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Journal of
Computational and Theoretical Nanoscience
Vol. 8, 1–9, 2011
The Analytical Study of Electronic and
Optical Properties of Pyramid-Like and
Cone-Like Quantum Dots
V. Lozovski and V. Piatnytsia∗
Department of Mathematics, Theoretical Physics and Computer Sciences, Institute of High Technologies,
National Taras Shevchenko University of Kyiv, Kyiv, 03022, Ukraine
The analytical expressions for electron wave function of pyramid-like and cone-like quantum dot
located on the semiconductor surface are obtained. The energy spectrum of electron states in the
quantum dots is calculated. The dependence of energy levels on the angle at the vertex of an
edge-shaped nanodots is studied. The distribution of electron density for different electronic states
is calculated for each case. The dependence of function of linear response to the external field on
vertex angle is calculated in the frame of proposed approach. The local field distributions for both
pyramid- and cone-like particles are calculated.
Keywords: Analytical Study, Quantum Dot, Pyramid Shape, Cone Shape, Energy Levels,
Probability Density Distribution, Effective Susceptibility, Lamb Shift, Local-Field.
Due to heavy growth of nano-technologies the study of
effects of spatial quantization uses in the practice now.
There are numerous papers devoted to studies of quantumdimensional effects (see, for example, the monograph and
reviews).1–5 Modeling of electron properties of quantumdimension structures customary as a rule is performed for
the particles of simple geometric shape.6–9 The technologies obtaining quantum dots (especially, quantum dots situated at the surface), however, allow to obtain the quantum
dots of different shapes.10–12 For example, QD obtained
by MBE usually are pyramid-like.13 Nowadays the technique of obtaining the cone-shaped particles at the surface
of Si by scanning of the surface with laser beam of high
intensity were reported in Refs. [14–15]. As a result, the
different types of cone-shaped particles were obtained—as
needle-like (with small vertex angle), as nano-hills (with
large vertex angle).16 Modeling of electronic properties of
edge-like quantum dots is a nontrivial problem. One can
point out only several papers in which the electron states
of cone-like quantum dots were calculated.17–19 It should
be noted that direct numerical calculations were used in
these works, except, may be Ref. [19] where the analytical
solution of Schrödinger equation for cone-shape restricted
∗
Author to whom correspondence should be addressed.
J. Comput. Theor. Nanosci. 2011, Vol. 8, No. 11
potential as the infinite series of the complicated functions, which expressed via Bessel functions was found. In
contrast pyramid-shaped QDs widely investigated problem
but only by the numerical methods. Approach based on
numerical solutions of Schrödinger equation sometimes is
not convenient for studying some electronic properties of
quantum dots. It is clear that for practical use one needs
to have simple analytical expressions for wave functions
of electronic states.
2. PYRAMID-LIKE QD. MODEL AND MAIN
EQUATIONS
Let one consider pyramid-like small particle (Fig. 1). Let
one supposes that confinement potential has very high
wells. This assumption will be correct for quantum dots
embedded in wide band gap matrix or for quantum dot
situated in the vacuum at the surface of solid. It is clear
that in the case of high barrier the solution is not strongly
differ from solution for infinite well. Especially when one
needs to know the ground state energy level and few lowest excited levels. Strongly speaking the electron energy
in quantum dot consists of several terms. Namely, U r =
Uconf r + Ustrain r + UC r, where Uconf r is confinement
potential, Ustrain r is strain-induced potential, UC r is
Coulomb interactions between electrons. In the present
1546-1955/2011/8/001/009
doi:10.1166/jctn.2011.1965
1
RESEARCH ARTICLE
1. INTRODUCTION
The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots
y
x
φ
h
z
P
RESEARCH ARTICLE
Fig. 1.
l
Geometry of pyramid-like QD problem.
work one considers the particles with characteristic dimensions about 10 nm. Because of doped semiconductor characterized by electron concentration about 1016 –1018 sm−3 ,
one can suppose that there is not more than one electron
per the particle. This means, that electron–electron interaction can be neglected in the present consideration. Then,
one should to consider electron states for a single electron
inside the particle. One can point to rather large number of
the works in which the studies of influences of mechanical
stresses on the electronic properties of quantum dots were
considered.21–25 It was established that the stresses can distort the structure of energy levels of electron in quantum
dot. These distortions are depended on the value and distribution of mechanical stresses, say on shape and dimension
of a particle. One can think that developing of regular analytical approach to calculate the electron energy structure
of cone-shaped quantum dot with taking into account the
stress potential is impossible at this time. Then, one should
to restrict oneself by consideration of a one side of the
problem, however the strain induced distortions of electron
potential energy usually can be accounted in the frame of
perturbation theory. As a result, in this work one did not
take into account the effects of mechanical stresses. On
the other hand, one will be needed to take into account
the strain-induced effects when the concrete experiment
would like to describe. Then, one consider electron states
inside potential box having pyramid-like shape and characterize by infinitely high potential barrier. This model, for
example, can be used for semiconductor pyramid-like and
cone-like quantum dots which is located at the dielectric
flat surface. The restrictive surface of the quantum dot in
this case is specified by expressions
⎧
z = ax
x > 0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪z = −ax x < 0
⎪
⎪
⎪
⎨
(1)
z = ay
y > 0
⎪
⎪
⎪
⎪
⎪
⎪
z = −ay y < 0
⎪
⎪
⎪
⎪
⎪
⎩
z ∈ 0 h
−1
where a = tan /2, with vertex angle of pyramid. The solution of Schrödinger equation inside the
2
Lozovski and Piatnytsia
pyramid-shaped domain [Eq. (1)] even for infinitely high
potential wells is rather hard problem due to complicated
boundary conditions. From the other hand, one can simplify the problem, to solve Schrödinger equation in the
domain restricted by right parallelepiped.26 In this case the
domain formed by coordinate lines of coordinate system
and boundary conditions has a simple form. Thus, one can
propose the approach which could transform the pyramidlike domain to the parallelepipedal one. To realize this
idea one proposes the new variables u v w according to
(Fig. 2)
⎧
⎧
⎪
⎪
u
=
x/z
⎪
⎪
⎪
⎪x = uw
⎪
⎪
⎨
⎨
(2)
v = y/z ⇒
y = vw
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩w = z
⎩z = w
In the new variables u v w the domain P has a form of
right parallelepiped of rib length 2/a and altitude h and
describes with the next expressions
⎧
⎪
u ∈ −1/a 1/a
⎪
⎪
⎪
⎨
(3)
v ∈ −1/a 1/a
⎪
⎪
⎪
⎪
⎩w ∈ 0 h
Then, Laplace operator in these variables can be written
in the form26
1 2
u
uvw = 2
+
w u2 u2 + v2 + 1 u
1 2
u
+ 2
+
w v2 u2 + v2 + 1 v
1
2 + 2 2
w
(4)
w
w u + v2 + 1 w
As a result, Schrödinger equation inside the domain P will
be written as
2
−
= E
(5)
2m uvw
One can easily see that Eq. (5) has a form of Helmholtz
equation
uvw + k2 = 0
(6)
w
z
P
P'
Transformation
y
x
u
v
Fig. 2. Visualisation of mathematical transformation.
J. Comput. Theor. Nanosci. 8, 1–9, 2011
Lozovski and Piatnytsia
The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots
√
with k = 2mE/. Let we suppose that the quantum dot
has a pyramid shape with small vertex angle < 90 . This
approximation means, that u v 1. Then, Eq. (6) can
be reduced to equation with separate variables. Then, one
can write the equations for each of components of wave
function separately
⎧ 2
⎪
⎪
+ ku2 u = 0
+u
⎪
⎪
⎪
u2
u u
⎪
⎪
⎪
⎪
⎨ 2
(7)
+v
+ kv2 v = 0
⎪
v2
v v
⎪
⎪
⎪
⎪
⎪
⎪
1 k2 + k2
⎪
⎪
w 2 w + k2 − u 2 v w = 0
⎩ 2
w w
w
w
Let one consider the first equation of the system above.
The solution in this direction is not trivial. Albeit one can
use the simplification. Let one suppose that 2 u /
u2 u
u /
u. This supposition has a sense because u 1.
It means that the first derivative of wave function on the
variable u should to be not so large. This assumption one
can verify when the solution of equation will be obtained.
Using this approximation one can neglect the second term
in the first line of Eq. (7) and obtain
2 + ku2 u = 0
(8)
u2 u
u = cCosku u + dSinku u
(9)
To find the solution (eigenstates) of the first equation of
Eq. (7) in domain u ∈ −1/a 1/a, one have to fulfill
boundary conditions
⎧
⎨u −a−1 = 0
(10)
⎩ −1 u a
= 0
C
w w = √ J√k2 +k2 − 3 kw
un
vm 4
w
As a result, the wave function of pyramid-like quantum
dot in Cartesian coordinates looks like
C
x y z = √P J knmj z
z
x
−1
× sin kun a cos kun
z
x
+ cos kun a−1 sin kun
z
y
× sin kvm a−1 cos kvm
z
y
+ cos kvm a−1 sin kvm
z
Now one can verificate the supposition discussed above.
Let one set solution Eq. (10) under first and second derivative orders and compare. One obtains the condition ku > u
which is satisfied because of ku ≈ a, u ≈ a−1 and a 1.
It is clear that solution for v direction is similar to function of u direction
v v = C2 Sin kvm a−1 Cos kvm v
+ Cos kvm a−1 Sin kvm v kvm = ma/2 (12)
J. Comput. Theor. Nanosci. 8, 1–9, 2011
(14)
with constant CP being founded from wave function
x y z dx dy dz = 1, where
normalization
Vpyramid
integration
is
over
volume
of the pyramid, and =
2 + k2 − 3/4. To find the eigenvalues and eigenfunckun
vm
tions
it is necessary to fulfill the boundary condition
P = 0, for two lateral directions which were found
above. For third direction one should to solve the system
of equations
⎧
kun = na/2
n = 1 2 ⎪
⎪
⎪
⎪
⎨
m = 1 2 kvm = ma/2
(15)
⎪
⎪
⎪
⎪
⎩J√
knmj h = 0
k2 +k2 − 3
un
After substitution of explicit form of wave function
[Eq. (9)] in Eq. (10) one obtains the equation for eigenstates finding Sinknu 2a−1 = 0. Thus the u direction wave
function can be written as
u u = C1 Sin kun a−1 Cos kun u
+ Cos kun a−1 Sin kun u kun = na/2 (11)
(13)
vm
4
The conditions of solution existence give us the energy
spectrum of electron in the pyramid-like quantum dot
2
2 knmj
Enmj =
(16)
2meff
where knmj is the wave vector satisfying system of
Eq. (15). It is interesting to examine the dependence of
electron energy structure on the shape of QD, particularly
the alternation of the spectrum when shape of the particle
changes from the sharp to the blunt form. And it is easy to
investigate this idea in our analytical solution. The calculations are performed for GaAs QDs located in the dielectric or wide-band semiconductor. Then, the dependence of
electron eigenenergies in the pyramid-like quantum dot on
the vertex angle is shown in Figure 3 for two different
cases. Panel “a” corresponds to the case when the volume
3
RESEARCH ARTICLE
The solution of Eq. (8) is well known
As one can see the structure of third equation of the
system [Eq. (7)] is similar to standard Bessel function
differential equation. After usual polynomial substitution
w = w s w s = −1/2 it reduces to the ordinary cylindrical equation for the function w. It is well known that
the solution the equation has a simple form26
The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots
222
(a)
1.5
Energy, eV
113
231
1.2
131
112
121
111
0.3
35
40
45
50
55
60
65
Vertex angle, φ0
222
(b)
1.8
113
231
Energy, eV
1.5
131
1.2
221
112
0.9
121
0.6
111
0.3
35
40
45
50
55
60
65
Vertex angle, φ0
RESEARCH ARTICLE
the wave functions obtained earlier with numerical calculations (see, for example Ref. [29]). This can serve as
evidence of truth of proposed analytical approach, which
one will use for calculation of electron energy structure of
pyramidal-shaped quantum dot.
221
0.9
0.6
Fig. 3. Dependence of electron spectrum the single pyramid-like GaAs
quantum dot on the vertex angle. Panel “a” corresponds to the case when
the volume of quantum dot is the constant (r0 = 5 nm, a0 = 50 ). The
panel “b” corresponds to the case when the radius of the base of quantum
dot is the constant. The set of quantum numbers nmj > are given at the
curves.
of quantum dot is constant r0 = 5 nm, 0 = 50 ) with
alternation of vertex angel. Panel “b” corresponds to the
case when the base edge of the pyramid is constant and
vertex angel changes. As one can see, the energy structure
strongly depends on the vertex angle. One should note that
there are some values of the angle when the energy levels
degenerate. Obtained structure of spectrum with analytical method is similar to calculated with numerical method
presented in Refs. [13, 27] earlier.
The distribution of probability density ∗ = f x y z
of electron location inside the particle is shown in Figure 4
for quantum dot with vertex angle = 60 for different
states to which are manifested at the bottom of each of the
figure. One should note that in three-dimension probability density distribution are belt with the surface of ∗ =
const, with the constant chosen that inside the bodies the
probability to find the electron is more than 0.9 (Fig. 4
panel “a”). The probability densities depicted as a function ∗ = f x 0 z in cross-section by plane XOZ are
shown in the panel “b.” One should point once more that
spatial structure of obtained wave functions are similar to
4
Lozovski and Piatnytsia
3. CONE-LIKE QD. MODEL AND MAIN
EQUATIONS
Cone-shaped quantum dot energy structure calculation is
the aim of this section. To solve this problem one will
use the approach of analytical calculation developed for
pyramid-like QD. Then, one can describe the cone-like QD
domain as
⎧
⎨z = a x 2 + y 2
(17)
⎩
z ∈ 0 h
with a = tan−1 /2, and the vertex angle of the cone.
It is clear that coordinate transformations should be the
similar to the transformations used in the previous consideration. But in contrast to the case of pyramid-shaped
QD, the present problem has an axial symmetry, that is
why the transformation domain is a cylinder. Then in the
new variables the domain C’ has a form of right cylinder
of radii and altitude h and is described with the next
expressions
⎧
u = cos
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨v = sin
(18)
√
⎪
⎪
⎪ = u2 + v2
⎪
⎪
⎪
⎪
⎩
w ∈ 0 h
where = 0 1/a. Then, Laplace operator in variables
given by [Eq. (18)] can be written in the form
2
1 2
1
1
w = 2
+
+
+
w 2
2 + 1 2 2
1
+ w2
(19)
w
w
2
2
w +1
To solve Schrödinger equation in this case one should
to simplify the problem. Namely, to obtain the analytical solution of the problem one needs to suppose that the
quantum dot has a cone shape with vertex angle < 90 .
In this case parameter will change in the range of 0 ≤
< 1. This fact allows one to neglect certain of the terms
in wave equation, namely, the term proportional to 2 can
be neglected in comparison with 1 and term proportional
to can be neglected as compared with 1/. Then, one can
write 1/2 + 1 ≈ 1 and 1/ + /2 + 1 ≈ 1/. As a
result, the wave equation can be reduced to equation with
J. Comput. Theor. Nanosci. 8, 1–9, 2011
Lozovski and Piatnytsia
The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots
|111>
(a)
|112>
y
x
φ
z
h
C
|211>
r
|311>
Fig. 5. Geometry of cone-like QD problem.
C1 eim . The solutions of second and third equations of
system [Eq. (21)] can be presented via√Bessel
function,
namely,
=
C
J
k
and
=
C
/
w
J
kw,
with
2 m w
3
|112>
|111>
0
0,1719
0,3438
0,5156
0,6875
0,8594
1,031
1,100
z , nm
8
6
4
2
–4
–2
0
2
8
z , nm
(b)
6
4
2
–4
4
–2
2
4
|311>
|211>
8
6
6
z , nm
8
4
4
2
2
–4
–2
0
2
4
–4
x , nm
–2
0
2
4
x , nm
Fig. 4. The probability density of electron in quantum dot (a) for different eigenstates which are shown in the brackets. In the panel (b) the
cross-sections by plane XOZ of the probability density of electron are
shown.
with constant CC being founded from wave function normalization. To find the wave functions satisfying the
boundary condition C = 0, one should to solve the system of equations
⎧
⎪
m = 0 ±1 ±2 ⎪
⎪
⎨ kmi Jm
= 0
(24)
a
⎪
⎪
⎪
⎩J√ 2 3 k h = 0
mij
k −
i
separate variables. It means that one can write the equations for the each component and w w of
wave function
w = w w
separately
⎧ 2
⎪
⎪
+ m 2 = 0
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎨ 2
1 m2
2
= 0
+
−
+
k
⎪
2
2
⎪
⎪
⎪
⎪
⎪
⎪
k2
1 w
⎪
⎪
2
2
⎩ 2
w
+ k − 2 w = 0
w w
w
w
(20)
(21)
It is clear that the solution of the first equation of
the system [Eq. (21)] has an exponential form =
J. Comput. Theor. Nanosci. 8, 1–9, 2011
4
One can easily find numerically the energy spectrum from
the system [Eq. (24)] it expresses via zeros of different
orders Bessel functions. The dependence of energy levels of electron in the cone-like quantum dot on the vertex angle is shown in Figure 6. The spectrum of electron
energy inside the cone-like quantum dot in general is similar to pyramid-shaped (extremely degenerated along the
shape alternation).
The distribution of probability density ∗ = f x y z
of electron location inside the cone-like particle is shown
in Figure 7 for quantum dot with vertex angle = 60 . It
can be easily seen that the probability density has cylindrical symmetry. One should note that in three-dimension
pictures of probability density distribution are belt with
the surface of ∗ = const, with the constant chosen that
inside the bodies the probability to find the electron is
more than 90%.
5
RESEARCH ARTICLE
z , nm
0
x , nm
x , nm
= k2 − 3/4. Thus, the solution of wave equation for
cone-like quantum dot has a form26
C
w = √ J kwJm k eim = k2 − 3/4
w
(22)
This solution in Cartesian coordinate system can be rewritten as
2
CC
x + y 2 im
x y z = √ J kzJm k
e z
z
y
= arctan
(23)
x
The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots
4. EFFECTIVE SUSCEPTIBILITY OF A
SINGLE PYRAMID-LIKE AND CONE-LIKE
QUANTUM DOT
The optical properties of the system with quantum dot
can be easily studied in the frame of effective susceptibility concept.28 This concept assumes calculation of optical
response to the external field which can be performed with
taking into account self-acting field.29 In other words, one
shall take into account processes of action of the local
field caused by electron transition currents on the electron
inside the quantum dot. Because this local field is defined
by induced inside the particle currents of electron transitions, this local field is the field of self action. As a result,
there exist effects caused by local field action or local-field
effects. For example, the Lamb shift of electron energy levels and broadening of the energy levels are caused by local
field effect.29 General scheme of calculation of effective
susceptibility consists in finding self-consistent local field,
which acts at arbitrary point inside the quantum dot. This
(a) 1.5
Energy, eV
self-consistent field obeys Lippmann-Schwinger equation
Ei R = Ei0 R − i0 dR Gij R R ×
V
dR jk R R Ek R (25)
V
where Gij R R is electrodynamical Green function,
which describes the electrodynamical properties of the
medium in which the quantum dot is embedded. When one
supposes that the quantum dot is situated at the surface of
dielectric, one can use the Green function for two homogeneous isotropic semi-spaces with the flat interface.30 Supposing that the electron transitions between the ground
state and few excited states mainly contribute in the paramagnetic part of the response and taking into account that
(a)
013
112
|011>
|111>
|012>
|021>
021
1.2
RESEARCH ARTICLE
Lozovski and Piatnytsia
211
0.9
012
111
0.6
(b)
35
40
45
50
55
60
65
Vertex angle, φ0
z, nm
013
112
4
2
021
1.5
0
211
–4
–2
0
2
8
6
4
2
0
–4
4
–2
x , nm
1.2
012
011
0.3
35
40
45
50
55
60
65
Vertex angle, φ0
Fig. 6. Dependence of eigenenergies on the vertex angle of the single
cone-like Si quantum dot located at the dielectric surface. Panel “a” corresponds to the case when the volume of quantum dot is the constant.
The panel “b” corresponds to the case when the radius of the base of
quantum dot is the constant. The set of quantum numbers mij > are
given at the curves.
2
4
2
4
|021>
8
8
6
6
z, nm
0.6
0
x , nm
|012>
111
0.9
z, nm
Energy, eV
|111>
1,000
0,9000
0,7500
0,6000
0,4500
0,3000
0,1500
0
6
(b) 1.8
6
|011>
8
z, nm
011
0.3
4
2
4
2
0
–4
–2
0
x , nm
2
4
0
–4
–2
0
x , nm
Fig. 7. The probability density of electron in quantum dot (a) for different eigenstates which are shown in the brackets. In the panel (b) the
cross-sections by plane XOZ of the probability density of electron are
shown.
J. Comput. Theor. Nanosci. 8, 1–9, 2011
Lozovski and Piatnytsia
The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots
only few (∼1) electrons can be localized inside the particle, one can write the linear response to the local field
in the form.30 31 The diamagnetic term in linear response
is proportional to electron number32 inside the object and
can be neglected because of about one electron per the
quantum dot in the case under consideration.
ji0n Rjjn0 R i ij R R =
n + i + E0 − En
jin0 Rjj0n R (26)
−
+ i − E0 + En
where
e 0∗
(27)
i n − n i 0∗
2im
is the current when electron transmits from state 0 to
state n characterizing by energy levels E0 and En , respectively. Damping constant describes the life time of electron in the energy level. The linear response to the local
(total) field can be rewritten in the more convenient form
i a ji0n Rjjn0 R ij R R =
n n
(28)
2En − E0 an = 2
+ i2 + En − E0 2
ji0n = −
Then, knowledge of effective susceptibility allows us to
write the mesoscopical current inside the particle induced
by external field. It means that one obtains the local field
at arbitrary point of the system
Ei R = Ei0 R − i0
× dR Gij R R Xjk R Ek0 R V
(35)
Now we introduce the calculations performed for the conelike quantum dot because of transitions intensities for
lowest energy levels are comparable against pyramid-like
nanoparticle where probability of transitions extremely differs. The dependence of effective susceptibility on the
energy and vertex angle is given in Figure 8 panel ‘a’.
The calculations were performed for damping constant
(a)
with
n =
dRjin0 REi R Jin0 =
V
RESEARCH ARTICLE
Substitution of [Eq. (28)] into [Eq. (25)] gives us the integral equation with multiplicative kernel which can be easily solved. Following Refs. [29, 33], one obtains
−1
n = mn + an Nmn Jin0 Ei0
(29)
dRjin0 R (30)
V
and self-energy part
Nmn = 0 dR dR jim0 RGij R R jj0n R V
V
(31)
Because the dipole momentum induced at the particle can
be defined as
i dRji R
(32)
pi =
(b)
6
ImXzz arb. units
φ = 45º
5
φ = 55º
4
3
φ = 65º
V
and current density
ji R =
2
dR ij R R V
(33)
Ej R = Xij R Ej0 R Using obtained solution of Lippmann-Schwinger equation [Eq. (29)], one can write the effective susceptibility
of the quantum dot in the form.33 34
−1
1 Xij R =
an Ji0n mn + an Nmn Jj0m 2
n
(34)
J. Comput. Theor. Nanosci. 8, 1–9, 2011
1
0
0.6
0.8
1.0
1.2
1.4
1.6
hω, eV
Fig. 8. Panel (a) dependence of zz-component effective susceptibility
on energy and vertex angle. The transitions (013) → (011) and (021) →
(011) were taken into account. Panel (b) dependence of effective susceptibility zz-component on energy for cone particles with different
angles.
7
The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots
RESEARCH ARTICLE
= 005n0 . As it can be easily seen, the spectral dependence of Xzz is nontrivial because strong dependence of
energy levels on vertex angle (shape of quantum dot). Narrow < 52 and wide ≥ 60 cones are characterized
by two peaks curve of Xzz , but cones which vertex
angle lies in the range of 52 < < 60 is characterized
by single peak curve of Xzz . Cross sections of the
frequency dependence of imaginary part of Xzz at
different are shown in Figure 8 panel ‘b’. Then, the
change of vertex angle of cone quantum dot can carry
out drastic change of optical properties of the system with
quantum dot.
The knowledge of effective susceptibility allows us to
calculate the local field distributions within the quantum dots and outside of the particles for both pyramid
shaped and cone shaped quantum dots, which are shown in
Figures 9 and 10. The strange on the face of it behaviour
of local field as function of coordinates is observed in
Figures 9 and 10. Indeed, the field enhancement at the top
of the cone could it might be observed. But one can see,
that the maximum of local field intensity is inside the cone
not at the vertex. This fact is very similar to near-field
image of pyramid-like quantum dot,35 where the local field
intensity is not located at the vertex.
As it was mentioned above, the self-action processes
lead to change of energy levels of the quantum dot, socalled Lamb shifts. The Lamb shift for electron transi(a)
1,000
12
0,8333
9
z, nm
6
0,6667
3
0,5000
0
–3
–9
–6
–3
0
3
6
9
x, nm
(b)
1,000
12
0,8500
z, nm
9
6
0,7000
3
0,5500
0
–3
–9
–6
–3
0
3
6
9
x, nm
Fig. 9. Cross sections (y = 0, panel “a” and y = r, panel “b”) of localfield distribution in the pyramid-like quantum dot.
8
Lozovski and Piatnytsia
Δ /ωα
0.15
0.12
0.09
0.06
40
45
50
55
60
ϕº
Fig. 11. Dependence of lamb shift on the vertex angle.
tion between states 011 and 012 (energy of transition
equals to ) dependence on the vertex angle is shown in
Figure 11. One can see that relative values of Lamb shifts
can achievement up to 14%. One should note that unlike to
Refs. [33, 36] contribution of interaction between the particle and surface in the case of cone quantum dot is smaller
than the self-action via vacuum contribution. Other electron transitions are characterized by smaller Lamb shifts.
5. SUMMARY AND DISCUSSIONS
The semi-analytical method for calculation of electron
properties of edge-shaped quantum dots is proposed in the
work. The analytical form of wave functions of electron
in pyramid-like and cone-like quantum dots was obtained.
One should to note that the analytical method of solving Schrödinger equation proposed in this work was used
for calculation of energy structure of electron states in the
edge-like quantum dots. The dependence of the electron
energy levels on the vertex angle of the particle was analysed. The energy levels and its dependences on the vertex
angle have the similar form as reported in Refs. [13, 27]
calculated by direct numerical methods. This fact allows
us to hope that proposed here analytical method gives the
true results for cone-shaped quantum dot too. The distributions of electron densities within the particles are obtained.
The structure of wave functions is the same as the symmetry of coordinate systems (Cartesian for pyramid-like and
cylindrical for cone-like QDs).
The distributions of electronic transition currents which
define paramagnetic part of electric response of quantum
dots were calculated. The self-consistent approach to calculation of optical response of the quantum dots was used.
The effective susceptibility of cone quantum dot was calculated and its dependence on quantum dot shape (vertex angle) was firstly demonstrated. The self-action effects
in particle, lead to energy levels shifts—so called Lamb
shifts.29 It was established that relative Lamb shift in the
J. Comput. Theor. Nanosci. 8, 1–9, 2011
Lozovski and Piatnytsia
The Analytical Study of Electronic and Optical Properties of Pyramid-Like and Cone-Like Quantum Dots
case under consideration can achieve of values more than
10% for transition between ground and first excited state.
It was established that the main contribution to the shift
give self-action via vacuum. The self-action effects by
exchange of virtual photons with the substrate give smaller
contribution. On the base of obtained effective susceptibility, the local field distributions were calculated. The both
electronic and local-field properties are strongly dependent
on the shape of the particle (vertex angle, in the case under
consideration). Indeed, the currents of electron transitions
J nm R and the self-energy part Nnm are defined by the
shape of the quantum dot. Moreover, one should note that
the self-energy part is defined by particle shape via transition currents as integration over the volume of the particle
[Eq. (20)].
References
Received: 19 January 2011. Accepted: 26 February 2011.
J. Comput. Theor. Nanosci. 8, 1–9, 2011
9
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