Confined and interface phonons in combined cylindrical nanoheterosystem
The spectra of all types of phonons existing in a complicated combined nanoheterosystem consisting of three cylindrical quantum dots embedded into the cylindrical quantum wire placed into vacuum are studied within the dielectric continuum model. It is shown that there are confined optical (LO) and...
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nasplib_isofts_kiev_ua-123456789-1214532025-06-03T16:27:04Z Confined and interface phonons in combined cylindrical nanoheterosystem Обмеженi та iнтерфейснi фонони в комбiнованих цилiндричних наносистемах Makhanets, O.M. Voitsekhivska, O.M. Gryschyk, A.M. The spectra of all types of phonons existing in a complicated combined nanoheterosystem consisting of three cylindrical quantum dots embedded into the cylindrical quantum wire placed into vacuum are studied within the dielectric continuum model. It is shown that there are confined optical (LO) and interface phonons of two types: top surface optical (TSO) and side surface optical (SSO) modes of vibration in such a nanosystem. The dependences of phonon energies on the quasiwave numbers and geometrical parameters of quantum dots are investigated and analysed. У моделi дiелектричного континууму вивчаються спектри всiх типiв фононiв, що iснують у складнiй комбiнованiй наногетеросистемi, яка складається з трьох цилiндричних квантових точок, умiщених в цилiндричний квантовий дрiт, який знаходиться у вакуумi. Показано, що у такiй наносистемi є обмеженi оптичнi (LO) та iнтерфейснi фонони двох типiв: вершиннi поверхневi оптичнi (TSO) та бiчнi поверхневi оптичнi (SSO) моди коливань. Дослiджено залежнiсть енергiй фононiв вiд хвильових чисел та геометричних параметрiв квантових точок. 2006 Article Confined and interface phonons in combined cylindrical nanoheterosystem / O.M. Makhanets, O.M. Voitsekhivska, A.M. Gryschyk // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 719–724. — Бібліогр.: 9 назв. — англ. 1607-324X PACS: 68.65.Hb, 68.65.La, 74.25.Kc DOI:10.5488/CMP.9.4.719 https://nasplib.isofts.kiev.ua/handle/123456789/121453 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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English |
| description |
The spectra of all types of phonons existing in a complicated combined nanoheterosystem consisting of three
cylindrical quantum dots embedded into the cylindrical quantum wire placed into vacuum are studied within
the dielectric continuum model. It is shown that there are confined optical (LO) and interface phonons of two
types: top surface optical (TSO) and side surface optical (SSO) modes of vibration in such a nanosystem. The
dependences of phonon energies on the quasiwave numbers and geometrical parameters of quantum dots
are investigated and analysed. |
| format |
Article |
| author |
Makhanets, O.M. Voitsekhivska, O.M. Gryschyk, A.M. |
| spellingShingle |
Makhanets, O.M. Voitsekhivska, O.M. Gryschyk, A.M. Confined and interface phonons in combined cylindrical nanoheterosystem Condensed Matter Physics |
| author_facet |
Makhanets, O.M. Voitsekhivska, O.M. Gryschyk, A.M. |
| author_sort |
Makhanets, O.M. |
| title |
Confined and interface phonons in combined cylindrical nanoheterosystem |
| title_short |
Confined and interface phonons in combined cylindrical nanoheterosystem |
| title_full |
Confined and interface phonons in combined cylindrical nanoheterosystem |
| title_fullStr |
Confined and interface phonons in combined cylindrical nanoheterosystem |
| title_full_unstemmed |
Confined and interface phonons in combined cylindrical nanoheterosystem |
| title_sort |
confined and interface phonons in combined cylindrical nanoheterosystem |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2006 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121453 |
| citation_txt |
Confined and interface phonons in combined cylindrical nanoheterosystem / O.M. Makhanets, O.M. Voitsekhivska, A.M. Gryschyk // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 719–724. — Бібліогр.: 9 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT makhanetsom confinedandinterfacephononsincombinedcylindricalnanoheterosystem AT voitsekhivskaom confinedandinterfacephononsincombinedcylindricalnanoheterosystem AT gryschykam confinedandinterfacephononsincombinedcylindricalnanoheterosystem AT makhanetsom obmeženitainterfeisnifononivkombinovanihcilindričnihnanosistemah AT voitsekhivskaom obmeženitainterfeisnifononivkombinovanihcilindričnihnanosistemah AT gryschykam obmeženitainterfeisnifononivkombinovanihcilindričnihnanosistemah |
| first_indexed |
2025-12-01T02:11:17Z |
| last_indexed |
2025-12-01T02:11:17Z |
| _version_ |
1850270106163085312 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 4(48) , pp. 719–724
Confined and interface phonons in combined cylindrical
nanoheterosystem
O.M.Makhanets, O.M.Voitsekhivska, A.M.Gryschyk
Fedkovych Chernivtsi National University, 2, Kotsyubinskoho Str., Chernivtsi, 58012, Ukraine∗
Received March 31, 2006, in final form June 2, 2006
The spectra of all types of phonons existing in a complicated combined nanoheterosystem consisting of three
cylindrical quantum dots embedded into the cylindrical quantum wire placed into vacuum are studied within
the dielectric continuum model. It is shown that there are confined optical (LO) and interface phonons of two
types: top surface optical (TSO) and side surface optical (SSO) modes of vibration in such a nanosystem. The
dependences of phonon energies on the quasiwave numbers and geometrical parameters of quantum dots
are investigated and analysed.
Key words: nanoheterosystem, phonon, energy spectrum
PACS: 68.65.Hb, 68.65.La, 74.25.Kc
1. Introduction
Recently, the complicated combined nanoheterosystems containing different spatial combinati-
ons of quantum dots (QD), wires (QW) and wells have been intensively researched. Such systems
have already been created experimentally and have good prospects of being utilized in the devices
of physical, biomedical and optical electronics [1]. For example, in the systems containing two or
more QD’s an electron practically without any loss of energy can be localized in different quantum
dots due to the anticrossing effect. Such systems create a single quabit and can be used as an
elementary base of modern quantum computer [2–4].
The theoretical investigations of combined nanoheterosystems are only at the very start. Their
importance is obvious but there are a priori some difficulties arising due to the complicated bound-
ary conditions for the wave functions of quasiparticles (electrons, holes, excitons) and polarization
potentials of free vibrations (phonons). In paper [5] the energy spectrum and life times of quasipar-
ticles in the open QD embedded into the cylindrical QW have been studied within the framework
of the effective mass approximation using the scattering matrix method. It was shown that the
energy loss of the quasiparticles inside the QD arises due to their tunneling through the potential
barrier into the external medium. It is clear that the change of the quantum state of the quasi-
particle (electron, hole, exciton) is accompanied by the processes of creation and annihilation of
phonons. Consequently, there is an additional channel of energy relaxation and finally it causes the
change in the quasiparticle energy spectrum.
This paper presents theoretical investigations of the phonon spectrum in combined nanoheterosys-
tems consisting of three QD’s embedded into the cylindrical semiconductor QW placed into the
medium. The study is performed within the dielectric continuum model widely applied in other
papers [6,7] and the results of which are in good agreement with the experimental data.
2. Theory
The spectra of all types of phonons existing in the complicated combined nanoheterosystem
consisting of three cylindrical quantum dots (CdS “1”, HgS “0”, CdS “1”) embedded into the
∗E-mail: theorphys@chnu.cv.ua
c© O.M.Makhanets, O.M.Voitsekhivska, A.M.Gryschyk 719
O.M.Makhanets, O.M.Voitsekhivska, A.M.Gryschyk
Figure 1. Geometrical scheme of combined cylindrical nanosystem.
cylindrical quantum wire (HgS “2”) placed into the external medium (vacuum “3”, figure 1) are
under study. The radii of all QD’s and QW are assumed to be equal (ρ0), the height of the internal
QD is ∆0 and that of the external is ∆1, the dielectric constant
εi(ω) = εi∞
ω2 − ω2
Li
ω2 − ω2
Ti
, i = 0, 1, 2, (1)
where εi∞ – is high frequency dielectric constant of the ith part of nanosystem, ωLi and ωTi – are
the frequencies of longitudinal and transversal optical phonons of its bulk analogue and ε3 = 1. As
it follows from the dielectric continuum model the phonon spectrum for such a system is obtained
by combining the electrostatic equations and getting the equation [6]
εi(ω)∇2Φ(~r) = 0, (2)
where Φ(~r) is the potential of phonon polarization field. It is obvious that there are two possible
solutions to this equation defining the spectra of confined and interface phonons which are further
observed separately.
Confined phonons
It is clear that at the condition
εi(ω) = 0, ∇2Φ(~r) 6= 0 (3)
and taking into account equation (1) one can obtain the spectrum of confined optical (LO) phonons
ΩLi = ~ωLi, i = 0, 1, 2, (4)
the energies of which are equal to the energies of the longitudinal phonons of the corresponding
bulk analogues of the nanosystem parts.
According to the symmetry of the system the polarization potential of the LO phonons can be
chosen as
Φm(~r) = Jm(kρ)f(z)eimϕ , (5)
where
f(z) =
{
A+
0 cos(q0z)
A−
0 sin(q0z)
, z 6
∆0
2
,
A+
1 cos(q1z) + A−
1 sin(q1z), ∆0
2
< z 6
∆0
2
+ ∆1 ,
A+
2 cos(q2z) + A−
2 sin(q2z), z > ∆0
2
+ ∆1 .
(6)
Here m is magnetic quantum number, qi and k = Xm
n ρ−1
0 , – are the quasiwave numbers, Xm
n is n-th
root of m-th order Bessel function and A±
i are unknown coefficients determined by the boundary
conditions of electrostatic: the continuity of the polarization potential Φm(~r) and normal terms of
electric displacement ~D at the media interfaces.
720
Confined and interface phonons in combined cylindrical nanoheterosystem
Interface phonons
The spectra of interface phonons due to the equation (2) are obtained from the Laplace’s equation
∆Φ(~r) = 0 (7)
the solution of which in the cylindrical coordinates (ρ, ϕ, z) can be generally written as
Φ(~r) = ϕ(ρ)F (z)eimϕ. (8)
It is clear that the solutions for the four planes are quite different from that of the solutions
at the side walls. One knows [6] that depending on the boundary conditions for the ϕ(ρ) and
F (z) functions there are two types of interface phonon modes: top surface optical (TSO) modes,
whose amplitude decreases away from the four planes; and side surface optical (SSO) modes, whose
amplitude decreases away from the side walls.
Spectrum of TSO phonons
For the TSO modes the function has to describe the non-decaying potential in the plane per-
pendicular to the OZ axis and the decaying one along this axis. Thus, they are chosen in the
form
ϕ(ρ) =
{
Jm(qρ), ρ 6 ρ0
Nm(qρ), ρ > ρ0
, (9)
F (z) =
{
B+
0 ch(qz)
B−
0 sh(qz)
, z 6
∆0
2
B+
1 ch(qz) + B−
1 sh(qz), ∆0
2
< z 6
∆0
2
+ ∆1
B2e
−qz, z > ∆0
2
+ ∆1
. (10)
Again, from the boundary conditions for polarization potential F (z) and normal terms of elec-
tric displacement ~D at z = ∆0
2
and z = ∆0
2
∆1 it is obtained B+
i = 0, B−
i 6= 0 (antisymmetric TSO
modes) or B−
i = 0, B+
i 6= 0 (symmetric TSO modes). As a result we get a system of four equations
for determining the frequencies of symmetric (ω+) and antisymmetric (ω−) TSO phonons. The de-
tailed analytical solution of the similar system has been performed in paper [8] where the equation
was obtained for defining the TSO phonon spectra but there were no numeric calculations for real
nanosystems. We are going to further analyse the results of computer calculations performed for
the system under research.
Spectrum of SSO phonons
It is clear that for the SSO modes the F (z) function has to describe the non-decaying potential
along the OZ axis and the decaying ϕ(ρ) in the plane perpendicular to this axis. Taking into
account the symmetry, the polarization potential is expressed as
Φ(~r) =
{
D±
i Im(kρ)C±
s (kz) eimϕ, ρ 6 ρ0 ,
C±
i Km(kρ)C±
s (kz) eimϕ, ρ > ρ0
(11)
herein
C±
s (kz) =
{
cos (kz),
sin (kz).
(12)
The unknown coefficients are found during the second quantization of the phonon field, Im(kρ)
and Km(kρ) – are the m-th order modified Bessel functions of the first and the second kind. The
frequencies of SSO phonons are determined by the boundary conditions for polarization potential
Φ(~r) and normal terms of electric displacement at ρ = ρ0.
D±
i Im(kρ)
∣
∣
ρ=ρ0
C±
s (kz) eimϕ = C±
3 Km(kρ)
∣
∣
ρ=ρ0
C±
s (kz) eimϕ ,
εi D
±
i I ′m(kρ)
∣
∣
ρ=ρ0
C±
s (kz) eimϕ = ε3C
±
3 K ′
m(kρ)|ρ=ρ0
C±
s (kz) eimϕ, (13)
721
O.M.Makhanets, O.M.Voitsekhivska, A.M.Gryschyk
here ε3 = 1 since the external medium is vacuum. Taking the ratio of these equations one can write
εi
I ′m(kρ0)
Im(kρ0)
=
K ′
m(kρ0)
Km(kρ0)
. (14)
Solving the latter and using equation (1) a dispersional equation is obtained for the frequencies
of SSO modes in i-th part of nanosystem
ωi(kρ0) =
√
(P (kρ0) − εi∞)(ω2
TiP (kρ0) − ω2
Liεi∞)
P (kρ0) − εi∞
, (15)
P (kρ0) =
Im(kρ0)
Km(kρ0)
mKm(kρ0) − kρ0Km+1(kρ0)
mIm(kρ0) + kρ0Im+1(kρ0)
. (16)
The equations (15),(16) prove that the energies of the side surface phonons create the bands
over the magnetic quantum number m.
3. Analysis of the results
The results of the computer calculations performed with the material parameters of β-HgS and
β-CdS [9] of the interface phonon energies are shown in figures 2–4. All of them prove that the
spectra of interface phonons depend on the geometric parameters of nanosystem and 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 , and ΩLHgS ,
ΩTHgS shown in the figures by dashed lines).
0
4
8
12
16
20
25
30
35
40
45
50
55
0,00
0,04
0,07
a
0 , a
HgS k,
/a H
gS
S
S
O
,
e
V
0,00 0,02 0,04 0,06 0,07 0,09
20
25
45
50
55
60
0,08 0,09
54,5
55,0
55,5
L HgS
T HgS
L CdS
T CdS
b
S
S
O
,
m
e
V
k, /a
HgS
m=3
m=2
m=1
m=0
Figure 2. Evolution of SSO phonon energy on the quasiwave number and radius of HgS QD.
(a): for m=0 and (b): on the quasiwave number for m=0,1,2,. . . at ρ0 = 10aHgS.
Figure 2a presents the dependence of SSO phonon energies on the radius of QW ρ0 and quasi-
wave number k. One can see that there are two modes of SSO phonons arising due to the two
sets of interfaces HgS/vacuum and CdS/vacuum. It is clear that the dispersion over the quasiwave
number is rather weak. There are the sets of modes with different magnetic quantum numbers
(figure 2b) and the energies of SSO phonons which are almost the same as for the energy with
m=1.
Figures 3a,b,c,d shows the evolution of TSO phonon energy spectra as a function of the
quasiwave number q for different thicknesses (∆0) of QD HgS at a fixed thickness of QD CdS
(∆1 = 5aCdS). When the QD HgS is absent ∆0 = 0 (figure 3a) there are four modes of TSO vibra-
tions: two modes with positive (symmetric ω+) and two modes with negative (antisymmetric ω−)
dispersion. Their existence is caused by the presence of two interfaces between QD CdS and QW
HgS. The arising of HgS QD (∆0 = 5aHgS, figure 3c; ∆0 = 10aHgS, figure 3d) and the increase of
its thickness brings about the appearance of four new modes (caused by two additional interfaces)
722
Confined and interface phonons in combined cylindrical nanoheterosystem
0,00 0,02 0,04 0,06 0,07 0,09
20
30
40
50
60
a
T CdS
L CdS
T HgS
L HgS
HgS HgSCdS
2
1
q, /a
HgS
T
S
O
,
m
e
V
0,00 0,02 0,04 0,06 0,07 0,09
20
30
40
50
60
b
HgS HgSHgS CdSCdS
1 10
T
S
O
,
m
e
V
q, /a
HgS
0,00 0,02 0,04 0,06 0,07 0,09
20
30
40
50
60
c
HgSCdSHgS CdSHgS
1 10
T
S
O
,
m
e
V
q, /a
HgS
0,00 0,02 0,04 0,06 0,07 0,09
20
30
40
50
60
d
HgSHgS CdS CdSHgS
1 10
T
S
O
,
m
e
V
q, /a
HgS
Figure 3. Dependence of TSO phonon energy on the quasiwave number at a fixed thickness of
CdS QD (∆1 = 5aCdS) for different thicknesses of HgS QD. (a): ∆0 = 0aHgS, (b): ∆0 = 100aHgS,
c: ∆0 = 5aHgS,d: ∆0 = 10aHgS.
0,00 0,02 0,04 0,06 0,07 0,09
20
30
40
50
60
a
T
S
O
,m
e
V
T CdS
L CdS
T HgS
L HgS
CdS HgSHgS CdSHgS
0 11
q, /a
HgS
0,00 0,02 0,04 0,06 0,07 0,09
20
30
40
50
60
b
HgS HgSHgS CdSCdS
1 10
T
S
O
,
m
e
V
q, /a
HgS
0,00 0,02 0,04 0,06 0,07 0,09
20
30
40
50
60
c
HgS HgSCdS CdSHgS
1 10
T
S
O
,
m
e
V
q, /a
HgS
0,00 0,02 0,04 0,06 0,07 0,09
20
30
40
50
60
d
1 10
- +8 8
CdS CdSHgS HgSHgS
T
S
O
,
m
e
V
q, /a
HgS
Figure 4. Dependence of TSO phonon energy on the quasiwave number at a fixed thickness of
HgS QD (∆0 = 10aHgS) for different thicknesses of CdS QD. (a): ∆1 = 1aCdS, (b): ∆1 = 10aCdS,
(c): ∆0 = 20aCdS,d: ∆0 = 40aCdS.
723
O.M.Makhanets, O.M.Voitsekhivska, A.M.Gryschyk
with different dispersion. At ∆0 → ∞ (figure 3b) eight modes of TSO vibrations again degenerate
into four. This is clear because the case of ∆0 → ∞ with the thickness ∆1 of QD CdS (figure 3b)
is almost equivalent to the case of ∆0 = 0 (figure 3a) but with the thickness of 2∆1 of CdS QD.
The evolution of TSO phonon spectra as function of quasiwave number with the increase of QD
CdS thickness at a fixed thickness of QD HgS (∆0 = 10aHgS) is shown in figures 4a,b,c,d. It is clear
that there are also eight modes of TSO vibrations just like in the previous case. At the increase of
∆1 four of them are degenerated into two (figure 4d), because the energies of the symmetric and
antisymmetric vibrations the potential of which is localized at the plus or minus infinity coincide.
The obtained information about the frequencies of all types of phonons existing in the combined
cylindrical nanoheterosystem and their dispersion laws would be further used for the investigation
of electron-, hole- and exciton-phonon interaction.
References
1. Siegel R.W. Assembling Materials From Nanoscale Building Blocks. Proc. E–MRS, 25, 2004.
2. Hayashi T., Fujisawa T., Cheong H.D., Jeong Y.H., Hirayama Y., Phys. Rev. Lett., 2003, 91, 226804.
3. Fujisawa T., Hayashi T., Cheong H.D., Jeong Y.H., Hirayama Y., Physica E (Amsterdam), 2004, 21,
1046.
4. Vorojtsov S., Mucciolo E.R., Baranger H.U., Phys. Rev. B, 2005, 71, 205322.
5. Tkach N.V., Makhanets A.M., FTT, 2005, 47, 550 (in Russian).
6. Wai-Sang Li, Chuan-Yu Chen, Physica B, 1997, 375.
7. Wang X.F., Lei X.L., Phys. Rev. B, 1994, 49, 4780.
8. Tkach N.V., Makhanets A.M., Zharkoy V.P., J. of Phys. Stud., 2005, 9, 163 (in Ukrainian).
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Обмеженi та iнтерфейснi фонони в комбiнованих
цилiндричних наносистемах
О.М.Маханець, О.М.Войцехiвська, А.М.Грищук
Чернiвецький нацiональний унiверситет iм. Юрiя Федьковича, вул. Коцюбинського 2, 58012,
Чернiвцi
Отримано 31 березня 2006 р., в остаточному виглядi – 2 червня 2006 р.
У моделi дiелектричного континууму вивчаються спектри всiх типiв фононiв, що iснують у складнiй
комбiнованiй наногетеросистемi, яка складається з трьох цилiндричних квантових точок, умiщених
в цилiндричний квантовий дрiт, який знаходиться у вакуумi. Показано, що у такiй наносистемi є
обмеженi оптичнi (LO) та iнтерфейснi фонони двох типiв: вершиннi поверхневi оптичнi (TSO) та бi-
чнi поверхневi оптичнi (SSO) моди коливань. Дослiджено залежнiсть енергiй фононiв вiд хвильових
чисел та геометричних параметрiв квантових точок.
Ключовi слова: наносистема, фонон, енергетичний спектр
PACS: 68.65.Hb, 68.65.La, 74.25.Kc
724
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