A theoretical model for exciton binding energies in rectangular and parabolic spherical finite quantum dots
. Using the variational method in real space and the effective-mass theory, we present quite an advanced semi-analytic approach susceptible for calculating the binding energy Eb of Wannier excitons in semiconductor quantum dot structures with rectangular and parabolic shapes of the confining pote...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2012
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irk-123456789-1187282017-06-01T03:05:34Z A theoretical model for exciton binding energies in rectangular and parabolic spherical finite quantum dots Taqi, A. Diouri, J. . Using the variational method in real space and the effective-mass theory, we present quite an advanced semi-analytic approach susceptible for calculating the binding energy Eb of Wannier excitons in semiconductor quantum dot structures with rectangular and parabolic shapes of the confining potential in the so-called strong-confinement regime. Illustration is given for CdS, ZnSe, CdSe, GaAs structures of crystallites for both rectangular and parabolic quantum dots, and it displays a very good agreement between the experimental and theoretical results reported in literature. 2012 Article A theoretical model for exciton binding energies in rectangular and parabolic spherical finite quantum dots / A. Taqi, J. Diouri // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 4. — С. 365-369. — Бібліогр.: 9 назв. — англ. 1560-8034 PACS 71.35.-y; 73.21.Fg, La http://dspace.nbuv.gov.ua/handle/123456789/118728 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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| description |
. Using the variational method in real space and the effective-mass theory, we
present quite an advanced semi-analytic approach susceptible for calculating the binding
energy Eb of Wannier excitons in semiconductor quantum dot structures with rectangular
and parabolic shapes of the confining potential in the so-called strong-confinement
regime. Illustration is given for CdS, ZnSe, CdSe, GaAs structures of crystallites for both
rectangular and parabolic quantum dots, and it displays a very good agreement between
the experimental and theoretical results reported in literature. |
| format |
Article |
| author |
Taqi, A. Diouri, J. |
| spellingShingle |
Taqi, A. Diouri, J. A theoretical model for exciton binding energies in rectangular and parabolic spherical finite quantum dots Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Taqi, A. Diouri, J. |
| author_sort |
Taqi, A. |
| title |
A theoretical model for exciton binding energies in rectangular and parabolic spherical finite quantum dots |
| title_short |
A theoretical model for exciton binding energies in rectangular and parabolic spherical finite quantum dots |
| title_full |
A theoretical model for exciton binding energies in rectangular and parabolic spherical finite quantum dots |
| title_fullStr |
A theoretical model for exciton binding energies in rectangular and parabolic spherical finite quantum dots |
| title_full_unstemmed |
A theoretical model for exciton binding energies in rectangular and parabolic spherical finite quantum dots |
| title_sort |
theoretical model for exciton binding energies in rectangular and parabolic spherical finite quantum dots |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2012 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118728 |
| citation_txt |
A theoretical model for exciton binding energies in rectangular
and parabolic spherical finite quantum dots / A. Taqi, J. Diouri // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 4. — С. 365-369. — Бібліогр.: 9 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT taqia atheoreticalmodelforexcitonbindingenergiesinrectangularandparabolicsphericalfinitequantumdots AT diourij atheoreticalmodelforexcitonbindingenergiesinrectangularandparabolicsphericalfinitequantumdots AT taqia theoreticalmodelforexcitonbindingenergiesinrectangularandparabolicsphericalfinitequantumdots AT diourij theoreticalmodelforexcitonbindingenergiesinrectangularandparabolicsphericalfinitequantumdots |
| first_indexed |
2025-07-08T14:32:39Z |
| last_indexed |
2025-07-08T14:32:39Z |
| _version_ |
1837089592881184768 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 365-369.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
365
PACS 71.35.-y; 73.21.Fg, La
A theoretical model for exciton binding energies in rectangular
and parabolic spherical finite quantum dots
A. Taqi*, J. Diouri
Faculté des sciences. Département de Physique, Université Abdelmalek Essaadi,
BP 2121, Tétouan, Morocco
*E-mail: abtaqi@yahoo.fr
Abstract. Using the variational method in real space and the effective-mass theory, we
present quite an advanced semi-analytic approach susceptible for calculating the binding
energy EB of Wannier excitons in semiconductor quantum dot structures with rectangular
and parabolic shapes of the confining potential in the so-called strong-confinement
regime. Illustration is given for CdS, ZnSe, CdSe, GaAs structures of crystallites for both
rectangular and parabolic quantum dots, and it displays a very good agreement between
the experimental and theoretical results reported in literature.
Keywords: exciton, binding energy, rectangular quantum dot, parabolic quantum dot.
Manuscript received 07.04.12; revised version received 19.09.12; accepted for
publication 17.10.12; published online 12.12.12.
1. Introduction
Recently, excitons in quantum dots have attracted more
and more interest and have become the centre of
attention of many experimental and theoreticals studies
[1], because their original properties allow many
interesting applications, namely: producing artificial
atoms and molecules, single-electron transistors, and
quantum dot lasers (see [2] and references cited therein).
In theory, as direct solving the Hamiltonian is rather
complicated and practically impossible, several attempts
have been made to solve specific related problems. To
determine the binding energy EB, the variational method
is commonly used with different formulations depending
on the choice of a trial wave function. On the other hand,
progress in experimental techniques has shown that the
confinement in GaAs/GaAlAs quantum dots is
approximately parabolic [3].
With regard to these important developments, we
started looking for a simplified formulation, making
possible a rapid and rather precise determination of
exciton properties for rectangular and parabolic
confining potential. Hence, we began with the usual
approximation, e.g. the effective mass one, which
enabled us to establish, in the framework of the
variational method, general formulae for calculating the
expected values of the exciton binding energy in terms
of the characteristic parameters of the structure: dot
radius R0, effective masses me and mh as well as potential
profiles Ve (re) and Vh (rh) for electron and hole forming
the exciton. To illustrate them, the formulae were
applied to CdS, CdSe, ZnSe/ZnS and GaAs structures
for rectangular and parabolic quantum dots (RQD and
PQD) shapes and gave very good results.
2. Theoretical model
1) Basic equations
Let us consider a heterostructure consisting of a single
quantum dot of type I. The confining potential is of
rectangular and parabolic shapes, say Ve (re) for electrons
and Vh (rh) for holes. Then, the Hamiltonian of one
electron-hole pair in the effective mass approximation
[4] is as follows:
),()()(),( hechheeheex rrVrHrHrrH , (1)
where He (re) and Hh (rh) are the single-particle
contributions for the electron and hole, respectively,
he
hec rr
e
rrV
2
),( is the electron-hole Coulomb
interaction he rr
, and stands for the dielectric
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 365-369.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
366
constant. The single-particle Hamiltonian )( ii rH
(i = e, h) is defined as:
)(
)(2
)(
2
iii
ii
iii rV
rm
rH
, (2)
where the first term is the kinetic energy for a particle
with the effective mass mi, and Vi (ri) corresponds to the
confining potential.
The problem consists in finding the eigenfunction
),( he rr
and the eigenvalue E of exH for the ground
state.
The binding energy EB is then related to E by:
EEEE heB , (3)
where Ee and Eh are solutions of the one-particle
problem with the ground state i (i = e, h):
iiii EH . (4)
The general method of solution is to apply the
variational principle with a trial wave function of the
following form:
he . (5)
In the strong confinement regime, the confinement
effect dominates; the well dissociates the electron-hole
pair, and the spatial correlation between the electron and
hole is little. Then, we can choose the ansatz as:
he rr
exp , is the variational parameter.
2) Solution
The eigenvalue E of exH for the ground state follows
simply as the expectation value:
exH
E . (6)
Letting D ,
the numerator N of Eq. (6) is given by
ehheche CAAVHHN ,
(7)
where Ae,h and Ceh can be written as:
hehe
he
hhee
he
hehe drdrrr
r
rr
m
DEA 22
2
,
22
,
22
,, )()(
2
)4(
=
D
m
DE
he
he 2
,
2
,
2
, (8)
ceh VC
hehe
rr
hechhee drdrrrerrVrr
he
22
2
222 ),()()()4(
. (9)
In spherical harmonics, the term of electron-hole
interaction can be expressed as (see for example, Marin
et al. [5]):
),(),(
)12(
1
4
1 *
0
1 hh
m
lee
m
l
l
l
lm
l
l
he
YY
r
r
lrr
,
where )( rr is the smaller (greater) of re and rh .
Then,
i
he rr
hheel
l
l
l
lm
eh
err
r
r
l
e
C
2
22
1
0
2
)()(
)12(
14
hehehehh
m
lee
m
l drdrddrrYY 22),(),(
*
,
where )( he dd denotes the solid angle for the
electron (hole).
As )(or he does not depend on the angles
),( ee )),((or hh ,
00
0
0 4),(),(4
),(
*
*
mleeeee
m
l
eee
m
l
dYY
dY
and
.4),(),(4
),(
00
0
0
*
*
mlhhhhh
m
l
hhh
m
l
dYY
dY
Thus, for the 1s state:
hehe
rr
hheeeh drdrrrerr
r
RaC
he
22
2
22*2 )()(
1
)4(2
,
(10)
where a and R are Bohr radius
2
2
e
and binding
energy
2
2
)(2 a
of the bulk exciton, respectively.
The binding energy is obtained by maximising the
expectation value
ex
heB
H
EEE )( (11)
with respect to .
Then, )(BE can be simply written in the form:
λ
a
I
C
aRE
*
*
B
2
0
2)(λ (12)
with
,)(
1
)(
1
)(
2
2
2
2
00
2
h
rr
hh
r h
h
rr
hh
r
e
eee
drerf
r
drerf
r
drrfC
he
e
hee
(13)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 365-369.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
367
drxrfrfrfxrfdxeI hehe
x
0
2222
0
2
0 )()()()( ,
(14)
where )()( iiiii rrrf and hehe mmmm /µ is the
reduced mass of the exciton.
3. Applications
For illustration and with the aim of testing the validity of
this model, we have calculated the exciton ground state
energy in rectangular and parabolic quantum dots and
compared the results with available existing data [5-7].
The agreement was very good.
1) Rectangular quantum dot
The confining potential that we assume as a spherical
quantum well-like potential defined by
iiii VRrrV )()( 0 , where is the step function and
Vi – barrier height.
The dependence of mi on ri arises from the fact that
the particles have different effective masses depending
on their location, inside or outside the dot. Then, the
one-particle problem was solved by computing the
solution of the following implicit eigenvalue equation
for the spherical symmetry quantum dot energies [4]
11)cotg( 000 Rk
m
m
RkRk ou
ou
in
inin . (15)
The associated wave functions are given by:
i
i
ou
i
ii
i
i
in
i
i
i
ii
r
rk
RrB
r
rk
rR
A
r
)exp(
)sin(
4
)(
0
0
(16)
with )(2 ;2 ii
ou
i
ou
ii
in
i
in
i EVmkEmk . (17)
The constants Ai and Bi are determined by
normalization requirements and are equal to:
2
1
0
2
00
2
)(sin
4
)2sin(
2
ou
i
in
i
in
i
in
i
i
k
Rk
k
RkR
A and
)sin()exp( 00 RkRkB in
i
ou
ii , (18)
where Rydberg units are used )1( 0 m .
To check the accuracy of our model in spherical
rectangular quantum dots, we compare our results
plotted in Fig. 1 and in Table with those of refs. [5] and
[6], respectively.
In Fig. 1, we display variation of the exciton
ground state energy in CdS crystallites as a function of
the dot radius R0. The dashed lines represent theoretical
prediction made by [5] and based on the effective-mass
approximation model in the single-band scheme and the
variational method, the higher curve corresponds to
V e= Vh = 2.5 eV and 1
in
ou
m
m
and the lower one to
0.475 eV and 1
in
ou
m
m
. The solid lines represent our
theoretical model, the higher curve corresponds to
Ve = Vh = 2.5 eV and 1
in
ou
m
m
, while the lower one – to
0.475 eV and 1
in
ou
m
m
. In both cases, the results were
evaluated for CdS material parameters me = 0.18m0,
mh = 0.53m0, 1
in
ou
m
m
and 5.5 , m0 – free-electron
mass.
From the analysis of this figure, we conclude the
following.
- The results obtained using our method are found to
be slightly higher and closer to the experimental
values reported in ref. [5] and represented in Fig. 1
by the symbols – circles and triangles.
- A decrease of dot radius is accompanied by an
increase of overlap integral of the electron and hole
wave functions; which causes an increase in the
exciton ground state energy.
- The exciton ground state energy is found to reduce
with decreasing the barrier height. This is due to
increased penetration of the wave function into the
barrier with a resulting lower Coulomb attraction.
In Table, we present our results for the exciton
binding energy EB in ZnSe/ZnS quantum dot to compare
them with the theoretical results of Jia-Lin Zhu et al. [6].
In this work, the authors used the variational method
with introduction of effective electron potentials. We
used the same parameters therein (me = 0.16m0,
mh = 0.61m0, Ve = 3279 meV, 7.8 for ZnSe and me =
0.27m0, mh = 0.96m0, Ve = 860 meV for ZnS). The
agreement between this method and our results is quite
good.
Table. Binding energy EB of excitons in ZnSe/ZnS quantum
dots as a function of the dot radius R0 with Ve = 3279 meV
and Vh = 860 meV.
Binding energy EB (meV)
R0 (Å)
our results ref. [6]
22 115.665 114.7
25 103.565 103.9
34 78.821 81.21
42 65.016 66.89
56 49.762 51.99
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 365-369.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
368
Radius (A)
ex
ci
to
n
en
er
gy
(e
V
)
0.0
0.5
1.0
1.5
5 10 15 20 25 30 35 40
Fig. 1. Exciton ground state energy in CdS crystallites as a
function of the dot radius R0: [5] (dots); this work (full curves).
Radius (A)
0
50
100
150
5 10 15 20 25 30
0.8
1.0
1.2
mou/min =
B
in
di
ng
e
ne
rg
y
(m
eV
)
Fig. 2. Exciton binding energy in CdSe quantum dots as a
function of the radius R0 for three different values of the
effective-mass ratio inside and outside the cristallite.
Fig. 2 illustrates behaviour of the ground state
exciton binding energy in CdSe quantum dots as a
function of the radius R0 for three values of the effective-
mass ratio
in
ou
m
m
= 0.8, 1.0 and 1.2. The calculations were
performed with the following parameters: me = 0.13m0,
mh = 0.4m0, Ve = Vh = 1.3 eV and 6.10 [5].
For these three values of the effective mass ratio
in
ou
m
m
considered in this work, we found that the exciton
binding energy EB has a maximum at a critical dot
radius: on increasing
in
ou
m
m
, the critical dot radius is
decreased, and the maximum binding energy is
increased. For a certain value of R0, these three values
become equal. We conclude that the exterior medium in
which the crystallites are embedded considerably
modifies behaviour of the exciton binding energy. On
the other hand, we observe that the exciton binding
energy increases as the radius decreases, reaching the
maximum at the dot radius ≈10.5 Å, and then diminishes
to a limited value corresponding to a particular radius of
the well, for which it is possible to find the free electron
and hole energy level [7]. Note that for narrower dots,
only confinement influences the increase of the exciton
binding energy. Furthermore, as R0 increases, the exciton
binding energy approaches the energy of the unconfined
two-dimensional exciton.
2) Parabolic quantum dot
In this work, we apply the model developed in Section 3
to calculate the ground state energy of excitons in a single
parabolic quantum dot. The studied structure consists in a
type I heterostructure with parabolic potential profiles for
electrons and holes described as 22
2
1
)( iiii rmrV [8, 9],
where
2
0R
[8], while R0 is the quantum dot radius,
and µ – reduced mass of the exciton. Then, the ground
state solutions of one-particle problem for the parabolic
quantum dots are: 2exp)( iiii rr with the energy
2
3
iE , where
2
i
i
m
.
We have applied our model to calculate the exciton
ground state energy for GaAs/GaAlAs parabolic
quantum dots. By way of comparison, we have referred
to the article of S. Jasiri et al. [8] and used the same
parameters therein (me = 0.067m0, mh = 0.377m0,
Eg= 1520 meV, and 1.13 , m0 is the free-electron
mass). In this work, the authors used “perturbative-
variational calculations”. The results obtained by this
method (full curve) are very close to ours (dots) for all
the range of R0 values, as it is shown in Fig. 3.
4. Conclusion
A new investigation of exciton properties in rectangular
and parabolic quantum dots has been performed using
the advanced analytical calculations. Basic equations are
derived in the framework of the commonly used
approximations allowing a relatively rapid and rather
precise determination of the exciton binding energy. The
Radius (A)
ex
ci
to
n
en
er
gy
(
m
eV
)
0
40
80
120
50 100 150 200
Fig. 3. Exciton ground state energy in GaAs parabolic
quantum dots as a function of the dot radius R0: [7] (full
curve); this work (dots).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 365-369.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
369
formulation applied to quantum-dot systems for RQD
and PQD gives good results and may be easily extended
to any given material.
References
1. M. El-Said // Sci. Technol. 9, p. 272-274
(1994).
2. G. Cantle, D. Ninno and G. Iadonosi // J.
Phys.: Condens. Matter, 12, p. 9019-9036
(2000).
3. K. Brunner, U. Bockelmann, G. Abstreiter,
M. Walther, G. Böhm, G. Tränkele and
G. Weimann // Phys. Rev. Lett. 69, p. 3216
(1992).
4. P.G. Bolcatto and R.C. Proetto // J. Phys.:
Condens. Matter, 13, p. 319-334 (2001).
5. J.L. Marin, R. Riera and S.A. Cruz // J. Phys.:
Condens. Matter, 10, p. 1349-1361 (1998).
6. Zhu Jia-Lin, Zhu Shaofeng, Zhu Ziqiang, Y.
Kawazoe and T. Yao // J. Phys.: Condens.
Matter, 10, p. L583-L587 (1998).
7. C.A Moscoso-Moreno, R. Franco, and
J. Solva-Valencia // Revista Mexicana de
Fisica, 53 (3), p. 189-193 (2007).
8. S. Jasiri, and R. Bennaceur // Semicond. Sci.
Technol. 9, p. 1775-1780 (1994).
9. T. Garm // J. Phys.: Condens. Matter, 8,
p. 5725-5735 (1996).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 4. P. 365-369.
PACS 71.35.-y; 73.21.Fg, La
A theoretical model for exciton binding energies in rectangular
and parabolic spherical finite quantum dots
A. Taqi*, J. Diouri
Faculté des sciences. Département de Physique, Université Abdelmalek Essaadi,
BP 2121, Tétouan, Morocco
*E-mail: abtaqi@yahoo.fr
Abstract. Using the variational method in real space and the effective-mass theory, we present quite an advanced semi-analytic approach susceptible for calculating the binding energy EB of Wannier excitons in semiconductor quantum dot structures with rectangular and parabolic shapes of the confining potential in the so-called strong-confinement regime. Illustration is given for CdS, ZnSe, CdSe, GaAs structures of crystallites for both rectangular and parabolic quantum dots, and it displays a very good agreement between the experimental and theoretical results reported in literature.
Keywords: exciton, binding energy, rectangular quantum dot, parabolic quantum dot.
Manuscript received 07.04.12; revised version received 19.09.12; accepted for publication 17.10.12; published online 12.12.12.
1. Introduction
Recently, excitons in quantum dots have attracted more and more interest and have become the centre of attention of many experimental and theoreticals studies [1], because their original properties allow many interesting applications, namely: producing artificial atoms and molecules, single-electron transistors, and quantum dot lasers (see [2] and references cited therein). In theory, as direct solving the Hamiltonian is rather complicated and practically impossible, several attempts have been made to solve specific related problems. To determine the binding energy EB, the variational method is commonly used with different formulations depending on the choice of a trial wave function. On the other hand, progress in experimental techniques has shown that the confinement in GaAs/GaAlAs quantum dots is approximately parabolic [3].
With regard to these important developments, we started looking for a simplified formulation, making possible a rapid and rather precise determination of exciton properties for rectangular and parabolic confining potential. Hence, we began with the usual approximation, e.g. the effective mass one, which enabled us to establish, in the framework of the variational method, general formulae for calculating the expected values of the exciton binding energy in terms of the characteristic parameters of the structure: dot radius R0, effective masses me and mh as well as potential profiles Ve (re) and Vh (rh) for electron and hole forming the exciton. To illustrate them, the formulae were applied to CdS, CdSe, ZnSe/ZnS and GaAs structures for rectangular and parabolic quantum dots (RQD and PQD) shapes and gave very good results.
2. Theoretical model
1) Basic equations
Let us consider a heterostructure consisting of a single quantum dot of type I. The confining potential is of rectangular and parabolic shapes, say Ve (re) for electrons and Vh (rh) for holes. Then, the Hamiltonian of one electron-hole pair in the effective mass approximation [4] is as follows:
)
,
(
)
(
)
(
)
,
(
h
e
c
h
h
e
e
h
e
ex
r
r
V
r
H
r
H
r
r
H
+
+
=
,
(1)
where He (re) and Hh (rh) are the single-particle contributions for the electron and hole, respectively,
h
e
h
e
c
r
r
e
r
r
V
r
r
-
e
-
=
2
)
,
(
is the electron-hole Coulomb interaction
(
)
h
e
r
r
r
r
¹
, and
e
stands for the dielectric constant. The single-particle Hamiltonian
)
(
i
i
r
H
(i = e, h) is defined as:
)
(
)
(
2
)
(
2
i
i
i
i
i
i
i
i
r
V
r
m
r
H
+
Ñ
-Ñ
=
h
,
(2)
where the first term is the kinetic energy for a particle with the effective mass mi, and Vi (ri) corresponds to the confining potential.
The problem consists in finding the eigenfunction
)
,
(
h
e
r
r
r
r
Y
and the eigenvalue E of
ex
H
for the ground state.
The binding energy EB is then related to E by:
E
E
E
E
h
e
B
-
+
=
,
(3)
where Ee and Eh are solutions of the one-particle problem with the ground state
i
y
(i = e, h):
i
i
i
i
E
H
y
=
y
.
(4)
The general method of solution is to apply the variational principle with a trial wave function of the following form:
l
l
f
y
y
=
y
h
e
.
(5)
In the strong confinement regime, the confinement effect dominates; the well dissociates the electron-hole pair, and the spatial correlation between the electron and hole is little. Then, we can choose the ansatz
l
f
as:
÷
÷
ø
ö
ç
ç
è
æ
l
-
-
=
f
l
h
e
r
r
exp
,
l
is the variational parameter.
2) Solution
The eigenvalue E of
ex
H
for the ground state follows simply as the expectation value:
l
l
l
l
y
y
y
y
=
ex
H
E
.
(6)
Letting
l
l
y
y
=
D
,
the numerator N of Eq. (6) is given by
eh
h
e
c
h
e
C
A
A
V
H
H
N
+
+
=
y
y
+
y
y
+
y
y
=
,
(7)
where Ae,h and Ceh can be written as:
h
e
h
e
h
e
h
h
e
e
h
e
h
e
h
e
dr
dr
r
r
r
r
r
m
D
E
A
2
2
2
,
2
2
,
2
2
,
,
)
(
)
(
2
)
4
(
òò
¶
f
¶
y
y
p
+
=
l
h
=
D
m
D
E
h
e
h
e
2
,
2
,
2
l
+
h
,
(8)
=
y
y
=
c
eh
V
C
EMBED Equation.3 h
e
h
e
r
r
h
e
c
h
h
e
e
dr
dr
r
r
e
r
r
V
r
r
h
e
2
2
2
2
2
2
)
,
(
)
(
)
(
)
4
(
l
-
-
y
y
p
òò
.
(9)
In spherical harmonics, the term of electron-hole interaction can be expressed as (see for example, Marin et al. [5]):
)
,
(
)
,
(
)
1
2
(
1
4
1
*
0
1
h
h
m
l
e
e
m
l
l
l
l
m
l
l
h
e
Y
Y
r
r
l
r
r
f
q
f
q
+
p
=
-
å
å
¥
=
-
=
+
>
<
r
r
,
where
)
(
>
<
r
r
is the smaller (greater) of re and rh .
Then,
´
y
y
´
´
+
e
p
-
=
ò
å
å
W
l
-
-
+
>
<
¥
=
-
=
i
h
e
r
r
h
h
e
e
l
l
l
l
l
m
eh
e
r
r
r
r
l
e
C
2
2
2
1
0
2
)
(
)
(
)
1
2
(
1
4
h
e
h
e
h
e
h
h
m
l
e
e
m
l
dr
dr
d
d
r
r
Y
Y
W
W
f
q
f
q
´
2
2
)
,
(
)
,
(
*
,
where
)
(
h
e
d
d
W
W
denotes the solid angle for the electron (hole).
As
)
(or
h
e
y
y
does not depend on the angles
)
,
(
e
e
f
q
EMBED Equation.3 ))
,
(
(or
h
h
f
q
,
0
0
0
0
4
)
,
(
)
,
(
4
)
,
(
*
*
m
l
e
e
e
e
e
m
l
e
e
e
m
l
d
Y
Y
d
Y
d
d
p
=
W
f
q
f
q
p
=
=
W
f
q
ò
ò
and
.
4
)
,
(
)
,
(
4
)
,
(
0
0
0
0
*
*
m
l
h
h
h
h
h
m
l
h
h
h
m
l
d
Y
Y
d
Y
d
d
p
=
W
f
q
f
q
p
=
=
W
f
q
ò
ò
Thus, for the 1s state:
h
e
h
e
r
r
h
h
e
e
eh
dr
dr
r
r
e
r
r
r
R
a
C
h
e
2
2
2
2
2
*
2
)
(
)
(
1
)
4
(
2
l
-
-
>
y
y
p
-
=
òò
,
(10)
where
*
a
and R are Bohr radius
m
e
2
2
e
h
and binding energy
2
2
)
(
2
*
m
a
h
of the bulk exciton, respectively.
The binding energy is obtained by maximising the expectation value
l
l
l
l
y
y
y
y
-
+
=
l
ex
h
e
B
H
E
E
E
)
(
(11)
with respect to
l
.
Then,
)
(
l
B
E
can be simply written in the form:
ú
ú
û
ù
ê
ê
ë
é
÷
÷
ø
ö
ç
ç
è
æ
-
=
λ
a
I
C
a
R
E
*
*
B
2
0
2
)
(
λ
(12)
with
,
)
(
1
)
(
1
)
(
2
2
2
2
0
0
2
ú
ú
ú
û
ù
+
+
ê
ê
ë
é
=
l
-
¥
+
l
-
-
+¥
ò
ò
ò
h
r
r
h
h
r
h
h
r
r
h
h
r
e
e
e
e
dr
e
r
f
r
dr
e
r
f
r
dr
r
f
C
h
e
e
h
e
e
(13)
[
]
dr
x
r
f
r
f
r
f
x
r
f
dx
e
I
h
e
h
e
x
ò
ò
+¥
+¥
l
-
+
+
+
=
0
2
2
2
2
0
2
0
)
(
)
(
)
(
)
(
,
(14)
where
)
(
)
(
i
i
i
i
i
r
r
r
f
y
=
and
(
)
h
e
h
e
m
m
m
m
/
µ
+
=
is the reduced mass of the exciton.
3. Applications
For illustration and with the aim of testing the validity of this model, we have calculated the exciton ground state energy in rectangular and parabolic quantum dots and compared the results with available existing data [5-7]. The agreement was very good.
1) Rectangular quantum dot
The confining potential that we assume as a spherical quantum well-like potential defined by
i
i
i
i
V
R
r
r
V
)
(
)
(
0
-
q
=
, where
q
is the step function and Vi – barrier height.
The dependence of mi on ri arises from the fact that the particles have different effective masses depending on their location, inside or outside the dot. Then, the one-particle problem was solved by computing the solution of the following implicit eigenvalue equation for the spherical symmetry quantum dot energies [4]
(
)
1
1
)
cotg(
0
0
0
R
k
m
m
R
k
R
k
ou
ou
in
in
in
+
-
=
.
(15)
The associated wave functions are given by:
(
)
(
)
ú
ú
û
ù
-
-
q
+
ê
ê
ë
é
+
-
q
p
=
y
i
i
ou
i
i
i
i
i
in
i
i
i
i
i
r
r
k
R
r
B
r
r
k
r
R
A
r
)
exp(
)
sin(
4
)
(
0
0
(16)
with
)
(
2
;
2
i
i
ou
i
ou
i
i
in
i
in
i
E
V
m
k
E
m
k
-
=
=
. (17)
The constants Ai and Bi are determined by normalization requirements and are equal to:
2
1
0
2
0
0
2
)
(
sin
4
)
2
sin(
2
-
ú
ú
û
ù
ê
ê
ë
é
+
-
=
ou
i
in
i
in
i
in
i
i
k
R
k
k
R
k
R
A
and
)
sin(
)
exp(
0
0
R
k
R
k
B
in
i
ou
i
i
=
,
(18)
where Rydberg units are used
)
1
(
0
=
=
h
m
.
To check the accuracy of our model in spherical rectangular quantum dots, we compare our results plotted in Fig. 1 and in Table with those of refs. [5] and [6], respectively.
In Fig. 1, we display variation of the exciton ground state energy in CdS crystallites as a function of the dot radius R0. The dashed lines represent theoretical prediction made by [5] and based on the effective-mass approximation model in the single-band scheme and the variational method, the higher curve corresponds to V e= Vh = 2.5 eV and
1
=
in
ou
m
m
and the lower one to 0.475 eV and
1
=
in
ou
m
m
. The solid lines represent our theoretical model, the higher curve corresponds to Ve = Vh = 2.5 eV and
1
=
in
ou
m
m
, while the lower one – to 0.475 eV and
1
=
in
ou
m
m
. In both cases, the results were evaluated for CdS material parameters me = 0.18m0, mh = 0.53m0,
1
=
in
ou
m
m
and
5
.
5
=
e
, m0 – free-electron mass.
From the analysis of this figure, we conclude the following.
· The results obtained using our method are found to be slightly higher and closer to the experimental values reported in ref. [5] and represented in Fig. 1 by the symbols – circles and triangles.
· A decrease of dot radius is accompanied by an increase of overlap integral of the electron and hole wave functions; which causes an increase in the exciton ground state energy.
· The exciton ground state energy is found to reduce with decreasing the barrier height. This is due to increased penetration of the wave function into the barrier with a resulting lower Coulomb attraction.
In Table, we present our results for the exciton binding energy EB in ZnSe/ZnS quantum dot to compare them with the theoretical results of Jia-Lin Zhu et al. [6]. In this work, the authors used the variational method with introduction of effective electron potentials. We used the same parameters therein (me = 0.16m0, mh = 0.61m0, Ve = 3279 meV,
7
.
8
=
e
for ZnSe and me = 0.27m0, mh = 0.96m0, Ve = 860 meV for ZnS). The agreement between this method and our results is quite good.
Table. Binding energy EB of excitons in ZnSe/ZnS quantum dots as a function of the dot radius R0 with Ve = 3279 meV and Vh = 860 meV.
R0 (Å)
Binding energy EB (meV)
our results
ref. [6]
22
115.665
114.7
25
103.565
103.9
34
78.821
81.21
42
65.016
66.89
56
49.762
51.99
Radius (A)
exciton
energy
(meV)
0
40
80
120
50100150200
Radius (A)
exciton
energy (
eV)
0.0
0.5
1.0
1.5
510152025303540
Fig. 1. Exciton ground state energy in CdS crystallites as a function of the dot radius R0: [5] (dots); this work (full curves).
Radius (A)
0
50
100
150
51015202530
0.8
1.0
1.2
m
ou
/m
in
=
Binding energy (meV)
Fig. 2. Exciton binding energy in CdSe quantum dots as a function of the radius R0 for three different values of the effective-mass ratio inside and outside the cristallite.
Fig. 2 illustrates behaviour of the ground state exciton binding energy in CdSe quantum dots as a function of the radius R0 for three values of the effective-mass ratio
in
ou
m
m
= 0.8, 1.0 and 1.2. The calculations were performed with the following parameters: me = 0.13m0, mh = 0.4m0, Ve = Vh = 1.3 eV and
6
.
10
=
e
[5].
For these three values of the effective mass ratio
in
ou
m
m
considered in this work, we found that the exciton binding energy EB has a maximum at a critical dot radius: on increasing
in
ou
m
m
, the critical dot radius is decreased, and the maximum binding energy is increased. For a certain value of R0, these three values become equal. We conclude that the exterior medium in which the crystallites are embedded considerably modifies behaviour of the exciton binding energy. On the other hand, we observe that the exciton binding energy increases as the radius decreases, reaching the maximum at the dot radius ≈10.5 Å, and then diminishes to a limited value corresponding to a particular radius of the well, for which it is possible to find the free electron and hole energy level [7]. Note that for narrower dots, only confinement influences the increase of the exciton binding energy. Furthermore, as R0 increases, the exciton binding energy approaches the energy of the unconfined two-dimensional exciton.
2) Parabolic quantum dot
In this work, we apply the model developed in Section 3 to calculate the ground state energy of excitons in a single parabolic quantum dot. The studied structure consists in a type I heterostructure with parabolic potential profiles for electrons and holes described as
2
2
2
1
)
(
i
i
i
i
r
m
r
V
W
=
[8, 9], where
2
0
R
m
=
W
h
[8], while R0 is the quantum dot radius, and µ – reduced mass of the exciton. Then, the ground state solutions of one-particle problem for the parabolic quantum dots are:
(
)
2
exp
)
(
i
i
i
i
r
r
a
-
µ
y
with the energy
W
=
h
2
3
i
E
, where
W
=
a
h
2
i
i
m
.
We have applied our model to calculate the exciton ground state energy for GaAs/GaAlAs parabolic quantum dots. By way of comparison, we have referred to the article of S. Jasiri et al. [8] and used the same parameters therein (me = 0.067m0, mh = 0.377m0, Eg= 1520 meV, and
1
.
13
=
e
, m0 is the free-electron mass). In this work, the authors used “perturbative-variational calculations”. The results obtained by this method (full curve) are very close to ours (dots) for all the range of R0 values, as it is shown in Fig. 3.
4. Conclusion
A new investigation of exciton properties in rectangular and parabolic quantum dots has been performed using the advanced analytical calculations. Basic equations are derived in the framework of the commonly used approximations allowing a relatively rapid and rather precise determination of the exciton binding energy. The formulation applied to quantum-dot systems for RQD and PQD gives good results and may be easily extended to any given material.
References
1. M. El-Said // Sci. Technol. 9, p. 272-274 (1994).
2. G. Cantle, D. Ninno and G. Iadonosi // J. Phys.: Condens. Matter, 12, p. 9019-9036 (2000).
3. K. Brunner, U. Bockelmann, G. Abstreiter, M. Walther, G. Böhm, G. Tränkele and G. Weimann // Phys. Rev. Lett. 69, p. 3216 (1992).
4.
P.G. Bolcatto and R.C. Proetto // J. Phys.: Condens. Matter, 13, p. 319-334 (2001).
5. J.L. Marin, R. Riera and S.A. Cruz // J. Phys.: Condens. Matter, 10, p. 1349-1361 (1998).
6. Zhu Jia-Lin, Zhu Shaofeng, Zhu Ziqiang, Y. Kawazoe and T. Yao // J. Phys.: Condens. Matter, 10, p. L583-L587 (1998).
7. C.A Moscoso-Moreno, R. Franco, and J. Solva-Valencia // Revista Mexicana de Fisica, 53 (3), p. 189-193 (2007).
8. S. Jasiri, and R. Bennaceur // Semicond. Sci. Technol. 9, p. 1775-1780 (1994).
9. T. Garm // J. Phys.: Condens. Matter, 8, p. 5725-5735 (1996).
�
Fig. 3. Exciton ground state energy in GaAs parabolic quantum dots as a function of the dot radius R0: [7] (full curve); this work (dots).
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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