Exciton in closed and opened quantum dot
The theory of exciton spectrum in spherically symmetric states for the three- shell closed spherical quantum dot is proposed. The evolution of the exciton spectrum while varying the outer well thickness from zero (stationary spectrum of single closed spherical quantum dot) to infinity (quasistationa...
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nasplib_isofts_kiev_ua-123456789-1179142025-02-23T18:23:48Z Exciton in closed and opened quantum dot Екситон у вiдкритiй та закритiй квантовiй точцi Tkach, M.V. Seti, Ju.O. The theory of exciton spectrum in spherically symmetric states for the three- shell closed spherical quantum dot is proposed. The evolution of the exciton spectrum while varying the outer well thickness from zero (stationary spectrum of single closed spherical quantum dot) to infinity (quasistationary spectrum of a single open spherical quantum dot) is investigated. The mechanism of damping (semiwidth) of quasistationary states due to the redistribution over the energy levels of probability of exciton location in the space of two inner shells of nanosystem is studied. It is shown that the three shell closed spherical quantum dot of a rather big thickness of the outer well quite sufficiently and exactly reflects the basic properties of the quasistationary exciton spectrum in a single open spherical quantum dot. Вперше запропонована теорiя екситонного спектра в сферично-симетричних станах тришарової закритої сферичної квантової точки (СКТ). Дослiджено еволюцiю спектра при змiнi товщини зовнiшнього шару-ями вiд нуля (стацiонарний спектр простої закритої СКТ) до безмежностi (квазiстацiонарний спектр простої вiдкритої СКТ). Дослiджено механiзм виникнення затухання (пiвширини) квазiстацiонарних станiв причиною якого є перерозподiл по енергетичних рiвнях ймовiрностей перебування екситона у межах двох внутрiшнiх шарiв наносистеми. Показано, що тришарова закрита СКТ з достатньо великою товщиною зовнiшньої ями з довiльною необхiдною точнiстю вiдтворює основнi властивостi квазiстацiонарного екситонного спектра у простiй вiдкритiй СКТ. This work was financially supported by MPZ PP “Kolos”. 2007 Article Exciton in closed and opened quantum dot / M.V. Tkach, Ju.O. Seti // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 23-31. — Бібліогр.: 11 назв. — англ. 1607-324X DOI:10.5488/CMP.10.1.23 PACS: 71.15.Dx, 73.21.La https://nasplib.isofts.kiev.ua/handle/123456789/117914 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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The theory of exciton spectrum in spherically symmetric states for the three- shell closed spherical quantum dot is proposed. The evolution of the exciton spectrum while varying the outer well thickness from zero (stationary spectrum of single closed spherical quantum dot) to infinity (quasistationary spectrum of a single open spherical quantum dot) is investigated. The mechanism of damping (semiwidth) of quasistationary states due to the redistribution over the energy levels of probability of exciton location in the space of two inner shells of nanosystem is studied. It is shown that the three shell closed spherical quantum dot of a rather big thickness of the outer well quite sufficiently and exactly reflects the basic properties of the quasistationary exciton spectrum in a single open spherical quantum dot. |
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Tkach, M.V. Seti, Ju.O. |
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Tkach, M.V. Seti, Ju.O. Exciton in closed and opened quantum dot Condensed Matter Physics |
| author_facet |
Tkach, M.V. Seti, Ju.O. |
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Tkach, M.V. |
| title |
Exciton in closed and opened quantum dot |
| title_short |
Exciton in closed and opened quantum dot |
| title_full |
Exciton in closed and opened quantum dot |
| title_fullStr |
Exciton in closed and opened quantum dot |
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Exciton in closed and opened quantum dot |
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exciton in closed and opened quantum dot |
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Інститут фізики конденсованих систем НАН України |
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2007 |
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https://nasplib.isofts.kiev.ua/handle/123456789/117914 |
| citation_txt |
Exciton in closed and opened quantum dot / M.V. Tkach, Ju.O. Seti // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 23-31. — Бібліогр.: 11 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT tkachmv excitoninclosedandopenedquantumdot AT setijuo excitoninclosedandopenedquantumdot AT tkachmv eksitonuvidkritijtazakritijkvantovijtočci AT setijuo eksitonuvidkritijtazakritijkvantovijtočci |
| first_indexed |
2025-11-24T09:08:29Z |
| last_indexed |
2025-11-24T09:08:29Z |
| _version_ |
1849662174203478016 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 1(49), pp. 23–31
Exciton in closed and opened quantum dot
M.V.Tkach, Ju.O.Seti
Fedkovych Chernivtsi National University, 2, Kotsyubinskoho St., Chernivtsi, 58012, Ukraine∗
Received September 4, 2006
The theory of exciton spectrum in spherically symmetric states for the three- shell closed spherical quantum
dot is proposed. The evolution of the exciton spectrum while varying the outer well thickness from zero (stati-
onary spectrum of single closed spherical quantum dot) to infinity (quasistationary spectrum of a single open
spherical quantum dot) is investigated. The mechanism of damping (semiwidth) of quasistationary states due
to the redistribution over the energy levels of probability of exciton location in the space of two inner shells
of nanosystem is studied. It is shown that the three shell closed spherical quantum dot of a rather big thick-
ness of the outer well quite sufficiently and exactly reflects the basic properties of the quasistationary exciton
spectrum in a single open spherical quantum dot.
Key words: spherical quantum dot, quasistationary state, exciton energy spectrum
PACS: 71.15.Dx, 73.21.La
1. Introduction
The present-day experimental possibilities of growing the multilayer nanoheterosystems, specif-
ically, the technique of ion substitution, permit the fabrication and study of both closed and open
quantum dots [1–3]. An important feature of the open system compared to the closed one is the
possibility to control the additional channel of energy relaxation of qusiparticles due to their tran-
sition from the quantum dot to the outer medium. This makes it possible to develop ultra fast
response detectors of radiation in the desired spectral region by choosing the proper geometric and
physical parameters of the open multilayer quantum dots.
The theory of the spectra and wave functions of electrons and holes in single and multilayer
spherical quantum dots (SQDs) embedded into an external medium is developed for both closed [2–
4] and open [5,6] nanoheterosystems. Regarding the theory of excitons or the theory of interaction of
electrons, holes, and excitons with phonons in closed SQDs, it is quite well developed in the context
of various models for the phonon subsystem (especially, the model of dielectric continuum) for
closed systems [7,8]. The results of the theory are in satisfactory agreement with the experimental
data [2,3].
In solving the problems of interaction of quasiparticles with quantum fields (phonons), the
mathematical approaches of quantum mechanics [9] and the method of secondary quantization
together with the Green functions theory [10] perform well for a closed system, but do not find
direct use for the open one (with quasistationary states, in which the quasiparticles have a finite
lifetime).
There is no theory of electron–phonon and exciton–phonon interactions in the open SQDs,
since the wave functions of these quasiparticles in the quasistationary states are not involved in
the complete system of normalized functions [9]. This fact does not allow us to use the methods
of the quantum field theory studying the spectra renormalized by the interaction.
Studying a single open SQD, the basic theoretical difficulties can be avoided by treating the
corresponding three-shell closed SQD with a very large width of the outer spherical well. Herein
it is shown that at the increase of the outer well thickness, the exciton states of closed SQD
cease to be stationary and when the thickness tends to infinity, the spectrum gradually becomes a
quasistationary one.
∗E-mail: theorphys@chnu.cv.ua
c© M.V.Tkach, Ju.O.Seti 23
M.V.Tkach, Ju.O.Seti
2. Exciton Hamiltonian, spectrum and wave functions in closed
and open SQD
The exciton energy spectrum and wave functions in semiconductor three-shell closed SQD with
two potential wells and one barrier are under study. Geometrical parameters of the system are: the
radius of inner well (r0), barrier thickness (∆1), and thickness of outer shell-well (∆2). It is clear
that at ∆2 → 0 the three-shell nanosystem transits into the single closed SQD and at ∆2 → ∞
– into the single open SQD . Within the effective mass approximation and rectangular potential
model for the electron and hole in spherical coordinate system (with the beginning in the center of
SQD), as it is usually performed for the systems with close magnitudes of lattice constants for the
composing parts of nanosystem [2–4] and assuming the electron and hole (p=e,h) effective masses,
potential energies and their dielectric constants to be known
mp(rp) =
{
m0,2
m1,3
, Up(rp) =
{
0,
U1,3
, ε(r) =
{
ε0, rp < r0, r1 < rp < r2
ε1, r0 6 rp 6 r1
(1)
the Hamiltonian of exciton in three-shell closed SQD is obtained as
H(c) = H(c)
e (~re) + H
(c)
h (~rh) + V (~re, ~rh). (2)
Herein, index (c) means that the system is closed and for the open system we shall further use
index (o). The Hamiltonians of uncoupling electron (p = e) and hole (p = h) have the form
H(c)
p (rp) = −~
2
2
~∇ 1
m
(c)
p (rp)
~∇ + U (c)
p (~rp) (3)
and the potential energy of interacting quasiparticles in the system with dielectric constant ε(r) is
V (~re, ~rh) = − e2
ε(~re, ~r)|~re − ~rh|
. (4)
The Schrodinger equation with the exciton Hamiltonian (2) is not solved exactly but for the
uncoupling electron and hole, the equation
H(c)
p (~rp)Ψ
(c)
p (~rp) = E(c)
p (~rp)Ψ
(c)
p (~rp) (5)
taking into account the spherical symmetry, has the exact solution obtained in [11].
Ψ(c)
p (~rp) ≡ Ψ
(c)p
`m (~rp) = R
(c)p
` (rp)Y`m(θp, ϕp), ` = 0, 1, 2 . . . , m = 0,±1,±2 . . . , (6)
where Y`m(θp, ϕp) is the spherical function and R
cp
` (rp) are the radial functions expressed through
the Bessel and Neumann functions.
The fitting conditions for the wave functions and currents of probability densities continuity at
all nanosystem interfaces together with the normalizing condition
R
cp
i` (rp)|rp=ri
= R
cp
i+1` (rp)|rp=ri
,
1
mpi
dR
p
i`(rp)
drp
∣
∣
∣
∣
rp=ri
=
1
mpi+1
dR
p
i+1`(rp)
drp
∣
∣
∣
∣
rp=ri
,
∞
∫
0
|Rcp
` (rp)|2r2
pdrp = 1 (7)
determine ([11]) the electron (Ece
n`) and hole (Ech
n`) energy spectra in the bound states (U < Up).
Using the information on the spectra and wave functions of uncoupling electron and hole,
one can obtain the low energy part of exciton spectrum (En
n′ ) in spherically symmetric states
24
Exciton in closed and opened quantum dot
(` = `
′
= 0) within the framework of perturbation theory. According to this method, the exciton
wave functions are written as
Ψn
n′ (~re, ~rh) = Ψce
n (~re)Ψ
ch
n′ (~rh) = (4π)−1Rce
n (re)R
ch
n′ (rh). (8)
The stationary exciton energy spectrum
En
n′ = Eg0 + Ee
n + Eh
n′ + ∆En
n′ (9)
is defined by the magnitude of the forbidden gap energy (Eg0) of the well, electron (Ee
n) and hole
(Eh
n′ ) energies in the corresponding states and interaction energy
∆En
n′ = −
∫∫
e2d3~red
3~rh
ε(~re, ~rh)|~re − ~rh|
|Ψn(~re)|2|Ψn′ (~rh)|2. (10)
The exciton in an open nanosystem has the Hamiltonian (3), but here the outer medium turns
out to be the well and there is no barrier at all. Thus, in the Hamiltonian of the open nanosystem,
the effective masses and potential energies are as follows:
m(o)
p (rp) =
{
mp0,2
mp1
, U (o)
p (rp) =
{
Up0,2 = 0 rp < r0, r1 < rp < r2
Up1 r0 6 rp 6 ∞ . (11)
Now, the quasistationary exciton spectrum and its wave functions can be established within the
S-matrix method using the general perturbation theory [9]. In this approach, the wave functions
of spherically symmetric quasistationary states of uncoupling electron and hole are written as
Ψ(0)
p (~r) = (4π)−1/2R(o)
p (Kprp). (12)
Their analytical expressions are presented in [11].
The exciton quasistationary spectrum and wave functions for the open SQD are obtained within
the framework of the perturbation theory method [9]. Herein, it is assumed that to a certain
magnitude of the exciton energy (Eex) of continuum spectrum, there corresponds an infinite number
of electron and hole energies satisfying the condition
Eex = (Ee ± E
′
) + (Eh ∓ E
′
), 0 6 E
′
6 min(Ee, Eh). (13)
Thus, the radial part of exciton wave function R
(o)
ex is infinitely degenerated over the energy and
can be determined by superposition
R(o)
ex (E, re, rh) =
1
∫
0
R(o)
e (Ex, re)R
(o)
h (E(1 − x), rh)dx. (14)
The exciton wave functions in resonance spherically symmetric quasistationary states with the
energy E
(o)N
N ′ = Ee
N + Eh
N ′ are as follows:
ΨN
N ′
(
E
(o)N
N ′ , ~re, ~rh
)
= (4π)−1R(o)
ex
(
E
(o)N
N ′ , re, rh
)
=
= (4π)−1
1
∫
0
R
(o)e
N
(
E
(o)N
N ′ x, re
)
R
(o)h
N ′
(
E
(o)N
N ′ (1 − x), rh
)
. (15)
Finally, according to the general theory [9], the resonance quasistationary exciton energies
(E
(o)N
N ′ ) and semiwidths (ΓN
N ′ ) of the respective bands in the open SQD are fixed by the expressions
E
(o)N
N ′ = Eg0 + E
(o)e
N + E
(o)e
N ′ + ∆E
(o)N
N ′ , Γ
(o)N
N ′ = Γ
(o)e
N + Γ
(o)e
N ′ + γ
(o)N
N ′ , (16)
25
M.V.Tkach, Ju.O.Seti
where
E
(o)p
N = RepolSp(kNp), Γ
(o)p
N = −2 Impol Sp(kNp) (17)
are the resonance energies and semiwidths of quasistationary bands of uncoupling electron and
hole, respectively ([11]). The magnitudes ∆E
(o)N
N ′ and γ
(o)N
N ′ are the corrections to the energy and
semiwidths of exciton bands due to the interaction between electron and hole and are defined by
∆E
(o)N
N ′ + iγ
(o)N
N ′ = − e2
ε0
√
4π
(Re − 2i Im)
∞
∫
0
∞
∫
0
r2
er
2
hR2
ex
(
E
(o)N
N ′ , re,h
)
(
r−1
e , re > rh
r−1
h , rh > re
)
dredrh
×
∞
∫
0
∞
∫
0
r2
er
2
hR2
ex
(
E
(o)N
N ′ , re,h
)
dredrh
−1
. (18)
Thus, the quasistationary exciton spectrum in an open SQD is established.
3. Evolution of exciton energy spectrum in spherically symmetric states
from closed to open SQD
The theory developed in the previous section allows us to study the exciton spectrum in spheri-
cally symmetric states (` = 0) depending on the thickness (∆2) of the external well of three-
shell closed SQD. We can examine in detail the variation of exciton spectrum from stationary for
the closed to the quasistationary for the open SQD. The numerical calculations were performed
for the nanoheterosystem HgS/CdS/HgS/CdS with the parameters: me
HgS = 0.036, me
CdS = 0.2,
Ue = 1350 meV, mh
HgS = 0.044, mh
CdS = 0.7, Uh = 650 meV. These parameters satisfy the require-
ments of rectangular potential model due to the slight difference between the lattice constants of
composition materials (aHgS = 5, 851 Å; aCdS = 5, 818 Å).
The evolution and properties of exciton spectrum is mainly caused by the spectra of uncou-
pling electron and hole. Therefore, figure 1 shows the results of calculation of the dependencies of
stationary state energies (E
e,h(c)
n0 ) on the thickness (∆2) for small (a) and big (c) magnitudes and
the probability of the location of these quasiparticles in the corresponding states
W
e,h(c)
n0 =
r0+∆1
∫
0
∣
∣
∣
R
e,h(c)
n0 (r)
∣
∣
∣
2
r2dr (19)
for small (b) and big (d) thicknesses, respectively. Herein figure 1e presents the probabilities of
electron and hole location (W
e,h(o)
n0 ) in the space of the open QD. In figure 1 one can see that
the properties of electron and hole spectra are qualitatively equal and the quantitative difference
is caused by the difference between their effective masses and potentials. The main features of
the spectra are as follows. For the small thicknesses (∆2) of the order of r0 + ∆1 (figure 1) the
stationary state energies (E
e,h(c)
n0 ) are decreasing with the increase of thickness. Herein, in the region
of the energies corresponding to the energy levels of quasiparticle in the single SQD, one can see
the anticrossings (“bottlenecks”) arising due to the spherical symmetry of both inner and outer
potential wells. Figure 1b presents the probabilities of the location of quasiparticles in stationary
states of SQD at ∆2 = 100aCdS. In the figure one can see that in the regions of anticrossings the
probabilities are big (∼1) while in the other regions of energies they are small (∼0).
For the big magnitudes of thicknesses, when ∆2 > r0 + ∆1 (figures 1c, d) the dependencies
of E
e,h(c)
n0 and W
e,h(c)
n0 on ∆2 are very different from the previous case (figures 1a, b). Here, at
the increase of ∆2, the energies E
e,h(c)
n0 are also decreasing (figure 1c) but the “bottlenecks” are
almost invisible. Nevertheless, the “memory” of the system about the location of anticrossings in
the energy scale is already saved only in the probabilities (W
e,h(c)
n0 ) with the maxima just in the
vicinity of the former anticrossings of quasistationary energy states of the open system (figure 1e).
26
Exciton in closed and opened quantum dot
0 20 40 60 80100
0
100
200
300
400
500
600
E
e n
0
,
m
eV
∆
2
, a
HgS
a
10000 10100
50
55
220
230
520
530
540
0,02 0,04 0,06 0,081
0
100
200
300
400
500
600
E
(o)e
1 0
(o)e
1 0
∆
2
, a
HgS
E
(o)e
2 0
(o)e
2 0
E
(o)e
3 0
(o)e
3 0
E
e
n 0E
e
n 0E
e
n 0
W
e
n 0
W
e
n 0
b c d
50
55
0,5 1,0 1,5 2,0
220
230
0,1 0,2 0,3
520
530
540
0,1 0,2 0,3
E
e
n 0
W
e
n 0
*10
7
e
500
400
300
200
100
0
0 20 40 60 80 100
∆
2
, a
HgS
E
h n
0
,
m
eV
a
30
10000 10100
158
156
402
400
398
0,05 0,10 0,15
500
400
300
200
100
0
1
E
(o)h
1 0
(o)h
1 0
E
(o)h
2 0
(o)h
2 0
E
(o)h
3 0
(o)h
3 0
E
h
n 0
W
h
n 0
E
h
n 0E
h
n 0
W
h
n 0
b c d
∆
2
, a
HgS
30
1 2 3 4 5
158
156
0,0 0,5 1,0 1,5
402
400
398
0,0 0,5 1,0 1,5
E
h
n 0
W
h
n 0
*10
7
e
Figure 1. Dependencies of the energies E
e,h(c)
n0 (a, c) on the thickness ∆2 and probabilities
W
e,h(c)
n0 (d) and W
e,h(o)
n0 on the energies.
Herein, the distribution of probabilities (W
e,h(c)
n0 ) over the energies have the quasi-Lorentz character
(figure 1d), similar to the analogous continuous distributions of probabilities (W
e,h(o)
n0 (Ee,h)) in the
open system (figure 1e).
Finally, in figures 1c, d, e one can see that at ∆2 = 10000aCdS among all the levels numerated
27
M.V.Tkach, Ju.O.Seti
0,0
0,5
1,0
W
h
(
c
)
n
0
∆
2
=100 a
a
W
h (c)
10
E
h (c)
10
h
0,0
0,5
1,0
W
e (c)
10
E
e (c)
10
W
e
(
c
)
n
0
e
0,0
0,5
W
10 (c)
10
E
10 (c)
10
W
n
0
(
c
)
n
0
ex
0,00
0,25
0,50
W
h
(
c
)
n
0 W
h (c)
10
E
h (c)
10
∆
2
=1000 a
b
0,00
0,25
0,50
W
e (c)
10
E
e (c)
10
W
e
(
c
)
n
0
0,0
0,1
0,2
W
10 (c)
10
E
10 (c)
10
W
n
0
(
c
)
n
0
0,00
0,05
0,10
∆
2
=10000 a Γ
h (c)
10
W
h (c)
10
E
h (c)
10
W
h
(
c
)
n
0
c
0,00
0,05
0,10
Γ
e (c)
10
W
e (c)
10
E
e (c)
10W
e
(
c
)
n
0
0,000
0,005
Γ
10 (c)
10
W
10 (c)
10
E
10 (c)
10W
n
0
(
c
)
n
0
20 25 30 35 40
0
5
Γ
h (o)
10
W
h (o)
10
E
h (o)
10
W
h
(
o
)
n
0
*1
0
7
E
h
, meV
d
∆
2
→∞
40 45 50 55 60 65
0
5
Γ
e (o)
10
W
e (o)
10
E
e (o)
10
W
e
(
o
)
n
0
*1
0
7
E
e
, meV 70 75 80 85 90 95
0
9
Γ
10 (o)
10
W
10 (o)
10
E
10 (o)
10
E
ex
, meV
W
n
0
(
o
)
n
0
*1
0
1
2
Figure 2. Dependencies of probabilities W
e(c)
n , W
h(c)
n , W
n(c)
n and W
e(o)
1 , W
h(o)
1 , W
1(o)
1 on the
energies for different thicknesses ∆2.
by n quantum number, there are three levels fixed by the number N=1,2,3 , corresponding to the
stationary states with maximum probabilities (W
e,h(c)
N ). These energies (E
e,h(c)
N ) almost coincide
with the resonance levels of the open SQD (E
e,h(o)
N ). Due to the expression
lim
∆2→∞
E
e,h(c)
N = E
e,h(o)
N ,
it is convenient to call these levels of closed system the “former resonance” ones. Figures 1d, 1 e
also proves that the widths of energy bands of closed SQD are almost equal to the widths of the
quasistationary bands, i.e. lim
∆2→∞
Γ
e,h(c)
N = Γ
e,h(o)
N .
The analysed behavior of the electron and hole spectra makes it possible to clarify the formation
and evolution of exciton spectrum from the closed to the open SQD. The example of exciton
28
Exciton in closed and opened quantum dot
spectrum and the formation of quasistationary bands at the variation of the thickness of SQD
outer well (∆2) is presented in figures 2a–d. Figures 2a–c show that at the increase of ∆2 (at
r0 = 20aHgS, ∆1 = 1aCdS), the distribution of probabilities of electron (W e
n), hole (W h
n ) and exciton
(Wn
n ) location in two inner shells of closed SQD in the vicinity of their lowest former resonance
energies is (Ē
e(c)
1 , Ē
h(c)
1 , Ē
1(c)
1 ). Here (figure 2d) one can also see the distribution of probabilities
(WN
N ) of exciton location in the same inner shells in the states with the energies E
h(o)
1 , E
e(o)
1 and
E1
1 = E
h(o)
1 + E
e(o)
1 from the vicinity of the lowest resonance state (without taking into account
the energy of interaction ∆E1
1 and width of the forbidden gap Eg0
).
0 10000 20000 30000 40000 50000
0
-2
-4
-6
-8
-10
∆
2
, a
HgS
∆
Ε
1 1
,
m
e
V
∆
1
=1,5
∆
1
=2
∆
1
=1
a
0 10000 20000 30000 40000 50000
0
5
10
15
20 Γ
3
3
(c)
Γ
2
2
(c)
Γ
1
1
(c)
∆�������
b
Γ
Ν Ν
,
m
e
V
Figure 3. Dependence of binding energy (∆E
1
1) at different thicknesses ∆2 (a) and semiwidth
of discrete band (ΓN
N ) at ∆1 = aHgS on the thickness ∆2.
Figure 2a proves that at the thickness ∆2 of the order of r0+∆1 (∆2 = 100aCdS), the stationary
spectra of all three quasiparticles are discrete with the big (∼1) probabilities in the vicinity of
resonance energies and small magnitudes of probability in a wider scale. At this thickness (∆2), for
every quasiparticle (electron, hole, exciton), there does not exist the conception of discrete band
semiwidth as an interval of energies where there are even two states around the resonance one
satisfying the conditions
2W
e,h
n±1 > W e,h
n = W
e,h
N=1 , 2Wn±1
n′
±1
> Wn
n′ = WN=1
N ′=1
.
At ∆2 = 1000aCdS the semiwidths of discrete bands do not arise yet, but their quasi-Lorentz
contour becomes visible for all three quasiparticles. At ∆2 = 1000aCdS (figure 2c) all discrete bands
are characterized by semiwidths (Γ
h(c)
1 ,Γ
e(c)
1 ,Γ
1(c)
1 ) and the former resonance energies (Ēh
1 , Ēe
1, Ē
1
1).
All these magnitudes for closed SQD almost do not differ from the resonance energies (Eh
1 , Ee
1, E
1
1)
and semiwidths (Γh
1 ,Γe
1,Γ
1
1) of the open SQD (figure 2d). Herein, Γ1
1 = Γ
h(o)
1 + Γ
e(o)
1 .
Figure 3a presents the dependencies of electron and hole interaction energy (∆Ē1
1) in the former
resonance states Ne = Nh = 1 on the thickness (∆2) at a different thickness of the barrier (∆1). In
29
M.V.Tkach, Ju.O.Seti
the figure one can see that at the increase of ∆2, the magnitude of ∆Ē1
1 decreases. The calculation
of the interaction energy ∆Ē1
1 for the open SQD proves that at the varying of barrier thickness
(∆1) in the range of several lattice constants, the magnitude of ∆Ē1
1 is not bigger than 0.01 meV,
since it is negligibly small.
Figure 3b presents the dependence of semiwidth (Γ̄n
n) of discrete exciton bands on the thickness
of the external well (∆2) at r0 = 20aHgS, ∆1 = 1aCdS. From this figure it is clear that the three
discrete exciton bands start to be characterized by corresponding semiwidths when the thickness
∆2 becomes bigger than 103aCdS. Herein, Γ̄N
N for the closed system are always bigger than the
respective semiwidths (ΓN
N = Γh
N + Γe
N ) for the open SQD. At the increase of ∆2, the Γ̄N
N values
asymptotically tend to ΓN
N , as it is supposed to be from physical considerations.
4. Conclusions
The theory of exciton spectrum in three-shell closed SQD has been developed for the first time.
It permits to study the process of the formation and peculiarities of the evolution of spherically
symmetric quasistationary states at the increase of the thickness of the outer well up to the sizes
at which the SQD can be assumed as the open system.
It is established that for rather big thicknesses (∆2 � r0 + ∆1) of the outer well of the closed
three-shell SQD, in the vicinity of the resonance exciton energies (EN
N ′ = Eh
N ′ + Ee
N ) of the single
open SQD (close to the respective energies of the single closed SQD), the discrete exciton bands
begin to form (with the former resonance energies (ĒN
N ′ ) and semiwidths (Γ̄N
N ′ )). At the increase
of the thickness (∆2), the ĒN
N ′ and Γ̄N
N ′ magnitudes asymptotically tend to the resonance energies
EN
N ′ and semiwidths ΓN
N ′ of exciton quasistationary states in a single open SQD.
Finally, it is worth noting that as far as the set of wave functions for the three-layer closed SQD
is complete and orthonormalized, one can establish the theory of electron- and exciton-phonon
interaction in the open SQD by choosing a rather big thickness of the outer well (∆2), ensuring the
required exactness of the resonance energy (EN
N ′ ) and semiwidth (ΓN
N ′ ) of quasistationary spectrum
of a single open SQD. This is the subject of our further investigation.
Acknowledgements
This work was financially supported by MPZ PP “Kolos”.
References
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3. Mews A., Kadavanich A.V., Banin U., Alivisator A.P. Phys. Rev. B, 1996, 53, 13242.
4. Tkach M., Golovatskioe N., Mikhaleva M., Fartushinskioe R., Fiz. Tverd. Tela (St. Petersburg), 2001,
43, 1315 [Phys. Solid State, 2001, 43, 1370].
5. Tkach M., Golovatskioe N. Fiz. Tverd. Tela (St.Petersburg), 1999, 41, 2081 [Phys. Solid State, 1999,
41, 1911].
6. Tkach M., Holovatsky V., Voitsekhivska O. Physica E, 2001, 11, 17.
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9. Baz A.I., Zeldovich Ya.B., Perelomov A.M. Scattering,Reactions and Decays in Nonrelativistic Quan-
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1966.
10. Abrikosov A.A., Gorkov L.P., Dzyaloshinskioe I.E. Methods of Quantum Field Theory in Statistical
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[Semiconductors, 2006, 40, 9].
30
Exciton in closed and opened quantum dot
Екситон у вiдкритiй та закритiй квантовiй точцi
М.В.Ткач, Ю.О.Сетi
Чернiвецький нацiональний унiверситет iм. Юрiя Федьковича,
вул. Коцюбинського 2, 58012, Чернiвцi
Отримано 4 вересня 2004 р.
Вперше запропонована теорiя екситонного спектра в сферично-симетричних станах тришарової
закритої сферичної квантової точки (СКТ). Дослiджено еволюцiю спектра при змiнi товщини зов-
нiшнього шару-ями вiд нуля (стацiонарний спектр простої закритої СКТ) до безмежностi (квазiста-
цiонарний спектр простої вiдкритої СКТ). Дослiджено механiзм виникнення затухання (пiвширини)
квазiстацiонарних станiв причиною якого є перерозподiл по енергетичних рiвнях ймовiрностей пе-
ребування екситона у межах двох внутрiшнiх шарiв наносистеми. Показано, що тришарова закрита
СКТ з достатньо великою товщиною зовнiшньої ями з довiльною необхiдною точнiстю вiдтворює
основнi властивостi квазiстацiонарного екситонного спектра у простiй вiдкритiй СКТ.
Ключовi слова: сферична квантова точка, квазiстацiонарний стан, екситонний енергетичний
спектр
PACS: 71.15.Dx, 73.21.La
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