Solid State Communications 135 (2005) 144–149
www.elsevier.com/locate/ssc
Confinement of polar optical phonons in AlN/GaN superlattices
S.K. Medeirosa,b, E.L. Albuquerqueb,*, G.A. Fariasa, M.S. Vasconcelosc,
D.H.A.L. Anselmoa,d
a
Departamento de Fı́sica, Universidade Federal do Ceará, Campus do Pici Caixa Postal 6030, 60451-900 Fortaleza-CE, Brazil
b
Departamento de Fı́sica, Universidade Federal do Rio Grande do Norte 59072-970 Natal-RN, Brazil
c
Departamento de Ciências Exatas, Centro Federal de Educação Tecnológica do Maranhão 65025-001 São Luı́s-MA, Brazil
d
Departamento de Fı́sica, Universidade do Estado do Rio Grande do Norte 59600-900 Mossoró-RN, Brazil
Received 11 November 2004; received in revised form 24 December 2004; accepted 16 February 2005 by J.A. Brum
Available online 4 March 2005
Abstract
We study the optical-phonon spectra in heterojunctions fabricated from III to V nitride materials (GaN and AlN). We are
concerned with the quaternary superlattice structure, namely, /substrate/AlN/AlxGa1KxN/GaN/AlxGa1KxN/./, where the
substrate is here considered to be a transparent dielectric medium like sapphire. We make use of a model based on the Frölich
Hamiltonian, taking into account the macroscopic theory developed by Loudon, known as the continuum dielectric model. The
optical-phonon modes are modelled considering only the electromagnetic boundary conditions (including retardation effects),
in the absence of charge transfer between ions. Numerical results of the confined optical-phonon spectra are presented,
characterizing three distinct optical-phonon classes designated as propagate (PR), interface (IF) and half-space (HS) modes.
Furthermore, due to the dielectric anisotropy presented in the nitrides, some additional peculiarities will be presented, like
dispersive confined modes.
q 2005 Elsevier Ltd. All rights reserved.
PACS: 63.20.Pw; 63.22.Cm; 68.65.Cd; 71.55.Eq
Keywords: A. Nanostructures; A. Semiconductors; D. Optical properties; D. Phonons
1. Introduction
Complex structures, with dimensions of nanometers,
composed by semiconductor materials have been a topic
of a lot of theoretical and experimental researches, due
to their technological potential application (for a review
see Ref. [1]). Among its applications, we stress the
enhancement of semiconductor lasers performance
through a better understanding of the confinement of
polar optical-phonons in the heterojunctions, single and
* Corresponding author. Tel.: C55 84 2153793; fax: C55 84
2153791.
E-mail address: eudenilson@dfte.br (E.L. Albuquerque).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2005.02.027
multiple quantum well structures, and superlattices [2].
For example, recent experimental studies in the quantum
heterostructures based on the wide-band-gap group-III
nitrides led to a light emission diode (LED) that emits
high-brightness blue light, and laser diodes emitting
green light [3–5].
Progress in microfabrication techniques enables us to
create various kinds of hetero-epitaxial interfaces between
two dissimilar and yet closely lattice-matched semiconductors, and facilitate the dramatic reduction in extrinsic
interface defects detrimental to the electron mobility and
other device parameters. In such heterojunctions, since
experimental reality is approaching theoretical models and
assumptions, detailed analysis and precise predictions are
unprecedentedly made possible [6]. This is particularly true
S.K. Medeiros et al. / Solid State Communications 135 (2005) 144–149
in the case of the nitrides heterojunctions, which are very
promising for the fabrication of resonant tunnelling diodes
for high-speed and high-power applications.
Investigations about optical-phonons in nitride semiconductors have been done due to the large gap characteristic featured by these semiconductors and their potential
technological applications in electronics and optoelectronics
devices [7]. Tailoring of optical-phonon modes in nanoscale
semiconductors was already proposed, suggesting that
confined phonon effects can be used to tune quantum-well
intersubband lasers [8].
The III–V nitride materials can crystallize in both
hexagonal wurtzite or cubic zinc-blend structures [1]. The
wurtzite crystals have a different unit cell structure (four
atoms per unit cell with nine optical and three acoustical
phonons for a given wavevector), as well as a lower
symmetry when compared to the cubic zinc-blende
counterpart, leading to a different carrier-phonon interaction. Although the theory of optical-phonon confinement
has been so far treated in detail for the cubic structures [9],
its understanding for the wurtzite crystals has been
primitive. Moreover, the optical-phonon confinement in
the latter materials has more complex nature than those in
the former one, and important physical properties, not
completely understood yet [10].
The hexagonal wurtzite structures are uniaxial crystals
with the optical axis coinciding with the cartesian z-axis,
which is perpendicular to the hexagons (forming the xyplane). Due to the crystal’s anisotropy, the frequency of the
polar phonons polarized along the optical axis is different
from that polarized in the xy-plane [11]. In these
polarization directions, only a group of three bulk polar
optical-phonons, among the nine optical-phonon modes, are
Raman and infrared active in the irreducible representation
of A1(z) (z-axis) and E1(xy) (xy-plane) at the G point. Two of
them are extraordinary waves associated with z- and xypolarized vibrations. The z-polarized mode has A1(z)
symmetry, while the xy-polarized one has E1(xy) symmetry.
The other one is an ordinary wave, which is always
transverse and polarized in the xy-plane, with E1(xy)
symmetry.
Optical vibration properties in heterojunctions were
extensively investigated both theoretically [12] and experimentally [13] during the past two decades. In particular, the
interface optical-phonon modes have been found to play a
dominant role in the electron–phonon interactions in
quantum wells and superlattices [14]. Considering a AlN/
GaN superlattice with sharp interfaces, it is well known that
its localized optical-phonon modes are bounded by the
longitudinal (LO) and transverse (TO) bulk optical-phonon
modes of its constituents, whereas the presence of
inhomogeneities such as surface, interface or defect layer
obviously will change this spectrum [15].
It is our aim in this work to investigate the polar opticalphonon spectra in multilayer structures composed of
AlN/GaN layers arranged in a periodical fashion. We
145
make use of a model based on the Frölich Hamiltonian,
taking into account the macroscopic theory developed by
Loudon to calculate the dispersion relation for the opticalphonon, which fulfills electromagnetic boundary conditions.
We consider also a transfer-matrix treatment to simplify the
algebra, which would be otherwise quite complicated, that
allows one to obtain a neat analytical expressions for the
phonon dispersion relation.
2. Theory
We now present our theory to study polar optical-phonon
modes in heterostructures composed of wurtzite GaN and
AlN, forming the quaternary superlattice structure, namely,
/substrate/AlN/AlxGa1KxN/GaN/AlxGa1KxN/./, where the
substrate is here considered to be a transparent dielectric
medium like sapphire. Although a comprehensive description of the phonon modes would require a full microscopic
lattice-dynamic calculation beyond the scope of this work,
we make use of the so-called dielectric continuum model,
which fulfills electromagnetic boundary conditions. We
justify its use because, besides its simplicity, it describes the
properties of dimensionally confined optical-phonons in
many electronic and optoelectronics devices fabricated from
semiconductor nanostructures, including quantum wells,
superlattices, quantum wires and quantum dots [16].
Within the macroscopic dielectric continuum model
approach, the fields associated with the polar optical modes
in each layer must satisfy Maxwell’s equations in the
retardation regime. Due to the symmetry, a matrix form of
the dielectric tensor 3(u) in each medium is given by
0
1
3t ðuÞ
0
0
B
C
0 A
3t ðuÞ
3ðuÞ Z @ 0
0
0
3s ðuÞ
Here 3t(u) and 3s(u) are the dielectric functions perpendicular and parallel to the z-axis, respectively. They are
given by:
3t ðuÞ Z 3t ðNÞ
3s ðuÞ Z 3s ðNÞ
u2 K u2LO;E1
u2 K u2TO;E1
u2 K u2LO;A1
u2 K u2TO;A1
(1)
(2)
where 3t(u) [3s(u)] is the high-frequency dielectric
constant perpendicular [parallel] to the z-axis, and uTO,X
(uLO,X), with XZA1(z) and E1(xy), is the transverse optical
(longitudinal optical) phonon angular frequency for the
mode X.
Solving Maxwell’s eqs., the electromagnetic fields
associated with the optical-phonon modes in a given layer
j are given by:
146
S.K. Medeiros et al. / Solid State Communications 135 (2005) 144–149
Exj Z Aj expðikjz zÞ C Bj expðKikjz zÞ
(3)
Ezj Z ½kjz qx =ð3sj u2 =c2 K q2x Þ½Aj expðikjz zÞ
K Bj expðKikjz zÞ
(4)
we can then rewrite (8)–(15) in matrix form as:
Hyj Z ½K30 u3sj kjz =ð3sj u2 =c2 K q2x Þ½Aj expðikjz zÞ
K Bj expðKikjz zÞ
(5)
kjz Z ð3tj =3sj Þ½3sj u =c
2
K q2x 1=2
(6)
As an example, let us first consider an isolated GaN
(AlN) layer, occupying the region zZ0 to zZKL, and
surrounded by vacuum (3Z1). Applying Maxwell’s boundary conditions (continuity of the tangential and normal
~ and D,
~ respectively), at the interfaces (zZ0
component of E
and zZKL), the dispersion relation for the optical-phonons
can be described by
3 q C ikz
ik K 3s q
expð2ikz LÞ s
Z z
ikz K 3s q 3s q C ikz
M1 jAn1 i ¼ N2 jAn2 i
M2 jAn2 i ¼ N3 jAn3 i
with qx being the common in-plane wavevector, and
2
each medium j at the nth layer the two-component column
vector
" n#
Aj
jAnj i Z
(16)
Bnj
M3 jAn3 i ¼ N2 jAn2 i
(17)
M2 jAn2 i ¼ N1 jAnþ1
1 i
where we have defined the matrices
!
fj
fjK1
Mj Z
gj fj Kgj fjK1
N j Z
1
1
gj Kgj
(18)
(19)
(7)
Using (17) it is easy to deduce that
with q2Zu2/c2Kq2x and kz is given by (6).
n1 i;
jA1nC1 i Z TjA
(20)
K1 K1 K1 T Z N K1
1 M2 N 2 M3 N 3 M2 N 2 M1
3. Results and discussions
Now we turn to the quaternary superlattice
/substrate/AlN/AlxGa1KxN/GaN/AlxGa1KxN/./. The unit
cell of the superlattice has thickness LZd1Cd3C2d2, where
dj is the thickness of the jth layer. For the superlattice bulk
modes, the electromagnetic field equations, defined by (3)–
(5), together with the boundary conditions of the nth unit
cell, i.e. the interfaces zZnLCd1 (AlN/AlxGa1KxN), zZ
nLCd2 (AlxGa1KxN/GaN), zZnLCd3 (GaN/AlxGa1KxN)
and zZ(nC1)L (AlxGa1KxN/AlN) yield, after a bit of
algebra:
where the matrix T is called a transfer matrix because it
relates the electrical (and hence the magnetic) field
amplitudes at a point in cell n to the equivalent point in
cell nC1. Taking into account the translational symmetry of
the system, by using Bloch’s ansatz, we obtain the following
eigenvalue equation:
n1 i Z expðiQLÞjAn1 i
TjA
(21)
where Q is the Bloch’s wavevector. Consequently, as T is a
unimodular matrix, (its determinant is equal to unity), the
dispersion relation for the superlattice optical-phonon mode
is simply given by:
An1 f1 C Bn1 f1K1 Z An2 C Bn2
(8)
cosðQLÞ Z ð1=2ÞTr ðTÞ
g1 ðAn1 f1 K Bn1 f1K1 Þ Z g2 ðAn2 K Bn2 Þ
(9)
We now introduce an external surface to the superlattice by considering it truncated at zZ0 with the half-space
zO0 filled by sapphire, whose frequency-independent
dielectric constant is denoted by 3s. This semi-infinite
superlattice does not possess full translational symmetry in
the z-direction, and therefore we may no longer assume
Bloch’s ansatz as in the bulk case. On the other hand, this
new interface allows the appearance of surface modes.
Instead of (22) we have
An2 f2 C Bn2 f2K1 Z An3 C Bn3
(10)
g2 ðAn2 f2 K Bn2 f2K1 Þ Z g3 ðAn3 K Bn3 Þ
(11)
An3 f3 C Bn3 f3K1 Z An2 C Bn2
(12)
g3 ðAn3 f3 K Bn3 f3K1 Þ Z g2 ðAn2 K Bn2 Þ
(13)
An2 f2 C Bn2 f2K1 Z A1nC1 C B1nC1
(14)
g2 ðAn2 f2 K Bn2 f2K1 Þ Z g1 ðAnC1
K BnC1
1
1 Þ
(15)
Here, gjZi3sj/kj and fjZexp(ikjdj).
Introducing a transfer-matrix formalism by defining for
cosðbLÞ Z ð1=2ÞTr ðTÞ
(22)
(23)
with Re(b)O0, as the condition for a localized mode.
The relevant electromagnetic fields in the region
occupied by sapphire (zO0) have the form:
Ex ðzÞ Z C expðKaS zÞ
(24)
S.K. Medeiros et al. / Solid State Communications 135 (2005) 144–149
Hy ðzÞ Z ðiu30 3S =aS ÞC expðKaS zÞ
(25)
where C is a constant, and aS Z ½q2x K 3S u2 =c2 1=2 .
Since, we now have to consider the extra boundary
conditions for the new interface at zZ0 sapphire/AIN
interface, assuming that layer AIN is the outermost layer in
the superlattice), this impose a further constraint in (23)
which enables us eventually to determine the attenuation
factor b.
Next, from the electromagnetic boundary conditions at
zZ0, we find after some algebra the implicit dispersion
relation for the surface optical modes, i.e.:
T11 C T12 l Z T22 C T21 lK1
(26)
with
l Z ð3S g1 C aS Þ=ð3S g1 K aS Þ
(27)
and Tmn (with m, nZ1, 2) are elements of the transfer matrix
Once this equation is solved, we can obtain a value for b
T.
which must satisfy (23) together with the requirement
Re(b)O0 to ensure localization.
Now we present the numerical results obtained for the
optical-phonon spectra in quaternary nitride superlattices.
To do so, we have used the following physical parameters
(all frequencies in units of cmK1):
(i) for GaN [17]: uLO,A1Z734, uTO,A1Z532, uLO,E1Z
741, uTO,E1Z559, and 3NZ5.35;
(ii) for AIN [18]: uLO,A1Z893, uTO,A1Z614, uLO,E1Z916,
uTO,E1Z673, and 3NZ4.84;
(iii) for AlxGa1KxN, with xZ0.15 [19]: uLO,A1Z772,
uTO,A1Z544, uLO,E1Z783, uTO,E1Z570, and 3NZ
5.20.
The thickness of all layers dj (jZ1, 2, 3, 4) are
considered to be equal to 20 nm, in such a way that the
size of the superlattice unit cell is LZ80 nm. For the sharp
heterointerface system under consideration here, there are
two distinct optical phonon classes, designated as interface
(IF) and half-space (HS). They are dependent on the
ordering of various phonon energy spectra in the nitride
materials 1 (AlN), 2 (Alx,Ga1KxN) and 3 (GaN) [20,21]. The
interface modes are the evanescent modes with maximum
amplitude at the interfaces. They appear when kjz (jZ1, 2
and 3), given by (6), is purely imaginary for both the well
and barriers, i.e. when 3tj!3sj%0. In addition we must
have 3sj!3sj 0 %0, for jsj 0 . The half-space modes are
interface modes which behaves as the nominal bulk modes
as z/GN. Besides these modes, one can identify also
a high frequency propagating mode (PR), which is a
phonon mode that propagates between the adjacent
materials, provided their dielectric properties do not
differ substantially and their dispersion curves overlap
[22,23].
Fig. 1 shows the optical-phonon (bulk and surface
modes) spectrum as a function of the reduced dimensionless
147
in-plane wavevector qxd1, d1, being the thickness of the AlN
layers (20 nm). The almost vertical light line, defined as uZ
cqx, is shown by a thin straight line. As we can see, the bulk
modes have two well-defined branches separated by a small
gap equal to the difference between the transverse optical
frequencies of AlN, namely uTO,E1 and uTO,A1, respectively.
These bulk bands are alternately bounded by the curves
QLZ0 and p. For small qxd1, the high-frequency bulk
branch is in the range uTO,E1 (AlN)%u%uLO,E1(GaN),
while the low-frequency one lies in the region uTO,A1
(GaN)%u%uTO,A1(AlN) for the same qxd1. The highfrequency PR mode emerges from the light with energyz0.12 eV, and then evolves toward the longitudinal optical
frequencies of AlN, namely uLO,A1. There are three IF
phonon modes for two distinct frequency ranges, all of them
belonging to the LO range. One of them lies between the
AlN longitudinal optical frequency uLO,A1 and the GaN
longitudinal optical frequency uLO,E1. The remaining two
phonon modes are between the longitudinal optical
frequencies of GaN, namely uLO,E1 and uLO,A1. Different
of the binary superlattice GaN/AlN, the IF phonon modes in
quaternary superlattices are not degenerate, even for large
in-plane wavevector qxd1. On the other hand, there is
one HF mode which propagates above the bulk bands,
emerging from the bulk band at qxd1z0.2, and then evolves
separately from it in a frequency range uTO,E1 (AlN)!u!
uLO,A1 (GaN). The remaining HF modes belong to the TO
range, two of them between the AlxGa1KxN E1 (TO) and the
GaN E1 (TO) frequencies, while the last one lies in the
frequency range defined by the A1 symmetry of GaN and
AlxGa1KxN layers.
Consider now the finite superlattice structure case,
obtained from the infinite one by truncating it at zZ0 and
zZpL, p being an integer and L the size of the superlattice
unit cell. It is surrounded by the isotropic media E (which
can be vacuum) and F (as a substrate, considered to be, as
before, the sapphire), which have dielectric constants 3V and
3S, respectively.
Just as in the semi-infinite case, we cannot use Bloch’s
theorem to relate the amplitude in one layer to that in
another one through the envelope function exp(imQL),
with m being the difference of the cells involved. Instead,
by an extension of results in the previous section, we have
to employ the envelope functions exp(KmbL) and exp[K
(pKm)bL], which are defined to correspond to localization of a surface mode at the top and bottom surfaces,
respectively. In the case of the bulk mode, one should
replace b by KiQ.
Let us assume, to simplify the algebra of the problem,
that the coefficients Aj and B j (jZ1, 2, 3) of the
electromagnetic fields which are related to the envelope
function exp(KmbL), are independent of the coefficients Aj0
and Bj0 of the electromagnetic fields associated with the
envelope function exp[K(pKm)bL]. This assumption
enables us to relate these coefficients in the eigenvalue
148
S.K. Medeiros et al. / Solid State Communications 135 (2005) 144–149
Fig. 1. Optical phonon’s spectra for a semi-infinite /substrate/AlN/AlxGa1KxN/GaN/AlxGa1KxN/./ superlattice as a function of the
dimensionless factor qxd1. The light line is defined by uZcqx.
equation of the transfer matrix T at the nth cell, i.e.
½T K expðKbLÞjAnj i Z 0
(28)
½T K expðbLÞjAj0 ni Z 0
(29)
tanhðpbLÞ Z
3s1 ð3E C 3F ÞðK K K 0 Þ
2
0
3s1 ð1 K KÞð1 K K Þ K 3E 3F ð1 C KÞð1 C K 0 Þ K 3s1 ð1 K KK 0 Þð3E
Hence we deduce:
B1 Z KA1 ;
B10 Z K 0 A10
(30)
with
K Z ½expðKbLÞ K T11 =T12
Also, K 0 is given by a similar expression to K, provided
we replace exp(KbL) by exp(bL) in (31).
Straightforward use of Maxwell boundary conditions
yields, after a heavy algebraic manipulation, the implicit
dispersion relation given by [24]:
(31)
K 3F Þ
(32)
This equation is particularly convenient to employ, as far
as numerical calculations are concerned, because the
number of cells in the finite superlattice, p, appears only
in the argument of the hyperbolic tangent function.
The optical-phonon dispersion curves, obtained from
(32), is shown in Fig. 2 considering the number of unitary
cells pZ20. As there is no difference in the bulk modes,
Fig. 2. Optical phonon’s surface modes dispersion curves considering a finite superlattice of thickness 20L, L being the size of the superlattice’s
unit cell.
S.K. Medeiros et al. / Solid State Communications 135 (2005) 144–149
149
besides the quantization of the bulk modes in this case due to
the finite thickness of the structure, we have decided to
present in Fig. 2 only the surface modes which differ from
the ones depicted in Fig. 1, namely:
although with their own peculiarities, as it is suggested by
recent Raman measurements [27].
(a) There is a new high-frequency phonon PR mode
emerging from the light line with energyz0.114 eV.
(b) The other PR mode leaves the light line with energy
closed to the longitudinal optical frequencies of AlN at
A1 symmetry, in contrast with its semi-infinite counterpart which evolves towards uLO,A1(AlN), tending to the
longitudinal optical (LO) frequency of AlxGa1KxN at E1
symmetry for a larger value of qxd1.
(c) The high-frequency IF phonon mode is slighted shifted
to the LO frequency of AlN (A1 symmetry), when
compared with its semi-infinite one.
(d) The high-frequency HF mode now belongs to the LO
range, emerging from the bulk band at qxd1z0.12, and
then evolves separately from it in the frequency range
[uLO,E1(GaN), uLO,E1(AlxGa1KxN)].
(e) The other HF mode, as in the semi-infinite case, belong
to the TO range but it lies in the frequency range defined
by the A1 symmetry of AlN and E1 symmetry of
AlxGa1KxN layer.
Acknowledgements
In summary we have calculated the dispersion relation of
the polar optical-phonon modes in wurtzite AlN/AlxGa1Kx
N/GaN/AlxGa1KxN/superlattices within the dielectric continuum approach. Although the low-frequency interface
optical-phonon modes for a GaN quantum well is sensitive
to the strain effects (for higher-frequency the influence of
the strain can be ignored) [25] our calculations, which were
performed for unstrained material, give a good qualitative
insight into the polar optical-phonon spectra in semi-infinite
(Fig. 1) and finite (Fig. 2) nitride superlattices. We have
neglected the 2.4% lattice mismatch which exists between
the materials [15]. Furthermore, due to the dielectric
anisotropy presented in the nitrides, the confined modes
found here are dispersive.
Although the theoretical predictions for the hexagonal
GaN structures considered here were not yet been observed
experimentally, optical properties of cubic GaN samples
deposited by plasma-assisted MBE on (001) GaAs substrates, with and without doping with carbon atoms, were
the focus of recent experimental work [26]. Contrary to their
hexagonal counterparts, the cubic structures can be grown
free from modulation, making them a good candidate for
further investigations on the optical-phonon spectra in these
compounds, using the same theoretical model employed
here. For this case, the expected spectra should have some
resemblance compared with the hexagon structures,
The authors would like to acknowledge the financial
support provided by the Brazilian Research Agencies
CAPES-Procad, CNPq-NanoSemiMat and Finep-CTInfra.
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