Fine structure of excited excitonic states in quantum disks

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by  M. M. Glazov

Fine structure of excited excitonic states in quantum disks
M. M. Glazov,1 E. L. Ivchenko,1 R. v. Baltz,2 and E. G. Tsitsishvili2
1A.F. Ioffe Physico-Technical Institute RAS, 194021 St.-Petersburg, Russia
2Universit¨at Karlsruhe, 76128 Karlsruhe, Germany
5 Wereportonatheoreticalstudyofthefinestructureofexcitedexcitoniclevelsinsemiconductor
0 quantum disks. A particular attention is paid to the effect of electron-hole long-range exchange
0 interaction. We demonstrate that, even in the axisymmetric quantum disks, the exciton P-shell is
2 split into three sublevels. The analytical results are obtained in the limiting cases of strong and
n weak confinement. A possibility of exciton spin relaxation dueto theresonant LO-phonon-assisted
a coupling between the P and S shells is discussed.
J

6
2 I. INTRODUCTION similarly to4 if one includes an admixture of light-hole
states into the heavy-hole exciton wave function.

] Weconsideraquantumdiskformedbya2Dharmonic
In the envelope-function approximation, the excitonic
l
al states in semiconductors and semiconductor quantum potential V(ρe,ρh) = Aeρ2e +Ahρ2h in a quantum well
grown along the z-axis. Here the 2D vector ρ deter-
h dots can be classified by referring to the orbital shape e,h
- of the exciton envelope function (S-, P-, D-like)and the mines the in-plane position of an electron or a hole, and
s A are positive constants. Assuming that the confine-
e characteroftheelectronandholeBlochfunctions(bright e,h
m and dark states). For example, the state P y means a ment along the growth direction is stronger than both
x
| i the quantum-disk and Coulomb potentials the envelope
brightP -shell exciton with the electron-hole dipole mo-
. x
t for the electron-holepair wavefunctioncanbe written as
a ment directed along the y axis. Theoretically, the fine
m structureofexcitedstates ofzero-dimensional(0D)exci-
- tonshasbeenstudiedforexcitonslocalized,respectively, Ψ(re,rh)=ψ(ρe,ρh)ϕe(ze)ϕe(zh), (1)
d by rectangular islands in a quantum-well structure1,
n quantum disks with a Gaussian lateral potential2 and where ϕe,h(ze,h) are the respective z-envelopes of elec-
o lens-shaped quantum dots3. However, up to now the tronandhole, and ψ(ρe,ρh) is anin-plane wavefunction
c
splitting of excited-exciton levels has been analyzed for of exciton calculated with allowance for both Coulomb
[
a fixed orbital shell, e.g., the P shell, which is valid in interaction and quantum disk potential. The form of
x
1 the case of strongly anisotropic confinement so that the the function ψ(ρe,ρh) depends on the relationship be-
v orbitalsplittingoftheP-likeshellsP andP exceedsby tween the potential radius and 2D-exciton Bohr radius,
5 x y
3 far the exchange-interaction energy. Here we show that, aB. In the weak confinement regime, i.e. in a quantum
6 foraxiallysymmetricorsquarequantumdots,onehas,in diskwiththediameterexceedingaB,thetwo-particleen-
1 addition, to take into account the exchange-interaction- velope probe function ψ(ρe,ρh) is a product F(R)f(ρ)
0 induced mixing between the excitonic states P y and oftwofunctionsdescribing,respectively,thein-planemo-
x
5 P x . Moreover, for the first time we focus o|n thie in- tion of the exciton center of mass R = (X,Y) and the
y
0 |terpliaybetweentheexchangeinteractionandanisotropic relativeelectron-holemotion,ρ=ρe ρh. Onthe other
t/ shape of the dot and develop an analytic theory in the hand, in small quantum disks with −strong confinement
a particularcasewhere the orbitalandexchangesplittings where single-particle lateral confinement dominates over
m
are comparable. the Coulomb interaction, the pair envelope can also be
- presented as a product of two functions, ψ (ρ )ψ (ρ ),
d e e h h
butheretheydescribetheindependentin-planelocaliza-
n
o II. ELECTRON-HOLE EXCHANGE tion of an electron and a hole.
c INTERACTION IN QUANTUM DISKS For the exciton envelope functions presented in the
v: form (1), the matrix element Hn(l′onng) of long-range ex-′
i change interaction taken between the exciton states n
X The two-particle excitonic wave function can be writ-
and n is written as follows
ten as a linear combination of products Ψ (r ,r )s,j ,
ar wfuhnecrteio|nss,,jsi aisnda jpraordeutchteoefletchteroenleactnrdonhoalsnejdspheinolehinBd|ilcoecsih, 2π1κ∞ (cid:18)me~0|pE0g|(cid:19)2Z dKKαKKα′ψn∗′(K)ψn(K), (2)
Ψ(r ,r ) is the envelope function, and r is the elec-
e h e,h e e
tron (hole) 3D radius-vector. In the following, for the where the exciton-state index n includes the dipole mo-
sakeofsimplicity, we concentrateonheavy-holeexcitons mentα,κ∞ isthehigh-frequencydielectricconstant,m0
withj = 3/2andthelong-rangemechanismofelectron- isthefreeelectronmass,Eg isthebandgap,p0 isthein-
holeexcha±ngeinteractionwhichallowstodiscussonlythe terband matrix element of the momentum operator,and
brightexcitonicstates s,j withs+j = 1ortheirlinear we introduced the 2D Fourier-transform
| i ±
combinations α with the dipole moment α=x,y. The
| i ψ(K)= dRe−iKRψ(R,R)
short-range mechanism may be taken into consideration Z
e
2
of the function ψ(ρ ,ρ ) at the coinciding coordinates, to introduce, as a parameter describing the anisotropy,
e h
ρ =ρ R. a half of the P -P splitting, E , calculated neglecting
e h x y anis
≡
theexchangeinteraction. Itsvaluedependsonthemodel
of exciton quantization. If an electron and a hole are
III. P-ORBITAL EXCITONS IN quantized independently (a a ) then, for the P ,S
B e h
AXIALLY-SYMMETRIC DISKS excited state, we have ≪ | i
We startwith anaxially-symmetricquantum disk and d 2~
E = (6)
calculate the fine structure of the P-orbital exciton level anis am a2
e
and then proceed to a slightly anisotropic disk. The
straightforwardcalculationshowsthatthelong-rangeex- and, for the Se,Ph state, Eanis differs from Eq. (6)
| i
changeHamiltonian(2)hasthefollowingnon-zeromatrix by the replacement me mh. Here Se,Ph means
→ | i
elements an exciton formed by a Se-shell electron and a Ph-shell
hole. If exciton is quantized as a whole then in Eq. (6)
hPxx|H(long)|Pxxi=hPyy|H(long)|Pyyi=λ , (3) one should replace me by the exciton translational mass
M =m +m and a by a =a(µ/M)1/4.
e h exc
Figure 1 shows splitting of the P-shell bright-exciton
P y (long) P y = P x (long) P x =η ,
h x |H | x i h y |H | y i level as a function the ratio ξ = Eanis/∆. For small
valuesofξ thesplittingofthedoublet 2isproportional
±
hPyx|H(long)|Pxyi=hPxy|H(long)|Pyxi=µ , ttoheξP2.-shIenlltehxecitliomnitstaotfesstfroornmgtawnoisdootruobpleyt,s,EaPnis,α≫an∆d,
x
| i
where λ = 3η = 3µ. According to Eq. (3) and in agree- P ,α (α=x,y), separated by 2E and each split by
y anis
| i
ment with the angular-momentum considerations, the ∆.

P-shell of the bright exciton in an axisymmetric quan-
tum disk is split into three sublevels, see Fig. 1. The (P , x)
4 x
outermost sublevels labelled 0U, 0L are nondegenerate
and characterized by a zero total angular-momentum z-
2 (P , y)
component. The central doubly-degenerate sublevel cor- 0U x
respondstothe angular-momentumcomponent±2. The D / 0 +2
intersublevel energy spacing, ∆, equals to 2η =2µ. E -2
The splitting ∆ depends on the character of exciton -2 0L (P , y)
confinementinadisk. LetusassumeA =~2/2m a4, y
e,h e,h
where ais the disk effective radius,m arethe electron
e,h -4
and hole effective masses. It follows then that in the (P , y)
y
strong confinement regime, a aB, one has 0.0 0.5 1.0 1.5 2.0 2.5 3.0
≪
3√2π e~p 2 x=E /D
∆= | 0| . (4) anis
16a3κ (cid:18)m E (cid:19)
b 0 g
FIG. 1: Interplay between the in-plane anisotropy and ex-
In the opposite limiting case a aB where the exciton changeinteraction. Theexciton sublevelenergyE isreferred
≫
is quantized as a whole we obtain to theenergy of theuniaxial exciton ±2.
3√π e~p 2
0
∆= | | (5)
2a a2κ (cid:18)m E (cid:19)
exc B b 0 g
with a = a(µ/M)1/4 being the radius of exciton in- V. RESONANT P-S EXCITONIC POLARONS
exc
plane confinement, M =m +m and µ=m m /M.
e h e h
Finally, we briefly discuss the P-S excitonic polaron
which is formed as a result of the resonant coupling of
IV. P-SHELL EXCITONS IN ANISOTROPIC the P- and S-like levels by a longitudinal optical (LO)
QUANTUM DISKS phonon5. We use an approachdeveloped in Ref.6 to cal-
culatetheLO-assistedresonantcouplingbetweenS-and
We introduce a slightly elliptical lateral potential re- P-shell electron states confined in a quantum dot. For
placingthediskradiusabytheeffectiveradiia =a d, the Fr¨olich interaction, the constant of exciton-phonon
x
−
a = a+d along the x and y axes, respectively, and as- P-S coupling is given by
y
suming d a. At zero d the exciton P-states are par-
≪
tially split by the exchange interaction. The anisotropy 2πe2~Ω (q) 2
of a quantum disk results in a full removal of the degen- γ =v I . (7)
u Vκ∗ (cid:12) q (cid:12)
eracy and formation of four sublevels. It is convenient u Xq (cid:12) (cid:12)
t (cid:12) (cid:12)
(cid:12) (cid:12)
3
HereΩistheLO-phononfrequency,V isthe 3Dnormal- tial linear polarization is being rapidly lost and the ex-
ization volume, κ∗−1 =κ−1 κ−1, and citon photoluminescence is practically depolarized. On
∞ − 0
the contrary, if ∆ γ then the polarization relaxes on
≪| |
(q)= dr dr Ψ (r ,r )Ψ (r ,r ) eiqre eiqrh . a much longer time scale and can be well preserved.
e h p e h s e h
I Z −
(cid:0) (cid:1) The work is supported by RFBR, “Dynasty” founda-
Depolarization of the linearly-polarized P-S excitonic tion–ICFPM,theCenterforFunctionalNanostructures
polaronexcitedviatheP-shellisdeterminedbytheratio of the Deutsche Forschungsgemeinschaft within project
of γ and the exchange splitting ∆. If ∆ γ the ini- A2 and by the programs of Russian Academy of Sci.
≫ | |
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