Study of an Acceptor Impurity Located at the Center of a Sperical Nanoheterostructure
Using the spherical 4x4 Hamiltonian, the discrete states of a hydrogenic acceptor impurity in a spherical GaSb/AlSb nanoheterostructure with various quantum-dot sizes are determined. The energies obtained are compared with those calculated without consideration of the complex structure of the valenc...
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irk-123456789-134062010-11-09T12:01:57Z Study of an Acceptor Impurity Located at the Center of a Sperical Nanoheterostructure Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. Shakleina, I.O. Наносистеми Using the spherical 4x4 Hamiltonian, the discrete states of a hydrogenic acceptor impurity in a spherical GaSb/AlSb nanoheterostructure with various quantum-dot sizes are determined. The energies obtained are compared with those calculated without consideration of the complex structure of the valence band. The calculations are carried out for both finite and infinite potentials at the heterostructure interface. The selection rules are found for intraband optical transitions between hole levels. The average distances and the transition probabilities for holes are determined as functions of the quantum-dot sizes. Для сферичної наногетероструктури GaSb/AlSb, використовуючи сферичний гамiльтонiан 4x4, визначено дискретнi стани водневоподiбної акцепторної домiшки для рiзних розмiрiв квантової точки. Проведено порiвняння визначених енергiй з вiдповiдними енергiями, що одержанi без урахування складної структури валентної зони. Обчислення проведено як для скiнченного, так i для нескiнченного потенцiалу на межi гетероструктури. Встановлено правила добору для внутрiшньозонних мiжрiвневих оптичних переходiв дiрки. Визначено середнi вiдстанi та ймовiрностi переходiв дiрки як функцiї розмiрiв квантової точки. 2010 Article Study of an Acceptor Impurity Located at the Center of a Sperical Nanoheterostructure / V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko., I.O. Shakleina // Укр. фіз. журн. — 2010. — Т. 55, № 3. — С. 326-334. — Бібліогр.: 25 назв. — англ. 2071-0194 PACS 71.55.-i; 73.21.La; 79.60.Jv http://dspace.nbuv.gov.ua/handle/123456789/13406 en Відділення фізики і астрономії НАН України |
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Наносистеми Наносистеми Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. Shakleina, I.O. Study of an Acceptor Impurity Located at the Center of a Sperical Nanoheterostructure |
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Using the spherical 4x4 Hamiltonian, the discrete states of a hydrogenic acceptor impurity in a spherical GaSb/AlSb nanoheterostructure with various quantum-dot sizes are determined. The energies obtained are compared with those calculated without consideration of the complex structure of the valence band. The calculations are carried out for both finite and infinite potentials at the heterostructure interface. The selection rules are found for intraband optical transitions between hole levels. The average distances and the transition probabilities for holes are determined as functions of the quantum-dot sizes. |
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Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. Shakleina, I.O. |
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Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. Shakleina, I.O. |
| author_sort |
Boichuk, V.I. |
| title |
Study of an Acceptor Impurity Located at the Center of a Sperical Nanoheterostructure |
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Study of an Acceptor Impurity Located at the Center of a Sperical Nanoheterostructure |
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Study of an Acceptor Impurity Located at the Center of a Sperical Nanoheterostructure |
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Study of an Acceptor Impurity Located at the Center of a Sperical Nanoheterostructure |
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Study of an Acceptor Impurity Located at the Center of a Sperical Nanoheterostructure |
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study of an acceptor impurity located at the center of a sperical nanoheterostructure |
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Відділення фізики і астрономії НАН України |
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2010 |
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Study of an Acceptor Impurity Located at the Center of a Sperical Nanoheterostructure / V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko., I.O. Shakleina // Укр. фіз. журн. — 2010. — Т. 55, № 3. — С. 326-334. — Бібліогр.: 25 назв. — англ. |
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2025-07-02T15:24:00Z |
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1836549240132730880 |
| fulltext |
NANOSYSTEMS
326 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
STUDY OF AN ACCEPTOR
IMPURITY LOCATED AT THE CENTER
OF A SPHERICAL NANOHETEROSTRUCTURE
V.I. BOICHUK, I.V. BILYNSKYI, R.YA. LESHKO, I.O. SHAKLEINA
Ivan Franko Drohobych State Pedagogical University
(3, Stryiska Str., Drohobych 82100, Ukraine; e-mail: leshkoroman@ mail. ru )
PACS 71.55.-i; 73.21.La;
79.60.Jv
c©2010
Using the spherical 4× 4 Hamiltonian, the discrete states of a hy-
drogenic acceptor impurity in a spherical GaSb/AlSb nanoheterostruc-
ture with various quantum-dot sizes are determined. The energies
obtained are compared with those calculated without considera-
tion of the complex structure of the valence band. The calculations
are carried out for both finite and infinite potentials at the het-
erostructure interface. The selection rules are found for intraband
optical transitions between hole levels. The average distances and
the transition probabilities for holes are determined as functions
of the quantum-dot sizes.
1. Introduction
Recently, low-dimensional heterostructures have become
a subject of both theoretical and experimental researches
in solid-state physics. A considerable attention is given
to studying quantum dots (QDs). Since the charge car-
riers possess a completely discrete spectrum in them,
QDs have improved optical properties, and they are re-
searched to be applied in diode lasers, amplifiers, and
biological sensors.
Many works are devoted to the study of properties of
electrons in QDs. Since the conduction band in the ma-
jority of crystals under consideration can be described
by a parabolic dispersion law, effective masses were in-
troduced, and a simple Schrödinger equation was de-
rived. The use of such an approximation and the ap-
plication of the model of dielectric continuum in theo-
retical works provide good agreement with experimental
data. On the basis of these principles, exact solutions
of the Schrödinger equation were found for QDs with
symmetric shapes, the calculations of electron discrete
states were carried out, and the probabilities of inter-
level transitions were found [1, 2]. The influence of po-
larization charges [3, 4] and deformation [5, 6] at the
heterostructure interfaces on the electron energy spec-
trum was studied, and the structure of energy subbands
in QD arrays was determined [7]. Impurities in QDs
can considerably change the localized states and, con-
sequently, the QD properties. In works [8–12], the first
theoretical researches of impurity donor states in spher-
ical QDs were carried out, and the exact solutions of the
Schrödinger equation with the Coulomb potential inter-
action between particles were obtained. In work [13], it
was shown that taking the exact solution of the Poisson
and Schrödinger equations for a hydrogenic donor impu-
rity into account somewhat changes the electron spec-
trum in comparison with the results of works [8–12].
The valence band of many semiconductors is degener-
ate. Work [14] is one of the first works, where general
spherically symmetric solutions for even and odd hole
states with the total angular momentum f were obtained
in the framework of the multiband crystal model. The
shallow acceptors in massive semiconductors were stud-
ied in work [15], where, on the basis of works [14, 16, 17],
the hole Hamiltonian in the spherical approximation was
derived, and calculations for acceptors were carried out
in the cases of strong and weak spin-orbit interaction.
The hole quantization and the absorption edge in spher-
ical microcrystals of semiconductors with a complicated
band structure were described in work [18]. In work
[19], the solutions of the Schrödinger equation for QDs
were obtained in the framework of the spherical approx-
imation, and the probability of optical transitions was
analyzed in the cases of finite and infinite band disconti-
nuities at the heterostructure interfaces. Again, in work
STUDY OF AN ACCEPTOR IMPURITY LOCATED
[20], in the framework of the four-band model, the ex-
citon states in spherical QDs and the influence of an
electric field on them were studied, the Stark effect was
analyzed, and the influence of an electric field on the
matrix elements of the dipole moment was described,
the analysis being carried out in the framework of the
model of infinite band discontinuity at the heterostruc-
ture interfaces. On the other hand, in work [21], the
8 × 8 Hamiltonian was obtained, the boundary condi-
tions were formulated, and the quasiparticle spectra were
calculated numerically for the model of finite band dis-
continuity.
The presence of donor and acceptor impurities in QDs
can considerably change localized states. Though the
calculations for acceptor states were carried out in var-
ious works – e.g. in works [15, 22] for massive crys-
tals and in work [23] for thin films – the acceptor states
and their influence on QD properties still remain insuf-
ficiently studied up to now. Therefore, the aim of this
work was to analyze the influence of dimensions of a
spherical QD, with an acceptor impurity and without
it, on the hole energy spectrum, taking a complicated
structure of the valence band into account. Here, we
will also calculate the average distances for the hole,
find the selection rules for optical transitions between
the hole levels resulted from the size quantization, and
determine the probabilities of such transitions. Specific
calculations were carried out for a spherical GaSb/AlSb
nanoheterosystem.
2. Statement of the Problem and Its Solution
2.1. 4 × 4 Hamiltonian
Let us consider a nanoheterosystem GaSb/AlSb with a
spherical QD of radius a which contains an acceptor im-
purity at the center. We suppose that the valence band
in this heterostructure is four-time degenerate at the
point k = 0, because it is formed by crystals with a wide
energy gap and the strong spin-orbit interaction, so that
the conduction and spin-split bands can be neglected.
Therefore, certain difficulties arise at the solution of the
Schrödinger equation. However, the problem can be sim-
plified, if one neglects the corrugation of isoenergetic sur-
faces in the k-space (the spherical approximation).
Let the energy axis in the valence band be directed
“downward”. Then, the spherical Hamiltonian – in the
effective mass approximation, without taking the con-
duction and spin-split bands into account, and assuming
that the hole can be assigned the spin j = 3/2 – looks
like
H =
1
2
(
γ1 +
5
2
γ
)
p2 − γ ( pJ)2 + Π (r) , (1)
where
Π (r) = V (r) + U (r) (2)
is the potential energy; γ = 1/5 (3γ3 + 2γ2); γ1, γ2, and
γ3 are the Luttinger parameters which are designated as
follows for different heterostructure regions:
(γ1 γ2 γ3) =
{ (
γin
1 γin
2 γin
3
)
, r ≤ a,
(γout
1 γout
2 γout
3 ) , r > a,
and J = iJx + jJy + kJz is the operator of spin moment
3/2. On the basis of the Poisson equation solution, the
energy of interaction between an acceptor ion and a hole
is given by the formula
V (r) =
− 1
εinr
− εin − εout
εinεouta
, r ≤ a,
− 1
εoutr
, r > a,
(3)
and the potential energy associated with a band disconti-
nuity (the confinement potential) is selected in the form
of a spherical rectangular potential well:
U (r) =
{
0, r ≤ a,
U0, r > a.
(4)
In all formulas, we use the system of units with m0 = 1,
~ = 1, and e = 1. The spherically symmetric Hamilto-
nian (1) commutes with the operator of total angular
momentum F = L + J, in which the orbital moment L
characterizes the “macroscopic” orbital motion which is
described by the effective mass method.
The wave function that describes a hole in the presence
of an acceptor can be expressed as a four-component
column function, and the even and odd states can be
separated [14, 18]:{
ψI = RIa,2 (r) Φ(4)
f−3/2 (θ, ϕ) +RIa,1 (r) Φ(4)
f+1/2 (θ, ϕ) ,
ψII = RIIa,2 (r) Φ(4)
f−1/2 (θ, ϕ) +RIIa,1 (r) Φ(4)
f+3/2 (θ, ϕ) .
(5)
The functions Φ(4) are four-component spinors that cor-
respond to the spin j = 3/2. In the spinors, the quantum
numbers f and l are connected as follows:
Φ(4)
f−3/2 (θ, ϕ) , f = l + 3
2 ; Φ(4)
f+1/2 (θ, ϕ) , f = l − 1
2 ;
Φ(4)
f−1/2 (θ, ϕ) , f = l + 1
2 ; Φ(4)
f+3/2 (θ, ϕ) , f = l − 3
2 .
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 327
V.I. BOICHUK, I.V. BILYNSKYI, R.YA. LESHKO et al.
Depending on the f - and l-values, the functions ψI and
ψII are even or odd. Substituting functions (5) into the
Schrödinger equation with Hamiltonian (1), multiplying
by the corresponding conjugate spinors, and integrat-
ing over the angular variables, we obtain two systems
of coupled differential equations of the second order for
the even and odd states. These two systems are written
down in the form of a matrix equation
γ1
2
(
− (1 + C1)Δl C2A−l+1A
−
l+2
C2A+
l+1A
+
l − (1 + C3)Δl+2
)
×
×
(
Ria,2
Ria,1
)
+ Π (r)
(
Ria,2
Ria,1
)
= Ea
(
Ria,2
Ria,1
)
,
(6)
where i = I for even states and i = II for odd ones.
The coefficients in Eq. (6) are written down, by using
6-j symbols
C1 = C1 (f, l) = µ
√
5 ( −1)3/2+l+f ×
×
{
l l 2
3/2 3/2 f
}√
2l (2l + 1) (2l + 2)
(2l + 3) (2l − 1)
,
C2 = C2 (f, l) = µ
√
30 ( −1)3/2+l+f ×
×
{
l + 2 l 2
3/2 3/2 f
}√
(l + 1) (l + 2)
2l + 3
,
C3 = −C1, (C1)
2 + (C2)
2 = µ2, C2/µ > 0, µ =
2γ
γ1
.
The operators in Eq. (6) are defined by the relations
A+
l = − ∂
∂r
+
l
r
, A−l =
∂
∂r
+
l + 1
r
,
Δl =
∂2
∂r2
+
2
r
∂
∂r
− l (l + 1)
r2
. (7)
Equation (6) with potential (2) cannot be solved ex-
actly. Therefore, the approximate methods of quantum
mechanics are used.
2.2. Discrete spectrum of a hole in a QD
without impurity
Let us find the solution of the Schrödinger equation for
a QD charge without impurity. In this case,
Π (r) = U (r) , Ea = Ei,
(
Ria,2 (r)
Ria,1 (r)
)
=
(
Ri2 (r)
Ri1 (r)
)
.
Then, Eq. (6) allows exact solutions which can be ex-
pressed in terms of of Bessel functions of the first kind
and modified Bessel functions of the second kind. The
solutions can be found for both coordinate regions:
1) at r ≤ a,
Ri,in2 (r) = B1
C in
1 − µin√
(µin)2 −
(
C in
1
)2
Jl+ 1
2
(
kir√
1−(µin)2
)
√
r
+
+B2
C in
1 + µin√
(µin)2 −
(
C in
1
)2
Jl+ 1
2
(
kir√
1+(µin)2
)
√
r
, (8)
Ri,in1 (r) = B1
Jl+ 5
2
(
kir√
1−(µin)2
)
√
r
+
+B2
Jl+ 5
2
(
kir√
1+(µin)2
)
√
r
, (9)
where ki =
√
2
γin
1
Ei;
2) at r > a,
Ri,out
2 (r) = D1
−Cout
1 + µout√
(µout)2 − (Cout
1 )2
Kl+ 1
2
(
λir√
1−(µout)2
)
√
r
+
+D2
−Cout
1 − µout√
(µout)2 − (Cout
1 )2
Kl+ 1
2
(
λir√
1+(µout)2
)
√
r
, (10)
Ri,out
1 (r) = D1
Kl+ 5
2
(
λir√
1−(µout)2
)
√
r
+
+D2
Kl+ 5
2
(
λir√
1+(µout)2
)
√
r
, (11)
where λi =
√
2
γout
1
(U0 − Ei).
Knowing the exact solutions, it is possible to use the
boundary conditions to determine the hole energy and
the average distance
〈r〉 =
∫
drr2
(∣∣Ri2 (r)
∣∣2 +
∣∣Ri1 (r)
∣∣2) r. (12)
328 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
STUDY OF AN ACCEPTOR IMPURITY LOCATED
2.3. Boundary conditions
In order to obtain the boundary conditions for a spheri-
cal QD, two conditions were used: the continuity of the
radial wave function and the normal component of the
probability density flux vector through the QD spheri-
cal surface. The continuity of the wave functions at the
heterostructure interface gives two equations which are
written down in the form of a column matrix:(
Ri,in2 (r)
Ri,in1 (r)
)∣∣∣∣
r=a
=
(
Ri,out
2 (r)
Ri,out
1 (r)
)∣∣∣∣
r=a
. (13)
To determine the normal component of the probability
density flux vector, we calculated the normal compo-
nent of the velocity operator vector Vr = r/r (∂H/∂p)
which is proportional to the normal component of the
probability density flux vector. Then, the operator is
written down in the spinor representation for even and
odd states. Using the explicit form of operators, two
more conditions used for matching the wave function are
obtained:
a) for even states,(
TI,in
11 TI,in
12
TI,in
21 TI,in
22
)(
RI,in2 (r)
RI,in1 (r)
)∣∣∣∣∣
r=a
=
=
(
TI,out
11 TI,out
12
TI,out
21 TI,out
22
)(
RI,out
2 (r)
RI,out
1 (r)
)∣∣∣∣∣
r=a
;
(14)
b) for odd ones,(
TII,in
11 TII,in
12
TII,in
21 TII,in
22
)(
RII,in2 (r)
RII,in1 (r)
)∣∣∣∣∣
r=a
=
=
(
TII,out
11 TII,out
12
TII,out
21 TII,out
22
)(
RII,out
2 (r)
RII,out
1 (r)
)∣∣∣∣∣
r=a
,
(15)
where the following notations for the operators were in-
troduced:
TI
11 =
bI1
r
+ bI2
∂
∂r
− 3gf,p4
χI1
r
;
TI
12 = −bI3Q
(−1)
f+1/2; TI
21 = bI3Q
(1)
f−3/2;
TI
22 = −b
I
1
r
+ bI4
∂
∂r
+ gf,p−8
χI2
r
;
TII
11 =
bII1
r
+ bII2
∂
∂r
− 3gf,p4
χII1
r
;
TII
12 = −bII3 Q(−1)
f+1/2; TII
21 = bII3 Q(1)
f−3/2;
TII
22 = −b
II
1
r
+ bII4
∂
∂r
+ gf,p−8
χII2
r
;
Q(m)
l = −m ∂
∂r
+
l + 1/2 (1−m)
r
;
and the coefficients look like
bI1 =
3 (2f − 3)
4f
γ, bI2 =
2f − 3
2f
γ + γ1,
bI3 =
γ
2
√
3
f
√
4f (f + 1)− 3, bI4 = −2f − 3
2f
γ + γ1,
χI1 =
(
f − 3
2
)
(1− 5γ + γ1) ,
χI2 =
1
6
(2f + 9) ( −1 + 5γ − γ1) ,
bII1 =
3 (2f + 5)
4 (f + 1)
γ, bII2 = − 2f + 5
2 (f + 1)
γ + γ1,
bII3 = γ
√
3
√
1− 3
2f + 2
√
1 +
1
2f + 2
,
bII4 =
2f + 5
2 (f + 1)
γ + γ1,
χII1 =
(
f +
5
2
)
( −1 + 5γ − γ1) ,
χI2 =
1
6
(2f − 7) (1− 5γ + γ1) ,
gf,pn = p (f + 1/2− p n/2) ,
and p = ±1 for even and odd states, respectively.
Hence, we determine the conditions the radial wave
functions for even and odd states must satisfy. Using
the boundary conditions (13), (14), or (15), we obtain
a linear homogeneous system of four equations for the
coefficients B1, B2, D1, and D2. By zeroing the deter-
minant of this system, we obtain an equation for finding
the hole energy.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 329
V.I. BOICHUK, I.V. BILYNSKYI, R.YA. LESHKO et al.
Fig. 1. Hole energies calculated in the finite potential model (solid
curves): (1 ) 1S3/2, (2 ) 1P3/2, and (3 ) 1P5/2; (4 ) hole energy in
the ground state 1S3/2 calculated in the infinite potential model
(dotted curve)
2.4. Energy spectrum of an acceptor impurity
As was already marked above, Eq. (6) with the poten-
tial energy (2) cannot be solved exactly. Therefore, we
present the wave function of an acceptor in the form of
an expansion in the functions obtained for the problem
without impurity:(
Ria,2 (r)
Ria,1 (r)
)
=
∑
n
an
(
Ri2,n (r)
Ri1,n (r)
)
, (16)
where n is the specific hole state. The expansion was
carried out in terms of states with identical parity at
the fixed quantum numbers f and l. When substitut-
ing expression (16) into the Schrödinger equation (6),
it is necessary to take into account that
(
Ri2,n
Ri1,n
)
are
the characteristic functions of the radial impurity-free
Hamiltonian, and Ein is its eigenvalue. Multiplying the
result by the conjugate row vector
(
Ri∗2,m Ri∗1,m
)
and
integrating over the coordinate r, we obtain a homoge-
neous system of equations for the coefficients an,∑
n
((
Ein − Ea
)
δmn + Vmn
)
an = 0, (17)
where Vmn =
∫
drr2V (r)
[(
Ri2,m
)∗
Ri2,n +
(
Ri1,m
)∗
Ri1,n
]
.
By equating the determinant of the system to zero, we
obtain an equation, from which the acceptor energy is
determined.
2.5. Probabilities of interlevel transitions
In the dipole approximation, the probability of optical
transitions is proportional to the squared matrix element
of the dipole moment of interlevel transitions. The cal-
culation of the angular part of integrals and the analysis
of the radial part of the matrix element enabled us to es-
tablish that the following transitions between states are
possible:
1) even–even, if f ′ − f = ±1 and M ′f −Mf = 0;
2) odd–odd, if f ′ − f = ±1 and M ′f −Mf = 0;
3) even–odd, if f ′ − f = 0 and M ′f −Mf = 0.
Here, Mf is the quantum number that corresponds to
the projection of the total angular momentum. Knowing
the states between which the transitions are possible,
one can immediately calculate the square of the matrix
element of the dipole moment of interlevel transitions:
|Dmn|2 =
∣∣∣∣∫ dr (ψm)+ rψn
∣∣∣∣2 . (18)
Using the formulas presented above, we calculated the
energy spectrum of an acceptor hole, the average dis-
tances, and the probabilities of interlevel transitions.
3. Analysis of the Results Obtained
All calculations were carried out for a GaSb/AlSb het-
erosystem, the crystal parameters of which were reported
in works [15, 24].
In Fig. 1, the dependences of the ground – 1S3/2
(f = 3/2, l = 0) – and excited – 1P3/2 (f = 3/2, l = 1)
and 1P5/2 (f = 5/2, l = 1) – state energies of the hole in
a spherical QD without impurity on the QD radius are
depicted. As one would expect, a reduction of the QD
radius enhances the spatial restriction of the hole; there-
fore, its energy grows. The figure demonstrates that the
ground state energy calculated in the framework of the
finite-potential model lies below its counterpart calcu-
lated for the infinite potential. However, for extremely
large radii, this energy difference vanishes, because the
influence of heterostructure interfaces becomes infinitesi-
mal. The figure also makes it evident that the calculated
energy levels are arranged in the following order: 1S3/2,
1P3/2, and 1P5/2.
If the complicated band structure is not taken into
account, only the heavy- or light-hole band is chosen
for calculations, and the effective mass for, respectively,
heavy, mh, or light, ml, holes is introduced – this was
done many times and in plenty of works; see, e.g., works
330 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
STUDY OF AN ACCEPTOR IMPURITY LOCATED
Fig. 2. Energies of the hole ground state calculated in the frame-
work of different models
[7, 25] – we obtain a simple equation −1
2
∇ 1
m{
h
l
}∇+ U (r)
Ψ = EΨ,
the solutions of which are spherical Bessel and spherical
Hankel functions of the first kind.
For the sake of comparison between the results of cal-
culations, we show the plots for the ground state energy
of a hole in Fig. 2 which were calculated by considering
the 4 × 4 Hamiltonian and the boundary conditions of
work [19] (curve 1 ), the 4 × 4 Hamiltonian and bound-
ary conditions (13) and (14) (curve 2 ), the heavy-hole
band (curve 3 ), and the light-hole band (curve 4 ). The
calculations were carried out in the case of a finite poten-
tial at the heterointerface. One can see that the energy
obtained in the one-band approximation for heavy holes
is lower than that calculated in the four-band model.
The energy of light holes is naturally higher than the
energy of heavy holes. In the case where the boundary
conditions of work [19] are used (they were obtained by
neglecting the non-diagonal terms in the radial Hamil-
tonian), the hole energies obtained for radii a ≥ 150 Å
become identical to those obtained in the case of the
exact boundary conditions (curves 1 and 2 coincide).
However, at small radii, the energies under the indi-
cated boundary conditions are different, because, when
the QD sizes become smaller, the heterostructure inter-
faces affect the energy spectrum more strongly. For small
radii, the hole energy calculated in work [19] tends to the
heavy-hole energy for a QD of the GaSb/AlSb heterosys-
tem. Hence, the boundary conditions of work [19] can be
Fig. 3. Ground state energies of an acceptor impurity hole for the
one-band (curves 1 and 2 ) and four-band (curve 3 ) models
used for the determination of the hole energy in a wide
enough region of variation of the QD radius, a ≥ 150 Å.
In this article, however, we use the boundary conditions
(13)–(15) for further calculations in the framework of the
four-band model.
We also studied the dependence of the acceptor en-
ergy on the QD radius. For the specific radius a and
quantum numbers f and l, expansion (16) contains the
same number of terms, as the number of hole states in
the impurity-free case with the same parameters a, f ,
and l for a finite potential at the heterointerface. The
same number of terms was taken for the infinite-potential
model.
If a complicated structure of the valence band is ne-
glected, and the heavy- and light-hole bands are con-
sidered separately, we obtain the Schrödinger equation,
the solutions of which for an impurity can be presented
in the form of Whittaker and Coulomb functions [11–
13]. In Fig. 3, we give the acceptor ground state energy
which was calculated for a finite potential at the het-
erointerfaces and taking the heavy-hole band (curve 1 ),
the light-hole band (curve 2 ), the 4 × 4 Hamiltonian,
and boundary conditions (13) and (14) (curve 3 ) into
consideration.
Similarly to what occurs for a hole without acceptor
impurity, taking the heavy-hole band into account only
underestimates the acceptor state energy, and the con-
sideration of the light-hole band overestimates it. The
energy of the acceptor impurity ground state calculated
within the four-band model approaches the correspond-
ing energy in a massive crystal (horizontal dotted line)
[15] for large QD radii. The consideration of the exact
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 331
V.I. BOICHUK, I.V. BILYNSKYI, R.YA. LESHKO et al.
Fig. 4. Energy spectra of an acceptor impurity hole for finite (solid
curves) and infinite (dotted curves) potentials at the heterointer-
face
Fig. 5. Dependences of the ground and excited acceptor hole state
energies on the dielectric permittivity of the matrix at the fixed
QD radius a = 100 Å
potential energy of interaction between the hole with the
impurity ion (formula (3)) gives rise to a nonmonotonous
dependence of the energy on the QD radius in all three
models. First, a reduction of QD dimensions is accom-
panied by a slight energy decrease, due to the growth
of the effective potential well, whereas a subsequent re-
duction leads to the energy growth, because the spatial
confinement of the hole prevails over the increase of the
effective potential well.
In the framework of the four-band model, besides the
ground hole state, we also calculated the excited states
Fig. 6. Dependences of the average acceptor hole distances on the
QD radius
for finite and infinite potentials at the medium interface.
The results of calculations are graphically demonstrated
in Fig. 4. The behavior of the excited states is similar
to that of the ground one. That is, at large QD radii,
the energies of states tend to the corresponding energies
in a massive GaSb crystal; a size reduction leads to an
insignificant smooth fall of the energy, due to an increase
of the effective potential well depth followed by a drastic
growth as a result of the spatial confinement prevalence
over the growth of the effective potential well depth.
We also calculated the dependences of the energies of
acceptor levels on the dielectric permittivity of the ma-
trix at the constant QD radius a = 100 Å and a finite
potential at the heterointerfaces (Fig. 5). The points
of intersection with the vertical straight line 1 corre-
spond to the dielectric permittivity of AlSb, and those
with the straight line 2 do to the case where the dielec-
tric permittivities of the QD and the matrix are iden-
tical. From the figure, one can see that a decrease of
εout leads to a reduction of the acceptor impurity en-
ergy; it is associated with the growth of the parameter(
εin − εout
)
/(εinεouta), which governs the magnitude of
the effective potential well.
To analyze the spatial distribution of holes, we calcu-
lated the average distances, as functions of the QD ra-
dius (Fig. 6). The figure demonstrates that, at large QD
radii, the average distances of the acceptor hole tends to
the corresponding values for a massive GaSb crystal. A
decrease of the radius localizes the electron in the QD, so
that the average distances also decrease. However, for
the three states under consideration and the QD radii
a > 20 Å, i.e. in the range where the effective mass ap-
332 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
STUDY OF AN ACCEPTOR IMPURITY LOCATED
proximation is valid, the average distances are less that
the radius, which means that the hole is located in the
QD with a higher probability.
When studying the physical processes which are re-
sponsible for the light absorption and emission, it is im-
portant to know the square of the matrix element of the
dipole moment of hole interlevel transitions. We calcu-
lated this quantity for the transitions between the ac-
ceptor hole states 1S3/2, 1P3/2, and 1P5/2. In Table, the
results of corresponding calculations are quoted. One
can see that the QD radius growth increases the value of
|Dmn|2 for all calculated states. In addition, the transi-
tions 1S3/2 ↔ 1P5/2 turn out by an order of magnitude
more probable than the transitions 1S3/2 ↔ 1P3/2. The
level 1P3/2 is characterized by a large lifetime in com-
parison with other excited states. Proceeding from this
fact, we suppose that, at experimental measurements of
absorption and luminescence spectra, the Stokes shift
is to be observed, the magnitude of which should de-
pend on the QD radius and the dielectric permittiv-
ity of the matrix. For example, for a QD in the het-
erosystem GaSb/AlSb with an acceptor impurity, the
Stokes shift Δλst = 78.72 µm for the size a = 50 Å, and
Δλst = 144.83 µm for a = 100 Å. If the dielectric per-
mittivity of the matrix is supposed to be εout = 5, we
obtain that Δλst = 80.59 and 149.31 µm for the QD
size a = 50 and 100 Å, respectively.
4. Conclusions
In this work, on the basis of the four-band model, a sys-
tem of coupled differential equations of the second order
was derived for a spherical heterostructure GaSb/AlSb
and used to calculate the hole energy. A comparison of
the results obtained with those calculated for the one-
band model was carried out. The consideration of a
complicated valence band spectrum gave rise to a lower
hole energy than that of the light-hole band only, and
Squares of matrix elements of the dipole moment of in-
terlevel hole transitions |Dmn|2 (in Å 2 units) for various
QD radii
1S3/2 ↔ 1P3/2 1S3/2 ↔ 1P5/2
a Å (M ′
f = −3/2, (M ′
f = −1/2, (M ′
f = −3/2, (M ′
f = −1/2,
Mf = −3/2) Mf = −1/2) Mf = −3/2) Mf = −1/2)
and and and and
(M ′
f = 3/2, (M ′
f = 1/2, (M ′
f = 3/2, (M ′
f = 1/2,
Mf = 3/2) Mf = 1/2) Mf = 3/2) Mf = 1/2)
50 12.6704 1.4078 115.6610 173.4910
100 14.4008 1.6890 305.0300 457.5450
150 23.2407 2.5823 502.6240 753.9360
to a higher hole energy than the consideration of the
heavy-hole band only. In view of the exact hole solu-
tions in a spherical QD and the exact solution of the
Poisson equation, the potential energy of interaction be-
tween the hole and the impurity center is written down,
and the approximate wave function of acceptor is con-
structed. The behavior of the average acceptor hole dis-
tances at the variation of the QD radius is investigated,
which enabled us to reveal that the hole is located in
the QD with a higher probability at all QD radii, for
which the effective mass approximation is valid. The
selection rules are found for the intraband level-to-level
optical hole transitions. The square of the matrix ele-
ment of the dipole moment of interlevel hole transitions
was calculated, which made it possible to establish that
the transitions between states 1S3/2 and 1P3/2 are less
probable than those between states 1S3/2 and 1P5/2.
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Translated from Ukrainian by O.I. Voitenko
ДОСЛIДЖЕННЯ АКЦЕПТОРНОЇ ДОМIШКИ У ЦЕНТРI
СФЕРИЧНОЇ НАНОГЕТЕРОСТРУКТУРИ
В.I. Бойчук, I.В. Бiлинський, Р.Я. Лешко, I.О. Шаклеiна
Р е з ю м е
Для сферичної наногетероструктури GaSb/AlSb, використову-
ючи сферичний гамiльтонiан 4×4, визначено дискретнi ста-
ни водневоподiбної акцепторної домiшки для рiзних розмiрiв
квантової точки. Проведено порiвняння визначених енергiй з
вiдповiдними енергiями, що одержанi без урахування складної
структури валентної зони. Обчислення проведено як для скiн-
ченного, так i для нескiнченного потенцiалу на межi гетеро-
структури. Встановлено правила добору для внутрiшньозон-
них мiжрiвневих оптичних переходiв дiрки. Визначено сере-
днi вiдстанi та ймовiрностi переходiв дiрки як функцiї розмiрiв
квантової точки.
334 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
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