Properties of interface phonon spectra in complicated cylindrical nanosystem
The spectra of two types of interface phonons (top and side optical) are studied within the framework of dielectric continuum model for combined nanoheterosystems consisting of semiconductor cylindrical quantum dots inside the cylindrical quantum wire placed into dielectric or semiconductor medium....
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| Cite this: | Properties of interface phonon spectra in complicated cylindrical nanosystem / O.M. Voitsekhivska, A.M. Gryschyk // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 17-22. — Бібліогр.: 9 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1179132025-02-09T17:32:47Z Properties of interface phonon spectra in complicated cylindrical nanosystem Властивостi спектру iнтерфейсних фонононiв у складних цилiндричних наносистемах Voitsekhivska, O.M. Gryschyk, A.M. The spectra of two types of interface phonons (top and side optical) are studied within the framework of dielectric continuum model for combined nanoheterosystems consisting of semiconductor cylindrical quantum dots inside the cylindrical quantum wire placed into dielectric or semiconductor medium. The dependencies of both types of interface phonon energies on the quasiwave numbers and geometric parameters of nanosystem are calculated and analysed. У межах моделi дiелектричного континууму вивчаються спектри двох типiв iнтерфейсних фононiв (вершинних i бiчних оптичних), що iснують у комбiнованiй наногетеросистемi, яка складається з напiвпровiдникової квантової точки всерединi цилiндричного квантового дроту, помiщеного у дiелектричне або напiвпровiдникове середовище. Отримано i проаналiзовано залежностi енергiй обохтипiв iнтерфейсних фононiв вiд квазiхвильових чисел i геометричних параметрiв наносистем. 2007 Article Properties of interface phonon spectra in complicated cylindrical nanosystem / O.M. Voitsekhivska, A.M. Gryschyk // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 17-22. — Бібліогр.: 9 назв. — англ. 1607-324X DOI:10.5488/CMP.10.1.17 PACS: 68.65.Hb, 68.65.La, 74.25.Kc https://nasplib.isofts.kiev.ua/handle/123456789/117913 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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The spectra of two types of interface phonons (top and side optical) are studied within the framework of dielectric continuum model for combined nanoheterosystems consisting of semiconductor cylindrical quantum dots inside the cylindrical quantum wire placed into dielectric or semiconductor medium. The dependencies of both types of interface phonon energies on the quasiwave numbers and geometric parameters of nanosystem are calculated and analysed. |
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Voitsekhivska, O.M. Gryschyk, A.M. |
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Voitsekhivska, O.M. Gryschyk, A.M. Properties of interface phonon spectra in complicated cylindrical nanosystem Condensed Matter Physics |
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Voitsekhivska, O.M. Gryschyk, A.M. |
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Properties of interface phonon spectra in complicated cylindrical nanosystem |
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Properties of interface phonon spectra in complicated cylindrical nanosystem |
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Properties of interface phonon spectra in complicated cylindrical nanosystem |
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Properties of interface phonon spectra in complicated cylindrical nanosystem |
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Properties of interface phonon spectra in complicated cylindrical nanosystem |
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properties of interface phonon spectra in complicated cylindrical nanosystem |
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Інститут фізики конденсованих систем НАН України |
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Properties of interface phonon spectra in complicated cylindrical nanosystem / O.M. Voitsekhivska, A.M. Gryschyk // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 17-22. — Бібліогр.: 9 назв. — англ. |
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Condensed Matter Physics |
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AT voitsekhivskaom propertiesofinterfacephononspectraincomplicatedcylindricalnanosystem AT gryschykam propertiesofinterfacephononspectraincomplicatedcylindricalnanosystem AT voitsekhivskaom vlastivostispektruinterfejsnihfonononivuskladnihcilindričnihnanosistemah AT gryschykam vlastivostispektruinterfejsnihfonononivuskladnihcilindričnihnanosistemah |
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Condensed Matter Physics 2007, Vol. 10, No 1(49), pp. 17–22
Properties of interface phonon spectra in complicated
cylindrical nanosystems
O.M.Voitsekhivska, A.M.Gryschyk
Fedkovych Chernivtsi National University, 2, Kotsyubinskoho Str., Chernivtsi, 58012, Ukraine∗
Received September 4, 2006
The spectra of two types of interface phonons (top and side optical) are studied within the framework of
dielectric continuum model for combined nanoheterosystems consisting of semiconductor cylindrical quantum
dots inside the cylindrical quantum wire placed into dielectric or semiconductor medium. The dependencies of
both types of interface phonon energies on the quasiwave numbers and geometric parameters of nanosystem
are calculated and analysed.
Key words: nanoheterosystem, phonon, quasiwave number, energy
PACS: 68.65.Hb, 68.65.La, 74.25.Kc
1. Introduction
For the past few years there has been observed a rapid development of experimental and theo-
retical investigations not only of the nanoscale objects (quantum dots, wires and wells) but their
spatial combinations as well [1]. The experimental treatment of the quantum objects exhibiting
the variety of shapes such as sphere, cylinder or rectangle, have caused the creation of rather
complicated combined constructions having dimensions within a few nanometers. There unique
properties are used in different modern devices, i.e. quantum computers [2,3].
The theory of quasiparticles spectra in single quantum dots, wires and wells embedded into
the dielectric or semiconductor external medium have been established for the past few years [4–6]
within the effective mass approximation and rectangular potentials for the electron, hole, exciton
and dielectric continuum model for the phonons. The investigations of physical properties, such
as carrier relaxation and transportation, linear and non-linear optical characteristics of quantum
dots, the interaction between quasiparticles and phonons are carried out both experimentally and
theoretically.
The complicated nanoconstructions created experimentally, contain different spatial combina-
tions of quantum dots, wires and wells [1]. These objects are actively studied theoretically, in
spite of the mathematical difficulties arising due to the complicated fitting conditions for the wave
functions of quasiparticles and polarization potentials of free vibrations.
It is obvious that in order to profoundly understand the properties of any nanoconstruction
one has to perform a precise description of all optical phonon modes, study the quasiparticles
spectra and establish the Hamiltonian of interaction between quasiparticles and phonons. Wai-
Sang Li and Chuan-Yu Chen [7] have studied a cylindrical quantum dot in vacuum and derived
confined LO phonons and two types of interface phonon modes with their eigenfunctions using the
dielectric continuum approach. The analytical investigations of phonon spectra in three cylindrical
quantum dots in the cylindrical quantum wire placed into the semiconductor external medium
were performed in [8]. Both references present only the analytical derivations without any numeric
calculations and analysis of phonon spectra for the specific nanoconstructions.
In this paper the energies and dispersion laws of interface vibrations are studied for the two
types of a combined nanoheterosystem consisting of semiconductor cylindrical quantum dot embed-
∗E-mail: theorphys@chnu.cv.ua
c© O.M.Voitsekhivska, A.M.Gryschyk 17
O.M.Voitsekhivska, A.M.Gryschyk
ded into the cylindrical semiconductor quantum wire placed into the semiconductor or dielectric
external medium (water). The phonon spectrum is treated within the framework of the dielectric
continuum model widely applied for low-dimensional crystal structures [6–9] and the results of
which are in good correlation with the experimental data.
2. Theory of interface phonon spectra
The interface phonon spectra are studied for two types of a combined nanosystem consisting of
a semiconductor cylindrical quantum dot (HgS “0”) embedded into the cylindrical quantum wire
( CdS “1”) placed into the semiconductor ( ZnS “2”, figure 1a) or dielectric (water “2” , figure 1b)
outer medium. The radii of a quantum dot (QD) and quantum wire (QW) are considered to be
equal (ρ0), the QD height is ∆0 and the dielectric constant of every i-th part of nanosystem is
assumed to be:
εi(ω) = εi∞
ω2 − ω2
Li
ω2 − ω2
Ti
, i = 0, 1, 2, . . . , (1)
where εi∞ is a high frequency dielectric constant, ωLi and ωTi are the frequencies of longitudinal
and transversal optical phonons of the respective bulk crystals. In case the external medium is
dielectric, ε2 = εd = 1.78. The phonon spectrum for such a system is obtained within the dielectric
continuum model combining the electrostatic equations [6] and getting
εi(ω)∇2Φ(~r) = 0, (2)
where Φ(~r) is the potential of phonon polarization field. It is clear that there are two possible so-
lutions to this equation defining the spectra of confined and interface phonons which are observed
hereafter. It is well known that the first solution of equation (2) proves that the frequencies of con-
fined phonons are equal to the frequencies of longitudinal phonons of the corresponding composing
parts of a nanosystem [6–8]. The second solution: ∆Φ(~r) = 0, determines the spectra of interface
phonons. According to the cylindrical symmetry of the system, the polarization potential can be
chosen as
Φ(~r) = ϕ(ρ)F (z)eimϕ. (3)
“0”
“1”“1”
“2”
“3”
0
0
HgSCdS CdS
ZnS(a) water(b)
ZnS(a) water(b)
Figure 1. a,b. Geometrical scheme of combined nanosystem.
Obviously, the solutions for the two planes are quite different from the solutions at the side
walls. Depending on the boundary conditions for the F (z) and ϕ(ρ) functions ([7]) there are two
types of interface phonon modes: top surface optical (TSO) modes with the amplitude decreasing
away from the two planes and side surface optical (SSO) modes with the amplitude decreasing
away from the side walls.
Top surface optical (TSO) modes
It is clear that for the TSO modes, the ϕ(ρ) function should describe the non-decaying potential
in the plane perpendicular to the OZ axis and decaying F (z) along this axis. Consequently,
ϕ(ρ) =
{
Jm(qρ),
Nm(qρ),
ρ 6 ρ0 ,
ρ > ρ0 ,
(4)
18
Properties of interface phonon spectra
F (z) =
{
B+
0 ch(qz) + B−
0 sh(qz),
B1e
−qz,
|z| 6 ∆0/2 ,
|z| > ∆0 .
(5)
From the boundary conditions for the polarization potential and normal terms of electric dis-
placement at z = ±∆0/2, there are obtained antisymmetric (ω−) or symmetric (ω+) TSO modes.
As a result, one can get a system of two equations
{
B±
0 Csh±(qz) = B1e
−qz ,
±ε0B
±
0 Sch±(qz) = −ε1B1e
−qz,
(6)
where
Csh±(qz) =
{
Ch(qz), “ + ” ,
Sh(qz), “ – ” ,
Sch±(qz) =
{
Sh(qz), “ + ” ,
Ch(qz), “ – ” .
(7)
The condition of non-trivial solutions of equations (6) leads to the equation fixing the frequencies
of TSO phonons
g2ω
4 − g1ω
2 + g0 = 0, (8)
where
g0 = αω2
T1ω
2
L0 − βω2
T0ω
2
L1, g2 = α − β,
g1 = α(ω2
T1 + ω2
L0) − β(ω2
T0 + ω2
L1),
α = ±
Sch±(gz)
Csh±(gz)
, β = −
ε1∞
ε0∞
. (9)
We are going to discuss the results of computer calculations performed for the specific nanosys-
tem hereafter.
Side surface optical (SSO) modes
For the SSO modes the F (z) function should describe the non-decaying potential along the OZ
axis and decaying ϕ(ρ) in the perpendicular plane. Thus, it is convenient to take as follows:
Φ(~r) =
{
Im(kρ) (D+ cos(kz) + D− sin(kz)) eimϕ,
Km(kρ) (C+ cos(kz) + C− sin(kz)) eimϕ,
ρ 6 ρ0
ρ > ρ0
. (10)
The unknown coefficients are found during the second quantization of the phonon field, Im(kρ)
and Km(kρ) is the m-th order modified Bessel functions of the first and second type. The frequencies
of SSO phonons are also determined by the boundary conditions for the polarization potential and
normal terms of electric displacement at the condition ρ = ρ0. The ratio of the respective equations
gives the transcendental equation
εi
I ′m(kρ0)
Im(kρ0)
= ε2
K ′
m(kρ0)
Km(kρ0)
, i = 0, 1. (11)
Solving the latter we get the dispersion equation for the frequencies of SSO modes.
When the quantum dot, wire and outer medium are all semiconductors, the frequencies are
fixed by the expression
ωi(kρ0) =
√
−bi ±
√
b2
i − 4aici
2ai
, (12)
where
ai = εi∞ − ε2∞P (kρ0),
bi = ε2∞P (kρ0)(ω
2
Ti − ω2
L2) − εi∞(ω2
T2 − ω2
Li),
ci = εi∞ω2
Liω
2
T2 − ε2∞P (kρ0)ω
2
L2ω
2
Ti ,
P (kρ0) =
Im(kρ0)
Km(kρ0)
×
mKm(kρ0) − kρ0Km+1(kρ0)
mIm(kρ0) + kρ0Im+1(kρ0)
. (13)
19
O.M.Voitsekhivska, A.M.Gryschyk
The expression (12) proves that there are four bands (formed over the magnetic quantum
number) of SSO modes for two side walls, produced by the interfaces between two semiconductor
media (dot/outer medium and wire/outer medium).
When the quantum dot and wire are semiconductors and the outer medium is dielectric, the
frequencies of SSO phonons for the i-th part of nanosystem are fixed by the expression
ωi(kρ0) =
√
εdω2
TiP (kρ0) − ω2
Liεi∞
εdP (kρ0) − εi∞
. (14)
The expression (14) proves that there are two bands of SSO modes for every side wall, produced
by the interface between semiconductor and dielectric media (dot/outer medium and wire/outer
medium). The results of the numeric calculations for the particular system are also analysed here-
after.
3. Discussion of the results
The results of the computer calculations of the TSO and SSO phonon energies are presented in
figures 2–4. The numeric calculations were performed for the cylindrical nanosystems CdS/HgS/CdS
placed into ZnS outer medium or into the water (figure 1a,b). The choice of these specific nanosys-
tems is conditioned, on the one hand, by the dielectric continuum model and, on the other hand,
by the requirements of the effective mass approximation and rectangular potentials for the quasi-
particles (the close magnitudes of the lattice constants). The latter are important for the possible
future investigations of interaction between quasiparticles and phonons.
0,00 0,01 0,02 0,03 0,04 0,05
20
25
30
35
40
45
50
55
60
=50 a
HgS
=25 a
HgS
=10 a
HgS
T CdS
L CdS
T HgS
L HgS
q, /a
HgS
T
S
O
,m
e
V
Figure 2. Dependence of TSO phonon energy on the quasiwave number for the different thickness
of HgS QD (∆0 = 10aHgS, 25aHgS, 50aHgS).
It should to be mentioned that the behavior of the interface phonon spectra depends on the
geometrical parameters of a nanosystem and on the type of vibrations (SSO or TSO) but the
energies of all these phonon modes are always located between the energies of LO and TO phonons
of the respective bulk crystals (ΩLCdS
, ΩTCdS
, ΩLZnS
, ΩTZnS
, and ΩLHgS
, ΩTHgS
shown in the figures
by dashed lines).
In figure 2 there is shown the evolution of TSO phonon energy spectra as a function of
quasiwave number q (in units of lattice constant of HgS crystal) for different thicknesses (∆0 =
10aHgS, 25aHgS, 50aHgS) of QD HgS. From the figure it is clear that there are four modes of TSO
vibrations: two vibrations with positive (symmetric) and two vibrations with negative (antisym-
metric) dispersion shown by the respective curves. They are caused by the presence of two plane
interfaces between QD HgS and QW CdS. The dependence on the quasiwave number is essential
only for the small magnitudes of the latter and then the curves are smoothly tending to saturation.
This fact makes it easier to study the interaction between quasiparticles and phonons that can be
assumed as dispersionless. At a fixed quantum number, the energies of TSO phonons are bigger
20
Properties of interface phonon spectra
for the bigger thickness of HgS QD for the modes with a positive dispersion and smaller for the
modes with negative dispersion.
0,00 0,01 0,02 0,03 0,04 0,05
20
30
40
50
60
aHgSCdS
ZnS
CdS
ZnS
T
S
O
,
m
e
V
k, /a
ZnS
L ZnS
L CdS
L HgS
T ZnS
T CdS
T HgS
0,00 0,01 0,02 0,03 0,04 0,05
20
30
40
50
60
bHgSCdS
H 02
H 02
CdS
L HgS
T HgS
L CdS
T CdS
S
S
O
,
m
e
V
k, /a
HgS
Figure 3. Dependence of SSO phonon energy on the quasiwave number for magnetic quantum
number m = 0(−), m = 1(. . . ).
In figure 3a,b there is shown the dependence of SSO phonon energies on the quasiwave num-
ber for two nanoconstructions: CdS/HgS/CdS placed into ZnS outer medium (figure 3a) and
CdS/HgS/CdS placed into the water (figure 3b). The difference is obvious because, as it is proved
by the formulas (12)–(14), in figure 3a, one can see four bands of SSO phonon energies: two of
them arise due to the side interface HgS/ZnS and the other two arise due to the interface CgS/ZnS.
In figure 3b there are two bands of SSO phonon energies: one band is produced by the interface
HgS/H2O and the other is produced by the interface CdS/H2O. The bands are formed over the
magnetic quantum number (m). Computer calculations show that the energies of SSO phonons
0 2 4 6 8 10
25
45
50
55
60
L HgS
T HgS
L CdS
T CdS
k=0.07 /a
k=0.02 /a
k=0.01 /a
S
S
O
,
m
e
V
0
,a
HgS
Figure 4. Dependence of SSO phonon energy on the radius of nanoconstruction.
21
O.M.Voitsekhivska, A.M.Gryschyk
with m > 1 are almost the same as for the energy with m = 1, since in the figure there are shown
the dependences only for m = 0 and m = 1. The dispersion of the energies on the quasiwave num-
ber is noticeable only for the small magnitudes of the latter and then the curves are also tending
to saturation.
In figure 4 there is shown the dependence of SSO phonon energies on the radius of nanocon-
struction (ρ0) for different magnitudes of the quasiwave number k. The figure proves that the
dependence on the radius is rather weak. The SSO modes also form the respective bands over the
magnetic number but in the figure there are presented the results calculated only for m = 0 in
order not to overload the figure because the dependencies are quite similar to the ones shown in
figure 3a,b.
The obtained frequencies of TSO and SSO phonons and their dispersion laws will be further used
for the study of interaction between quasiparticles and these types of vibrations in the combined
cylindrical nanosystems.
References
1. Siegel R.W. Assembling Materials From Nanoscale Building Blocks. – In: Proc. E–MRS, 2004, p. 24–25.
2. Fujisawa T., Hayashi T., Cheong H.D., Jeong Y.H., Hirayama Y. Physica E (Amsterdam), 2004, 21,
1046.
3. Vorojtsov S., Mucciolo E.R., Baranger H.U. Phys. Rev. B, 2005, 71, 322 .
4. Tkach M., Holovatsky V., Voitsekhivska O. Physica E (Amsterdam), 2001, 11, 177.
5. Tkach M., Holovatsky V., Votsekhivska O., Mykhalyova M., Fartushunsky R. Physica Status Solidi,
2001, 225, 331.
6. Tkach M.V. Quasiparticles in nanoheterosystems. Quantum dots and wires, Chernivtsi (Ruta), 2003,
312.(in Ukrainian)
7. Wai-Sang Li, Chuan-Yu Chen Physica B, 1997, 375.
8. Tkach N.V., Makhanets A.M., Zharkoy V.P. Journal of Physical Studies, 2005, 9, 163 (in Ukrainian)
9. Chen R., Lin P.L., George T.F. Phys. Rev. B, 1990, 41, 1439.
Властивостi спектру iнтерфейсних фонононiв у складних
цилiндричних наносистемах
О.М.Войцехiвська, А.М.Грищук
Чернiвецький нацiональний унiверситет iм. Юрiя Федьковича,
вул. Коцюбинського 2, 58012, Чернiвцi
Отримано 4 вересня 2006 р.
У межах моделi дiелектричного континууму вивчаються спектри двох типiв iнтерфейсних фононiв
(вершинних i бiчних оптичних), що iснують у комбiнованiй наногетеросистемi, яка складається з
напiвпровiдникової квантової точки всерединi цилiндричного квантового дроту, помiщеного у дiеле-
ктричне або напiвпровiдникове середовище. Отримано i проаналiзовано залежностi енергiй обох
типiв iнтерфейсних фононiв вiд квазiхвильових чисел i геометричних параметрiв наносистем.
Ключовi слова: наносистема, фонон, квазiхвильове число, енергiя
PACS: 68.65.Hb, 68.65.La, 74.25.Kc
22
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