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Phys. Status Solidi B 252, No. 10, 2296–2303 (2015) / DOI 10.1002/pssb.201552156
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basic solid state physics
A study of the piezoelectric
properties of semipolar (112̄2)
GaN/AlN quantum dots
T. D. Young*,1 , G. Jurczak1 , A. Lotsari2 , G. P. Dimitrakopulos2 , Ph. Komninou2 , and P. Dłużewski1
1
Division of Computational Methods, Institute of Fundamental Technological Research of the Polish Academy of Sciences,
ul. Pawińskiego 5b, 02-106 Warsaw, Poland
2
Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Received 30 March 2015, revised 16 June 2015, accepted 16 June 2015
Published online 17 July 2015
Keywords III–V semiconductors, AlN, GaN, nanostructures, piezoelectric properties, quantum dots
∗
Corresponding author: e-mail tyoung@ippt.pan.pl
GaN quantum dots grown in (112̄2)-orientated AlN are studied. The (112̄2)-nucleated quantum dots exhibit rectangularor trapezoid-based truncated pyramidal morphology. Another
quantum dot type orientated on (101̄1) is reported. Based on
high-resolution transmission microscopy and crystal symmetry,
the geometry of (101̄1)-orientated quantum dots is proposed. A
piezoelectric model is used within a finite element method to
determine and compare the strain-state and electrostatic potential associated with the quantum dot morphology and an estimation of the band-edge energy is made. We report on some
novel properties of the (101̄1)-orientated quantum dot, including mixed strain-states and strain-state bowing.
© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction The commercial success of type
III-nitrides in the last 20 years has been the result of efforts
to solve basic problems in their growth and characterization.
Most research efforts in the field of wurtzite III-nitrides are
aimed at the enhancement of optoelectronic devices [1, 2].
The band gap of III-nitrides, ranging from 1.9 to 6.2 eV, covers the visible spectrum and the near-ultraviolet region and
are thus considered to be candidates for the blue-ultraviolet
region of the optical spectrum [3]. Additionally, III-nitrides
posses other useful physical properties such as, piezoelectricity and pyroelectricity, making them candidates for other
devices including mechanical, electrical, and optical sensors
[4]. In principle, the optimal optical wavelengths can be
achieved by forming binary or ternary alloys of Al, Ga, and In
nitrates and/or by doping with impurities such as Si, Ge, Se,
O, Mg, Be, and Zn [5]. As it turns out, the range of attainable
wavelengths is reduced in III-nitride devices grown in the
[0001]- or [0001̄]-direction due to the presence of an internal
polarized electrostatic field, that shifts the electron and hole
states to opposite ends of the structure [6]. Due to dielectric
and strain mismatch at heterojunctions, complemented by a
built-in spontaneous polarization, a strong electrostatic field
is induced in the system. Therefore, the resultant combined
electric field can reach several MV/cm in the direction
corresponding to the orientation of the c -axis. This reduces
the electron-hole wavefunction overlap which in turn reduces
recombination efficiency, ie., the Quantum Confined Stark
Effect (QCSE) [7], which redshifts the luminescence so, it
is therefore a significant challenge to obtain optical emission in the ultraviolet range. To enhance the optoelectronic
properties and to control the wavelength of the emitted light
Quantum Dots (QDs), systems are used instead of Quantum Wells (QWs). The employment of QDs nanostructures
either along the polar [0001]-axis, or various semipolar or
nonpolar growth orientations, is an effective way to trap the
carriers, thus increasing the radiative recombination rate and
ultimately the internal quantum efficiency of the device.
Studies on the effect of the polarization potential in the electronic structure of polar and nonpolar GaN/AlN QDs showed
that modifications in size, shape, and arrangement can improve the carrier recombination rate. Smaller QD sizes resulted in better confinement in the nonpolar case than in polar
nanostructures [8]. A detailed analysis of the built-in potential and electronic structure of that system was presented in
[9], where suppression of the electric field was reported for
the nonpolar case. Moreover, it was found that the precise
geometry of the QDs and also the Coulomb interactions between charge carriers have to be taken into consideration in
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order to accurately model their optical properties [10]. The
shape of the QDs is critical and more important when QDs
of approximately same volume are being compared due to
the change the different shapes induces in the electrostatic
potential within the QDs [10].
In an attempt to overcome difficulties with the QCSE and
to design improved optoelectronic devices, directions other
than [0001] are the subject of intensive research as candidate
structures that are free from inhibiting fields [11–14]. Recently, there has been much interest in GaN/AlN heterostructures grown on sapphire substrate either on a-, r- or m-plane
orientations because of their potential for fabrication of a
more effective light-emitting device. Unfortunately, an additional challenge to the development of technological devices based on semipolar crystal growth arises from the fact
that it often results in a very high density of stacking faults
and defects, to the order of 108 −1010 cm−2 , that promote
non-radiative electron–hole recombination affecting optical
efficiency and device lifetime [15]. The first results on aplane nonpolar structures were presented in Ref. [16, 17] and
shortly after structures grown on the m-plane were presented
in Refs. [18–20]. Since then, suppression of the built-in electric field for semipolar structures was confirmed in a number of recent experiments [21, 22]. The majority of growth
and experimentation concerning nitride-based heterostructures has concentrated on QWs [23–25] and more recently
QDs have gained increasing attention [26–30]. Growth of
GaN/AlN QDs by Plasma-Assisted Molecular-Beam Epitaxy
(PAMBE) with a reduced internal electric field has recently
been demonstrated [26]. Photoluminescence (PL) measurements indicated an attenuation of the internal electric field,
and in particular, the PL energy peak was systematically
found to reside above the GaN band gap, which suggests
a reduction of the QCSE [31]. Similar results were obtained
in the case of GaN/AlGaN QD nanostructures where, in the
semipolar case, the PL emission is strongly shifted toward
the UV with simultaneous reduction of the PL peak (at full
width at half maximum) compared to the polar ones as a
result of the reduction of the QCSE [28, 32]. On an alternative substrate, asymmetric nonpolar GaN/AlN QDs grown
on 6H-SiC exhibited a preferential polarization of the emission [33]. Reduced electric field and reduced radiative lifetime (112̄2) GaN/Al0.5 Ga0.5 N QWs and QDs have also been
reported [25, 30]. PL studies of nonpolar InGaN QDs on rplane sapphire have shown exciton lifetimes shorter than the
equivalent polar ones, a result that suggests suppression of
the electric field and stronger confinement [34]. Other work
based on Inx Ga1−x N/GaN QWs [23] and QDs [35] have confirmed a reduction of the electric field in QWs fabricated on
various semipolar facets. In particular, epitaxial growth of
(112̄2) GaN/Al0.5 Ga0.5 N QDs [27–29] has shown PL shifted
to higher energies toward the near ultra-violet. Recent work
has demonstrated active III-nitride semipolar light-emitting
diodes and lasers in the green and blue region of the spectrum
[36–38]. Analysis of the internal built-in polarization potential of nonpolar InGaN/GaN QDs showed strong reduction
with increasing Ga content [39]. Comparison between polar
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and semipolar InGaN/GaN QDs showed blue-shifted emission of the PL for the semipolar nanostructures compared
to the polar ones, but this was attributed to lower indium
incorporation in the semipolar case [35].
The fundamental optical eigenmodes that are of interest
are ultimately influenced by the underlying strain distribution and internal electric field and therefore accurate characterization of these fields is crucial for optical device design
[40, 41]. Recently, theoretical investigations of semipolarand nonpolar-orientated structures were attempted with the
aim of describing the internal strain state and electrostatic
polarization. In general, these works are either an analytic
estimate of the internal electric field [42–44] or numerical
simulations [27, 45–48]. Numerical schemes are attractive,
since they allow for the determination of internal fields in a
relatively straightforward way and are well suited for treating
multi-dimensional systems such as quantum dots (QDs).
This contribution is motivated by recent growth experiments on a heterostructure consisting of GaN QDs in (112̄2)
AlN [27]. The geometrical shapes of the (112̄2) QDs were
determined from observations based on High-Resolution
Transmission Electron Microscopy (HRTEM) and Z-contrast
imaging, as well as by exploiting knowledge of the underlying wurtzite crystal symmetry. The delimiting morphology
was determined to be either rectangular-based or trapezoidbased truncated pyramids. Aside from the QDs reported
there, an additional type of QD with a different morphology
lying on (101̄1) was also observed due to nucleation at facets
introduced by ascending threading dislocations. By use of
quantitative analysis of electron microscopy analysis and Zcontrast images of the heterostructure, the morphology of a
(101̄1)-orientated QD is proposed. A piezoelectric model is
then taken into finite element (FE) analysis in order to estimate the relaxed strain state, electrostatic potential, and the
band-edge structure of a set of isolated (101̄1) QDs.
2 Experimental observations Semipolar GaN QD
super-lattices grown by PAMBE on (112̄2) AlN were analyzed. The super-lattice consists of 20 QD periods with 10 nm
AlN spacers deposited on m-plane sapphire substrate. The
growth details are given elsewhere [27]. TEM samples were
prepared with tripod polishing followed by Ar+ ion milling.
Electron microscopy experiments (both conventional TEM
and HRTEM) were performed using a 200 kV JEOL 2011
microscope (0.19 nm point resolution, Cs = 0.5 nm). Information on the local atomic chemistry of the GaN/AlN QD
layers was extracted by Z-contrast imaging using a 200 kV
JEOL JEM 2200FS. A semipolar QD nucleated on the (112̄2)
plane is shown in Fig. 1.
A detailed structural analysis of the rectangular and
trapezoidal semipolar (112̄2) QDs, to be simulated in the
next sections, was given in Ref. [27]. In addition to these
dots, another semipolar type of QD, oriented along (101̄1),
was also observed in our samples. Information derived from
HRTEM and Z-contrast images suggests that these QDs exhibit a trapezoidal-projected shape. Figure 2a shows a crosssectional HRTEM image along the [112̄3̄] zone axis of these
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T. D. Young et al.: Piezoelectric properties of semipolar (112̄2) GaN/AlN quantum dots
Figure 1 HRTEM image of a semipolar QD nucleated on the
(112̄2) growth plane projected on the [112̄3̄] direction.
investigated nanostructures. QD facet traces form ∼ 40◦ to
50◦ angles to the (101̄1) growth plane. The STEM image of
Fig. 2b shows the trapezoid-projected QDs along the [11̄00]
zone axis. We note here that the (101̄1) plane is inclined in
this projection direction. We also note that the (101̄1) growth
plane is normal to the (12̄10) mirror symmetry plane. Thus,
the possible shape of the mentioned QD should be in agreement to this symmetry operation. A matching shape of a kitebased QD delimited by two {11̄01} and two {112̄0} facets is
presented in Fig. 3, in which the cross-sectional views along
the [112̄3̄] and the [11̄00] directions are given. This shape is
bound by two planes symmetric to (12̄10), i.e., (11̄01) and
(011̄1), and the facet traces form angles of ∼ 40◦ to the growth
plane, in a cross-sectional view along [11̄00] .
The GaN/AlN and GaN/AlGaN systems exhibit similarities regarding QD size and shape when viewed along [11̄00]
although one side facet may vary in orientation, i.e., the
(112̄6) plane has been determined in the case of AlN matrix
instead of (112̄3) for the AlGaN matrix [8]. In the GaN/AlN
case, nucleation on different planes other than (112̄2) was
also reported by Dimitrakopulos et al.. InGaN/GaN QDs
were also nucleated in (112̄2) as well as on inclined planes,
similar to the GaN/AlN case. However, their shape was different since the InGaN QDs were not strongly facetteed but
instead exhibited a lenticular shape with a large base and
small height [35]. Also note, that the morphology of semipolar QDs is different from that of polar QDs. QDs grown
along the [0001] typically form the geometry of a truncated
hexagonal-based pyramid, which reflects the hexagonal symmetry of the (0001)-plane [49]. Semipolar QDs are also pyramidal, which is an effect of the growth process, but with a
different base geometry due to crystal orientation effects and
Figure 2 (a) HRTEM image of (101̄1)-orientated semipolar QDs
in a cross-sectional view along [112̄3̄] direction with projected facet
traces forming angles ∼ 40◦ and 50◦ to the growth plane; and (b)
STEM image along the [11̄00] zone-axis showing the trapezoidal
QDs nucleated on the (101̄1) plane.
reduced base-plane symmetry. This in turn affects the strain
distribution and alignment of the polarized piezoelectric field
[45, 48]. It is thus to be expected, and here it will be shown,
that the piezoelectric properties of semipolar QDs are rather
different from those of polar QDs.
3 Piezoelectric model of semipolar quantum
dots The electrostatic potential induced in wurtzite heterostructures consists of two components; namely, piezoelectric and spontaneous polarization. In the case of heterostructures grown in the [0001] (polar) direction, the effects of
piezoelectric and spontaneous polarization are strictly additive. In semipolar heterostructures, the additive effect can be
significantly reduced [48]. To apply the piezoelectric model
to an arbitrary chosen crystallographic orientation, a combination of two planar tensor rotations are used to align the
c-axis of the model. In this sense, our approach follows a
similar line to the works of Park and his co-workers [45, 46].
These transformations are applied to each of the materialdependent tensors of moduli.
The numerical simulations start from a standard piezoelectric model, that couples elastic stress with electric displacement [49]. Using Einstein notation with Latin indices,
Figure 3 Proposed morphology of the (101̄1)-orientated semipolar QDs: (a) a view along the [112̄3̄] zone-axis; (b) a view along the
[11̄00] zone-axis; and (c) a plan view. The (101̄1) surface is indicated as a top facet while the bottom one is (1̄011̄). The QDs side facets
are delimited by the (11̄01), (011̄1), (112̄0), and (21̄1̄0) planes.
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Figure 4 FE meshes used to represent three geometrical forms of truncated-pyramid GaN QDs (green cells) nucleated in (112̄2) AlN (red
cells). (a) (112̄2)-orientated rectangular-based, (112̄2)-orientated trapezoid-based, and (c) (101̄1)-orientated kite-based QD. The meshes
are truncated to the immediate vicinity of the QD and for visualization the capping layer has been removed.
the constitutive equation set describing stress σ, and electric
displacement D, is written as the simultaneous pair:
σij = Cijkl εkl − ekij Ek ,
(1a)
Dk = ekij εij + ki Ei + P .
sp
k
(1b)
To take into account spontaneous polarization that arises
in wurtzite structures, a vector term P sp was added to the
electric displacement (1b). Additional symbols are materialdependent tensors of moduli; namely, the crystal stiffness C,
the dielectric permittivity , and the piezoelectric coefficients
e that couple elastic and electric fields [50].
The total lattice strain εlat observed in experiment is a
state of equilibrium between the strain arising from lattice
mismatch εmis and elastic relaxation of the structure. The
character of deformation results in a decomposition of the
total lattice strain into a lattice mismatch and elastic strain
ε = εlat − εmis . The lattice mismatch, expressed with respect
to relaxed bulk AlN, is defined by the following:
ε
mis
ij
=
aijGaN − aijAlN
aijAlN
(2)
δij ,
where a11 = a22 = a and a33 = c are lattice parameters of
the wurtzite hexagonal structure. The piezoelectric model
includes a definition of the lattice strain and electric field
from the relations:
εlat
kl =
1
(∇l uk + ∇k ul )
2
and
Ek = −∇k φ ,
(3)
where u and φ are the displacement field and the electrostatic
potential, respectively.
Boundary constraints on the domain Ω and on the surface
∂Ω are written as follows:
ui = ũi
φ = φ̃
on Ω
and
σij · n̂j = 0i
Di · n̂i = 0
on ∂Ω,
(4)
where n̂ is a vector normal to the surface ∂Ω. These zerovalued boundary constraints reduce the problem to that of
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an isolated system. The set of Eqs. (1)–(4), together with the
equilibrium conditions, div σ = 0 and div D = 0, compose a
boundary-value problem that was solved numerically by an
FE method [51].
4 Numerical results In these numerical simulations,
we make use of irregular finite elements that together conform to the QD morphology proposed in Section 2 and illustrated in Fig. 4. The particular geometries considered are as
follows: (i) a (112̄2) nucleated rectangular-based truncated
pyramid with base edges lengths of 25 nm along [11̄00] and
18 nm along [112̄3̄], and height 3.9 nm (Fig. 4a); (ii) a (112̄2)
nucleated trapezoid-based truncated pyramid with parallel
base edges of length 11 and 22 nm along [11̄00], an orthonormal base length of 15 nm along [112̄3̄], and height 3.5 nm
(Fig. 4b); and (iii) a (101̄1)-orientated kite-based truncated
pyramid with orthodiagonals equal to 10.3 nm along [1̄21̄0]
and 22 nm along [1̄012], and height 4.3 nm (Fig. 4c). For
the (112̄2) QDs, the alignment of the c-axis is found by an
anticlockwise planar rotation of ∼ 58◦ around [11̄00] and
for the (101̄1) QD, the orientation is found by an anticlockwise planar rotation of ∼ 62◦ around [1̄21̄0] . Note, that the
trapezoid-based QD in cross-sectional view along (11̄00), resembles the cross-sectional morphology of semipolar (112̄2)
QDs that have been reported elsewhere [28]. However, in
that report the QDs were found to form closely connected
islands, rather than the more discrete QDs found in our sample (cf. Fig. 2 and Ref. [27]). It is worth emphasizing, that in
this study isolated semipolar (112̄2) GaN/AlN QDs are being
considered; making a direct comparison with experimental
observations somewhat tricky.
Surface plots of the elastic strain (εxx ) and the electrostatic potential for different QD geometries are illustrated
in Fig. 5. Let us consider the (112̄2) QDs first (Fig. 4a and
b). The elastic strain distribution in (11̄00) is illustrated in
columns one and two of Fig. 5. After the relaxation process,
both dots similarly show an almost homogeneous compression internally and some extension in the matrix local to the
QD. Areas of high strain are located in the corner edges on
the (112̄2) base of the dots where the material is not permitted to fully relax due to the sharp geometry of this region.
The electrostatic potential is a dipole that is characteristic of
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Figure 5 Cross-sections of three QDs nucleated in (112̄2) GaN/AlN showing the elastic strain εxx (top) and the electrostatic potential φ
(bottom). From left to right, the cross-sections correspond to: the rectangular-based QD along (11̄00), trapezoid-based QD along (11̄00),
and kite-based QD along (1̄21̄0).
wurtzite heterostructures, in which a positive pole appears on
side of the quantum dot and a negative pole on the opposite,
and where their alignment is defined by the orientation of the
c-axis.
Now, let us consider the (11̄01)-orientated kite-based
QD (Fig. 4c). The elastic strain and potential distributions
are illustrated in the third column of Fig. 5. After the relaxation process, there is a irregularly pronounced nonhomogeneous distribution of the elastic strain inside the QD. In
the rectangular- and trapezoid-based QD, the elastic strain is
exclusively compressive and changes to a region of extension on the borders between the QD and matrix. The case
of the kite-based QD is quite different and regions of small
extension (∼0.8%) are found within the QD volume. This
“mixed strain state” most probably results from the presence
of a sharp, elongated corner formed by the two facets (11̄01)
and (011̄1), that join in the (1̄21̄0) symmetry plane (refer to
Fig. 3). A similar, weaker effect is seen on the other side of
the quantum dot where the facets (21̄1̄0) and (112̄0) meet in
a less sharp and less elongated way. Note the peak of extension (∼ 1.8 %) in the matrix near to the hat of the truncated
pyramid, corresponding to another sharp corner joining three
facets. The peaks of the electric dipole are orientated along
the c-axis, i.e., along [1̄012] and elevated at ∼62◦ . It is interesting to note, that although the negative part of the potential
lies on the interface between the QD and the matrix, the positive part of the potential lies inside the QD.
The upper panel of Fig. 6 shows line scans of elastic
and lattice strain sweeped through the z-axis and the center of the QD. This axis is also the growth axis of the QD
(cf. Fig. 4). The qualitative behavior of the elastic relaxation
through the z-direction of each of the QDs are characteristic of [0001] grown-truncated pyramidal QD structures [52].
The kite-based QD exhibits some interesting detail; namely,
(i) a mixed strain-state; and (ii) a “strain-state bowing” effect. The first was discussed in some detail above, as it was
© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
first noted to occur for the xx-component of strain on (1̄21̄0)
(see discussion surrounding Fig. 5). Here it is visible in the
elastic strain component εzz as a change in sign of the strain
state at z ≈ 1.3 nm. The second effect manifests itself as a
undulating wave-like quality in the strain state as the sweep
progresses through the QD. It can be hypothesized that this is
also connected to the presence of sharp corners and sharply
sloping facets and that the effect is being carried deep into
the QD. This idea is supported by the wave-like quality of
the strain distribution in Fig. 5. Sweeps of the electrostatic
potential through each of the three Cartesian directions is
illustrated in the lower panel of Fig. 6. The electrostatic potential is strongest in the yz-plane since this is the plane on
which the axis of spontaneous polarization lies.
The peak-to-peak values of the elastic strain are similar
among each of the QDs considered here and is comparable with an isolated polar QD [40]. Perturbations arise from
differences in crystallographic orientation and from the specific QD morphology (strain distribution). Significantly, the
electrostatic potential distribution in (101̄1) QD and its peakto-peak value are quite different. A noticeable change is observed along y-orientation, where a reversed polarity of the
potential is observed.
Figure 7 shows the variation of the conduction bandedge for electrons and valence band-edge for holes in the
effective mass approximation. The qualitative behavior of
the band-edge structure is similar regardless of the QD geometry. A minimum (maximum) in the conduction (valence)
band is apparent in the xy-plane and is more-or-less flat in
the x-direction. We remark that the minimum of the conduction band and maximum of the valance band in the [1̄012]direction is rather broad, indicating that the electron wavefunction would be smeared out in that direction.
5 Summary In this contribution, three GaN QDs with
different delimiting geometries nucleated in (112̄2) AlN were
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Phys. Status Solidi B 252, No. 10 (2015)
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Figure 6 Top panel: sweeps of the spherical components of the elastic strain ε in percent (solid lines) and corresponding components of
lattice strain εlat in percent (dotted lines) taken through the center of the QD in the z-direction. Horizontal bars denote the position of the
base of the quantum dot (at z = 0) and the hat of the quantum dot. The black arrows point to strain-state bowing modes (for a discussion
see the main text). Bottom panel: sweeps of the electrostatic potential φ taken through the center of the QD as a function of each of the
three Cartesian directions. From left to right: samples taken for the rectangular-, trapezoid-, and kite-based QDs.
presented. The geometric structure of (112̄2) QDs was taken
from Ref. [27]. The geometric structure of (101̄1)-orientated
QDs was proposed by exploiting information from HRTEM
and Z-contrast images the crystallographic symmetry of the
wurtzite Bravais lattice. To determine the relaxed strain state
and electrostatic potential, a piezoelectric model was taken
into FE analysis and from that an estimation of the bandedge structure was obtained. The particular morphology has
a notable affect on the piezoelectric properties of QDs. We
reported on some novel properties of a (101̄1)-orientated QD,
whose morphology was proposed to resemble that of a kitebased truncated pyramid. Those were as follows: (i) mixed
strain-states, where regions of compression and extension
were found to coexist within the QD; and (ii) strain-state bowing, in which the strain distribution forms wave-like patterns
within the QD. The positive pole of the electrostatic potential was found to reside within the QD due to the combined
piezoelectric effect. It was hypothesized that the orientation
of the facets and the sharpness at which edges are formed
are both responsible for the effects exhibited by that QD. It
Figure 7 Same as in the lower panel of Fig. 6, but here showing an estimation of the band-edge structure given as the sum of the bulk
conduction (valance) band edge and the electrostatic potential. The gray horizontal dotted line denotes the conduction (valance) band of
with respect to relaxed bulk GaN.
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is hoped that the study presented here will stimulate future
investigations into the character and trend of semipolar IIInitride QDs. Future work will focus on studying localization
of the electron-hole wavefunction overlap in such QDs.
Acknowledgements The authors thank Eva Monroy for
providing the PAMBE grown samples used in this study. This research was supported by the project N N519 647640 founded by
the Polish Ministry of Science and Higher Education. A. Lotsari
acknowledges support through the Fellowship of Excellence of the
Aristotle University of Thessaloniki Research Committee.
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