (39)
excitons with a nonzerorelative-motionangularmomen- h | i (cid:16) L (cid:17) (1+a2Xp2)2
tum.
L is the sample size, a = ~2ǫ /µ e2 is the 3D ex-
X sc X
citon Bohr radius, with µ−1 = m−1 +m−1, and ǫ is
X e h sc
thestaticsemiconductordielectricconstant,oneorderof
V. RESULTS AND DISCUSSIONS
magnitude larger in semiconductor samples than in vac-
uum.
To solve the Schr¨odinger equation (37) for the biex- The Schr¨odinger equation (37) can be cast into a gen-
citon binding energy δ , we use the normalized exciton eralized eigenvalue problem, with matrices spanned by
η
9
the k momentum. To solve it, we sample the k = k
| | 2
values with 150 mesh points in 2D and 100 in 3D, ac-
cordingtoki =u3i wheretheui’sareequallydistributed, 2D η=η0, mh/me=5
therebyallowingformoresamplinginthesmallk region. 2η〉| 1.5 η=η0, mh/me=50
3TDhebuuptp2e0ricnut2oDff bkmecaaxu(sientah−Xe1euxnciitto)niswtaavkeenfutnoctbieon10hains 0, p| η=η1, mh/me=50
=
a larger radial extension (in k space) in 2D than in 3D 0, r1 1 ~1/aXX
and also because the Coulomb interaction V decreases =
q 1
iatsh1a/sqiinn32DD,(si.eee.,Emqo.r(eAs.l6o)w).lythan the 1/q2 dependence 2〈L| r− 0.5
0
0 1 2 3 4 5
A. Biexciton binding energies for ground and p
excited states
FIG.9: (Coloronline)Plotofthegroundstate(η=η )wave
Solid curves in Figs. (7) and (8) show the biexciton 0
binding energies in 2D and 3D as a function of hole-to- function L2|hr−1 =0,r1 =0,p|ηi|2 as a function of p in a−X1
electronmass ratiom /m . The results are expressedin unit, when the mass ratio mh/me is equal to 5 and 50. Its p
h e extension scales as the inverse of the biexciton Bohr radius,
terms of the corresponding effective Rydbergs, namely
aXX. We also show the same quantity for the first excited
(2D) (3D) (2D) (3D) (3D)
RX and RX , with RX = 4RX and RX = state (η=η1) when mh/me =50.
(µ /m ǫ2 )13.6eV, where m is the free electron mass.
X 0 sc 0
The curves for the 2D and 3D binding energies are qual-
itatively similar. However, the values in 2D are signif-
10
icantly larger than those in 3D. This is physically ex-
pected since the reduction of dimensionality allows for 8 2D
2
much stronger Coulomb interactions and more localized η〉| 6
wave functions to enhance overlapping. We further no- p|
0, 4
tice that both, the 2D and 3D binding energies, have a =
1
minimuminthepositroniumlimit,i.e.,atmh/me =1;it 0, r 2
thenincreaseslogarithmicallyasthemassratioincreases, =1 0
−
unOtilurit rseastuulrtastegsivfoerslaarggeromuansds-srtaattioes.biexciton binding 〈 ln | r −2
(3D) −4
energy equal to 0.012R for m /m = 1 in 3D,
X h e
which accounts for only about 40% of the more accu- −6
0 0.1 0.2 0.3 0.4 0.5
rate variational results32,33, while it reaches 0.21R(3D) p
X
when m /m = 1000, which accounts for about 70%.
h e
In 2D, our calculated binding energies give the ground
state at 0.075R(2D) when m /m = 1, which accounts FIG.10: (Color online) Plot of ln|hr−1 =0,r1 =0,p|ηi|2 for
for about 50% oXf the best vahriateional result16, while it three unbound biexciton states in a semilog plot, the mass
reaches 0.44RX(2D) when mh/me = 1000, which accounts urantiito. Nbeoitnegthmaht/tmheew=av5efaunndcttiohneimsonmotenhteuremrepscsatlieldl ibnyaL−X21,
for about 80%. All this shows that our approach gives so the amplitude of the wave function is significantly larger
a much better ground-state energy when mh/me 1, thanforboundstates,butitspextensionsignificantlysmaller
≫
i.e., close to the hydrogen molecule limit, possibly be- for normalized functions.
cause the exciton wavefunctions areless deformedwhen
forming a molecule than in the case of lighter hole.
Oneimportantadvantageofthepresentapproachover monicpotentialwhichleadstoequalenergyspacingsbe-
variationalproceduresisthatitallowsreachingthebiex- tween eigenstates.
citon bound and unbound excited states as easily as the
ground state. Dashed curves in Figs. 7 and 8 show the
binding energiesofthe biexcitonboundstates in2D and
3D.Thenumberofboundstatesincreaseswiththe mass
B. Biexciton wave function
ratiom /m ,thisnumberreducingto1form /m .20
h e h e
in 2D and m /m . 30 in 3D. For a large mass ratio
h e
m /m = 1000, we find 9 bound states in 2D and 8 In a previous work on biexciton, we have shown that
h e
in 3D. The binding energy differences become smaller the wave function of a biexciton made of opposite-spin
for higher excited states, evidencing a difference in the carriers,with center-of-massmomentum K, relative mo-
exciton-exciton interaction compared to the usual har- tion index η, and electron and hole total spins S =
10
(S ,S ), splits as (see Eq. (12) in Ref. 31) We can compute ν ,ν ,pη,S from r ,r ,pη,S
e h i j −1 1
h | i h | i
through a double Fourier transform “in the exciton
hre1,re2,rh1,rh2|K,η,Si=hRXX|Kihr−1,r1,u|η,Si, sense” (see also Eq. (31) of Ref. 31), namely,
(40)
wmhe)reisRtXhXe =bie(xmcietroen1+cemnetrere2-o+f-mmhasrsh1c+oomrdhirnha2t)e/,2(rme+= hνi,νj,p|η,Si=Z dr−1dr1hνi|r−1ihνj|r1ihr−1,r1,p|η,Si,
h 1
(47)
r r andr =r r aretheelectron-to-holedis-
e2− h1 −1 e1− h2 which also reads as
tancesofthetwobrightexcitonshavingspins( 1),while
±
u=(mere2+mhrh1)/(me+mh)−(mere1+mhrh2)/(me+ hr−1,r1,p|η,Si= hr−1|νiihr1|νjihνi,νj,p|η,Si.
mh) is the distance between the center-of-masses of the νXi,νj
two bright excitons. (48)
For bound biexciton, the wave function Note that for r or r equal to 0, the ν exciton lev-
1 −1
r−1,r1,uη,S has an extension of the order of els that survive in the above ν sum are s-like states
h | i
the exciton size aX over r−1 and over r1, and an only. So, the νi,νj,pη,S function just is the function
extension of the order of the biexciton size aXX over u. φ(η,S)(ν ,ν ) ghiven in E| q. (i34). Since, when numerically
p i j
Bycontrast,forunboundbiexciton,theextensionoveru
solving the biexciton Schr¨odinger equation (37) for sin-
is as large as the sample size L, since unbound biexciton
glet state (S = 0), we have restricted the sum over ν to
resembles very much an exciton with another free
thegroundstateν ,wemustforconsistencyalsokeepν
0 0
exciton moving around anywhere in the sample. Thus,
only in the ν sum of Eq. (48).
in the case of bound states, dimensional arguments give Figure 9 shows r = 0,r = 0,pη,S = 0 2 for the
−1 1
the normalized relative motion wave functions through |h | i|
groundstate whenthe mass ratio is m /m =5,and for
h e
two bound states when the mass ratio is m /m = 50.
h e
1 = dr dr du r ,r ,uη,S 2 Note that Eq. (45) forces us to plot bound-state wave
1 −1 −1 1
Z |h | i| functions through L2 r = 0,r = 0,pη,S = 0 2 in
−1 1
aDaDaD 0,0,0η,S 2, (41) order to have a quant|hity independent of |sample siiz|e L.
≃ X X XX|h | i| For unbound states, the r =0,r =0,pη,S =0 2
−1 1
which leads to function is peaked on mom|henta p which dep|end on tih|e
η
unbound biexciton energies. Figure 10 shows that this
D
1
r =0,r =0,u=0η,S 2 , (42) peaked function broadens when the unbound biexciton
|h −1 1 | i| ≃(cid:18)a2XaXX(cid:19) energy increases, the broadening being due to exciton-
exciton interactions.
while, in the case of unbound states, a is replaced by
XX
L; so,
D
r =0,r =0,uη,S 2 1 . (43) C. Biexciton absorption spectrum in pump-probe
|h −1 1 | i| ≃(cid:18)a2 L(cid:19) experiment
X
To obtain r ,r ,pη,S , which is of physical rele- (i) Let us now first consider an initial state made of
−1 1
vance in phothon absorpt|ion,iwe perform a Fourier trans- onecircularlypolarizedphotonσ+ withmomentumQph
form as and frequency ω, and one exciton already present in the
sample, this (ν ,Q ) exciton having an opposite circular
0 i
polarization,σ . Afterphotonabsorption,thefinalstate
r ,r ,pη,S = du pu r ,r ,uη,S , (44) −
−1 1 −1 1
h | i Z h | ih | i containstwoelectron-holepairs,theircenter-of-massmo-
mentumbeingK =Q +Q . Thephotocreatedexciton
i ph i
where pu = eip·u/LD/2; so, the extension over p of interacts with the exciton present in the sample to pos-
h | i
r−1,r1,pη,S isoftheorderof1/aXX forboundstates sibly form a biexciton. Since we are mainly interested in
h | i
(see Fig. 9), and of the order of 1/L for unbound states low-lyingbiexcitonstates,weshallfocusonsingletstates
(seeFig.10). Thesamedimensionalargumentsthengive, (S =0). The Fermigoldenrulegivesthe photonabsorp-
in the case of bound states, tion as ( 2) times the imaginary part of the response
−
function to one photon (ω,Q ). This response function
r =0,r =0,p=0η,S 2 aXX D, (45) reads as (see Eq. (36) of Ref.p3h1)
|h −1 1 | i| ≃(cid:18)a2 L(cid:19)
X S (ω,Q ;Q )=
XX ph i
and, in the case of unbound states, f(η) (p )
XX i , (49)
r =0,r =0,pη,S 2 1 D. (46) Xη ω+Eν0,Qi −hEη+ (Qp4hM+XQi)2i+i0+
|h −1 1 | i| ≃(cid:18)a2X(cid:19) where pi =(Qi Qph)/2 is the relative motion momen-
−
tum ofthe (X,X)pair andE is the free excitonen-
ν0,Qi