Hole, impurity and exciton states in a spherical quantum dot
The 3x3 kp hole Hamiltonian for the wave-function envelopes (effective mass Hamiltonian) was used for calculation of discrete states of the hole and acceptor hydrogenic impurity in a spherical Si/SiO2 nanoheterostructure as a function of the quantum dot radius by neglecting the corrugation of consta...
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| Zitieren: | Hole, impurity and exciton states in a spherical quantum dot / V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13702: 1-12. — Бібліогр.: 26 назв. — англ. |
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irk-123456789-320462012-04-07T12:21:50Z Hole, impurity and exciton states in a spherical quantum dot Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. The 3x3 kp hole Hamiltonian for the wave-function envelopes (effective mass Hamiltonian) was used for calculation of discrete states of the hole and acceptor hydrogenic impurity in a spherical Si/SiO2 nanoheterostructure as a function of the quantum dot radius by neglecting the corrugation of constant-energy surfaces. A study was conducted in the case of finite potential well at the separation boundary of the nanoheterosystem. The dependence of the hole energy spectrum on polarization charges, which arise at the separation boundary of the media, and on the dielectric permittivity, was defined. Using the exact electron and hole solutions, the exciton wave-function was constructed and the exciton ground-state energy was defined. The theoretical results have been compared with experimental data. 2010 Article Hole, impurity and exciton states in a spherical quantum dot / V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13702: 1-12. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 71.55.-i, 73.21.La, 79.60.Jv http://dspace.nbuv.gov.ua/handle/123456789/32046 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The 3x3 kp hole Hamiltonian for the wave-function envelopes (effective mass Hamiltonian) was used for calculation of discrete states of the hole and acceptor hydrogenic impurity in a spherical Si/SiO2 nanoheterostructure as a function of the quantum dot radius by neglecting the corrugation of constant-energy surfaces. A study was conducted in the case of finite potential well at the separation boundary of the nanoheterosystem. The dependence of the hole energy spectrum on polarization charges, which arise at the separation boundary of the media, and on the dielectric permittivity, was defined. Using the exact electron and hole solutions, the exciton wave-function was constructed and the exciton ground-state energy was defined. The theoretical results have been compared with experimental data. |
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Article |
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Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. |
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Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. Hole, impurity and exciton states in a spherical quantum dot Condensed Matter Physics |
| author_facet |
Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. |
| author_sort |
Boichuk, V.I. |
| title |
Hole, impurity and exciton states in a spherical quantum dot |
| title_short |
Hole, impurity and exciton states in a spherical quantum dot |
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Hole, impurity and exciton states in a spherical quantum dot |
| title_fullStr |
Hole, impurity and exciton states in a spherical quantum dot |
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Hole, impurity and exciton states in a spherical quantum dot |
| title_sort |
hole, impurity and exciton states in a spherical quantum dot |
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Інститут фізики конденсованих систем НАН України |
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Hole, impurity and exciton states in a spherical quantum dot / V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13702: 1-12. — Бібліогр.: 26 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT boichukvi holeimpurityandexcitonstatesinasphericalquantumdot AT bilynskyiiv holeimpurityandexcitonstatesinasphericalquantumdot AT leshkorya holeimpurityandexcitonstatesinasphericalquantumdot |
| first_indexed |
2025-07-03T12:31:48Z |
| last_indexed |
2025-07-03T12:31:48Z |
| _version_ |
1836629002775691264 |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 1, 13702: 1–12
http://www.icmp.lviv.ua/journal
Hole, impurity and exciton states in a spherical
quantum dot
V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko∗
Department of Theoretical Physics, Ivan Franko Drohobych State Pedagogical University, 3 Stryiska Str.,
82100, Drohobych, Lviv region, Ukraine
Received September 7, 2009, in final form November 30, 2009
The 3x3 ~k~p hole Hamiltonian for the wave-function envelopes (effective mass Hamiltonian) was used for calcu-
lation of discrete states of the hole and acceptor hydrogenic impurity in a spherical Si/SiO2 nanoheterostruc-
ture as a function of the quantum dot radius by neglecting the corrugation of constant-energy surfaces.
A study was conducted in the case of finite potential well at the separation boundary of the nanoheterosystem.
The dependence of the hole energy spectrum on polarization charges, which arise at the separation bound-
ary of the media, and on the dielectric permittivity, was defined. Using the exact electron and hole solutions,
the exciton wave-function was constructed and the exciton ground-state energy was defined. The theoretical
results have been compared with experimental data.
Key words: impurity, multiband model, polarization charges
PACS: 71.55.-i, 73.21.La, 79.60.Jv
1. Introduction
Recently a study of properties of electrons and holes has been given a lot of attention in
quantum dots (QD’s). Since the conductive band of most semiconductors considered in this study
can be described by the parabolic dispersion relation, the effective masses were introduced and
the Schrödinger equation has been obtained. The use of such an approximation and the dielectric
continuum model yields good theoretical results that conform to experimental data. The presence
of impurities in QD’s can significantly change the localization states. First theoretical researches
of impurity donor states in QD’s were reported in [1–6], where exact solutions of the Schrödinger
equation with the Coulomb potential interaction between particles were obtained. It is shown in
work [6] that the exact solutions of the Poisson and Schrödinger equations for hydrogenic donor
impurity being taken into account changes to some extent the electron spectrum as compared with
results [1–5].
The valence band in many semiconductors is degenerate. Work [7] is one of the first works,
where the general spherically symmetrical solutions of the hole with a total angular momentum f
were obtained in a multiband model. The study of shallow acceptors in bulk semiconductors was
conducted in [8]. Using [9, 10], the hole Hamiltonian was derived there in the spherical approxi-
mation, and acceptor states were calculated in the cases of strong and weak spin-orbit interaction.
Electron-hole pairs were theoretically studied in the multiband effective-mass approximation
with the infinite potential well at the heteroboundary [11]. Single-particle states in spherical Si/SiO2
QD’s were studied in [12] , where the finiteness of crystals band discontinuity and exact boundary
conditions were already taken into account.
The presence of donor and acceptor impurities in QD’s can significantly change the localized
states. Although the calculations of acceptor states were performed, for example [8, 13] – in bulk
crystals, [14] – in thin films, today the acceptor states and their effect on the QD’s properties
have not been studied well enough. On this account, the aim of the present work is to study the
effect of QD size, matrix dielectric permittivity and polarization charges on the energy spectrum
∗
E-mail: leshkoroman@meta.ua
c© V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko 13702-1
http://www.icmp.lviv.ua/journal
V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko
of the hole, acceptor impurity and exciton. Specific calculations are performed for the Si/SiO2
nanoheterostructure. Theoretical results have been compared with experimental data.
2. Formulation of the problem and its solution
2.1. Hole Hamiltonian 3x3 and wave functions
We consider a spherical QD heterostructure. The radius of the QD is a. There is an acceptor
impurity in the center of the QD. Let the heterosystem be constructed of crystals with a large band
gap Eg and weak spin-orbit interaction ∆ [8]. Therefore, the conduction band can be neglected
and one can assume, that ∆ = 0. In order to simplify formulas, the system of units (m0 = 1,
~ = 1, e = 1) is used, and the energy is counted downward. Taking into account these remarks and
neglecting the corrugation of constant-energy surfaces (the so-called spherical approximation), we
write the kp-Hamiltonian for envelope functions in the form of [15]
H = −
[
(A+ 2B) p2 − 3B
(
p̃J̃
)2
]
+ Π (r) , (1)
where
A = −γ1
2
, B = −γ , µ =
2γ
γ1
, γ =
1
5
(3γ3 + 2γ2) ,
γ1, γ2, γ3 are the Luttinger parameters, which for different regions of the heterostructure are de-
noted as follows:
(γ1 γ2 γ3) =
{ (
γin
1 γin
2 γin
3
)
, r 6 a,
(γout
1 γout
2 γout
3 ) , r > a.
Further all notations with the index in will correspond to a QD, and those with the index out
– to the matrix. The operator J̃ of the angular momentum corresponds to unity spin. The hole
potential energy is given by
Π (r) = U (r) + Vc (r) + Vp (r) , (2)
where
U (r) =
{
0, r 6 a,
U0 r > a
(3)
is the potential energy caused by the band offset. Using the exact solution of the Poisson equation,
the potential energy of the hole-acceptor ion coupling is given by
Vc (r) = −
{
1/
(
εinr
)
+
(
εin − εout
)
/
(
εinεouta
)
, r 6 a,
1/ (εoutr) , r > a.
(4)
A charged particle will induce polarization charges at the separation boundary of the media. The
corresponding energy of the electron-polarization charges interaction will be determined like in [16]:
Vp (r) =
ω
4ε (r)
∞
∫
0
dt
[
th
(
t− a
r0
)
+
t
r0
sech2
(
t− a
r0
)]
1
t2 − r2
,
ε (r) =
εin + εout
2
[
1 − ω · th
(
r − a
r0
)]
, ω =
εin − εout
εin + εout
, (5)
where r0 is the thickness of the transition layer at the interface where the dielectric permittivity
depends on coordinates.
The wave function, which is the eigenfunction of the Hamiltonian (1), can be written as a
product of the radial function and the eigenfunction of the square of the total angular momentum
F2 =
(
J̃ + L̃
)2
. It is
Φl
f,M (θ, ϕ) =
l
∑
m=−l
1
∑
mj=−1
Cf,M
l,m;1,mj
Yl,m (θ, ϕ)χmj
, (6)
13702-2
Hole, impurity and exciton states in a spherical quantum dot
where f(f + 1), l(l+ 1), M , m, mj are the quantum numbers, which determine the eigenvalues of
the operators F2, L2, Fz, Lz, Jz respectively. Cf,M
l,m;1,mj
are the Clebsch-Gordan coefficients, χmj
are the spin functions, Yl,m are the spherical harmonics, being the eigenfunctions of L2. According
to the common rules of addition of the angular momenta, we write the most general expression of
solutions of the Schrödinger equation with the Hamiltonian (1) for three types of states [8, 12]:
ψf−1,f+1
f,M (~r) = Rf−1
f (r) Φf−1
f,M (θ, ϕ) +Rf+1
f (r) Φf+1
f,M (θ, ϕ) , (f > 1; l = f − 1, f + 1), (7)
ψf
f,M (~r) = Rf
f (r) Φf
f,M (θ, ϕ) , (f > 1; l = f) , (8)
ψ1
0,0 (~r) = R1
0 (r) Φ1
0,0 (θ, ϕ) , (f = 0; l = 1) . (9)
Inserting functions (7)–(9) into the Schrödinger equation with the Hamiltonian (1), which is
then multiplied by the corresponding conjugated spinors
(
Φl
f,M (θ, ϕ)
)+
, we get the following
equations for the radial functions of the above-mentioned states:
A
(
C1B
+
f−2B
−
f−1 C2B
−
f B−
f+1
C2B
+
f B+
f−1 C3B
−
f+2B
+
f+1
)
(
Rf−1
f
Rf+1
f
)
+ Π (r)
(
Rf−1
f
Rf+1
f
)
−E
(
Rf−1
f
Rf+1
f
)
= 0, (10)
A (1 − µ)
(
∆fR
f
f
)
+ Π (r)Rf
f −ERf
f = 0, (11)
A (1 + 2µ)
(
∆1R
1
0
)
+ Π (r)R1
0 −ER1
0 = 0. (12)
In equations (10)–(12) the following notations are introduced:
∆l =
∂2
∂r2
+
2
r
∂
∂r
− l (l + 1)
r2
, B+
l =
d
dr
− l
r
, B−
l =
d
dr
+
(l + 1)
r
,
C1 = 1 +
f − 1
2f + 1
µ , C2 = −3
√
f (f + 1)
2f + 1
µ , C3 = 1 +
f + 2
2f + 1
µ .
From the equations for the radial functions one can find the hole energies and wave functions.
2.2. Boundary conditions
In order to get the boundary conditions for the spherical QD, two conditions are used. They are
as follows: the continuity of the radial wave-function and the continuity of the normal component
of the probability density flux at the boundary of the QD. The continuity of wave function gives
(
Rf−1
f
)in
(
Rf+1
f
)in
∣
∣
∣
∣
∣
∣
∣
r=a
=
(
Rf−1
f
)out
(
Rf+1
f
)out
∣
∣
∣
∣
∣
∣
r=a
, (13)
(
Rf
f
)in
∣
∣
∣
∣
r=a
=
(
Rf
f
)out
∣
∣
∣
∣
r=a
,
(
R1
0
)in
∣
∣
∣
r=a
=
(
R1
0
)out
∣
∣
∣
r=a
.
In order to determine the normal component of the probability density flux, the normal component
of the velocity operator Vr = ~r/r (∂H/∂p̃), which is proportional to the normal component of the
probability density flux, is defined. Then, the operator was written in the spinor representation for
three types of states (7)–(9). Using the explicit form of the operators, three more conditions are
found (see Appendix)
(
Tin
11 Tin
12
Tin
21 Tin
22
)
(
Rf−1
f
)in
(
Rf+1
f
)in
∣
∣
∣
∣
∣
∣
∣
r=a
=
(
Tout
11 Tout
12
Tout
21 Tout
22
)
(
Rf−1
f
)out
(
Rf+1
f
)out
∣
∣
∣
∣
∣
∣
r=a
, (14)
13702-3
V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko
Ain
(
1 − µin
) d
dr
(
Rf
f
)in
∣
∣
∣
∣
r=a
= Aout
(
1 − µout
) d
dr
(
Rf
f
)out
∣
∣
∣
∣
r=a
, (15)
Ain
(
1 + 2µin
) d
dr
(
R1
0
)in
∣
∣
∣
∣
r=a
= Aout
(
1 + 2µout
) d
dr
(
R1
0
)out
∣
∣
∣
∣
r=a
. (16)
We use the following notations in equations (14)–(16):
T
{
in
out
}
11 = A
{
in
out
}
(
1 +
f − 1
2f + 1
µ
{
in
out
}
)
d
dr
+
3 (f − 1)
2 (2f + 1)
µ
{
in
out
}
r
,
T
{
in
out
}
12 =
3
√
f (f + 1)
2f + 1
B
{
in
out
}
(
d
dr
+
f + 2
r
)
,
T
{
in
out
}
21 =
3
√
f (f + 1)
2f + 1
B
{
in
out
}
(
d
dr
− f − 1
r
)
,
T
{
in
out
}
22 = A
{
in
out
}
(
1 +
f + 2
2f + 1
µ
{
in
out
}
)
d
dr
+
3
2
f + 2
2f + 1
µ
{
in
out
}
r
.
The hole energy spectrum can be determined using the solution of equations (10)–(12), when
the boundary conditions (13)–(16) are satisfied. If we denote the number of solution of the corre-
sponding equation system through n (n = 1, 2, . . . ), then apart from the above-mentioned quantum
numbers, the energies and wave-functions will depend on one more quantum number – n. There-
fore, functions (7)–(9) may be redenoted as ψf−1,f+1
f,M ≡ ψf−1,f+1
n,f,M ; ψf
f,M ≡ ψf
n,f,M ; ψ1
0,0 ≡ ψ1
n,0,0.
It will be used to analyze figure 1 and figure 2.
2.3. Hole energy spectrum without impurity
We assume Π (r) = U (r) + Vp (r). This is equivalent to the study of the hole states in a
spherical QD without acceptor impurity. The potential energy Vp (r) of the interaction of a hole
with polarization charges at the interface can be considered as perturbation, the effect of which will
be accounted for in first-order perturbation theory [16]. For the unperturbed system Π (r) = U (r).
In this case equations (10)–(12) have exact solutions given by the spherical Bessel functions of the
first kind and the modified spherical Bessel functions of the second kind
(
Rf−1
f
)in
= Din
1 jf−1
(
λinr/a
)
+Din
2 jf−1
(
λinβinr/a
)
,
(
Rf+1
f
)in
= −Din
1
√
f
f + 1
jf+1
(
λinr/a
)
+Din
2
√
f + 1
f
jf+1
(
λinβinr/a
)
,
(
Rf
f
)in
= Din
f,f jf
(
λinr/a
)
,
(
R1
0
)in
= Din
1,0 j1
(
λinβinr/a
)
, r 6 a, (17)
(
Rf−1
f
)out
= Dout
1 kf−1
(
λoutr/a
)
+Dout
2 kf−1
(
λoutβoutr/a
)
,
(
Rf+1
f
)out
= Dout
1
√
f
f + 1
kf+1
(
λoutr/a
)
−Dout
2
√
f + 1
f
kf+1
(
λoutβoutr/a
)
,
(
Rf
f
)out
= Dout
f,f kf
(
λoutr/a
)
,
(
R1
0
)out
= Dout
1,0 k1
(
λoutβoutr/a
)
, r > a, (18)
13702-4
Hole, impurity and exciton states in a spherical quantum dot
where
λin =
√
−Ea2
Ain (1 − µin)
, λout =
√
− (−E + U0) a2
Aout (1− µout)
, β
{
in
out
}
=
√
√
√
√
√
1 − µ
{
in
out
}
1 + 2µ
{
in
out
} .
Applying the boundary conditions to solutions (17)–(18), we determine the eigenvalues. The energy,
caused by the interaction of electron-polarization charges, is given by
Ep =
∫
d~r
(
ψl
f,M
)+
Vp (r)ψl
f,M . (19)
2.4. Acceptor energy spectrum
The total potential energy of the hole in the spherical QD with an acceptor impurity has the
form (2). If the term Vp (r) is neglected in zero-order approximation, then equations (11)–(12)
(second and third types of states) have exact solutions, and the effect of Vp (r), like in the previous
section, can be taken into account in first-order perturbation theory. Those two equations are
written for two regions.
If r 6 a , then
− 1
2bin
(
∆nR
in
)
− 1
εinr
Rin − ẼinRin = 0. (20)
For the second type of states bin = 1/
(
γin
1
(
1 − µin
))
, ∆n = ∆f , Rin =
(
Rf
f
)in
, and for the third
type of states bin = 1/
(
γin
1
(
1 + 2µin
))
, ∆n = ∆1, R
in =
(
R1
0
)in
. In both cases Ẽin = E + U∗ (a),
U∗ (a) =
(
εin − εout
)
/(εinεouta). Equation (20) is considered in two energy ranges: Ẽin < 0,
Ẽin > 0.
If Ẽin < 0, we introduce the following notations: ξin = αinr,
(
αin
)2
= −8binẼin, λin =
2bin/
(
εinαin
)
, Rin
(
ξin
)
= gin
(
ξin
)
/ξin. After performing some simple transformations, we obtain
the Whittaker equation
∂2
(∂ξin)
2 g
in +
[
−1
4
+
λin
ξin
− n (n+ 1)
(ξin)
2
]
gin = 0. (21)
The solution of equation (21) is represented by
gin
(
ξin
)
= Dine−ξin/2
(
ξin
)n+1
M
(
n+ 1 − λin, 2n+ 2, ξin
)
, (22)
where M (a, b, x) is the confluent hypergeometric function of the first kind [17].
Let us consider the other energy range: Ẽin > 0. By defining dimensionless values ξin = βinr,
(
βin
)2
= 2binẼin, δin = −bin/
(
εinβin
)
, Rin
(
ξin
)
= gin
(
ξin
)
/ξin equation (20) is transformed into
the Coulomb equation
∂2
(∂ξin)
2 g
in +
[
1 − 2δin
ξin
− n(n+ 1)
(ξin)
2
]
gin = 0. (23)
The solution of this equation, which satisfies the finiteness condition of the wave function, can be
represented by the Coulomb function
gin
(
ξin
)
= Din 2ne−πδin/2
∣
∣Γ
(
n+ 1 + iδin
)∣
∣
Γ (2n+ 2)
e−iξin (
ξin
)n+1
M
(
n+ 1 − iδin, 2n+ 2, 2iξin
)
, (24)
where Γ (z) is the Euler gamma-function [17].
When r > a, equations (11)–(12) can be written
− 1
2bout
(
∆nR
out
)
− 1
εoutr
Rout − ẼoutRout = 0.
13702-5
V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko
For the second type of states bout = 1/ (γout
1 (1 − µout)), ∆n = ∆f , Rout =
(
Rf
f
)out
, and for the
third type of states bout = 1/(γout
1 (1 + 2µout)), bout = 1/ (γout
1 (1 + 2µout)), Rout =
(
R1
0
)out
. In
both cases Ẽout = E − U0. After introducing dimensionless variables and other parameters
ξout = αoutr, (αout)
2
= −8boutẼout, λout = 2bout/ (εoutαout), Rout (ξout) = gout (ξout) /ξout,
we obtain the Whittaker equation. The solution of this equation, which satisfies the finiteness
condition of the wave function, in the limit r → ∞, takes the form [17]:
gout
(
ξout
)
= Dout exp (−ξout/2) (ξout)
−n
Γ (−n− λout)
∞
∫
0
dt exp
(
−ξoutt
)
t−n−λout−1(1 + t)−n+λout−1. (25)
From the boundary conditions and the normalization condition one can define the eigenvalues and
eigenfunctions of an acceptor impurity. The potential energy Vp (r) will be taken into consideration
in first-order perturbation theory.
Equation (10) for an acceptor impurity has no exact solution, even if we neglect the Vp (r) term.
Therefore, we will find its solution expanding the exact radial wave function over the functions of
the problem without an impurity (17)–(18):
(
Rf−1
f
)
a(
Rf+1
f
)
a
=
∑
i
ci
(
Rf−1
f
)
i(
Rf+1
f
)
i
. (26)
Inserting (26) into (10) and multiplying it by the conjugate row-vector
((
Rf−1
f
)∗
i′
(
Rf+1
f
)∗
i′
)
gives the homogeneous system of equations with coefficients ci:
∑
i
((Ei −Ea) δi′i + Vi′i) ci = 0, (27)
where Ei is the specific h hole energy level without an impurity, which is determined by solutions
(17)–(18), and
Vi′i =
∫
drr2 (Vc (r) + Vp (r))
[(
Rf−1
f
)∗
i′
(
Rf−1
f
)
i
+
(
Rf+1
f
)∗
i′
(
Rf+1
f
)
i
]
.
Equating the determinant of the homogeneous system (27) to zero, we get the equation to determine
the hole energy Ea with the acceptor presence. Using system (27) and the normalization condition
∑
i
|ci|2 = 1, one can define the eigenfunctions of an acceptor impurity.
2.5. Exciton ground-state energy
The ground-state energy of the electron-hole pair without an impurity can be calculated by
perturbation theory. The expression of the electron-hole coupling is given by [18]. The exciton wave
function was constructed as a product of the electron and hole wave functions
Ψex = ψ1,0,0 (~r1)ψ
0,2
1,1,M (~r2) , (28)
where ~r1, ~r2 are the electron and hole coordinates, respectively. The solutions of the electron
Schrödinger equation in the spherical QD, neglecting the effective mass anisotropy in the QD, were
defined similar to [18, 19]. The polarization charges at the separation boundary of the media were
taken into the consideration therein.
Therefore, adding the electron and hole energies and taking into account the electron-hole
interaction, the exciton energy was obtained.
13702-6
Hole, impurity and exciton states in a spherical quantum dot
3. The analysis of results
All results have been obtained for the spherical Si/SiO2 QD heterostructure. We take the basic
parameters of crystals, which form the heterostructure, to be identical to those in [12]: γ
(in)
1 = 4.22,
γ
(in)
2 = 0.53, γ
(in)
3 = 1.38, ε(in) = 11.4, γ
(out)
1 = 1/5, γ(out) = 0, ε(out) = 3.9, U0 = 4.3 eV. The
introduced isotropic effective electron mass in Si is considered equal to the mean value ofm∗
⊥ = 0.19
and m∗
‖ = 0.916 effective electron masses.
Figure 1 shows the dependences of the ground and lower hole energy levels on the QD size for
three types of states (7)–(9). Also, the effect of polarization charges at the interface was taken into
account. It is seen from figure 1 that with the decrease the QD radius, the hole energy increase is
caused by the enhancement of the effect of the spatial confinement and polarization charges at the
interface.
Figure 1. Dependence of the hole energy on the QD radius without (dashed curves) and with
(solid curves) taking into account polarization charges.
Figure 2. The acceptor hole energy spectrum.
13702-7
V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko
The acceptor impurity energy spectrum, with taking into account the total potential energy
(2), is represented in figure 2. Therein and further in the paper, the polarization term Vp(r) is
taken into consideration. The energy levels smoothly approach the values, which correspond to
the acceptor impurity hole levels in the bulk silicon crystal in the limit of a large QD radius. The
reduction of the QD radius causes two competing effects: the rise of the spatial confinement, which
increases the hole energy, and the rise of the quantum well depth U ∗ (a) (4), which in this case
decreases the hole energy. As is shown in [6], the effect of the effective potential well on the particle
energy is clear for the ground and first lower states. Therefore, the decrease of the QD radius
for ψ0,2
1,1,M (7) leads to the reduction of the particle energy. Moreover, the reason of such energy
behavior is the large hole effective mass in the matrix, which is taken mv = 5, like in [12]. As for
the states ψ1
1,1,M , ψ1
1,0,0 (8)–(9), the hole energy decreases slightly first (which is unnoticeable in
figure 2) and then rises.
In order to study the dependence of the acceptor energy levels on the matrix dielectric per-
mittivity and to clearly show the effect of the effective potential well on the hole spectrum, the
calculation of the acceptor ground-state energy is performed for different εout. The result of the
calculation is shown in figure 3.
Figure 3. The acceptor ground-state energy for different values of the matrix dielectric permit-
tivity.
If εout = 3.9, like in [12], then the effective potential well in the QD is negative and in its
absolute value is larger than the kinetic energy of the particle, and thus the total energy decreases.
In case the dielectric permittivity of the matrix εout = 4.6 [20], then contributes less to the total
energy, but when a 6 5 Å, the total energy begins to increase. If the matrix dielectric permittivity
εout = 7, then the particle energy is larger than that in the previous case. When a ≈ 25 Å,
the minimum is observed in figure 3. When the dielectric permittivity of the matrix equals the
QD dielectric permittivity εout = εin = 11.4, then U∗ (a) = 0 and all polarization corrections
equal zero. In that case we get a monotonous increase of the particle energy, if the QD radius
decreases. However, in all the cases for large QD radii the acceptor ground-state energy leads to
the corresponding energy in the bulk silicon crystal: E = −0.031 eV.
After having analysed the hole spectrum, one can consider the electron-hole bound state in the
QD. Figure 4 shows the dependences of the ground-state exciton energy as a function of the QD
radius. The experimental data obtained in [21–25], are presented too. As is seen from figure 4, our
calculation results are both in good qualitative and quantitative agreement with the experimental
data. Taking account of the multiband effective-mass approximation, interparticle interaction and
interaction of particles with polarization charges allows us to determine the dependence of the
exciton energy (denoted in figure 4 with “1”).
13702-8
Hole, impurity and exciton states in a spherical quantum dot
Figure 4. The exciton ground-state energy within (solid curves 1, 2) and without (dashed curve
3) the multiband effective-mass approximation. The ground-state electron-hole energy without
the electron-hole interaction (dotted curve 4).
If the dielectric permittivity of the matrix equals that of the QD, then the exiton energy will
be described by curve 2. It is found that the increase of the QD radius leads to the decrease of
the effect of the matrix properties (in particular, the dielectric properties). Therefore, curves 1 and
2 begin to approach each other if a > 40 Å. If the complex band spectrum is neglected and the
heavy hole effective mass is employed in calculations, similar to [26], then the exciton ground-state
energy will be described by a dashed curve 3. In addition, figure 4 shows the dependence of the
electron-hole ground-state energy without taking into account the electron-hole interaction (dotted
curve 4). Therefore, the exciton energy depends on the dimensional quantization of the electron
and hole energies in the QD and dielectric properties of the matrix. Besides, the calculations show
that taking account of the complex band structure numerically changes the results as compared
with the single valence-band model.
4. Summary
In the present work the hole and acceptor energy spectra in the QD have been studied. All
the energies in this paper are calculated, when the polarization charges are taken into account.
The calculation has been done in the multiband effective-mass approximation in case of the weak
spin-orbit interaction. Within the frames of this model and in the finite spherical potential-well
approximation, the exact hole solutions of all the radial equations have been obtained for the
heterostructure. As for an acceptor impurity, the exact solutions have been derived for two types
of states, which are described by a single differential equation. The matrix radial equation of
the acceptor impurity has been solved approximately, using the exact hole solutions without an
impurity. On the basis of exact electron and hole solutions, the exciton wave function has been
written, and the exciton ground-state energy has been defined. It is found that the difference
between the QD and matrix dielectric permittivity leads to an increase of the exciton energy. This
is in good agreement with the data of other theoretical and experimental works. It is shown that
the complex band structure being taking into account changes the exciton energy as compared
with the single valence-band model. The comparison of the obtained results with experimental
data has shown good numerical agreement.
13702-9
V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko
Appendix
For example, we briefly show the scheme of obtaining the boundary condition (14). In order to
find the normal component of the velocity operator
Vn =
~r
r
∂H
∂p̃
=
(
n+
∂H
∂p+
+ n−
∂H
∂p−
+ nz
∂H
∂pz
)
,
where n+ = (x+ iy) /r = −
√
2
√
4π/3Y1,1, n− = (x− iy) /r =
√
2
√
4π/3Y1,−1, nz = z/r =
√
4π/3Y1,0, we write (1) in the matrix form, using the following operators px = (p+ + p−)/2,
py = (p+ + p−)/(2i), p2 = p2
z + p+p−, where [p+,p−] = 0 property is taken into account, and
the explicit form of Ji is used
Jx =
1√
2
0 1 0
1 0 1
0 1 0
, Jy =
i√
2
0 −1 0
1 0 −1
0 1 0
, Jz =
1 0 0
0 0 0
0 0 1
.
It is found
Vn =
(Vn)11 (Vn)12 (Vn)13
(Vn)21 (Vn)22 (Vn)23
(Vn)31 (Vn)32 (Vn)33
, (A1)
where
(Vn)11 = (Vn)33 = −
(
A− B
2
)
(n+p− + n−p+) − 2 (A−B)nzpz ,
(Vn)12 = − [(Vn)21]
∗
= − (Vn)23 = [(Vn)32]
∗
=
3B√
2
(nzp− + nmpz) ,
(Vn)13 = − [(Vn)31]
∗
= 3Bnmp−,
(Vn)22 = − (A−B) (n+p− + n−p+) − 2 (A+ 2B)nzpz .
The normal component of the velocity operator is proportional to the normal component of the
probability density flux. Therefore, we write condition for the I-type of states in the form
(
Vin
n ψ
f−1,f+1
f,M
)∣
∣
∣
r=a
=
(
Vout
n ψf−1,f+1
f,M
)∣
∣
∣
r=a
. (A2)
Left and right parts of the previous equation are similar, that is why we continue to work only
with the left part and the index in is omitted:
Vnψ
f−1,f+1
f,M = Vn
(
Rf−1
f Φf−1
f,M +Rf+1
f Φf+1
f,M
)
. (A3)
The explicit form of spinors is
Φf−1
f,M =
1
√
2f (2f − 1)
√
(f +M) (f +M − 1)Yf−1,M−1
√
2 (f −M) (f +M)Yf−1,M
√
(f −M) (f −M − 1)Yf−1,M+1
, (A4)
Φf+1
f,M =
1
√
2 (f + 1) (2f + 3)
√
(f −M + 1) (f −M + 2)Yf+1,M−1
√
2 (f −M + 1) (f +M + 1)Yf+1,M
√
(f +M + 1) (f +M + 2)Yf+1,M+1
. (A5)
Operator Vn contains p±, pz. Those operators will operate on the function R (r) Yl,m (θ, ϕ).
Therefore, we derive the following relation for the spherical harmonics:
pz (RYlm) =−i
√
(l +m+ 1) (l−m+ 1)
(2l + 3) (2l+ 1)
Yl+1,m
(
B+
l R
)
−i
√
(l +m) (l−m)
(2l+ 1) (2l−1)
Yl−1,m
(
B−
l R
)
, (A6)
13702-10
Hole, impurity and exciton states in a spherical quantum dot
p+ (RYlm) =
√
(l +m+ 2) (l +m+ 1)
(2l+ 3) (2l+ 1)
Yl+1,m+1
(
B+
l R
)
−
√
(l−m) (l−m−1)
(2l+ 1) (2l−1)
Yl−1,m+1
(
B−
l R
)
,
(A7)
p− (RYlm) =−
√
(l−m+ 2) (l−m+ 1)
(2l + 3) (2l+ 1)
Yl+1,m−1
(
B+
l R
)
+
√
(l +m) (l +m−1)
(2l + 1) (2l−1)
Yl−1,m−1
(
B−
l R
)
.
(A8)
Insertion (A4)–(A5) into (A3), the three-component vector-column was obtained. This vector-
column was multiplied by the corresponding conjugated spinors
(
Φf−1
f,M
)+
and
(
Φf+1
f,M
)+
by turns.
Two expressions were obtained. Then, those expressions were integrated by the angular variables.
After simple, but very huge simplifications we obtain the condition (14). In the same manner the
equations (15)–(16) were derived too.
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20. Babichev A.P., Bratkovsky A.M., et al. Fizicheskii velichiny: Spravochnik, eds. Grigoryev I.S., Meili-
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13702-11
V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko
Дiрковi, домiшковi та екситоннi стани у сферичнiй квантовiй
точцi
В.I. Бойчук, I.В. Бiлинський, Р.Я. Лешко
Кафедра теоретичної фiзики, Дрогобицький державний педагогiчний унiверситет iм. I. Франка,
вул. Стрийська 3, Дрогобич, 82100
Для сферичної наногетероструктури Si/SiO2, нехтуючи гофрованiстю iзоенергетичних поверхонь в
k-просторi, використано дiрковий ~k~p гамiльтонiан 3x3 для огинаючих хвильових функцiй (гамiль-
тонiан ефективної маси) при обчислення дискретних станiв дiрки та водневоподiбної акцепторної
домiшки як функцiї розмiрiв квантової точки. Дослiдження проведено для скiнченного потенцiалу на
межi гетероструктури. Визначено залежнiсть енергетичного спектру дiрки вiд поляризацiйних за-
рядiв, що виникають на гетеромежах, та вiд дiелектричної проникностi матрицi. На основi точних
розв’язкiв для електрона та дiрки побудовано хвильову функцiю i визначено енергiю основного ста-
ну екситона. Проведено порiвняння з експериментальними даними.
Ключовi слова: домiшка, багатозонна модель, поляризацiйнi заряди
13702-12
Introduction
Formulation of the problem and its solution
Hole Hamiltonian 3x3 and wave functions
Boundary conditions
Hole energy spectrum without impurity
Acceptor energy spectrum
Exciton ground-state energy
The analysis of results
Summary
|