Theoretical derivation of the bi- and tri-molecular trion formation coefficients

0.23 MB·English

by  J. Berney

APS/123-QED
Theoretical derivation of the bi- and tri-molecular trion formation coefficients
J. Berney,1,2 M.T. Portella-Oberli,1 and B. Deveaud-Pl´edran1
1Institut de Photonique et Electronique Quantiques,
Ecole Polytechnique F´ed´erale de Lausanne (EPFL) CH1015 Lausanne, Switzerland
2Attolight S`arl, Chemin de la Raye 13, CH1024 Ecublens, Switzerland
(Dated: January 23, 2009)
A theoretical investigation of the trion formation process from free carriers in a single GaAs/
Al1−xGaxAsquantumwellispresented. Themechanismfortheformationprocessisprovidedbythe
9 interaction of the electrons and holes with phonons. The contributions from both the acoustic and
0 optical phonons are considered. The dependence of both bi-molecular and tri-molecular formation
0 rates on temperature is calculated. We demonstrate that they are equivalent for negatively and
2 positively charged excitons.

n
a PACSnumbers: 71.35.Cc,71.35.Ee,73.21.Fg,78.47.+p,78.67.De
J

3
I. INTRODUCTION II. BIMOLECULAR FORMATION OF
2
EXCITONS

]
i
c Insemiconductorwells,thephotoluminescencespec- Thedynamicsoftheexcitonformationisconsidered
s
- trum following the generation of electron hole plasma in the framework of the Boltzmann equation for a sys-
rl is dominated by an exciton line1,2,3,4,5,6. The for- tem containing free electrons, free holes, and excitons.
t mation of excitons in quantum wells has been ex- The residual Coulomb interaction between the free car-
m
tensively investigated both experimentally4,5,6,7,8,9 and riers is neglected, which is justified in the range of tem-
t. theoretically10,11,12. Recently, it was experimentally peratures and densities considered13. In this work we
a showed that the exciton formation is strongly density focus on the exciton formation mechanism and do not
m
andtemperaturedependent;itisabimolecularprocessin discuss the relaxation of the three species within their
- which an electron and a hole are bound by Coulomb in- respectivebands,theelectron-holescattering,andradia-
d
n teraction with the emission of the appropriate phonon8. tive recombination. We denote the occupation numbers
o This experimental result unambiguously confirmed the for electrons, holes, and excitons by fe(ke), fh(kh), and
c theoreticalpredictionforluminescence spectrumis mod- fX(kX), respectively, where ke, kh, and kX are the in-
[ ified: a chargedexciton resonance appears below that of planemomentaforelectrons,holes,andexcitons. Forthe
1 the exciton. bimolecular formation, following Piermarocchi et al.12,
v the scattering terms in the Boltzmann equation process
5 In a previous publication13, we showed experimen- reads
4
36 tpalallsymtahactatnhebedyrunlaemdibcsyoafesximciptolen,ratrtieoneqaunadtieolnecmtroodne-hl,oilne dfde(tke) =− wke,kh→kXfh(kh)fe(ke),
1. which we account for bimolecular formation of excitons (cid:18) (cid:19)form kXX,kh
from an electron-hole plasma, bimolecular formation of (1)
0
9 trions from excitons and free carriers and trimolecular where wke,kh→kX represents the probability per unit
0 formation from free carriers. Using only two fit param- time for a free electron and a free hole to bind to-
: eters, we were able to extract experimentally the depen- getherandformanexciton. Freecarriersthermalizevery
v
dence of both bi-molecular and tri-molecular formation quicklyincomparisonto the excitonformationtime, no-
i
X coefficients on temperature. tably through fast carrier-carrier scattering14. It is thus
assumedthatduringtheevolutionofthesystem,thefree
r
a electrons and holes are thermalized at the same temper-
In this paper, we propose a theoretical derivation
ature T . In the scattering term of Eq. (1), we use for
of these coefficients. We consider formation channels c
f (k ), and f (k ) equilibrium Boltzmann distribution
throughwhichtheformationchannelandshowthatthey e e h h
function at T . Consequently, by summing Eq. (1) over
correspond to experimental calculations. c
k , we obtain an adiabatic equation for the evolution of
e
the electronic density n = 1 f (k )
InSec. I,wefirstreproducetheresultsofPiermaroc- e S ke e e
chi et al.12 on exciton formation. We then extend the P
dn
formalism to the case of charged excitons; in Sec. II, e =− F(k )n n ≡−Cn n . (2)
X e h e h
dt
we calculate the bi-molecular formation and in Sec. III
XkX
the tri-molecular formation. In Sec. IV, we show how
these formation rates can be used to calculate formation ThecoefficientC isthebimolecularformationcoefficient,
dynamics at thermodynamical equilibrium. whichdependsonbothT andthelatticetemperatureT
c l
2
through the term of a finite square quantum well15
2π~2 2 1 Aαcos kzw(α)zα for |zα|< L2z
F(kX)=(cid:18)kBTc(cid:19) memhS χα(zα)= Bαexp(cid:16)−kzb(α)((cid:17)|zα|−Lz/2) for |zα|> L2z
× w e−(Ee(ke)+Eh(kh))/kBTc, (3) (cid:16) (cid:17) (7)
ke,kh→kX
with α =e,h and we use the simplest electron orbital
kXe,kh
function
where S denotes the QW surface area.
Free carriersare coupled to the excitonby a contin- ϕλX(r )= 2 e−rk/λX, (8)
uum of phonon states (q,qz) through a carrier-phonon k sπλ2X
interaction Hamiltonian H . A phonon can be
e/h−ph
emitted (+) or absorbed (-) in the formation process of whose in-plane Fourier transform is given by
the exciton. We calculate both case separatly using the
Fermi’s golden rule ϕλX = 8πλX2 1+(λ k)2 −3/2. (9)
k s S X
(cid:2) (cid:3)
2π ThevariationalparameterλX isassociatedwiththeBohr
wk±e,kh→kX = ~ radiusoftheexcitonintheQW.Eq.(5)canberewritten
qX,qz as
2
hk |⊗hn ±1|H |n i⊗|k i⊗|k i 1
X q,qz e/h−ph q,qz e h ΦX (z ,z )= dR dr φX(r ,z ,z )
×(cid:12) δ Ee(ke)+Eh(kh)−EX(kX)∓~ωph(q,qz) ,(cid:12) (4) ke,kh e h S Z k k k e h
(cid:12) (cid:12) ×eikX·Rke−i[Rk·(ke+kh)+rk·(βXke−αXkh)]
with E(cid:2)(k ), E (k ) and E (k ) the energy (cid:3)disper-
e e h h X X
tionoftheelectrons,holesandexcitonsrespectively,and =δkX−ke−khφXke−αXkX(ze,zh), (10)
~ω (q,q )theenergyoftheemitted(absorbed)phonon.
ph z where φX(z ,z ) = χ (z )χ (z )ϕλX is the in-plane
We firstbuildthe boundandunboundelectron-holepair k e h e e h h k
Fourier transform of the exciton envelope function. We
states |k i and |k i⊗|k i.
X e h can now construct the state of a single exciton with an
in-planemomentumk intheFermionicHilbertspaceof
X
electron-hole pairs. It is the superposition of wavefunc-
A. Bound and unbound exciton states
tions (10) with all electron momenta k and all electron
e
z and hole z coordinates, given by
Let (r ,z ) and (r ,z ) be the electron and hole e h
ek e hk h
tphoesiteixocnitovnectwoarvserfuesnpcetciotinv,elwyhaenredwΦeXh[(arveek,szeep)a,r(arthekd,zthh)e] |kXi= dzedzhφXαX∗kX+ke(ze,zh)
coordinatesin the QW plane (x-y) from the perpendicu- Xke Z
×cˆ† dˆ† |0i, (11)
larcoordinates(z). Denotingtheelectron(hole)in-plane −ke,ze kX+ke,zh
momenta k (k ), we write the in-plane Fourier trans-
e h wherecˆ† (dˆ† )istheelectron(hole)creationoper-
form of this function kX,ze kX,zh
atorwithin-planemomentumk andz (z )coordinate.
X e h
1
ΦX (z ,z )= dr dr ΦX[(r ,z ),(r ,z )] Similarly,wechooseplanewavesforthefreecarriers.
ke,kh e h S ek hk ek e hk h Thus the in-plane Fourier transformψα(z) of the carrier
Z k
×e−i(ke·rek+kh·rhk), (5) wavefunction takes the simple form
ψα(z)=χ (z), α=e,h. (12)
where S denotes the QW surface area. Transforming to k α
center-of-mass(CM)andrelativecoordinatesinthe QW The unbound electron-hole pair then reads
plane — R = α r +β r , r = r −r , where
k X ek X hk k ek hk
αX = me/MX, βX = mh/MX and me, mh, MX are |ke,khi= dzed zhψke∗e(ze)ψkhh∗(zh)cˆ†ke,zedˆ†kh,zh|0i.
the electron, hole and exciton in-plane effective mass —
Z
(13)
we can apply Bloch’s theorem and decompose the exci-
ton wavefunctioninto a free motion parteikX·Rk related
to the exciton in-plane momentum k and an envelope
X B. Carrier-phonon interaction Hamiltonian.
function. To facilitate the calculation, we use an enve-
lopefunctionseparableinz andr ,althoughitisstrictly
k
justifiable only for narrow well structures, We write the interaction Hamiltonian for a coupled
electron-phonon system in the notation of the second
φX(rk,ze,zh)=χe(ze)χh(zh)ϕλX(rk). (6) quantization16
tThheewcaovnefifnuenmcteionntfoufnacntieolnecstχroe(nz(eh)o(lχeh)(iznht)h)eisgrtoaukenndtsotabtee Hα−ph = Vqα,qz aˆq,qz +aˆ†−q,−qz ̺ˆα(q,qz) (14)
qX,qz (cid:16) (cid:17)
3
where aˆ† is the phonon creation operator. The where ǫ is the static dielectric constant and ǫ is the
q,qz 0 ∞
electron density operator ̺ˆ (r ,z) and its counterpart high frequency dielectric constant. We use the notation
α k
in Fourier space ̺ˆ (q,q ) are expressed on the basis a forthedeformation-potentialconstant(assumedtobe
α z α
φσk(rk,z,s)=eik·rkδ(z)ζσ(s) associated with a non-degenerate conduction or valence
band), ρ for the density of the crystal, e for the charge
0
(cid:8) ̺ˆσ(r ,z)= cˆσ† cˆσ e(cid:9)−i(k−k′)·rk, (15) of the electron and V for the volume of the sample. We
e k k,z k′,z
follow Einstein interpolation scheme, so that the disper-
k,k′
X sion is merely ωLA = v q2+q2 for LA phonons and
q s z
̺ˆσe(q,qz)= dz e−iqzzcˆσk†+q,zcˆσk,z, (16) ωqLO = ωLO for LO phonpons, vs standing for the Debye
k Z sound velocity and ω for the reststrahl frequency.
X LO
̺ˆσ(r ,z)=− dˆσ† dˆσ e−i(k−k′)·rk, (17)
h k k,z k′,z
k,k′
X
̺ˆσ(q,q )=− dz e−iqzzdˆσ† dˆσ . (18)
h z k+q,z k,z
k Z
X
Spin states ζ (s) = hσ|si have been intruduced for their
σ
will be necessary when we treat the trion formation.
Only longitudinal acoustical (LA) and longitudinal
opticalphonons (LO) couple significantlyto careers. We
express the coupling vertex functions Vα for both cou-
q,qz
pling
C. Matrix element calculation.
~(|q|2+q2)
Vα(LA) =ia z , (19)
q,qz αs 2ρ0Vωq,qz
2π~ω e2 1 1
Vα(LO) = q,qz − , (20) ThematrixelementsinEq. (4)arecalculated,mak-
q,qz s(|q|2+qz2)V (cid:18)ǫ0 ǫ∞(cid:19) ing use of Eq. (11), (13) and (14)
hk ,n ±1|H +H |k ,k ,n i
X q,qz e-ph h-ph e h q,qz
= dz dz dz′ dz′ dzϕX hn ±1|aˆ aˆ† |n iχ∗(z )χ (z′)χ∗(z )χ (z′)e−iq˜zz
e h e h αkX+k′e q,qz ±q˜,q˜z −q˜,−q˜z q,qz e e e e h h h h
qX˜,q˜zkX,k′eZ
× Ve h0|cˆ cˆ† cˆ cˆ† |0ih0|dˆ dˆ† |0i−Vh h0|cˆ cˆ† |0ih0|dˆ dˆ† dˆ dˆ† |0i .
q˜,q˜z −k′e,ze′ k+q˜,z k,z ke,ze kX+k′e,zh′ kh,zh q˜,q˜z −k′e,ze′ ke,ze kX+k′e,zh′ k+q˜,z k,z kh,zh
n o
Applying operators on the ground state
h0|cˆ cˆ† cˆ cˆ† |0i=δ δ(z−z′)δ δ(z−z ), h0|cˆ cˆ† |0i=δ δ(z −z′),
−k′e,ze′ k+q˜,z k,z ke,ze −k′e,k+q˜ e k,ke e −k′e,ze′ ke,ze ke,−k′e e e
h0|dˆ dˆ† dˆ dˆ† |0i=δ δ(z−z′)δ δ(z−z ), h0|dˆ dˆ† |0i=δ δ(z −z′),
kX+k′e,zh′ k+q˜,z k,z kh,zh kX+k′e,k+q˜ h k,kh h kX+k′e,zh′ kh,zh kh,kX+k′e h h
hn ±1|aˆ +aˆ† |n i= n + 1 ± 1 δ δ ,
q,qz q˜,q˜z −q˜,−q˜z q,qz q,qz 2 2 ∓q˜,q ∓q˜z,qz
q
we obtain
hk ,n ±1|H +H |k ,k ,n i
X q,qz e-ph h-ph e h q,qz
= n + 1 ± 1 δ Ve ϕX I (q )−Vh ϕX I (q ) , (21)
q,qz 2 2 ±q,ke+kh−kX q,qz −βXkX+kh e z q,qz αXkX−ke h z
q n o
where the integrals, in the orthogonaldirection are given by
I (q )= dz |χ (z )|2eiqzzα, α={e, h}. (22)
α z α α α
Z
4
Finally, if we choose the bound and unbound electron-hole pairs dispersion to be parabolic, the probability transi-
tion (4) reads
w± = 2π (n + 1 ± 1) Ve ϕX I (q )−Vh ϕX I (q ) 2
ke,kh→kX ~ q,qz 2 2 q,qz βXkX−kh e z q,qz αXkX−ke h z
qX,qz (cid:12) (cid:12)
(cid:12) ~2k2 ~2k2 ~2k2 (cid:12)
×δ e + h +E − X ∓~ω (q,q ) δ . (23)
2m 2m b 2M ph z ±q,ke+kh−kX
(cid:20) e h (cid:21)
D. LA phonons assisted formation and reexpressing the Dirac distribution as
Considering that the sample volume V = L S
z
is macroscopic the sum over the orthogonal phonon
wavevectorsmay be replaced by the integral
L
z
←→ dq (24)
z
2π
Xqz (cid:18) (cid:19)Z
~2k2 ~2k2 ~2k2
δ e + h +E − X ∓~v q2+q2
2m 2m b 2M s z
(cid:20) e h (cid:21)
q2+q2 p ~2k2 ~2k2 ~2k2
= z δ q −q(0)(k ,k ,k )) +δ q +q(0)(k ,k ,k ) θ ± e + h +E − X , (25)
|~v q | z z e h X z z e h X 2m 2m b 2M
p s z n h i h io (cid:20) (cid:18) e h (cid:19)(cid:21)
with
1 ~2k2 ~2k2 ~2k2 2
q(0)(k ,k ,k )= e + h +E − X −q2, (26)
z e h X s~2vs2 (cid:18)2me 2mh b 2M (cid:19)
makes the integration (24) trivial for LA phonons: E. LO phonons assisted formation
In the case of interaction with LO phonons, Eq. 23
becomes
2π~ω e2(1/ǫ −1/ǫ )
wke,kh→kX = 4~πL2πz 2ρV~vsq|2~v+sqqz(z(00))|2 wke,kh×→kϕXX=Xqz ~I (qL)O−Vϕ(qX2+∞qz2) I (0qn)q2,qz
× a ϕX I (q(0))−a ϕX I (q(0)) 2 βXkX−kh e z αXkX−ke h z
×(cid:26)(cid:16)(cid:12)(cid:12)(cid:12)neq,qβz(X0)k+X−1k(cid:17)h θe(cid:18)z~22mkee2 +h~22mαkhXh2k+X−Ekbe−h~22zMkX2(cid:12)(cid:12)(cid:12)(cid:19) where w×eδd(cid:12)(cid:12)(cid:20)ro~2p2mpkeee2d+th~2e2mkphh2ho+noEnba−bs~o22rMkpX2tio+n~pωaLrtO(cid:12)(cid:12),(cid:21)w,hic(h28is)
~2k2 ~2k2 ~2k2 negligible up to room temperature. For the calculation
+ n θ − e − h −E + X . (27)
q,qz(0) (cid:18) 2me 2mh b 2M (cid:19)(cid:27) ofthe excitonformationcoefficientC, it is convenientto
rewrite the Dirac distribution as
~2k2 ~2k2 ~2k2
δ e + h +E − X +~ω
b LO
2m 2m 2M
(cid:20) e h (cid:21)
M
This expression already includes the sum over absorbed = X δ k −k(0)(k ,k )) +δ k +k(0)(k ,k ) ,
andemitted phononcontributions. The phononin-plane ~ X X e h X X e h
momentumneedstobesubstitutedbyq =k +k −k . n h i h io(29)
e h X
5
100 Parameter Symbol Value Unit
Optical Phonons
Band gap energy Eg 1519 meV
10 Accoustical Phonons Electron effectivemass me 0.08 m0
Heavy hole effective mass mhh 0.17 m0
1 LO phonon energy ~ωLO 36 meV
]1-s 0.1 HStiagthicfrdeiqeuleecntcryicdcioenlescttarnict constant ǫǫ∞0 1102..8885
. 2m Crystal density ρ 5.3162 g·cm−3
[c 0.01 Sound velocity vs 4726.5 m·s−1
C 0.001 Conduction band deformation potential ae −7.0 eV
Valence band deformation potential a 3.5 eV
h
1e-04 Exciton bindingenergy ET 6.5 meV
Trion binding energy ET 1.77 meV
1e-05 Exciton Bohr radius λX 11.0 nm
0 0.02 0.04 0.06 0.08 0.01 Trion variational parameter #1 (X−) λT 15 nm
Trion variational parameter #2 (X−) λ′ 30.0 nm
1/ Tc [K-1] Trion variational parameter #1 (X+) λTT 16 nm
Trion variational parameter #2 (X+) λ′ 25.0 nm
T
FIG. 1: The exciton formation coefficient C as a function
of the carrier temperature Tc, at a fixed lattice temperature
T =10 K.Otherparameters are given in the text. TABLE I:GaAs Material Parameters
l
with whereF representstheprobabilityperunittime
kh,ke→kT
foraboundelectron-holepairandafreeelectrontobind
k(0)(k ,k )= 2MX ~2ke2 + ~2kh2 +E −~ω . , together and form an exciton. We assume that bound
X e h s ~2 (cid:18)2me 2mh b LO (cid:19) andunboundcarriersarethermalizedanduseBoltzmann
(30) distribution function fX(kX) and fe(ke) for exciton and
electron population, respectively. By summing Eq. (31)
overk ,weobtainanadiabaticequationfortheevolution
e
F. Numerical Results. of the electron density
If we change all the sum in Eq. 2, 3, 27 and 28 into dne =− F(k )n n ≡−A−n n . (32)
integrals, the bimolecular formation coefficient C can be dt T X e 2 X e
numerically calculated by Monte Carlo integration. In XkT
Fig.1,wereportC asafunctionof1/T forafixedlattice
temperature Tl =5 K, for a GaAs QWcof 80˚A. The two The coefficient A−2 is the bimolecular formation coeffi-
cient, which depends on both T and the lattice temper-
contributions from the acoustic and optical phonons are c
ature T through the term
shown separately. The acoustical phonon dominates for l
temperatures smaller than 40 K and does not depend
on Tl. We see that these results perfectly match those 2π~2 2 1
published by Piermarocchi et al.12. We show in Table I F(kT)= k T m m S wkX,ke→kT (33)
the numerical value of the different parameters entering (cid:18) B c(cid:19) X e kXX,ke
in the calculation. ×e−(EX(kX)+Ee(ke))/kBTc. (34)
WecalculatetheformationrateusingFermi’sgoldenrule
III. BIMOLECULAR FORMATION OF TRIONS
2π
We now extend our formalism to the derivation of w± =
thebimolecularformationoftrions. Werestrictourselves kX,ke→kT ~
tonegativelychargedexcitonsandwillgiveattheendof qX,qz
2
hk |⊗hn ±1|H |n i⊗|k i⊗|k i
this work some indication on how to retrieve their posi- T q,qz e/h−ph q,qz X e
tive counterpart. ×(cid:12)δ EX(kX)+Ee(ke)−ET(kT)∓~ωph(q,qz) , (cid:12)(35)
WewritethescatteringtermintheBoltzmannequa- (cid:12) (cid:12)
(cid:2) (cid:3)
tion process for the bimolecular formation of trions:
with E (k ), E (k ) and E (k ) the energy disper-
X X e e T T
tionofthe electrons,holesandexcitonsrespectively,and
df (k )
e e =− F f (k )f (k ), ~ω (q,q )theenergyoftheemitted(absorbed)phonon.
dt kX,ke→kT X X e e ph z
(cid:18) (cid:19)form kXT,ke We firstbuild the boundandunboundelectron-holepair
(31) states |k i and |k i⊗|k i.
T X e
6
A. Trion state. and the state of a single trion with an in-plane CM mo-
mentum k is constructed similarly to that of a single
T
The two electrons and the hole are positioned at exciton
(r ,z ), (r ,z ) and (r ,z ) respectively, while the
1k 1 2k 2 hk h
center-of-mass(CM)andrelativecoordinatesinthe QW
kS = dz dz dz φT∗ (z ,z ,z )
plane are now givenby Rk =αT(r1k+r2k)+βTrhk and T 1 2 h αTkT+k1,αTkT+k2 1 2 h
rihk =rik−rhk (i =1,2); αT =me/MT, βT = mh/MT (cid:12)(cid:12) (cid:11) ksX11,,sk22Z
and M is the trion mass. We consider a simple two
T ×ξ∗(s ,s )cˆs1† cˆs2† dˆ† |0i, (40)
parameter Chandrasekhar-type trial envelope function S 1 2 −k1,z1 −k2,z2 kT+k1+k2,zh
that was successively used to calculate trion-electron
scattering17: where we have added the spin index to the electron cre-
ation operatorand introduced ξ (s ,s )=hS|s ,s i the
S 1 2 1 2
φT(r ,r ,z ,z ,z ) projectionofagenericspinconfigurationoftwoelectrons
1hk 2hk 1 2 h
on the singlet spin configuration. The in-plane Fourier
=N χ (z )χ (z )χ (z )
T e 1 e 2 h h transform of the trion singlet wavefunction in Eq. (39)
[ϕλT(r )ϕλ′T(r )±ϕλ′T(r )ϕλT(r )], (36) is given by
1h 2h 1h 2h
where the + (−) sign applies to the singlet (triplet) spin
φT∗ (z ,z ,z )=N χ (z )χ (z )χ (z )
configuration and the trion wavefunction normalization k1,k2 1 2 h T e 1 e 2 h h
factor is given by × ϕλTϕλ′T +ϕλ′TϕλT , (41)
k1 k2 k1 k2
1 h i
N = , (37)
T 2(1±κ2) where ϕλ′T has already been defined in Eq. 9. The
k1
exciton-free electron state is given by
with p
κ= (λ4+λλλ′′)2. (38) |kX,ke′i= dzedzhdze′ φ∗αXkX+ke(ze,zh)ψk∗e′(ze′)
Xke Z
Its in-plane Fourier transform reads × cˆ†−ke,zedˆ†kX+ke,zhcˆ†ke′,ze′ |0i. (42)
B. Matrix element calculation.
ΦT (z ,z ,z )
k1,k2,kh 1 2 h
=δ φT (z ,z ,z ) (39)
kT−k1−k2−kh αTkT−k1,αTkT−k2 1 2 h We calculate the matrix elements in (35)
hkS;s′|⊗hn ±1|H |n +1i⊗|k ;s ;s i⊗|k ;s i
T h q,qz e/h−ph q,qz X 1 h 2 2
= dz dz dz dz dz′ dz′ dz′ n +1
1 2 h 1 2 h q,qz
Xk,s Xk1 kX′1,k′2sX′1,s′2Z p
×ξ∗(s′,s′)φX (z ,z )ψ (z )φT∗ (z′,z′,z′)eiqzz
S 1 2 αXkX+k1 1 h k2 2 αTkT+k′1,αTkT+k′2 1 2 h
× Ve h0|cˆ cˆ cˆ†cˆ cˆ†cˆ†|0ih0|dˆdˆ†|0i−Vh h0|cˆ cˆ cˆ†cˆ†|0ih0|dˆdˆ†dˆdˆ†|0i . (43)
q,qz 1 2 3 4 5 6 7 8 q,qz 1 2 5 6 7 3 4 8
h i
where we simplified the operators index using the following scheme
1=(−k′,z′,s′) 2=(−k′,z′,s′) 3=(k+q,z,s)
1 1 1 2 2 2
4=(k,z,s) 5=(−k ,z ,s ) 6=(k ,z ,s )
1 1 1 2 2 2
7=(k +k′ +k′,z′,s′) 8=(k +k ,z ,s )
T 1 2 h h X 1 h h
Using the electron, holes anti-commutation relations, the Fermi vacuum expectation value of the operators read
h0|cˆ cˆ cˆ†cˆ cˆ†cˆ†|0i=δ (δ δ −δ δ )−δ (δ δ −δ δ ) (44)
1 2 3 4 5 6 13 25 46 26 45 23 15 46 16 45
h0|dˆdˆ†|0i=δ (45)
7 8 78
h0|cˆ cˆ cˆ†cˆ†|0i=δ δ −δ δ (46)
1 2 5 6 25 16 15 26
h0|dˆdˆ†dˆdˆ†|0i=δ δ (47)
7 3 4 8 37 48
7
Using the later results in Eq. (43) gives
hkS;s′|⊗hn ±1|H |n i⊗|k ;s ;s i⊗|k ;s i
T h q,qz e/h−ph q,qz X 1 h 2 2
=[ξ∗(s ,s )−ξ∗(s ,s )]δ n +1/2±1/2 φX
S 2 1 S 1 2 sh,s′h q,qz αXkX+k1
q Xk1
× Ve I (q )[φT∗ +φT∗ ]−Vh I (q )φT∗ δ . (48)
q,qz e z αTkT−k2,αTkT+k1+q αTkT−k2+q,αTkT+k1 q,qz h z αTkT−k2,αTkT+k1 q,kX+k2−kT
n o
Finally, for parabolic electron, exciton and trion dispersion, the probability transition (35) is
2π
w± = (n + 1 ± 1)
kX,k2→kT ~ q,qz 2 2
qX,qz
2
× Ve I (q )[φT∗ +φT∗ ]−Vh I (q )φT∗
q,qz e z αTkT−k2,αTkT+k1+q αTkT−k2+q,αTkT+k1 q,qz h z αTkT−k2,αTkT+k1
(cid:12)(cid:12)Xk1 (cid:12)(cid:12)
(cid:12) ~2k2 ~2k2 ~2k2 (cid:12)
(cid:12) ×δ X + 2 +E − T ∓~ω (q,q ) δ (cid:12) , (49)
2M 2m T 2M ph z ±q,kX+k2−kT
(cid:20) X e T (cid:21)
where we averagedover the initial electron spin states and exciton angular momentum states.
IV. TRIMOLECULAR FORMATION OF k , we obtain an adiabatic equation for the evolution of
e
TRIONS the electron density
dn
Again, we write the scattering term in the Boltz- e =− F(k )n2n ≡−A−n2n . (51)
mann equation process dt T e h 3 e h
XkT
The coefficient A− is the trimolecular formation coeffi-
df (k ) 3
e e =− F f (k )f (k′)f (k ),cient, which depends on both T and the lattice temper-
dt ke,k′e,kh→kT e e e e h h c
(cid:18) (cid:19)form kXT,k′e ature Tl through the term
(50)
where F represents the probability per unit 2π~2 3 1
ke,k′e,kh→kT F(k )= w
time for two free electrons and one free hole to bind to- T k T m2m S3/2 ke,ke,k′h→kT
gether and form an exciton. We assume that bound and (cid:18) B c(cid:19) e h keX,k′e,kh
unboundcarriersarethermalizedanduseBoltzmanndis- ×e−(Ee(ke)+Ee(k′e)+Eh(kh))/kBTc. (52)
tributionfunctionf (k )andf (k )forexcitonandelec-
X e h h
tronpopulation,respectively. Bysumming Eq.(50)over WecalculatetheformationrateusingFermi’sgoldenrule
w± = 2π hk |⊗hn ±1|H |n i⊗|k i⊗|k′i⊗|k i 2
ke,k′e,kh→kT ~ T q,qz e/h−ph q,qz e e h
qX,qz(cid:12) (cid:12)
(cid:12) ×δ E (k )+E (k′)+E (cid:12)(k )−E (k )∓~ω (q,q ) , (53)
e e e e h h T T ph z
with E (k ), E (k ) and E (k ) the energy disperti(cid:2)on of the electrons, holes and excitons respective(cid:3)ly, and
X X e e T T
~ω (q,q )theenergyofthe emitted(absorbed)phonon. Finally,forparabolicelectron,excitonandtriondispersion,
ph z
the probability transition 53 becomes
2π
w± = (n + 1 ± 1)
ke,k′e,kh→kT ~ q,qz 2 2
qX,qz
2
× Ve I (q )[φT∗ +φT∗ ]−Vh I (q )φT∗
q,qz e z αTkT−k2,αTkT+k1+q αTkT−k2+q,αTkT+k1 q,qz h z αTkT−k2,αTkT+k1
(cid:12) (cid:12)
(cid:12)(cid:12)(cid:12) ×δ ~22mke2 + ~22mke′2 + ~22mkh2 +ET − ~2M2kT2 ∓~ωph(q,qz) δ±q,kX+(cid:12)(cid:12)(cid:12)k2−kT, (54)
(cid:20) e e h T (cid:21)
8
where we averagedover the initial electron spin states and exciton angular momentum states.
A. Numerical results
e+h↔X
x 10-12 x 10-22 X +e↔X− X+h↔X+
1.2 0.9
2e+h↔X− 2h+e↔X+
A (cm2ps-1) A (cm4ps-1)
2 3 The different formation rates for these populations
-1ps) 0.8 -1ps) 0.6 read
2m 4m FX =Cnp−γCK X, (55)
X
c c
A (2 0.4 A (30.3 FX− =A−Xn−A−K−X−, (56)
2 2 2 2
FX− =A−nnp−A−K−X−, (57)
3 3 3 3
00 0.05 0.1 00 0.05 0.1 F2X+ =A+2Xp−A+2K2+X+, (58)
1/Temperature [1/K] 1/Temperature [1/K] FX+ =A+npp−A+K+X+, (59)
3 3 3 3
FIG.2: Thebi-andtri-moleculartrionformationcoefficients where C, Aα and Aα are respectively the exciton,
Aα2 andAα3 asafunctionoftheinversecarriertemperatureas trion bimolec2ular and t3rion trimolecular formation rate
calculatedfromourmodelforLAphononassistedformation.
calculated in this article; E is the exciton binding
TheredcurveindicatesX+ formationandthebluecurveX− bX
energy;K , Kα, Kα the equilibrium coefficients. For
formation. X 2 3
a 2D system, they can be derived from the Boltzmann
distribution
In Fig. 2, we represent the results of our numeri-
cal calculation for LA phonon assisted formation. We n =2g d2ke−hbar2k2/2mαkBT =g mαkBTe−µα/KBT,
stress the fact that the results for LO phonons are or- α α 2π α 2π~2
Z
ders of magnitude smaller and consequently negligible (60)
for bi- and tri-molecular processes. This shows that the whereα=e,h,X,T,thefactorgα =2is thespindegen-
exclusive formation mechanism for trions is governed by eracy of the electron, hole, exction and trion in the non
LA phonon interaction. We predict a decrease of the bi- degenerate regime, where Boltzmann statistics applies.
or tri-molecular formation for raising temperatures. We Usingthatfactthatthechemicalpotentialoftheex-
alsodemonstratethatthebiandtri-molecularformation citon µX is relatedto the chemicalpotential of electrons
coefficientsfornegativelyandpositivelychargedexcitons µeandholesµhasµX =µe+µh+EX (EX isheredefined
are approximatively equal (A+ =A−) and (A+ =A−). as minus the exciton binding energy), one immediately
2 2 3 3
A more accurate calculation should rely on a bet- obtains the Saha relation for the exciton density
ter trion wavefunction. The Chandrashekar variational
np g g m m k T
function is mostlikely to simple to yield quantitative re- X =KX(T)= ge h me h2πB~2e−EX/kBT. (61)
sults. We note however that electron-trion scattering18 X X
will contribute to ionize trions. This effect will dramat- Similarly, the chemical potential of the trions is
ically increase at high temperatures. We consequently µT = 2µe +µh +ET (ET is here defined as minus the
propose that the electron-trion scattering reduce the ex- trion binding energy), so that Saha equations for trion
perimental values for the trion formation coefficients. bi-molecular formation are
Xn g2g m m k T
=K−(T)= e X X e B e−(ET−EX)/kBT
X− 2 g m 2π~2
V. FORMATION RATES CLOSE TO T X−
(62)
EQUILIBRIUM
Xp g2g m m k T
=K+(T)= h X X h B e−(ET−EX)/kBT
In this Section, we propose to derive formation X+ 2 g m 2π~2
T X+
rate for electrons, holes, excitons and trions assuming (63)
thermodynamical equilibrium. In this set of equations,
Finally, the following set of equation is infered for
we neglect both biexciton channels and Auger channels
tri-molecular formation:
because experiments on undoped samples demonstrate
that, at the densities considered in the present work, n2h g2g m2m k T 2
=K−(T)= e h e h B e−ET/kBT (64)
these channels are not significant8. X− 3 g m 2π~2
T X− (cid:18) (cid:19)
The dynamics of a plasma of electrons containing
nh2 g g2 m m2 k T 2
electrons(e),holes(h),excitons(X)andtrions(X+ and =K+(T)= e h e h B e−ET/kBT (65)
X−) is governedby the following five channels: X+ 3 gT mX+ (cid:18)2π~2(cid:19)
9
Such a set of equations, together with the equilib- of trions. We have shown that bi- and tri-molecular for-
riumdensities,allowtocomputethedynamicsofthedif- mationrateofnegativelyandpositivelychargedexcitons
ferent populations after non-resonant optical excitation. have similar orders of magnitude for densities that are
This may apply both to undoped8 as well as to doped used in the experiments. We have then developed the
quantumwells. Theresultsinthecaseofasampledoped set of relations and the equilibrium conditions allowing
with electrons will be detailed in another publication13. to calculate the dynamics of free carriers, excitons and
The experiments show that indeed, when the density of trions. Ourresultsareinveryreasonableagreementwith
electrons is sufficient and at excitation densities of the recentexperimentsandallowtoconfirmthat,indeed,tri-
orderof1010cm−2 andabove,thetrimolecularformation molecular trion formation may not be neglected in real
processoftrionshastobetakenintoaccounttoproperly samples.
reproduce the observed dynamics.
VI. CONCLUSION VII. ACKNOWLEDGMENTS
Inthispaper,wehavederivedtheappropriatemodel We wish to thank Fabienne Michelini, Michiel
for computing the rates for exciton and trion formation. Wouters, Christiano Ciuti, Carlo Piermarocchi and Vi-
For the case of trions, we have derived the equations for cenzo Savona for fruitful discussions. We also thank the
boththebi-andtri-molecularphonon-assistedformation Swiss National Fundation for funding.
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