Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect
Based on the macroscopic dielectric continuum model, the interface-optical-propagating (IO-PR) mixing phonon modes of a quasi-zero-dimensional (Q0D) wurtzite cylindrical quantum dot (QD) structure are derived and studied. The analytical phonon states of IO-PR mixing modes are given. It is found that...
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| Cite this: | Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect / L. Zhang // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13801: 1-14. — Бібліогр.: 46 назв. — англ. |
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| citation_txt | Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect / L. Zhang // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13801: 1-14. — Бібліогр.: 46 назв. — англ. |
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| description | Based on the macroscopic dielectric continuum model, the interface-optical-propagating (IO-PR) mixing phonon modes of a quasi-zero-dimensional (Q0D) wurtzite cylindrical quantum dot (QD) structure are derived and studied. The analytical phonon states of IO-PR mixing modes are given. It is found that there are two types of IO-PR mixing phonon modes, i.e. ρ-IO/z-PR mixing modes and the z-IO/ρ-PR mixing modes existing in Q0D wurtzite QDs. Each IO-PR mixing mode also has symmetrical and antisymmetrical forms. Via a standard procedure of field quantization, the Fröhlich Hamiltonians of electron-(IO-PR) mixing phonons interaction are obtained. The orthogonal relations of polarization eigenvectors for these IO-PR mixing modes are also displayed. Numerical calculations for a wurtzite GaN cylindrical QD are focused on the quantum size effect on the dispersive properties of IO-PR mixing modes. The results reveal that both the radial-direction size and the axial-direction size have great effect on the dispersive frequencies of the IO-PR mixing phonon modes. The limiting features of dispersive curves of these phonon modes are discussed in depth. The phonon modes "reducing" the behavior of wurtzite quantum confined structures have been explicitly observed in the systems. Moreover, the behaviors that the IO-PR mixing phonon modes in wurtzite Q0D QDs reduce to the IO modes and PR modes in wurtzite Q2D QW and Q1D QWR systems are profoundly analyzed both from the viewpoint of physics and mathematics. These results show that the present theories of polar mixing phonon modes in wurtzite cylindrical QDs are consistent with the phonon modes theories in wurtzite QWs and QWR systems. The analytical electron-phonon interaction Hamiltonians obtained here are useful in further analyzing the phonon effect on optoelectronic properties of wurtzite Q0D QD structures.
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Condensed Matter Physics 2008, Vol. 13, No 1, 13801: 1–14
http://www.icmp.lviv.ua/journal
Polar interface optical phonon states and their
dispersive properties of a wurtzite GaN quantum dot:
quantum size effect
L. Zhang∗
Department of Mechanism and Electronic, Guangzhou Panyu Polytechnic,
Guangzhou, 511483, People’s Republic of China
Received April 23, 2009, in final form September 2, 2009
Based on the macroscopic dielectric continuum model, the interface-optical-propagating (IO-PR) mixing
phonon modes of a quasi-zero-dimensional (Q0D) wurtzite cylindrical quantum dot (QD) structure are de-
rived and studied. The analytical phonon states of IO-PR mixing modes are given. It is found that there are
two types of IO-PR mixing phonon modes, i.e. ρ-IO/z-PR mixing modes and the z-IO/ρ-PR mixing modes
existing in Q0D wurtzite QDs. Each IO-PR mixing mode also has symmetrical and antisymmetrical forms.
Via a standard procedure of field quantization, the Fröhlich Hamiltonians of electron-(IO-PR) mixing phonons
interaction are obtained. The orthogonal relations of polarization eigenvectors for these IO-PR mixing modes
are also displayed. Numerical calculations for a wurtzite GaN cylindrical QD are focused on the quantum size
effect on the dispersive properties of IO-PR mixing modes. The results reveal that both the radial-direction
size and the axial-direction size have great effect on the dispersive frequencies of the IO-PR mixing phonon
modes. The limiting features of dispersive curves of these phonon modes are discussed in depth. The phonon
modes “reducing” the behavior of wurtzite quantum confined structures have been explicitly observed in the
systems. Moreover, the behaviors that the IO-PR mixing phonon modes in wurtzite Q0D QDs reduce to the
IO modes and PR modes in wurtzite Q2D QW and Q1D QWR systems are profoundly analyzed both from the
viewpoint of physics and mathematics. These results show that the present theories of polar mixing phonon
modes in wurtzite cylindrical QDs are consistent with the phonon modes theories in wurtzite QWs and QWR
systems. The analytical electron-phonon interaction Hamiltonians obtained here are useful in further analyzing
the phonon effect on optoelectronic properties of wurtzite Q0D QD structures.
Key words: wurtzite nitride quantum dots, phonon states, electron-phonon interactions, quantum size effect
PACS: 81.05.Ea, 78.67.Hc, 63.22.-m, 63.20.Kd
1. Introduction
Thanks to the excellent characteristics of wide band-gap, strong atomic bonding and high elec-
tronic mobility as well as high optical efficiency, GaN-based semiconductor materials are quite
attractive materials as a basis for the creation of reliable high-power or high-temperature elec-
tronic equipment and short wave-length optoelectronic devices [1–14]. Following this trend, the
investigations on various physical properties of nitride semiconductor low-dimensional quantum
structures, such as quasi-2-dimensional (Q2D) quantum wells (QWs) [2–8], quasi-1-dimensional
(Q1D) quantum wires (QWRs) [9–16] have become a hot topic. Among these research attempts,
the crystal lattice dynamical properties of GaN-based quantum structures have attracted special
attention both from theoretical and experimental viewpoint [2–16]. The driving force behind these
efforts lies in the evident fact that lattice vibrations have an important effect on the optoelectronic
and electronic properties of nitride low-dimensional quantum systems [6–13]. In fact, the phenom-
ena of phonon replicas in the emission spectra, the homogeneous broadening of excitonic line width
and the relaxations of hot carriers to the fundamental band edge are directly related to the lattice
vibration of nitride materials [14]. Hence, lattice dynamical properties of nitride low-dimensional
quantum structures, especially the Q2D QW and Q1D QWR structures have been intensively
studied for the last two decades [2–16].
∗E-mail: zhangli-gz@263.net
c© L. Zhang 13801-1
http://www.icmp.lviv.ua/journal
L. Zhang
With the technological advancement of the crystal growing, not only Q2D nitride QW and Q1D
nitride QWR structures, but also quasi-0-dimensional (Q0D) nitride quantum dots (QDs) can be
fabricated [17–27]. It is well known that group-III nitride usually crystallizes in the hexagonal
wurtzite structure, whose physical behavior is anisotropic in space. The previous works on the
wurtzite QW and QWR structures reveal that the lattice vibrating modes (i.e. phonon modes)
and electron-phonon interactions become more complicated with the increase of confined dimensi-
onality [2–10]. Though the bound electronic states, excitonic states, donor bound excitons, as well
as the nonlinear optical properties in the Q0D wurtzite QD have been widely investigated [23–27],
the polar optical phonon states and their coupling features with electrons in wurtzite Q0D QD
systems have rarely been reported due to the complexity of phonon modes in the structures origi-
nated from the high confined dimensionality and anisotropic wurtzite structure [28–30]. Fonoberov
et al. [28] derived an integral equation that defines interface optical (IO) and confined modes in
wurtzite nanocrystals. The spectrum of polar optical-phonon modes is calculated numerically and
discussed only on a wurtzite spheroidal nanocrystal. Chassaing and coworkers [29] analyzed the
surface optical phonon modes in a wurtzite cylindrical ZnO QD structure, and two types of surface
phonon modes, i.e. side surface phonon modes and top surface phonon modes are found. In their
studies, infinite height (radius) for the side (top) surface phonon modes were assumed. With the
aid of the DCM and Loudon’s uniaxial crystal model [31], we extended the works of polar optical
phonon modes from the wurtzite Q2D QWs and Q1D QWRs to the wurtzite Q0D QDs structures
[30]. However, the dispersive properties of phonon modes in the wurtzite QDs have not been di-
scussed, and electron-phonon interaction Hamiltonian has not been established so far. Thus, we
will investigate the polar phonon states and their dispersive properties of a wurtzite Q0D QD, and
we will lay emphasis on the quantum size effect of the dispersive spectra of polar phonon modes
in the Q0D structure in the present paper.
In fact, the size-dependence of phonon spectra of CdSe/Te and AlGaAs QDs have been widely
reported [32–39]. Paula’s group [32] experimentally studied the phonon spectra of CdTe QDs as a
function of the QD size by means of resonant Raman scattering measurements. Their results show
that, as the QD size decreases, the surface optical (SO) modes scattering intensity increases, but the
electron-phonon coupling decreases. Hwang et al. [33] and Baranov et al. [34] investigated the effect
of quantum size on the polar optical phonon modes in CdSe QDs. The blue-shift and broadening
of the SO phonon frequency were clearly observed as the QD size was reduced. Dzhagan et al.
[35] analyzed the size effects on the dispersive features of phonon modes in CdSe nanoparticles
by using the resonant Raman scattering technology. Recently, Lange et al. [36] experimentally
studied the geometry dependence of the first-order Raman scattering band of CdSe nanorod. They
observed an explicit frequency blue-shift of longitudinal optical (LO)-like phonon modes as the
diameter of the nanorod increases. Comas and coauthors [37] theoretically deduced and analyzed
the SO phonon modes in spherical nanostructured QDs and semiconductor quantum rods under
the standard dielectric continuum (DC) approach. Their discussions were mainly focused on the
dispersion of SO modes as functions of the QD size and dielectric constants εd of matrix. Vasilevskiy
[38] discussed the dispersion frequency of the dipolar vibrational modes versus the radius in a
CdSe QD embedded in different nonpolar matrix. Kanyinda-Malu and Cruz [39] investigated the
oscillation spectra of IO and LO phonon modes as a function of the radius in AlGaAs cylindrical
QDs.
However, up to now, there has been little research into the size dependence of polar vibration
spectra in wurtzite quantum systems [40], especially for the new synthesized wurtzite Q0D QDs [17–
23]. Furthermore, the calculations of polaronic effects in Q0D QD revealed that the quantum-size
can greatly effect the binding energy of the polarons in Q0D QD. Moreover, due to the reduction of
the dimensionality and the anisotropy of Q0D wurtzite structures, the properties of optical phonon
modes in wurtzite QDs should have more distinct phonon branches [24–29]. Hence, it is of vital
importance to investigate the size-dependence phonon spectra in Q0D wurtzite QDs.
The main accomplishments and significance of this work are as follows. (i) Based on the DC
model (DCM) and Loudon’s uniaxial crystal model, the explicit phonon states and dispersive
equations of important optical phonon modes, i.e. the IO-propagating (PR) phonon mixing modes
13801-2
Polar interface optical phonon states and their dispersive properties
in wurtzite cylindrical GaN-based QDs are given. The difference and relationship of the phonon
states as well as dispersive equations in wurtzite quantum structures with three different confined
dimensionality (i.e. QWs, QWRs and QDs) are profoundly analyzed both from the viewpoint of
physics and mathematics. (ii) Numerical calculations on the size-dependence dispersive relation of
the IO-PR mixing phonon modes are performed, and their characteristics are discussed in detail.
Both the radial- and axial- direction size effects are discussed. The limiting behavior of the IO-PR
mixing phonon modes as the radial- and axial- direction sizes approach infinity is profoundly ana-
lyzed from the viewpoint of physics and mathematics, and a detailed comparison with Q2D QWs
and Q1D QWRs [2–5, 9, 10] is also carried out. (iii) The orthogonal relation of polarization eigen-
vector of IO-PR mixing phonon modes are obtained, and the Fröhlich electron-phonon interaction
Hamiltonians are also deduced using the method of field quantization. The analytical electron-
phonon coupling functions are important and useful for further investigation of polaronic effect on
the electronic and optical properties in wurtzite Q0D QD structures. The paper consists of four
sections and these are as follows: in section 2, the phonon states of IO-PR mixing modes and their
dispersive equations are presented, and the orthogonal relation of polarization eigenvector as well
as the Fröhlich electron-phonon interaction Hamiltonians are deduced; in section 3, the numerical
calculations on the dispersive frequency of two types of IO-PR mixing modes are carried out and
discussed; and finally, we summarize the main results obtained in the paper in section 4.
2. Theory
It is well known that the wurtzite GaN QDs are frequently of different shapes and symmetries,
such as spherical-caps, spheroids, hexagonal pyramids and cylindroid structures [17–23], which
strongly depends on the material nature and the conditions of growth. For simplicity, we consider
a freestanding wurtzite cylindrical QD structure with radius R and height 2d along the z-direction.
The z-axis is taken to be along the direction of the c-axis of the wurtzite material and denotes
the radial- (axial-) direction as t (z). Thus, under the cylindrical coordinate, the heterointerfaces
of the wurtzite QD in z-direction are located at z = ±d, and in radius the heterointerface is at
ρ = R. The frequency-dependent dielectric functions in t-direction and z-direction are given by
εt(ω) = ε∞t
ω2 − ω2
t,L
ω2 − ω2
t,T
, εz(ω) = ε∞z
ω2 − ω2
z,L
ω2 − ω2
z,T
. (1)
Here ωz,L, ωz,T , ωt,L and ωt,T are the zone center characteristic frequencies of A1(LO), A1(TO),
E1(LO), and E1(TO) modes, respectively.
2.1. Phonon states of IO-PR mixing modes and their dispersive equations of GaN cylindri-
cal QDs
Under the DCM and considering the case of free oscillations (the charge density ρ0(r) = 0),
the electric displacement vector D of phonon modes in wurtzite QD satisfies the relation,
∇ ·D = −∇2[ε(ω)Φ(r)]
= −ε0
{
εt(ω)
[
1
ρ
∂
∂ρ
(ρ
∂
∂ρ
) +
1
ρ2
∂2
∂ϕ2
]
+ εz(ω)
∂2
∂z2
}
Φ(r) = 0. (2)
Based on the Loudon’s uniaxial crystal modes [31] and the Laplace equation (2) in wurtzite crystal,
it can be confirmed that there are four types of polar mixing phonon modes in wurtzite cylindrical
QD structures [30]. As the first step of solving the complicated mixing optical phonon modes in
wurtzite QDs, we will pay attention to the IO-PR mixing phonon modes hereinafter.
The IO-PR mixing mode is a mode which behaves as IO mode in t(z)-direction, and behaves
as PR mode in z(t)-direction. Considering the exchange of t-direction and z-direction, it is found
that the IO-PR mixing modes are also of two forms, i.e. the z−IO/ρ−PR and ρ−IO/z−PR mixing
13801-3
L. Zhang
modes. Under the cylindrical coordinates, the electrostatic potential functions of z−IO/ρ−PR
mixing modes are given by
Φz−IO/ρ−PR
m (r) = eimϕfPR(ρ)φIO(z),
fPR(ρ) =
{
a1Jm(kt1ρ) ρ 6 R
a2Jm(kt2ρ) + a3Ym(kt2ρ) ρ > R
,
φIO(z) =
b1 exp(kz2z) z < −d
b2 sinh(kz1z) |z| 6 d
−b1 exp(−kz2z) z > d
, AS
b1 exp(kz2z) z < −d
b2 cosh(kz1z) |z| 6 d
b1 exp(−kz2z) z > d
, S
. (3)
For the ρ−IO/z−PR mixing modes, their electrostatic potentials can be written as
Φρ−IO/z−PR
m (r) = eimϕf IO(ρ)φPR(z),
f IO(ρ) =
{
A1Im(kt1ρ) ρ 6 R
A2Km(kt2ρ) ρ > R
,
φPR(z) =
B1 exp(ikz2z) z < −d
B2 sin(kz1z) |z| 6 d
−B1 exp(ikz2z) z > d
, AS
B1 exp(ikz2z) z < −d
B3 cos(kz1z) |z| 6 d
B1 exp(ikz2z) z > d
, S
. (4)
In equation (3), Jm(x) and Ym(x) are the Bessel and Neumann functions of m-order, respectively.
In equation (4), Km(x) and Im(x) are the first- and second-kind modified Bessel functions of the
order m, respectively. Here ai (Ai) and bi (Bi) are coupling coefficients of phonon modes determined
by additional boundary conditions (BCs). Due to the complexity of the coupling coefficients, they
are not displayed completely here. Only some important coupling coefficients b2, B2 and B3 are
given in the Appendix [because the three coefficients appear in equations (10)–(15) and Fröhlich
electron-phonon interaction Hamiltonian (19)]. The other coefficients can be referred to in the
Appendixes of [5] and [41]. The symbols “AS” and “S” in equations (3) and (4) correspond to
the antisymmetrical solution and symmetrical solution, respectively. This treatment completely
satisfies the symmetry demand of the phonon potential in z-direction. According to the relationship
of Loudon’s uniaxial crystal model [31] and the Laplace equation (2) in the areas of the inner and
outer QD, the dependent relations of the phonon wave-numbers kuv (u = t, z; v = 1, 2) can be
chosen as
√
εti(ω)kti ±
√
εzi(ω)kzi = 0,
√
εz1(ω)kz1 ±
√
εz2(ω)kz2 = 0. (5)
It should be noted that the subscripts 1 and 2 in equations (3), (4) and (5) correspond to the GaN
material and vacuum dielectric environment, respectively. The reasonableness of equation (5) lies in
the fact that the present theories in wurtzite Q0D QDs can naturally reduce to the corresponding
results of Q1D QWR and Q2D QW structures well known under a certain condition, which will
be discussed in detail hereinafter.
Next, we deduce the dispersive equations of the IO-PR mixing phonon modes. For the
ρ−IO/z−PR mixing modes, using the continuity BCs of the potential functions and the elec-
tric displacement vector at the axial interfaces z = ±d and the radial interface ρ = R, one can
get the following two equations [5, 9], i.e.
kz1 = nπ/2d, n = ±1,±2, . . . (6)
13801-4
Polar interface optical phonon states and their dispersive properties
and
εt,1kt1Km(kt2R)[Im−1(kt1R) + Im+1(kt1R)] = −εt2kt2Im(kt1R)[Km−1(kt2R) + Km+1(kt2R)]. (7)
In equation (6), n taking even (odd) number corresponds to the symmetric (antisymmetric)
ρ−IO/z−PR mixing phonon modes. Connecting equations (5), (6) and (7), the dispersive fre-
quencies of ρ−IO/z−PR mixing modes can be worked out. In the same way, via the continuity
BCs of the potential functions and the electric displacement vector at the axial interfaces z = ±d
and the radial interface ρ = R, one can obtain two other dispersive equations (8) and (9) for the
z−IO/ρ−PR mixing modes [4, 10]. They are given by
kz1 =
{
arctanh
√
−εz2/εz1/d, S
arctanh
√
−εz1/εz2/d, AS
(8)
and
{kt2εt,2Jm(kt1R)[Jm−1(kt2R) − Jm+1(kt2R)]
+ kt1εt,1Jm(kt2R)[Jm+1(kt1R) − Jm−1(kt1R)]}Ym(kt2L)
= Jm(kt2L){kt1εt,1[Jm+1(kt1R) − Jm−1(kt1R)]Ym(kt2R)
+ kt2εt,2[Ym+1(kt2R) − Ym−1(kt2R)]Jm(kt1R)}. (9)
In the same way, “S” and “AS” in equation (8) also denote the symmetric and antisymmetric
z−IO/ρ−PR mixing modes. In equation (9), L is the maximum radial size of the nonpolar dielectric
environment (in general, L � R). The dispersive frequencies and properties of the z−IO/ρ−PR
mixing modes in GaN QDs can be completely obtained by numerically solving the equations (8)
and (9).
2.2. Free phonon fields and Fröhlich electron-phonon interaction Hamiltonians
To obtain the expressions for the Hamiltonian of the free phonon field and electron-phonon
interaction Hamiltonian, we first institute the orthogonal relationships of polarization vector of
IO-PR mixing phonon modes. Using the formula P = (1 − ε)/4π∇Φ(r), and via equation (3), we
obtain the orthogonal relations of polarization vector for z−IO/ρ−PR mixing modes, i.e.
∫
P
z−IO/ρ−PR∗
AS,m′ ·Pz−IO/ρ−PR
AS,m d3
r
=
|A0|2
16π
∫
ρdρdz{b2
2 sinh(kz1z)2(1 − εt1)
2k2
t1[J
2
m−1(kt1ρ) + J2
m+1(kt1ρ)]
+2J2
m(kt1ρ)(1 − εz1)
2k2
z1b
2
2 cosh(kz1z)2}δm′m (10)
and
∫
P
z−IO/ρ−PR∗
S,m′ ·Pz−IO/ρ−PR
S,m d3
r
=
|A0|2
16π
∫
ρdρdz{b2
2 cosh(kz1z)2(1 − εt1)
2k2
t1[J
2
m−1(kt1ρ) + J2
m+1(kt1ρ)]
+2J2
m(kt1ρ)(1 − εz1)
2k2
z1b
2
2 sinh(kz1z)2}δm′m (11)
for the antisymmetrical and symmetrical z−IO/ρ−PR mixing phonon modes, respectively. Based
on equation (4), the orthogonal relations of polarization vectors for symmetrical and antisymmet-
rical ρ−IO/z−PR mixing modes are unified as
∫
P
ρ−IO/z−PR∗
AS/S,m′
·Pρ−IO/z−PR
AS/S,m d3
r
=
|A0|2
16π
∫
ρdρdz{|B2 cos(kz1z) + B3 sin(kz1z)|2 (1 − εt1)
2k2
t1K
2
m(kt2R)[I2
m−1(kt1ρ) + I2
m+1(kt1ρ)]
+2K2
m(kt2R)I2
m(kt1ρ)(1 − εz1)
2k2
z1 |B2 sin(kz1z) + B3 cos(kz1z)|2}δm′m. (12)
13801-5
L. Zhang
When deducing the above orthogonal relations of polarization vectors (10)–(12), only the region
inside the cylindrical QDs is considered, and the region outside the QDs is neglected because the
polarization vectors in this region is null due to εd = 1. Furthermore, it is also observed from these
equations that only the azimuthal quantum number m is a good quantum number originating
from the symmetry of cylindrical QDs, which is obviously different from the situations of QWs and
QWRs systems [2–5, 9, 10, 29]. Choosing suitable normalization constants A0 (A0), i.e.,
|A0|2 =
{
1
2ω2
∫
ρdρdz
{
b2
2 sinh2(kz1z)εt1k
2
t1[J
2
m−1(kt1ρ) + J2
m+1(kt1ρ)]
+2J2
m(kt1ρ)εz1k
2
z1b
2
2 cosh2(kz1z)
}
}−1
(13)
for antisymmetrical z−IO/ρ−PR mixing modes,
|A0|2 =
{
1
2ω2
∫
ρdρdz{b2
2 cosh2(kz1z)εt1k
2
t1[J
2
m−1(kt1ρ) + J2
m+1(kt1ρ)]
+2J2
m(kt1ρ)εz1k
2
z1b
2
2 sinh2(kz1z)}
}−1
(14)
for symmetrical z−IO/ρ−PR mixing modes, and
|A0|2 =
{
1
2ω2
∫
ρdρdz{|B2 cos(kz1z) + B3 sin(kz1z)|2 εt1k
2
t1K
2
m(kt2R) [I2
m−1(kt1ρ) + I2
m+1(kt1ρ)]
+2K2
m(kt2R)I2
m(kt1ρ)εz1k
2
z1 |B2 sin(kz1z) + B3 cos(kz1z)|2}
}−1
(15)
for the symmetrical and antisymmetrical ρ−IO/z−PR mixing phonon modes, the polarization
vectors can be treated as orthogonal and complete sets, which can be used to express the free
phonon field HIO−PR of IO-PR mixing modes and the Hamiltonians He−ph of electron-(IO-PR)
mixing phonons interactions. In equation (13)–(15), εv1 (v = t, z) is the effective dielectric function
of the GaN material, which is defined as
εv =
(
1
εv − εv0
− 1
εv − εv∞
)−1
, v = t, z. (16)
Using the orthogonal relations (10)–(12) and following the quantization steps similar to those in
[2–5], one can obtain the free phonon Hamiltonian operators for the IO-PR mixing phonons, i.e.
HIO−PR =
∑
m,kz1
~ω
[
b†m(kz1)bm(kz1) +
1
2
]
, (17)
where b†m(kz1) and bm(kz1) are creation and annihilation operators for the IO-PR mixing phonons
of mth modes. They satisfy the commutative rules for bosons:
[bm(kz1),b
†
m′(kz1)] = δm′m,
[bm(kz1),bm′(kz1)] = [b†m(kz1),b
†
m′(kz1)] = 0. (18)
The interaction Hamiltonians of electron with the IO-PR phonon fields is read as He−ph =
−eΦIO−PR(r). Expanding ΦIO−PR(r) in terms of the normal modes, and after some trivial al-
gorithms, we get the electron-(IO-PR)phonons interaction Hamiltonians as
He−ph = −e
∑
m,kz1
(
~
8πω
)1/2
[
b†m(kz1) + bm(kz1)
]
ΦIO−PR
m (r)
= −
∑
m,kz1
ΓIO−PR
m,kz1
(ρ)ΓIO−PR
m,kz1
(z)
[
bm(kz1)e
imϕ + H.c.
]
, (19)
13801-6
Polar interface optical phonon states and their dispersive properties
where ΓIO−PR
m,kz1
(ρ) and ΓIO−PR
m,kz1
(z) are the coupling functions defined as
ΓIO−PR
m,kz1
(ρ) =
√
|Nm(kz1)|f IO,PR(ρ),
ΓIO−PR
m,kz1
(z) =
√
|Nm(kz1)|φIO,PR(z), (20)
with
|Nm(kz1)| =
√
~e2
ω
|A0| . (21)
In equation (20), the functions fSO,PR(ρ) and φSO,PR(z) are defined in equation (3) and equa-
tion (4).
3. Numerical results and discussion
In the present section, we will numerically discuss the quantum size effect on the dispersive
properties of IO-PR mixing phonon modes in a wurtzite GaN cylindrical QD. The physical param-
eters of the materials used in our calculations are listed in table 1.
Table 1. Zone-center energies (in cm−1) of polar optical phonons, dielectric constants of wurtzite
GaN material and dielectric matrix [3, 4].
A1(TO) E1(TO) A1(LO) E1(LO) ε∞ ε0 εd
532 559 734 741 5.35 9.2 1
Figure 1. Dispersive frequencies ω of the first four branches of ρ-IO/z-PR mixing modes (n =
1, 2, 3, 4) as a function of the half-height d of the wurtzite GaN cylindrical QDs when the radius
R = 4.8 nm and the azimuthal quantum number m = 0.
Let us first investigate the dispersive feature of the ρ-IO/z-PR mixing phonon modes. In figure 1,
the dispersive frequencies ω of the ρ-IO/z-PR mixing modes as a function of the half-height d of
the wurtzite GaN cylindrical QDs are plotted when the radius R takes 4.8 nm and the azimuthal
quantum number m is kept at 0. Only the first four branches (n = 1, 2, 3, 4) of ρ-IO/z-PR mixing
modes with m = 0 are depicted. In fact, for a certain half-height d and a given azimuthal quantum
13801-7
L. Zhang
number m, the equations (6) and (7) have infinite solutions for ω, which means that there exists
infinite branches of ρ-IO/z-PR mixing modes in the GaN QD structures. From the figure, it can
be seen that all the phonon branches start with the constant frequency value 715.42 cm−1, and
decrease monotonously to the characteristic frequency ωtT1 of GaN material with the increase of
d. The decline of low-order modes (n is small) is steeper than that of the high-order ones (n is
relatively large). Via a detailed analysis, it is found that the frequency value of 715.42 cm−1 is
just the root of equation εt1 = 1, and this equation also determines the limiting frequencies of
phonon modes for quite large wave-numbers in wurtzite QW and QWR structures [4, 5, 9]. This
interesting feature needs to be further explained. From a pure viewpoint of mathematics, as d → 0,
the phonon wave-numbers kz1 and ku2 (u = t, z) will approach the infinity via equations (6) and
(5). Based on the limiting relations of modified Bessel functions [42], i.e.
lim
x→∞
Im(x) = ex/
√
2πx,
lim
x→∞
Km(x) = e−x
√
π/
√
2x, (22)
it is easy to prove that, as ktv → ∞ (v = 1, 2), equation (7) will degenerate to the form of
εt1 = 1. This equation just gives the frequency of 715.42 cm−1. From a physical viewpoint, the
wave-lengths of mixing phonon modes become very short as kuv → ∞ (u = t, z; v = 1, 2), thus
the phonons cannot distinguish planar heterostructure and the curved cylindrical heterostructure
[43]. This directly results in the identical limiting frequency of phonon modes in wurtzite QW,
QWR and QD systems for very large wave-numbers [4, 5, 9]. On the other hand, we observe that
the dispersive curve of n = 1 ρ-IO/z-PR mode is cut off at about d = 19 nm, and that of n = 2
mode is cut off at about d = 36 nm. The other two mixing phonon modes with n = 3, 4 are also
cut off at two certain d. These are the typical “reducing” behaviors of confined phonon modes in
wurtzite quantum structures [2–10]. In fact, as ω is lower than ωtT1, the sign of εt1(ω)εz1(ω) will
become negative, thus the ρ-IO/z-PR mixing phonon modes cannot exist in this situation, and
they will reduce to the other phonon modes, such as the half-space modes or quasi-confined modes
[2–5, 9, 10].
Figure 2. Dispersive frequencies ω of the first five branches of ρ-IO/z-PR mixing phonon modes
as a function of the radius R when m = 0 and d = 2.4 nm.
The dispersive frequencies ω of ρ-IO/z-PR mixing phonon modes as a function of the radius R
of the wurtzite GaN QDs are depicted in figure 2. A certain azimuthal quantum number (m = 0)
and a certain half-height (d = 2.4 nm) are chosen to plot the figure. For clarity, only the first five
13801-8
Polar interface optical phonon states and their dispersive properties
branches of mixing phonon modes are shown here. It is explicitly seen that the frequency range of
the ρ-IO/z-PR mixing modes is [715.42 cm−1, ωtT1], which is exactly as the case in figure 1. All the
curves are the monotonic and decreasing functions of R. The “reducing” behavior of ρ-IO/z-PR
mixing phonon modes is also observed again in the figure, namely the curves are cut off at ω = ωtT1
as the radius R approaches 0. When R approaches ∞, the frequencies of all the ρ-IO/z-PR modes
converge to the constant frequency value of 715.42 cm−1. In fact, via the limiting relations (22) of
modified Bessel functions, the dispersive equation (7) can also degenerate to the form of εt1 = 1,
thus the limiting frequencies of ρ-IO/z-PR mixing phonon modes for very large R also converge to
the constant of 715.42 cm−1.
Figure 3. Dispersive curves of the z-IO/ρ-PR mixing modes as a function of d when R =
4.8 nm and m = 0. The symbols of S-i (AS-i) in the figure denote the i-order symmetrical
(antisymmetrical) z-IO/ρ-PR modes.
The dispersive curves of the z-IO/ρ-PR mixing modes as a function of the half-height d of the
GaN QDs are plotted in figure 3 when R and m are respectively fixed at 4.8 nm and 0. From the
figure, it is clearly seen that dispersive curves of z-IO/ρ-PR mixing phonon modes are distributed in
the frequency range of [ωtT1, ωzL1]. This range is divided into two ranges at ω = 706.03 cm−1. The
above range of [706.03 cm−1, ωzL1] is the frequency range of the antisymmetrical z-IO/ρ-PR mixing
modes, while neither range of [ωtT1,706.03 cm−1] is the frequency range of the symmetrical z-IO/ρ-
PR mixing modes. In the figure, the symmetrical (antisymmetrical) z-IO/ρ-PR modes are labeled
by S-i(AS-i). Based on equations (8) and (9), there are infinite branches of z-IO/ρ-PR mixing modes
in the symmetrical and antisymmetrical frequency ranges, and only the first fourteen branches of
z-IO/ρ-PR modes in each frequency range are shown. The index i in symbols of S/AS-i denotes the
node-number of the electrostatic potential of z-IO/ρ-PR modes in t-direction. Using this feature,
all the symmetrical and antisymmetrical z-IO/ρ-PR modes are labeled and distinguished. It is
necessary to briefly discuss the origin of the characteristic value of 706.03 cm−1. In fact, the
frequency of 706.03 cm−1 is just the solution of equation, εz1 = 1. As d → ∞, the wave-numbers
kz1 approach 0 via the relationship (8). In terms of the nature of Bessel function and Neumann
function [42]:
lim
x→0
Jm(x) ∝ 1
Γ(m + 1)
(x
2
)m
,
lim
x→0
Ym(x) ∝
{ − 2
π ln(x/2)−m m = 0
−Γ(m)
π (x/2)−m m > 0
, (23)
13801-9
L. Zhang
the dispersive equation (9) will reduce to the form of εz1 = 1 as the wave-numbers kt1 and kt2
approach 0. This distinctly explains the mathematic origin of the frequency values of 706.03 cm−1.
The profound physical origin lies in the fact that the z-IO/ρ-PR mixing phonon modes of GaN QDs
will reduce to the corresponding phonon modes in Q1D GaN QWRs [9, 10] as d → ∞, which will
be analyzed in detail hereinafter. We also notice that, from the figure, the lowest-order z-IO/ρ-PR
mode in high frequency range is AS-0, but the lowest-order z-IO/ρ-PR mode in low frequency
range is S-1. All the symmetrical (antisymmetrical) z-IO/ρ-PR modes are the monotonous and
incremental (decreasing) function of R. By contrast to the symmetrical z-IO/ρ-PR modes, the dis-
persions of antisymmetrical z-IO/ρ-PR modes are weaker. Moreover, the symmetrical z-IO/ρ-PR
modes explicitly indicate the “reducing” behavior of phonon modes in wurtzite confined systems,
i.e. they are cut off at the characteristic frequency ωtT1 of GaN at a series of certain values of d.
Figure 4. Dispersive frequencies ω of z-IO/ρ-PR mixing phonon modes as a function of R of the
GaN QDs when the azimuthal quantum number m = 0 and the half-height d = 2.4 nm. The
meanings of the symbols (S/AS-i) in the figure are the same as those in figure 3.
Figure 4 depicts the dispersive frequency ω of z-IO/ρ-PR mixing phonon modes as a function of
the radius R of the GaN QDs. Same as in figure 2, m = 0 and d = 2.4 nm are chosen when plotting
the figure. The meanings of the symbols (S/AS-i) in the figure are the same as those in figure 3.
It is observed that all the curves are located in two frequency ranges, i.e. the higher-frequency
range (antisymmetrical z-IO/ρ-PR modes) and the lower-frequency range (symmetrical z-IO/ρ-PR
modes), which is quite similar to the case in figure 3. The symmetrical z-IO/ρ-PR modes are more
dispersive than the antisymmetrical modes. Moreover, the dispersions of the lower-order (i of S-i is
small) symmetrical z-IO/ρ-PR modes are more explicit than those of the higher-order modes. With
the increase of R, the frequencies of antisymmetrical z-IO/ρ-PR modes nearly remain unchanged.
As stated in figure 3, the i of S/AS-i represents the node-numbers of phonon electrostatic potentials
in t-direction. From the figure, we find that, as R < 24 nm, only the symmetrical z-IO/ρ-PR modes
with the order higher than 10 (i > 10) appear. However, all the antisymmetrical z-IO/ρ-PR modes
(from zero-order to infinity order) can be observed (only the first ten branches antisymmetrical
modes are shown here).
Now let us briefly discuss the relation and difference between the present phonon modes theories
of wurtzite Q0D QDs and those of wurtzite Q1D (Q2D) QWR (QW) structures [2–5, 9, 10]. From
a purely mathematical viewpoint, as the half-height of GaN QDs d → ∞, the wave-numbers kz1 of
IO-PR mixing phonon modes will become continuous via equations (6) or (8). Under this condition,
13801-10
Polar interface optical phonon states and their dispersive properties
based on equation (5) one can get the relations:
kz1 = kz2 = kz ,
kti =
√
εzi(ω)/εti(ω)kz . (24)
Substituting the two conditions of equation (24) into the equations (7) or (9), equation (7) [equa-
tion (9)] will reduce to the form of dispersive equation which is exactly the same as the dispersive
equation of IO (PR) phonon modes in Q1D wurtzite QWRs [9, 10]. This distinctly explains the
fact, from a mathematic viewpoint, that the ρ−IO/z−PR (z−IO/ρ−PR) mixing phonon modes
in Q0D wurtzite QDs reduce to the IO (PR) phonon modes in Q1D wurtzite QWRs under the
condition of d → ∞. From a viewpoint of physics, as the height d approaches ∞ and the radius
R is kept at a finite value, the physical model of the wurtzite Q0D cylindrical QD structures will
naturally reduce to the wurtzite Q1D cylindrical QWR structures. This further proves the correct-
ness and reliability of the phonon modes theories in Q0D wurtzite QD systems established in the
present work.
On the other hand, as the height 2d takes a certain value and the radius R approaches infinity,
via the limiting relations of Bessel function and Neumann function [42]:
lim
x→∞
Jm(x) =
√
2
πx
cos
[
x − (m +
1
2
)
]
π
2
,
lim
x→∞
Ym(x) =
√
2
πx
sin
[
x − (m +
1
2
)
]
π
2
, (25)
the wave-number kti (i = 1, 2) will become continuous based on equation (9). From equations (5),
one can get the relations:
kt1 = kt2 = kt ,
kzi =
√
εti(ω)/εzi(ω)kt . (26)
Thus the equation (8) reduces to the dispersive equation of IO phonon modes in Q2D wurtzite
QWs. This mathematical result clearly illustrates that, when R → ∞, the z-IO/ρ-PR mixing
modes will reduce to the IO modes in Q2D wurtzite QWs [3, 4]. In the same way, based on the
equations (7) and (22), it can be easily proven that the ρ-IO/z-PR mixing modes of wurtzite QDs
will reduce to the PR modes of Q2D wurtzite QW structures [5] as R approaches ∞. These are
also natural results because the cylindrical QDs will degenerate into Q2D QWs when the radius
of the QDs approaches ∞.
Finally, we should point out that, due to the effective limit of DCM theories [44–46], the present
theoretical schemes and results are meaningful only as the height 2d and radius R of the QD are not
very small (such as 2d and R are over 4 nm, i.e. ten GaN monolayers). In addition, the analytical
expressions of Fröhlich interactions between electron with IO-PR mixing phonon modes obtained
in the present paper are very useful in investigating the polaronic effect on the physical properties
of the commonly used nitride-based devices, such as LEDs and LDs based in the Q0D wurtzite
QD systems [1].
4. Conclusions
In conclusion, important polar optical phonon modes, i.e. the IO-PR mixing modes in a wurtzite
cylindrical QD system have been investigated in the present work. The analytical phonon states
are obtained. It is found that there exist two types of IO-PR mixing phonon modes in wurtzite
cylindrical QD structures, namely the ρ-IO/z-PR mixing modes and the z-IO/ρ-PR mixing modes.
Each IO-PR mixing phonon mode also has two forms. One is symmetrical, and the other one is
antisymmetrical. Based on the method of field quantization, the Fröhlich Hamiltonian of electron-
(IO-PR) mixing phonons interactions is given. The orthogonal relations of polarization vectors
13801-11
L. Zhang
for these IO-PR mixing modes are derived. Numerical calculations on a wurtzite GaN cylindrical
QD are performed. The quantum size effect on the dispersive properties of the IO-PR mixing
phonon modes are emphasized in the calculations. The results reveal that both the sizes of radial-
direction R and the axial-direction d have a great effect on the dispersive frequencies of the IO-PR
mixing phonon modes. The limiting features of dispersive curves for these mixing phonon modes
are analyzed in depth from the mathematical and physical viewpoints. The “reducing” behavior of
phonon modes in wurtzite quantum confined structures has been explicitly observed. We also find
that, as the height or the radius of the Q0D wurtzite cylindrical QDs approaches infinity, both
types of the IO-PR mixing modes (i.e. the ρ-IO/z-PR modes and the z-IO/ρ-PR modes) will reduce
to the IO modes or PR modes in Q2D wurtzite QW and Q1D QWR structures. These reducing
behaviors of IO-PR mixing modes in wurtzite QDs have been profoundly analyzed both from the
viewpoint of physics and mathematics. This shows that the theories of mixing phonon modes in
wurtzite Q0D QDs established in the present paper are consistent with those in wurtzite Q2D QW
and Q1D QWR systems [2–5, 9, 10]. Therefore, the present theories and numerical results turn out
to be correct and reliable. We hope that the present work will stimulate further theoretical and
experimental investigations into lattice dynamical properties, as well as the device applications
based on the Q0D wurtzite QD systems.
Acknowledgements
The author acknowledges the kind help and valuable discussions of Prof. J.J. Shi (Peking
University). This work was supported by Science and Technology Project of Advanced Academy
of Guangzhou City under Grant No. 2060, P. R. China.
Appendix
The coupling coefficients b2 for the symmetrical and antisymmetrical z-IO/ρ-PR mixing phonon
modes are respectively defined as
b2 = e−kz2d/ cosh(kz1d), (27)
and
b2 = −e−kz2d/ sinh(kz1d). (28)
The coupling coefficients B2 and B3 of the symmetrical and antisymmetrical ρ-IO/z-PR mixing
phonon modes are complex quantities. Thus, they can be written as
Bi = BRi + iBIi , (29)
where the real quantities BRi and BIi denote the real part and the imaginary part of Bi, respecti-
vely. The coupling coefficients B2 are given by
BR2 = 2kz2εz2 cos(kz1d) sin2(kz1d)(k2
z2ε
2
z2 − k2
z1ε
2
z1)/D, (30)
and
BI2 = −2kz1k
2
z2ε
2
z2εz1 cos(2kz1d) sin(kz1d)/D. (31)
The coefficients B3 are given by
BR3 = kz2εz2 sin(kz1d)[k2
z2ε
2
z2 − k2
z1ε
2
z1 + cos(2kz1d)(k2
z2ε
2
z2 + k2
z1ε
2
z1)]/D, (32)
and
BI3 = 0. (33)
In equations (30)–(33), D is defined as
D = kz1kz2εz1εz2[sin(kz1d) − sin(3kz1d)][kz1εz1 cos(kz2d) sin(kz1d) − kz2εz2 cos(kz1d) sin(kz2d)].
(34)
13801-12
Polar interface optical phonon states and their dispersive properties
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Стани полярних iнтерфейсних оптичних фононiв у квантовiй
точцi вюрциту GaN та їхнi дисперсiйнi властивостi: вплив
квантового розмiру
Л. Жанг
Полiтехнiка Гуанчжоу Панью, Гуанчжоу 511483, Народна Республiка Китай
На основi макроскопiчної дiелектричної неперервної моделi отримано i дослiджено iнтерфейснi
оптичнi пропагаторнi (IO-PR) змiшанi фононнi моди цилiндричної квантової точки (QD) структури
вюрциту квазiнульової вимiрностi (Q0D). Даються аналiтичнi фононнi стани IO-PR змiшаних мод.
Знайдено, що є два типи IO-PR змiшаних фононних мод, а саме, ρ-IO/z-PR змiшанi моди i z-IO/ρ-PR
змiшанi моди, що iснують у квантових точках Q0D вюрциту. Кожна IO-PR мода має симетричну та
антисиметричну форми. За допомогою стандартної процедури квантування отримано гамiльтонiан
Фрьолiха для взаємодiї електрон-(IO-PR) змiшанi фонони. Продемонстровано також ортогональнi
спiввiдношення поляризацiйних власних векторiв для цих IO-PR змiшаних мод. Числовi обчислення
для цилiндричної QD вюрциту GaN зосереджуються на впливi квантового розмiру на дисперсiйнi
властивостi IO-PR змiшаних мод. Результати показують, що як радiально напрямлений розмiр, так i
аксiально напрямлений розмiр мають великий вплив на дисперсiйнi частоти IO-PR змiшаних фонон-
них мод. Детально обговорено обмеження властивостей дисперсiйних кривих цих фононних мод.
Фононнi моди, що “редукують” поведiнку квантово обмежених структур вюрциту були явно спосте-
реженi у цих системах. Крiм того, така поведiнка, що редукує IO-PR фононнi моди у квантових точках
вюрциту Q0D до IO мод i PR мод у системах вюрциту Q2D QW and Q1D QWR є глибоко проаналiзо-
вана з точки зору фiзики i математики. Цi результати показують, що дана теорiя полярних змiшаних
фононних мод у цилiндричних квантових точках вюрциту узгоджується з теорiями фононних мод
для систем вюрциту QWs i QWR. Отриманий аналiтичний гамiльтонiан електрон-фононної взаємо-
дiї є корисним при наступному аналiзi фононного впливу на оптоелектроннi властивостi квантової
точки структур вюрциту Q0D.
Ключовi слова: вюрцит нiтриднi квантовi точки, фононнi стани, електро-фононнi взаємодiї, вплив
квантового розмiру
13801-14
Introduction
Theory
Phonon states of IO-PR mixing modes and their dispersive equations of GaN cylindrical QDs
Free phonon fields and Fröhlich electron-phonon interaction Hamiltonians
Numerical results and discussion
Conclusions
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| id | nasplib_isofts_kiev_ua-123456789-32051 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T13:34:42Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Zhang, L. 2012-04-06T18:19:09Z 2012-04-06T18:19:09Z 2010 Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect / L. Zhang // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13801: 1-14. — Бібліогр.: 46 назв. — англ. 1607-324X PACS: 81.05.Ea, 78.67.Hc, 63.22.-m, 63.20.Kd https://nasplib.isofts.kiev.ua/handle/123456789/32051 Based on the macroscopic dielectric continuum model, the interface-optical-propagating (IO-PR) mixing phonon modes of a quasi-zero-dimensional (Q0D) wurtzite cylindrical quantum dot (QD) structure are derived and studied. The analytical phonon states of IO-PR mixing modes are given. It is found that there are two types of IO-PR mixing phonon modes, i.e. ρ-IO/z-PR mixing modes and the z-IO/ρ-PR mixing modes existing in Q0D wurtzite QDs. Each IO-PR mixing mode also has symmetrical and antisymmetrical forms. Via a standard procedure of field quantization, the Fröhlich Hamiltonians of electron-(IO-PR) mixing phonons interaction are obtained. The orthogonal relations of polarization eigenvectors for these IO-PR mixing modes are also displayed. Numerical calculations for a wurtzite GaN cylindrical QD are focused on the quantum size effect on the dispersive properties of IO-PR mixing modes. The results reveal that both the radial-direction size and the axial-direction size have great effect on the dispersive frequencies of the IO-PR mixing phonon modes. The limiting features of dispersive curves of these phonon modes are discussed in depth. The phonon modes "reducing" the behavior of wurtzite quantum confined structures have been explicitly observed in the systems. Moreover, the behaviors that the IO-PR mixing phonon modes in wurtzite Q0D QDs reduce to the IO modes and PR modes in wurtzite Q2D QW and Q1D QWR systems are profoundly analyzed both from the viewpoint of physics and mathematics. These results show that the present theories of polar mixing phonon modes in wurtzite cylindrical QDs are consistent with the phonon modes theories in wurtzite QWs and QWR systems. The analytical electron-phonon interaction Hamiltonians obtained here are useful in further analyzing the phonon effect on optoelectronic properties of wurtzite Q0D QD structures. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect Стани полярних інтерфейсних оптичних фононів у квантовій точці вюрциту GaN та їхні дисперсійні властивості: вплив квантового розміру Article published earlier |
| spellingShingle | Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect Zhang, L. |
| title | Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect |
| title_alt | Стани полярних інтерфейсних оптичних фононів у квантовій точці вюрциту GaN та їхні дисперсійні властивості: вплив квантового розміру |
| title_full | Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect |
| title_fullStr | Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect |
| title_full_unstemmed | Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect |
| title_short | Polar interface optical phonon states and their dispersive properties of a wurtzite GaN quantum dot: quantum size effect |
| title_sort | polar interface optical phonon states and their dispersive properties of a wurtzite gan quantum dot: quantum size effect |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32051 |
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