The document studies small excitonic complexes in a disk-shaped quantum dot using the Bethe-Goldstone equation. It examines systems with up to 12 electron-hole pairs. For symmetric configurations where the number of electrons equals the number of holes, it finds: 1) The triexciton and four-exciton system show weak binding or possible unbinding in the weak confinement regime. 2) Higher complexes beyond four pairs exhibit binding in the weak confinement regime. 3) The Bethe-Goldstone approach provides better energies than the BCS variational method in the weak confinement regime.

1. Physica E 8 (2000) 91–98
www.elsevier.nl/locate/physe
Small excitonic complexes in a disk-shaped quantum dot
Ricardo PÃ reza; b; ∗ , Augusto Gonzalezb
e
a Centro de Matemà ticas y FÃsica Teà rica, Calle E No. 309, Ciudad Habana, Cuba
a à o
b Universidad de Antioquia, A.A. 1226, MedellÃn, Colombia
Ã
Received 28 October 1999; accepted 18 November 1999
Abstract
Conÿned excitonic complexes in two dimensions, consisting of Ne electrons and Nh holes, are studied by means of Bethe–
Goldstone equations. Systems with up to 12 pairs, and asymmetric conÿgurations with Ne = Nh are considered. The weak
conÿnement regime gives indication of weak binding or even unbinding in the triexciton and the four-exciton system, and
binding in the higher complexes. ? 2000 Elsevier Science B.V. All rights reserved.
PACS: 71.35.E.e; 78.65.−w
Keywords: Quantum dots; Electron–hole systems; Bethe–Goldstone equations
1. Introduction citon peaks in photoluminescence have been identi-
ÿed in quantum dots [2– 4]. Recently, a confocal mi-
Although the excitonic matter as a research object croscopy technique has been applied to resolve the
has already a relatively long history, dating back to luminescence coming from a single self-assembled
pioneer theoretical ideas in the 1960s [1], the study quantum dot [5]. The power of the excitation laser is
of small excitonic complexes conÿned in regions of such that up to six pairs are excited simultaneously in
nanometer scale became possible only recently, and is the dot. Clearly distinct lines coming from transitions
closely related to dramatic advances in semiconductor between multiexcitonic complexes are identiÿed.
technology. The calculations presented in our paper, although
Due to the smallness of the exciton lifetime in semi- unrealistic, are inspired by the experimental work [5].
conductors (hundreds of picoseconds, sometimes even We address a di erent question, i.e. the dependence
greater), signatures of excitonic complexes are looked of the ground-state energy on the number of pairs.
for mainly in optical experiments. Exciton and biex- From this dependence, the lowest optical absorption
line related to the creation of an electron–hole pair
∗ Correspondence address. Centro de Matemà ticas y FÃsica
in the background of N photo-excited pairs may be
a
Teà rica, Calle E No. 309, Ciudad Habana, Cuba.
o
extracted.
E-mail addresses: rperez@cidet.icmf.inf.cu (R. PÃ rez), agon-
e On the other hand, small excitonic systems have
zale@cidet.icmf.inf.cu (A. Gonzalez). been widely studied theoretically in order to describe
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved.
PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 3 2 7 - 6

2. 92 R. PÃ rez, A. Gonzalez / Physica E 8 (2000) 91–98
e
the optical properties of bulk and low-dimensional ductor, ÿ = Ec =(˜!0 ), and Ec = me4 =(Ä3 ˜2 ) is the
semiconductors. To the best of our knowledge, one of characteristic Coulomb energy.
the most complete studies is the paper [6], in which The only parameter entering Hamiltonian (1) is
a variational approach is used to compute the prop- ÿ, which may be thought of as the inverse of the
erties of two- and tri-dimensional free (not conÿned) conÿnement strength. In the ÿ = 0 limit, we have
excitons with up to ÿve particles. Our work has also a picture of non-interacting fermions whose energy
a close connection with that paper. levels and one-particle quantum states are those
By using the Bethe–Goldstone (BG) equation to of the 2D harmonic oscillator. The electron (hole)
compute the ground-state energies, we are able to rise states are grouped into shells with “magic numbers”
the number of particles in the complex from a max- Ne ; Nh = 2; 6; 12; : : : : As ÿ grows, correlations be-
imum of ÿve in the variational approach [6] to 24. tween particles become more and more important and
Asymmetric systems with Ne = Nh are studied as well. the previous picture is modiÿed.
In the weak conÿnement regime, we can also address One way to go beyond the independent-particle pic-
the question about the stability of the free complexes. ture is the independent-pair approximation in which
And in fact, our results give indications that the free an exact treatment of the two-body correlations is
triexciton and the four-exciton system are unstable (or made. Its main component is the Bethe–Goldstone
weakly bound), whereas the higher complexes are sta- (BG) equation, extensively used in nuclear matter and
ble. These properties have a distinct trace in the optical ÿnite nucleus calculations [8,9].
absorption, and may be conÿrmed by the experiments. The BG equation applies only to fermionic sys-
The present paper is complementary to Ref. [7], in tems. It describes the motion of an independent pair
which a BCS variational approach is used to study of fermions in the system. The rest of the particles ex-
intermediate-size complexes, containing from 12 to erts an indirect in uence on the pair motion through
180 particles. For those sizes where both approaches the Pauli principle. The equation takes the form
overlap, as a rule BCS gives lower energy values for
strong conÿnement, whereas BG gives better energies (T1 + T2 + Q V ) =E ; (2)
in the weak conÿnement regime.
where and label the states of each fermion in the
pair. These states are below the Fermi level. The Ti
2. The Bethe–Goldstone equation are the one-particle terms in the Hamiltonian, V is the
two-body interaction potential (Coulomb), and Q is
We will study a two-dimensional model of Ne elec- a projection operator given by
trons and Nh holes conÿned by a parabolic potential in
the plane of motion, and interacting via pure Coulomb Q = | )( | + | )( |; (3)
;
potentials. Only one conduction and one valence band,
both ideally parabolic, will be considered, and the where the sum runs over states above the Fermi level.
masses of holes and electrons will be supposed to be Q projects a given function onto states over the Fermi
equal. By choosing the scales of distances and ener- level. (r1 r2 | ) is the unsymmetrized product of two
gies as ˜=m!0 and ˜!0 , we get the dimensionless non-interacting one-particle eigenfunctions (we use
Hamiltonian the notation given in Ref. [10]). E is the pair energy,
Ne pi2 r2 Nh pi2 r2 and – its wave function.
H= + i + + i
i=1 2 2 i=1 2 2 Eq. (2) is formally similar to a pair scattering equa-
tion, except for the presence of the projection opera-
Ne 1 Nh 1 Ne Nh 1 tor (and the fact that all of the states in the external
+ÿ + − ; (1) quadratic potential are bound states). The pair wave
i¡j rij i¡j rij i j rij
function is looked for in the form
where the ri are in-plane coordinates for the particles,
rij = |ri − rj |; !0 is the frequency of the conÿning =| )+ C | ) (4)
potential, Ä is the dielectric constant of the semicon- ;

3. R. PÃ rez, A. Gonzalez / Physica E 8 (2000) 91–98
e 93
and the total energy is computed from 700 unknowns each, and 15 nonlinear equations for
(0) the pair energies are solved.
E= + ; (5)
¡
where = E − (0) − (0) . The corrections coming 3. Symmetric (Ne = Nh ) systems
from the BG equations are proven to be equiva-
lent to summing up all the ladder diagrams in the In this section we present the results obtained for
linked-cluster expansion for the energy [11]. symmetric systems, where Ne = Nh = N . The param-
Multiplying Eq. (2) from the left by ( | or ( |, eter ÿ is varied in the interval (0; 2:5).
we get The ÿrst step in our Bethe–Goldstone calculation is
(0) (0) to deÿne ÿlled and empty levels, i.e the Fermi surface.
( + − E )C + ( |V | )C
; To this end, a Hartree–Fock (HF) calculation was im-
plemented. The harmonic-oscillator states were ÿlled
=−( |V | ); (6) in accordance with the HF results. We show them in
spectroscopic (nuclear) notation in Table 1. 2p− , for
(0) (0)
E = + + ( |V | ) example, means the second level with lz = −1. Se , and
Sh refer to the total electron and hole spins respec-
+ ( |V | )C : (7) tively, and L is the total angular momentum (along the
;
z-axis).
Eq. (6) may be seen as a linear system of equa-
tions for the coe cients C , from which we obtain 3.1. The biexciton
C = C (E ). Then, the transcendental equation (7)
is solved for E . For the biexciton, our starting conÿguration is one
One shall note that the Coulomb interaction does in which the ÿrst harmonic oscillator shell is ÿlled for
not change either the angular momentum of the pair both electron and holes (see Table 1). This state has
or the spin of the particles. The matrix elements are zero total angular momentum and total spin. In order
real and have the following properties: to write down the Bethe–Goldstone equations we shall
identify all possible pairs below the Fermi level. In
( |V | ) = ( |V | )=( |V | ): (8)
the e–e and h–h sectors there is only one pair, but we
Due to these properties, the matrix entering the linear have 4 e–h pairs. For a given initial pair, the number of
system is symmetric. ÿnal pair states above the Fermi level depends on the
In order to solve the BG equation we have to iden- number of harmonic oscillator shells, Nshell , included
tify, given an initial pair, all the possible ÿnal states in the calculations. Fig. 1 shows the results for the
above the Fermi level perserving angular momentum energy as a function of ÿ and Nshell . The solid line is
and spin. Then we solve the linear system and the tran- a two-point Padà approximant, which construction is
e
scendental equation to get the pair energy. One shall described below.
take into account that there are three kinds of pairs in A few remarks shall be made at this point. First,
the system, namely e–e, h–h and e–h. In the second the convergence is slow. Second, the BG results do
term of Eq. (5), ¡ means that for identical parti- not exactly reproduce the perturbative energies at
cles (e–e and h–h) a state | ) should be counted only small values of ÿ. It means simply that the charac-
once. For e–h pairs, however, we should take into ac- teristic distances are much smaller than the charac-
count the two possibilities, i.e. | e h ) and | h e ). teristic pair dimensions, and thus the independent
In our calculations, up to 272 harmonic-oscillator pair approximation breaks down. Fortunately, in this
one-particle states (16 shells) are included. Whenever strong-conÿnement regime the energy is dominated
possible, the e–h, spatial inversion and time-reversal by the one-particle energies, which are properly ac-
symmetries are used to reduce the actual number of counted for by the BG approach. Finally, the BG
equations to solve. For example, in the Ne = Nh = 6 energies seem to overestimate the actual binding
problem, 15 systems of linear equations with roughly energies at large ÿ.

4. 94 R. PÃ rez, A. Gonzalez / Physica E 8 (2000) 91–98
e
Table 1
Occupied electron and hole states for the BG calculations
Ne Nh Electrons Holes Se Sh L
2 2 1s2 1s2 0 0 0
1 1
3 3 1s2 1p1
+ 1s2 1p1
− 2 2
0
4 4 1s2 1p1 1p1
+ − 1s2 1p1 1p1
+ − 1 1 0
1 1
5 5 1s2 1p2 1p1
+ − 1s2 1p1 1p2
+ − 2 2
0
6 6 1s2 1p2 1p2
+ − 1s2 1p2 1p2
+ − 0 0 0
12 12 1s2 1p2 1p2 2s2 1d+ 1d−
+ −
2 2 1s2 1p2 1p2 2s2 1d+ 1d−
+ −
2 2 0 0 0
4 2 2 1p1 1p1
1s + − 1s 2 1 0 0
6 2 1s2 1p2 1p2
+ − 1s2 0 0 0
Fig. 1. Convergence of the biexciton energy as a function of the number of harmonic-oscillator shells included in the calculations.
The two-point Padà approximant for the ground-state
e oscillation, contributing with a one to the ground-state
energy is constructed in the following way [12,13]. energy.
The asymptotic expansions The two-point approximant is a rational function
interpolating between Eqs. (9) and (10):
E|ÿ→0 = b0 + b1 ÿ + b2 ÿ2 + O(ÿ3 ); (9)
b2 ÿ2 + p3 ÿ3 + p4 ÿ4
P(ÿ) = b0 + b1 ÿ + : (11)
1 + q1 ÿ + q2 ÿ2
E|ÿ→∞ = a0 ÿ2 + a2 + O(1=ÿ2 ) (10)
The values of the parameters are the following,
are used, where b0 = 4; b1 = −2:50662; b2 = p3 = 0:525372; p4 = −2:06239; q1 = 0:83554 and
−2:92 [7], and a0 = −2:1928 [6], a2 = 1. a0 is the q2 = 0:940527.
ground-state energy of the free biexciton in two di- As for electron quantum dots [13,14] or conÿned
mensions, and a2 comes from the center of mass charged bosons [15], we expect the error of the PadÃ
e

5. R. PÃ rez, A. Gonzalez / Physica E 8 (2000) 91–98
e 95
Fig. 2. HF, an improved BCS, and BG energies of the 6-exciton complex.
Fig. 3. Energies of the N -exciton systems as a function of ÿ.

6. 96 R. PÃ rez, A. Gonzalez / Physica E 8 (2000) 91–98
e
Fig. 4. Energy di erences EN +1 − EN for the smallest clusters studied.
approximant to be lower than a few percents, typically BG estimate slightly fails for small ÿ, as mentioned
3–5%. Thus, our BG estimate seems to reproduce the above. The relative error, however, is not signiÿcant.
actual biexciton energy. And, what is more important, this small-ÿ region cor-
responds to very strong conÿnements, i.e. very small
3.2. N = 3; 4; 5; 6 and 12 dot sizes, in which a basic assumption of our model
like the e ective mass approximation breaks down.
Next, we present results for N = 3; 4; 5; 6 and In the more physically interesting region, ÿ¿1, the
12, computed with Nshell = 16. The occupied BG approximation gives the lowest energies, mean-
harmonic-oscillator states are indicated in Table 1. ing that it takes proper account of two-particle corre-
Note that, in some cases, the HF approximation does lations.
not suggest a unique state, thus we performed BG The BG results are drawn in Fig. 3 in a “scaled”
calculations with di erent occupations, and select the form, i.e. Egs =N 3=2 versus ÿ=N 1=4 . This scaling comes
one corresponding to the minimal energy. For exam- from the dependence b0 ≈ 4 N 3=2 , b1 ≈ −0:96N 5=4 ,
3
ple, in the N = 3 system we performed calculations b2 ≈ −1:65N and a0 ≈ −N for large N [7], but it is
also for a conÿguration very similar to the one given nicely satisÿed even for the smallest systems. In this
in Table 1, but with total momentum L = 2. scaled drawing, the N = 4 cluster clearly distinguishes
To test the quality of our BG calculations in these as the less bound one.
larger systems, we show in Fig. 2 three di erent ap- The results at the largest values of ÿ, i.e. in the
proximations for the energy of the N = 6 complex. weak conÿnement regime, can be taken as indica-
The highest curve corresponds to the HF results. The tions of stability or instability of the free clusters. The
curve denoted BCS is in fact a Lipkin–Nogami calcu- triexciton seems to be unbound (like in three dimen-
lation with exact projection onto the N -pair sector by sions [16]), and the four-exciton system – evidently
means of a Monte-Carlo algorithm [7]. Note that the unbound. However, the larger clusters are likely to

7. R. PÃ rez, A. Gonzalez / Physica E 8 (2000) 91–98
e 97
Fig. 5. Energies of the Ne = 4; Nh = 2, and Ne = 6; Nh = 2 systems.
be stable. The situation may be analogous to nuclei, 4. Asymmetric cases
where there is a small instability island around atomic
number ÿve. We also studied non-symmetric or non-neutral
The di erences EN +1 − EN are shown in Fig. 4. We systems, in which Ne = Nh . In particular, the cases
note that these di erences enter in the expression Ne = 4, Nh = 2 and Ne = 6; Nh = 2. The results for
their ground-state energies are shown in Fig. 5.In a
e h
h = Egap + Ez + Ez + EN +1 − EN (12) non-neutral system, the energy is an increasing func-
tion of ÿ for low ÿ values, as for electrons [14], but
for weaker conÿnement the −ÿ2 biexcitonic contri-
for the frequency of light creating a new electron– bution dominates over the ÿ2=3 repulsion. A biexciton
hole pair in the background of N pairs. Egap is the plus remaining electrons shall be seen at very large ÿ.
semiconductor gap, and Ez h are the electron (hole)
e;
The Ne = 6, Nh = 2 system clearly reveals the change
conÿnement energies in the z-direction. in slope.
As follows from Fig. 4, the absorption peak corre-
sponding to Eq. (12) will exhibit an interesting be-
haviour as a function of the number of photoexcited 5. Conclusions
pairs, N . High values of N can be reached by rising
the laser excitation power. When N is around four, we In the present paper, we have shown that the
shall observe a highly blue-shifted line, followed by a Bethe–Goldstone equations may be used as a power-
red-shifted one as the power is further increased. The ful method to study small conÿned excitonic clusters.
conclusions coming from our oversimpliÿed model Pure electronic quantum dots may be studied as well,
have only a qualitative predictive power. The e ect and external electric and magnetic ÿelds can be eas-
is so pronounced, however, that we expect it may be ily included in the calculations. Larger unbalanced
observed in experiments. systems with only one exciton may be studied to

8. 98 R. PÃ rez, A. Gonzalez / Physica E 8 (2000) 91–98
e
obtain the optical absorption and photoluminescence [3] H. Drexler, D. Leonard, W. Hansen, J.P. Kotthaus,
of small electronic quantum dots. P.M. Petro , Phys. Rev. Lett. 73 (1994) 2252.
Our numerical results suggest a small instability [4] A. Kuther, M. Bayer, A. Forchel, A. Gorbunov,
V.B. Timofeev, F. Schafer, J.P. Reithmaier, Phys. Rev. B 58
island for the free clusters around Ne = Nh = 4. We (1998) 7508.
showed that this instability causes the appearance of [5] E. Dekel, D. Gershoni, E. Ehrenfreund, D. Spektor,
distinct absorption peaks as the laser excitation power J.M. Garcia, P.M. Petro , Phys. Rev. Lett. 80 (1998) 4991.
is raised. [6] J. Usukura, Y. Suzuki, K. Varga, Phys. Rev. B 59 (1999)
More realistic calculations, closely related to the 5652.
[7] B.A. Rodriguez, A. Gonzalez, L. Quiroga, R. Capote,
experimental results [5], are in progress. F.J. Rodriguez, Int. J. Mod. Phys. B 14 (2000) 71.
[8] H.A. Bethe, Ann. Rev. Nucl. Sci. 21 (1971) 93.
[9] A. de Shalit, H. Feshbach, Theoretical Nuclear Physics, Vol.
Acknowledgements 1, Wiley, New York, 1974.
[10] J.P. Blaizot, G. Ripka, Quantum Theory of Finite Systems,
The authors acknowledge support from the Univer- MIT Press, Cambridge, 1986.
[11] B.D. Day, Rev. Mod. Phys. 39 (1967) 719.
sidad de Antioquia, Medellin. Part of this work was [12] A.H. MacDonald, D.S. Ritchie, Phys. Rev. B 33 (1986) 8336.
done during a visit to the Abdus Salam ICTP under the [13] A. Gonzalez, J. Phys.: Condens. Matter 9 (1997) 4643.
Associateship Scheme and the Visiting Young Student [14] A. Gonzalez, B. Partoens, F.M. Peeters, Phys. Rev. B
Programme. Useful discussions with E. Lipparini are 56 (1997) 15 740.
also acknowledged. [15] A. Gonzalez, B. Partoens, A. Matulis, F.M. Peeters, Phys.
Rev. B 59 (1999) 1653.
[16] K. Varga, Y. Suzuki, Phys. Rev. A 53 (1996) 1907.
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