Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 2, pp. 387-392, Warsaw 2013

AN ANALYTICAL SOLUTION FOR CONICAL QUANTUM DOTS

Aleksander Muc, Aleksander Banaś, Piotr Kędziora

Cracow University of Technology, Institute of Machine Design, Kraków, Poland

e-mail: olekmuc@mech.pk.edu.pl; olek.banas@gmail.com; kedziora@mech.pk.edu.pl

In the paper, an analytical method of the solution of a governing nonlinear eigenproblem
is proposed. It can be directly applied into analysis of axisymmetric conical quantum dots
embedded in amatrix. Themethod is based on the use of variational formulation combined
with the method of the Rayleigh quotient and the series expansions. In order to explain
the form of the series expansions (the Bessel and sine series), the analytical solutions for
quantum dots are demonstrated and discussed. The solved example shows the efficiency of
the method.

Keywords: conical quantumdots, nonlinear eigenproblem, variational formulation,Rayleigh
quotient

1. Introduction

Quantum effects begin to dominate as the size of semiconductor structures approaches the elec-
tron de Broglie wavelength. In low-dimensional semiconductor nanostructures (LDSN), themo-
tion of electrons can be confined spatially, from one, two, and even three spatial directions. The
operation of semiconductor quantumdevices is based on the confinement of individual electrons
and holes in one spatial dimension (quantumwells), two spatial dimensions (quantumwires) or
three spatial dimensions (quantumdots –QDs). For instance the potential barriers, forming the
well are provided primarily by either free surfaces, which impose essentially infinite confinement,
or by sharply layered compositional differences. In the SiGe system, for example, the valence
bands of uniformbulkSi andGeare shifted in energyby0.46 eV.Phonon induced relaxation of a
charge to its ground state and other dynamics of excited charge carriers affects many important
characteristics of nanoscale devices such as switching speed, luminescence efficiency and car-
rier mobility and concentration. Therefore, a better understanding of the processes that govern
such dynamics has an important and fundamental technological implications. These phenomena
are investigated experimentally through intraband transitions induced by carrier-phonon in-
teractions in QDs and probed effectively by various time-resolved spectroscopy methods. Yet
fundamental understanding of the underlying physics responsible for carrier dynamics and the
specific role that phonons play in the relaxation mechanisms in QDs is still lacking. In such
nanostructures, the free carriers are confined to a small region of space by potential barriers,
and if the size of this region is less than the electron wavelength, the electronic states become
quantized at discrete energy levels. The ultimate limit of low dimensional structures is the qu-
antum dot, in which the carriers are confined in all three directions. Therefore, a quantum dot
can be thought of as an artificial atom.
Many of the previous numericalmodeling approaches for these quantum structures used spa-

tial discretisationmethods, such as the finite element or finite differencemethod (Grundmann et
al., 1995; Stier et al., 1999; Johnson andFreund, 2001). As an alternative, the boundary element
method was proposed byGeldbard andMalloy (2001). Voss (2006) employed the Rayleigh-Ritz
method to solve the nonlinear eigenvalue problemwhere the eigenstates of the electron in QDs
were derived with the use of the finite element method incorporated in the MATLAB package.



388 A.Muc et al.

LewYanVoon andWillatzen (2004, 2005) found analytical solutions for paraboloidal and ellip-
soidal QDs not embedded in the matrix. Various aspects of the evaluation of eigenenergies in
closed periodic systems of quantum dots were also discussed in the literature – see e.g. Refs Li
(2005) and Cattapan et al. (2009).
In the present paper, we intend to demonstrate the method of analytical solutions of the

nonlinear eigenvalue problem for conical quantum dots embedded in the matrix. In the mathe-
matical sense, the problem is described by the solution of the Helmholtz equation.

2. Governing relations

Let Ωq be a 3D domain occupied by the quantum dot, which is embedded in a bounded ma-
trix Ωm of a different material – Fig. 1. A typical example is an InAs quantum dot embedded
in a GaAs matrix. By adopting this continuum view of confinement in semiconductor quan-
tum devices, the spectrum of confined states available to individual electrons or holes can be
characterized by the steady state Schrdinger equation, given by

Hi(mi)Φi =EΦi (2.1)

where

Hi(mi)=−div
(

~
2

2mi
gradΦi

)

+Vi(x)Φi x∈Ω1∪Ω2

where Hi is the Hamiltonian function (operator), Φi means the wave function, E denotes the
energy level (the identical value for the matrix and the quantum dot), ~ is the reduced Planck
constant, andthe index i correspondsto thequantumdot (i=1;q) andto thematrix (i=2;m),
respectively. The electron effective mass mi is assumed to be constant on the quantum dot and
the matrix for every fixed energy level E and is taken as (Chuang, 1995)

1

mi(E)
=
P2i
~2

( 2

E+Eg,i−Vi
+

1

E+Eg,i−Vi+ δi

)

(2.2)

where the confinement potential Vi is piecewise constant, and Pi,Eg,i and δi are themomentum
matrix element, the band gap, and the spin–orbit splitting in the valence band for the quantum
dot (i= q) and the matrix (2 =m), respectively. The values of the above constants are given
in Table 1.

Fig. 1. Geometry of the quantum dot surrounded by the matrix

The aim of the study is to solve eigenvalue problem (2.1) and to determine the k-th eige-
nenergy. Since effective masses mi (2.2) are, in fact, functions of the eigenergies, problem (2.1)



An analytical solution for conical quantum dots 389

Table 1.Material properties of the quantum dot InAs and thematrix GaAs

Pi Eg,i Vi δi

i=1 (q) InAs 0.8503 0.42 0 0.48

i=2 (m) GaAs 0.8878 1.52 0.7 0.34

constitutes the nonlinear eigenvalue problem. Commonly, the wave function decays outside the
quantumdot and thematrix very rapidly, it is then reasonable to assumehomogeneousDirichlet
conditions, i.e.: Φi =0 on the outer boundaries. On the interface between the quantumdot and
the matrix, the Ben Daniel-Duke condition (Chuang, 1995) holds

1

m1

∂Φ1

∂n1
=
1

m2

∂Φ2

∂n2
(2.3)

Here n1 and n2 denote the outward unit normal on the boundary of Ω1 and Ω2, respectively.

3. Solution to the boundary value problem

Steady state Schrödinger equation (2.1), which governs the behavior of individual charge carriers
in strained devices, is in the form of the classical Helmholtz equation

div gradΦi+k
2
iΦi =0 k

2
i =
2mi(E−Vi)

~2
i=1,2 (3.1)

Let us assume rotational symmetry of the problem. Thus, in the cylindrical coordinateds the
wave function can be written as Φi(r,z,ϕ) =Ri(r)Zi(z)sin(lϕ), where l=0,±1,±2, . . . is the
electron orbital quantumnumber.Finally, after separation of the variables, Schrödinger equation
(3.1) obtains the form:

(1

r

∂

∂r

(

r
∂

∂r

)

+k2i +s
2
i −
l2

r2

)

Ri(r)= 0
∂2Zi(z)

∂z2
+s2iZi(z)= 0 i=1,2 (3.2)

and si are the separation constants. If we rescale the spatial coordinate r in Eq. (3.2) and

introduce a new variable x= r
√

k2i +s
2
i , the first equation can be reduced to the classical form

of the Bessel equation

(1

x

∂

∂x

(

x
∂

∂x

)

+1−
l2

x2

)

Xi(x)= 0 Xi(x)=Ri(r) (3.3)

Since theBessel function J−l(x) tends to infinity at x=0, the solutions of Schrödinger equation
(3.1) can be expressed as follows

Φi(r,z,ϕ) = [Mi sin(siz)+Nicos(siz)]Jl
(

r
√

k2i +s
2
i

)

sin(lϕ) i=1,2 (3.4)

where Mi and Ni are the unknown constants. Using the Dirichlet boundary condition
Φ1(r,0,) = 0, one can find that N1 = 0 and Φ2(r,z = z,ϕ) = 0, N2 = −M2 tan(s2z). The
third boundary condition Φ2(r= r,z,ϕ) = 0 leads to an infinite number of solutions being the

zeroth of the Bessel functions of the l-th order, i.e.: Jl
(

r
√

k22 +s
2
2

)

=0. Let us note that with

the use of that relation, it is possible to determine the value of the separation constant s2.



390 A.Muc et al.

4. Variational formulation

The analytical form of solution (3.4) is not very convenient for direct numerical computations
of eigenenergies due to the existence of boundary condition (2.3). Therefore, we adopt herein
the variational formulation. Multiplying (2.1) by Ψ in the Sobolev space H01(Ω),Ω=Ω1∪Ω2
and integrating by parts, one gets the variational form of the Schrödinger equation

a(Φ,Ψ;E) =
2
∑

i=1

∫

Ω

(

~
2

2mi
gradΦ·gradΨ+Vi(x)Φ·Ψ) dΩ=E

∫

Ω

Φ·Ψ dΩ=Eb(Φ,Ψ) (4.1)

If the quadratic form a(Φ,Φ;E) is a positive, monotonically decreasing function then a unique
positive solution exists

E(Φ)=
a(Φ,Φ;E)

b(Φ,Φ)
(4.2)

and the corresponding eigenvectors Φk are stationary elements of E
′(Φk). If the quadratic

form a does not dependent on E, then the above Rayleigh functional is just the well known
Rayleigh quotient. The evaluation of the Rayleigh functional is based on the discretization
method of the eigenvectors Φk. Since the solution to equations (3.3) can be represented in form
of the Bessel function, we propose to define the radial part of the eigenvectors in form of the
orthogonal Bessel functions

R(r)=
∞
∑

m=1

AmJl

(µ
(l)
m

r0
r
)

(4.3)

where the symbols µ
(l)
m denote the m-th zero of the Bessel function Jl at r= r0. The expansion

in form given by equation (4.3) is assumed to be valid in the whole space occupied by the
quantum dot and thematrix. In the z direction the function Z(z) can be expressed in form of
the classical Fourier series, i.e

Z(z)=
∞
∑

n=0

Cn sin
(nπ(z−z0)

z1−z0

)

(4.4)

where z1 and z0 denote the upper and lower boundaries of the quantum dot, respectively (see
Fig. 1). Similarly, as in Section 3, in the cylindrical system of coordinates, the representation of
the eigenfunctions can bedefined as themultiplication of series (4.3), (4.4) and function sin(lϕ).
In equation (4.1), the integration over the variables r,ϕ and z is separated.Using the properties
of the Bessel function, the numerator in equation (4.1) takes the form

a(Φ,Φ)=
2
∑

i=1

(

~
2

2mi
D1+ViD2

)

(4.5)

and

D2 =
r20dz

2
F(Am)

∞
∑

n=0

( nπ

z1−z0
Cn

)2

D1 =−dz
∞
∑

n=0

(Cn)
2
∞
∑

m,k=0

AmAk
µ
(l)
mµ
(l)
k

r20

r0
∫

0

dr Jl−1

(µ
(l)
k
r

r0

)

Jl+1

(µ
(l)
mr

r0

)



An analytical solution for conical quantum dots 391

whereas the denominator

b(Φ,Φ)=
r20dz

2
F(Am)

∞
∑

n=0

(Cn)
2 F(Am)=

∞
∑

m=0

[

AmJ
′

l(µ
(l)
m )
]2

dz=
z1−z0

2

(4.6)

The computation is carried out in the iterative way in two steps:

1. For the assumed value of the eigenenergy E0 the minimum of the functional a(Φ,Φ)−
−E0b(Φ,Φ) is searched for, with respect to the unknown coefficients Am and Cn

2. For the calculated values of Am and Cn, rationalmatrix eigenvalue problem (4.2) is solved
to determine a new eigenvalue E1.

The procedure is repeated until the required accuracy is reached. The computations are conduc-
ted with the use of the symbolic package Mathematica.

5. Numerical example

Figure 1 demonstrates the axisymmetrical conical quantum dot embedded in the matrix. The
radius and the height of theQTare equal to 10, whereas the radius and the height of thematrix
are equal to 40 and30, respectively. Thephysical properties of thematerials are given inTable 1.
The boundary conditions are assumed to be in the Dirichlet form, i.e. the wave functions are
equal to zero on the boundaries.
Table 2 shows results of the computations. The results are presented for the first six eige-

nvalues and compared also with the results available in the open literature.

Table 2.Eigenvalues for the quantum dot embedded in thematrix

l
Eigenvalue

present analysis Voss (2006)

0 0.254607 0.254585

1 0.387332 0.384162

0 0.466941 0.467239

2 0.502889 0.503847

0 0.560774 0.561319

1 0.599138 0.598963

3 0.622213 0.617759

l – the electron orbital quantum number

The computed eigenvalues show quite good agreement with the results obtained by Voss
(2006). In the presented computations, the maximal number of terms in expansion (4.3) was
equal to 50, and approximation (4.4) was cut off on 15 to 20 terms. It is worth to note that the
lowest values of the eigenvalues do not correspond to the lowest values of the electron orbital
quantum number.

6. Concluding remarks

Thepresentedmethodshowsthepossibilityof analytical computationsof thenonlinearboundary
valueproblemcharacterizedby theHelmholtz equations. In thisway, it is possible tofindnotonly
the energy spectrum and wave functions of an electron in a quantum dot but also the acoustic



392 A.Muc et al.

eigenfrequencies and eigenmodes of the pressure field inside an acoustic cavity. Eigensolutions
are presented in a convenient form of the series expansions. The analysis can be conducted for
an arbitrary form of the potential function, however having the axisymmetry with respect to
the axis of rotations.

Acknowledgment

The Polish Research Foundation PB 174/B/T02/2009/36 is gratefully acknowledged for financial

support.

References

1. Cattapan G., Lotti P., Pascolini A., 2009,A scattering-matrix approach to the eigenenergies
of quantum dots,Physica E, 41, 1187-1192

2. Chuang S.L., 1995,Physics of Optoelectronic Devices, JohnWiley & Sons, NewYork

3. Gelbard F., Malloy K.J., 2001, Modeling quantum structures with the boundary element
method, Journal of Computational Physics, 172, 19-39

4. Grundmann M., Stier O., Bimberg D., 1995, InAs/GaAs pyramidal quantum dots: Strain
distribution, optical phonons, and electronic structure,Physical Review B, 52, 11969-11981

5. Harrison P., 2000, Quantum Wells, Wires and Dots: Theoretical and Computational Physics,
JohnWiley & Sons, Chichester

6. JohnsonH.T., Freund L.B., 2001, The influence of strain on confined electronic states in semi-
conductor quantum structures, International Journal of Solids and Structures, 38, 1045-1062

7. Lew Yan Voon L.C., Willatzen M., 2004, Helmholtz equation in parabolic rotational co-
ordinates: application to wave problems in quantum mechanics and acoustics, Mathematics and
Computers in Simulation, 65, 337-349

8. Li Y., 2005, An iterative method for single and vertically stacked semiconductor quantum dots
simulation,Mathematical and Computer Modelling, 42, 711-718

9. Stier O., Grundmann M., Bimberg D., 1999, Electronic and optical properties of strained
quantum dots modeled by 8-band k ·p theory,Physical Review B, 59, 5688-5701

10. Voss H., 2006, Numerical calculation of the electronic structure for three-dimensional quantum
dots,Computer Physics Communications, 174, 441-446

11. Willatzen M., Lew Yan Voon L.C., 2005, Numerical implementation of the ellipsoidal wave
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1-18

Rozwiązanie analityczne dla stożkowych kropek kwantowych

Streszczenie

W pracy zaproponowano analityczną metodę do rozwiązania nieliniowego problemu własnego. Mo-
że ona być bezpośrednio zastosowana do analizy osiowosymetrycznych stożkowych kropek kwantowych
osadzonych w osnowie. Polega na wykorzystaniu wariacyjnego sformułowania w połączeniu z metodą
Rayleigha i rozwinięciem w szereg. Przedstawiono i przeanalizowano analityczne rozwiązania dla kro-
pek kwantowych z wykorzystaniem różnych form rozwinięć w szereg (Bessela i Fouriera). Rozwiązany
przykład pokazuje skuteczność przedstawionej metody.

Manuscript received June 10, 2012; accepted for print July 6, 2012