Henini M. Handbook of Self Assembled Semiconductor Nanostructures for Novel devices in Photonics and Electronics
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Growth and Characterization of Structural and Optical Properties 229
48. J. Coraux , V. Favre-Nicolin , M.G. Proietti , B. Daudin , and H. Renevier , Phys. Rev. B . 75 , 235312
( 2007 ) .
49. J. Coraux , B. Amstatt , J.A. Budagoski , E. Bellet-Amalric , J.-L. Rouvière , V. Favre-Nicolin ,
M.G. Proietti , H. Renevier , and B. Daudin , Phys. Rev. B . 74 , 195302 ( 2006 ) .
50. J. Coraux , H. Renevier , V. Favre-Nicolin , G. Renaud , and B. Daudin , Appl. Phys. Lett. 8 8 , 153125
( 2006 ) .
51. P. Waltereit , O. Brandt , A. Trampert , H.T. Grahn , J. Menniger , M. Ramsteiner , M. Reiche , and
K.H. Ploog , Nature 406 , 865 ( 2000 ) .
52. N. Akopian , G. Bahir , D. Gershoni , M.D. Craven , J.S. Speck , and S.P. DenBaars , Appl. Phys. Lett. 8 6 ,
202104 ( 2005 ) .
53. A. Cros , J.A. Budagosky , A. García-Cristóbal , N. Garro , A. Cantarero , S. Founta , H. Mariette , and
B. Daudin , Phys. Status Solidi B . 243 , 1499 ( 2006 ) .
54. J. Simon , N.T. Pelekanos , C. Adelmann , E. Martinez-Guerrero , R. André , B. Daudin , L.S. Dang ,
and H. Mariette , Phys. Rev. B . 68 , 035312 ( 2003 ) .
55. T. Bretagnon , P. Lefebvre , P. Valvin , R. Bardoux , T. Guillet , T. Taliercio , B. Gil , N. Grandjean ,
F. Semond , B. Damilano , A. Dussaigne , and J. Massies , Phys. Rev. B . 73 , 113304 ( 2006 ) .
56. F. Rol , S. Founta , H. Mariette , B. Daudin , L.S. Dang , J. Bleuse , D. Peyrade , J.-M. Gérard , and
B . G a y r a l , Phys. Rev. B . 75 , 125306 ( 2007 ) .
57. S. Kako , K. Hoshino , S. Iwamoto , S. Ishida , and Y. Arakawa , Appl. Phys. Lett. 85 , 64 ( 2004 ) .
58. R. Bardoux , T. Guillet , P. Lefebvre , T. Taliercio , T. Bretagnon , S. Rousset , B. Gil , and F. Semond ,
Phys. Rev. B . 74 , 195319 ( 2006 ) .
59. L. Besombes , K. Kheng , L. Marsal , and H. Mariette , Phys. Rev. B . 63 , 155307 ( 2001 ) .
60. I. Favero , G. Cassabois , R. Ferreira , D. Darson , C. Voisin , J. Tignon , C. Delalande , G. Bastard ,
Ph. Roussignol , and J.-M. Gérard , Phys. Rev. B . 68 , 233301 ( 2003 ) .
61. E. Peter , J. Hours , P. Senellart , A. Vasanelli , A. Cavanna , J. Bloch , and J.-M. Gérard , Phys. Rev. B.
6 9 , 041307(R) ( 2004 ) .
62. K. Huang and A. Rhys , Proc. R. Soc. London, Ser. A . 204 , 404 ( 1950 ) .
63. T. Takagahara , Phys. Rev. B . 60
, 2638 ( 1999 ) .
64. B. Krummheuer , V.M. Axt , and T. Kuhn , Phys. Rev. B . 65 , 195313 ( 2002 ) .
65. A.V. Kavokin , Phys. Rev. B . 50 , 8000 ( 1994 ) .
66. L.C. Andreani , G. Panzarini , and J.-M. Gérard , Phys. Rev. B . 60 , 13276 ( 1999 ) .
67. J. Hours , P. Senellart , E. Peter , A. Cavanna , and J. Bloch , Phys. Rev. B . 71 , 161306(R) ( 2005 ) .
68. S.L. Chuang and C.S. Chang , Phys. Rev. B . 54 , 2491 ( 1996 ) .
69. I. Vurgaftman , J.R. Meyer , and L.R. Ram-Mohan , J. Appl. Phys. 89 , 5815 ( 2001 ) .
70. J.S. Im , A. Moritz , F. Steuber , V. Härle , F. Scholz , and A. Hangleiter , Appl. Phys. Lett. 7 0 , 631
( 1996 ) .
71. A.J. Steckl , M. Garter , D.S. Lee , J. Heikenfeld , and R. Birkhahn , Appl. Phys. Lett. 75 , 2184 ( 1999 ) .
72. J. Heikenfeld , M. Garter , D.S. Lee , R. Birkhahn , and A.J. Steckl , Appl. Phys. Lett. 75 , 1189 ( 1999 ) .
73. H. Bang , S. Morishima , Z. Li , K. Akimoto , M. Nomura , and E. Yagi , J. Cryst. Growth 237–239 ,
1027 ( 2002 ) .
74. Y. Hori , X. Biquard , E. Monroy , D. Jalabert , F. Enjalbert , L.S. Dang , M. Tanaka , O. Oda , and
B. Daudin , Appl. Phys. Lett. 84 , 206 ( 2004 ) .
75. E.E. Nyein , U. Hömmerich , J. Heikenfeld , D.S. Lee , A.J. Steckl , and J.M. Zavada , Appl. Phys. Lett. 8 2 ,
1655 ( 2003 ) .
76. T. Andreev , Y. Hori , X. Biquard , E. Monroy , D. Jalabert , A. Farchi , M. Tanaka , O. Oda , L.S. Dang ,
and B. Daudin , Phys. Rev. B . 71 , 115310 ( 2005 ) .
77. Y. Hori , D. Jalabert , T. Andreev , E. Monroy , M. Tanaka , O. Oda , and B. Daudin , Appl. Phys. Lett. 8 4 ,
2247 ( 2004 ) .
78. T. Andreev , Y. Hori , X. Biquard , E. Monroy , D. Jalabert , A. Farchi , M. Tanaka , O. Oda , L.S. Dang ,
and B. Daudin , Phys. Rev. B . 71 , 115310 ( 2005 ) .
79. Y. Hori , T. Andreev , D. Jalabert , E. Monroy , L.S. Dang , M. Tanaka , O. Oda , and B. Daudin , Appl.
Phys. Lett. 88 , 053102 ( 2006 ) .
80. T. Andreev , N.Q. Liem , Y. Hori , M. Tanaka , O. Oda , D.L.S. Dang , B. Daudin , and B. Gayral , Phys.
Rev. B . 74 , 155310 ( 2006 ) .
CH006-I046325.indd 229CH006-I046325.indd 229 6/27/2008 4:22:29 PM6/27/2008 4:22:29 PM
Optical and Vibrational Properties of Self-assembled GaN Quantum Dots
Núria Garro , Ana Cros , Alberto García-Cristóbal , and Andrés Cantarero
Institute of Materials Science, University of Valencia (Spain)
7.1 Introduction
Quantum dots (QDs) based on group III nitrides (III-N) are expected to be the active medium of new
optoelectronic devices operating at high powers and high temperatures. Besides the well-known
advantages of their bulk and quantum well (QW) counterparts [1] , III-N QDs provide strong con-
fi nement of carriers in nearly perfect zero-dimensional boxes. Consequently, their optical response
is almost insensitive to the high dislocation density present in state-of-the-art epitaxial nitride
semiconductors. Furthermore, quantum effects provide new degrees of freedom for the design of
advanced devices. As an illustrative example of the potential of these nanostructures we high-
light that triggered single-photon emission has been achieved from GaN QDs at 200 K, a tempera-
ture easily reachable with thermo-electric cooling and much higher than temperatures required
for quantum information-processing schemes based on InAs QDs [2] . III-N QDs can be grown by
a variety of techniques, ranging from molecular beam epitaxy and metal-organic vapour phase
chemical deposition to other less extended methods, such as vapour–liquid–solid mechanism, reac-
tive laser ablation or inorganic pyrolysis [3] . While the mentioned growth schemes result in the for-
mation of QDs that nucleate randomly, other techniques aim for a more ordered disposition of the
dots. Selective growth of InGaN QDs on hexagonal uniform GaN pyramids was demonstrated by
Tachibana et al. [4] . Another approach takes advantage of the embedding of GaN quantum discs in
AlGaN nanocolumns grown on Si(111) substrates by plasma-assisted molecular beam epitaxy [5] .
More recently, QD arrays have been selectively fabricated by holographic lithography and subse-
quent ion etching of InGaN QW structures [6] . Other approaches envisage the one-dimensional
alignment of GaN dots when grown on vicinal substrates [7] . In the following, we concentrate on
those systems of dots that appear spontaneously during epitaxial growth without the need of arti-
fi cial post-processing, and designate them as self-assembled or self-organized quantum dots regard-
less of the mechanism responsible for their appearance.
Nowadays, research efforts on III-N QDs are mainly focused on the understanding and control
of their optical properties. These properties are strongly infl uenced by the existence of internal
electric fi elds of the order of several MV/cm. As a result of such built-in fi elds, the photolumi-
nescence spectrum of GaN QDs is often redshifted below the bulk GaN band gap energy. Taking
advantage of this shift, the emission of III-N QDs can sweep most of the visible spectrum. Photo-
and cathodoluminescence are widely spread techniques in the investigation of the optical prop-
erties of both QD ensembles and single QDs. The resolution of single GaN QD emission has only
been possible very recently thanks to the improvement in the control of the QD density and the
development of processing techniques [8, 9] . Raman spectroscopy has also been used in the
investigation of III-N QDs. It allows the analysis of strain, composition, size and orientation of
CHAPTER 7
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Optical and Vibrational Properties of Self-assembled GaN Quantum Dots 231
the constituting materials through the study of their atomic vibrations. The widespread use of
Raman scattering for nanostructure characterization is, however, limited by the small scattering
rates. As a consequence, the experiments are usually performed on samples where the structure
of interest is repeated over several periods. One way to overcome this limitation takes advantage
of the enhancement of the Raman scattering rate under resonant conditions. Experimental
efforts have to be complemented with theoretical modelling which is essential for understand-
ing the interplay between electric fi eld and quantum confi nement effects in these systems. The
results point out that the strain state of the system contributes signifi cantly to the internal elec-
tric fi eld and must be considered in full detail.
Several authors have reviewed in the past the properties of III-N QDs. Huang et al. have com-
piled a general description of growth methods and optical characterization of InGaN and GaN
QDs [3] . Other authors have given partial accounts summarizing their own work [10–13] .
However, being a fi eld of lively activity it is timely to present an updated review of the proper-
ties of self-assembled nitride quantum dots. The scope of this chapter is restricted to the studies
on GaN QDs, which present distinctive features as compared to InGaN QDs such as the different
emission spectral range and the absence of alloy and segregation effects. In particular, we focus
on the optical and vibrational properties of the dots, and pay special attention to the interplay
between the strain state of the dots, the magnitude of their internal electric fi eld and its effects
on their optical properties. We shall review results obtained mostly on wurtzite QDs, though the
properties of zinc-blende GaN QDs will be mentioned briefl y. The aim of this chapter is to give
an overview of the different results obtained by the various research teams working on the topic
rather than analyse in detail the peculiarities of one specifi c type of sample. In chapter 6, Daudin
and co-workers perform such an exhaustive review of the growth and structural characteristics
of the GaN QDs grown by plasma-assisted molecular beam epitaxy.
7.2 Growth and structural characterization
It is well known that one of the most important technical barriers that prevent achieving
high-quality nitride materials is the lack of adequate substrates for homoepitaxy. Lattice mis-
match and thermal expansion coeffi cient differences between nitride fi lms and the available sub-
strates induce large densities of structural defects (mainly threading dislocations, with densities
of the order of 10
10
cm
2
). Such diffi culties have been overcome by the growth of high-quality
QDs. In the following we give a brief overview of the most relevant growth techniques employed
for the spontaneous or self-assembled growth of GaN QDs. Special attention will be paid to epi-
taxial growth categorized into two different modes: Volmer–Weber (VW) mode corresponding to
three-dimensional (3D) island growth and Stranski–Krastanov (SK) mode, where the deposition
of a strained two-dimensional (2D) wetting layer (WL) is followed by elastic relaxation through 3D
islanding. The strain distribution in GaN QDs will be analysed in detail at the end of this section.
7.2.1 Growth and structural properties
Self-organized GaN QDs have been successfully grown by several epitaxial techniques like molec-
ular beam epitaxy (MBE) [14, 15] , metal-organic chemical vapour deposition [16, 17] (MOCVD)
or a combination of both techniques [18, 19] . Depending on the substrate material and orienta-
tion, both wurtzite [14, 16, 19] and zinc-blende [20] dots have been obtained. Dots with the wur-
tzite structure are typically grown on a buffer layer of AlN or Al
x
Ga
1
x
N deposited on sapphire
(Al
2
O
3
), 4H-SiC(0001), 6H-SiC(0001) or Si(111), while dots with the zinc-blende structure have
been obtained starting from a 3C-SiC(001) template grown by MOCVD on Si(001). Characteristic
values of QD sizes and densities obtained by the relevant research groups working in the fi eld are
displayed in Table 7.1 . Varying the growth techniques and conditions, the dot density may cover a
wide range (10
8
–10
11
cm
2
), although most of the samples obtained so far lie in the high-density
end of the range (10
10
–10
11
cm
2
).
Gallium nitride QDs were grown for the fi rst time by Tanaka and co-workers via MOCVD [16] .
They demonstrated that it is possible to take advantage of the antisurfactant properties of Si on
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232 Handbook of Self-assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
Table 7.1 Typical parameters of GaN self-assembled QDs grown by several authors. When there
is no direct report of the representative values of the dot height h and diameter D they have
been estimated from other data specifi ed by the authors.
Spacer h/D (nm) Dot density
(cm
2
)
Substrate T
s
(°C) Growth technique/
mode
Ref.
AlGaN 6/40
3 1 0
9
6H-SiC 1100 MOCVD/VW [16]
AlGaN
h 20–200
10
8
Al
2
O
3
1100 MOCVD/VW [21]
AlN 1.6/35
1.6 1 0
8
6H-SiC 1080 MOCVD/SK [17]
AlN 1.5–4/20 10
10
Al
2
O
3
960 MOCVD/SK [22]
AlN 2.5–4/20 10
11
Al
2
O
3
700 PAMBE/SK [23]
AlN 5/20
1–5 1 0
11
Si(111) 800 MBE/SK [15]
AlN 3/15 10
8
–10
11
Al
2
O
3
750 PAMBE/MSK [24]
AlN 4/40 10
10
–10
11
6H-SiC 750 PAMBE/MSK [25]
AlN 2.2/20 10
11
a -plane 6H-SiC 700 PAMBE/SK [26]
AlGaN
D 9.3
3 1 0
11
6H-SiC 600 GSMBE [18]
AlN 8/4–20
4 1 0
9
6H-SiC 700 VLS [19]
Al
x
Ga
1
x
N surfaces ( x ~ 0.15) to induce the formation of GaN islands. The introduction of the
antisurfactant (tetraethylsilane, TESi, with H
2
as the carrier gas) modifi es the nitride growth
kinetics so that the surface-free energy can be artifi cially controlled to change the growth mode
of GaN from 2D to 3D. The detailed study of the growth mechanism involved [27] indicates the
formation of a one monolayer thick Si–N mask (nano-mask) that infl uences the morphology of
the deposited GaN surface, giving rise to the formation of the quantum structures and contribut-
ing to the termination of threading dislocations in the GaN fi lm. In Fig. 7.1 a high-resolution
transmission electron microscopy (HRTEM) image illustrates the morphological characteristics
of the resulting dots [28] . The absence of a WL indicates that they are apparently grown on a
VW mode. They have a disk-like shape and approximate dimensions of 20 nm diameter and 5 nm
height. The density and size of the dots may be controlled by varying the growth temperature,
the feeding rate of TESi and the GaN coverage.
Figure 7.1 HRTEM image of GaN QDs grown epitaxially on an AlGaN layer by MOCVD [28] . With permission
from AIP© 2000.
A modifi cation of this method has been recently proposed by K. Pakula et al. [21, 29] . They
combined the antisurfactanct and etching properties of SiH
4
on Al
x
Ga
1
x
N to obtain QDs of
pyramidal shape with heights ranging from 20 to 200 nm and densities as low as 10
8
cm
2
which allow the study of the optical properties of single QDs without artifi cially masking the
samples. The use of antisurfactants may be advantageous for the growth of QDs in systems
where the lattice mismatch between the dot material and the substrate is very small and does not
allow an SK growth mode. In this case the resulting dots are to a large extent relaxed.
CH007-I046325.indd 232CH007-I046325.indd 232 6/27/2008 5:15:49 PM6/27/2008 5:15:49 PM
Optical and Vibrational Properties of Self-assembled GaN Quantum Dots 233
Alternatively, in the system GaN/AlN, where the in-plane lattice mismatch between the dot
and substrate materials is large enough (~2.4%), GaN QDs may be grown by MOCVD in the SK
mode [17] . The formation of GaN QDs on a very fl at surface of AlN results from a delicate bal-
ance between III/N ratio and growth temperature. Indeed, it has been found that the dot forma-
tion takes place under a very low III/N ratio, around 100 times smaller than that used in the
conventional two-dimensional MOCVD GaN growth. The experiments indicate that substrate
temperatures higher than 990°C inhibit the formation of dots due to the large migration and
evaporation rates of Ga atoms. The typical diameter and height of the QDs were of the order
of 20 and 2 nm, respectively. By this method it is possible to obtain dot densities as small as
10
8
cm
2
, and they can be increased up to 10
10
cm
2
by changing the growth conditions. Further
details can be found in the review paper of Arakawa and Sako [13] . More recently, MOCVD growth
of SK GaN QDs has also been reported by Grandjean and his collaborators at EPFL [22] .
The MBE growth of GaN QDs by the SK mode was initiated by Daudin and co-workers already
in 1997 [14] . They demonstrated that, under particular growth conditions, the deposition of a
few monolayers (MLs) of GaN on AlN(0001) by plasma-assisted MBE (PAMBE), where the active
nitrogen is provided by an RF plasma source, is followed by the relaxation of strain in the 2D layer
through 3D islanding, a process characteristic of the SK growth mode. Besides being governed by
strain relaxation, the growth of QDs by MBE is extremely sensitive to the III/N ratio. Nitrogen-
rich conditions lead to the formation of small islands on top of a 2 ML thick WL when the deposi-
tion thickness exceeds a critical value slightly over 2 ML (2.25 ML) [30] . The size and density of
the islands increase with GaN coverage until the density saturates, with the subsequent appear-
ance of a bimodal distribution of dot sizes. In this regime, dot densities around 10
11
cm
2
are
obtained. Metal-rich conditions, on the other hand, reveal the autosurfactant properties of Ga for
GaN growth. In these conditions, a dynamically stable Ga bilayer accumulates on top of the GaN
surface, inhibiting the 2D–3D transition [31] , much as happens during the growth of InAs on
GaAs where the island formation is suppressed under excess In conditions [32] . The formation of
the GaN QDs takes place after the subsequent evaporation of the Ga layer by growth interruption
under vacuum, a growth procedure known as modifi ed Stranski–Krastanov (MSK) [25] . This
method allows the control of the dot density in the 10
10
–10
11
cm
2
range. Much smaller densi-
ties have been achieved following a similar growth scheme by J. Brown et al. [24] . As shown in
Fig. 7.2 , a strong correlation between QD density and GaN coverage is observed for a growth rate
of 0.23 ML/s when the Ga/N fl ux ratio is in the Ga droplet regime at 750°C. Medium to high GaN
coverage (above 2.5 ML) results in a homogeneous density, whereas a density gradient across the
sample, reaching values as low as 3 1 0
8
cm
2
, has been observed for low GaN coverage.
0.10 (ML/s)
0.23 (ML/s)
2.0 2.5 3.0
GaN coverage (ML)
10
9
10
10
10
11
S-K density (cm
2
)
0.10 (ML/s) 0.23 (ML/s)
(a) (b)
250 nm
3.0 ML
2.6 ML
2.0 ML 2.2 ML
2.5 ML
3.0 ML
250 nm
Figure 7.2 (a) Atomic force micrographs of GaN/AlN QDs grown by PAMBE for different GaN coverage and
growth rates. (b) QD density as a function of GaN coverage [24] . With permission from AIP© 2004.
GaN QDs have also been grown using ammonia as the nitrogen active source for MBE on
Si(111) or sapphire substrates covered by a thick AlN buffer layer [15] . Similar to the MSK
growth, 3D islands are formed only if a growth interruption without ammonia fl ux is performed.
CH007-I046325.indd 233CH007-I046325.indd 233 6/27/2008 5:15:50 PM6/27/2008 5:15:50 PM
234 Handbook of Self-assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
A critical thickness of 3 ML GaN was determined for this growth method. The QD size can then
be easily controlled by varying the amount of GaN deposited with an upper limit of 12 ML to
avoid plastic relaxation. Although typical QD densities are of a few times 10
11
cm
2
, a recent
report has shown that some samples may present a gradient of densities across the wafer going
down to 10
9
cm
2
[9] .
The shape of the QDs obtained by SK growth is similar for the different growth methods (MOCVD
or MBE). They are truncated hexagonal pyramids with
{}1103
facets inclined 32° with respect
to the (0001) direction [13, 14, 33, 34] . In Fig. 7.3 the shape of MOCVD grown QDs obtained by
atomic force microscopy (AFM) and cross-sectional HRTEM is presented. The aspect ratio (height/
diameter) of the dots depends on GaN coverage and growth temperature. While in other material
systems the capping process strongly infl uences the shape and composition of the buried QDs, the
GaN/AlN dots grown in the SK mode do not exhibit signifi cant Ga–Al interdiffusion [36] . More
recently, it has also been shown that during the capping process with AlN, the GaN/AlN QDs only
experience a slight reduction of their volume, but do not change their shape [35] .
(a) AFM (b) HTEM
(c) Structure model
Z
X
Y
[0001]
[0110]
[2110]
α
Figure 7.3 (a) AFM image of an uncovered GaN QD grown in the SK growth mode by MOCVD. (b) Cross-
sectional HRTEM image of a covered QD. (c) Model for the QD structure. The angle α corresponds to 32° [13] . With
permission from Wiley.
The smoothing effect of the AlN cap layer allows growing superlattices containing many lay-
ers of GaN QDs that preserve an excellent crystal quality. Furthermore, in QD multilayer struc-
tures with a thin enough spacer, QDs display a vertical alignment along the growth direction as
shown in Fig. 7.4 . The value of the critical AlN spacer thickness for which the vertical correla-
tion between adjacent GaN dot layers is observed seems to depend not only on QD height, but
also on its diameter [37, 38] . According to a model developed by Tersoff et al . [39] , the verti-
cal correlation between adjacent layers of dots is thought to be mediated by the strain fi eld in
the spacer that results from the presence of the buried QDs. The modifi cations in the strain in
GaN/AlN QDs when going from an isolated dot to a multilayer sample will be analysed in section
7.2.2 in the light of the continuum elasticity model. Another effect of the piling-up process is the
homogenization of the dot sizes in successive dot planes accompanied by an average increase of
their diameter, see Fig. 7.4 . The spacer layer acts like a bandpass fi lter that prevents the formation
of QDs of smaller size. In Fig. 7.5 it is shown that, in correspondence to the diameter increase, a
decrease of the dot density takes place as the number of QD layers increases [37] .
CH007-I046325.indd 234CH007-I046325.indd 234 6/27/2008 5:15:50 PM6/27/2008 5:15:50 PM
Optical and Vibrational Properties of Self-assembled GaN Quantum Dots 235
Figure 7.4 HRTEM of vertically aligned GaN/AlN QDs grown by PAMBE in the SK mode at 720°C. The 2D GaN
WL is clearly visible [23] . With permission from AIP© 1998.
Island density (10
11
QD/cm
2
)
1.2
1.0
0.8
0.6
0.4
1 10 100
Number of periods
Figure 7.5 QD Areal density in the upper layer of GaN QD multilayer samples as a function of the number of
layers [37] . With permission from AIP© 2004.
Until very recently, the (0001) wurtzite direction has been preferentially chosen for the growth
of GaN/AlN QDs. Growth directions perpendicular to the polar c -axis may, however, pose inter-
esting advantages for the optical performance of the QDs, as will be shown in section 7.4.5.
Incidentally, these so-called “ non-polar ” GaN QDs have been grown on
()11 20
-oriented SiC
substrates [26] , and their growth dynamics are very different from those of the more common
“ polar ” QDs grown along the c -axis. Extensive details on the growth of these structures can be
found in chapter 6.
Other less extended growth methods have demonstrated the possibility of obtaining GaN
quantum dots without the use of surfactants or the need of high lattice–mismatch between the
GaN layer and the substrate. These techniques take advantage of the low melting point of Ga,
and rely on the deposition of Ga droplets on the substrate surface, which are subsequently nitri-
dated. Among other methods, we can mention the procedure developed by Kawasaki et al. [18]
using a gas-source MBE (GSMBE) chamber, and the vapour–liquid–solid mechanism (VLS)
proposed by Hu et al. [19] , see Table 7.1 .
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236 Handbook of Self-assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
7.2.2 Strain fi eld distribution
The detailed understanding of the strain distribution in SK self-assembled QDs is of paramount
importance for various reasons. In the fi rst place, the strain relaxation acts as the main driving
force in the self-organized growth itself. It is therefore important for understanding the equilibrium
dot shape and the spatial correlation in the positioning of the islands. The local strain at the sur-
face above the capped dots is also determinant for the vertical ordering of the subsequent dot lay-
ers [40] . Second, as in every pseudomorphic structure, the strain substantially modifi es the band
structure. And fi nally, in the particular case of III-N heterostructures, due to the large piezoelectric
effect, the strain results in high internal electric fi elds, which in turn affect strongly the electronic
and optical properties (see section 7.4.2). In addition, the study of the actual strain distribution in
a given sample allows quantifying its deviation from the expected elastic relaxation, and therefore
can shed some light on the presence of defects, such as dislocations, stacking faults, etc.
Information about the strain can be accessed experimentally through structural characteriza-
tion techniques (electron diffraction and transmission, X-ray diffraction, ion scattering) and opti-
cal probe techniques (most notably Raman and infrared spectroscopy). In the case of GaN/AlN
QDs, rather exhaustive studies of the strain by means of structural characterization techniques
have been performed, as will be described in detail in chapter 6. The corresponding light scat-
tering studies will be reviewed in section 7.3. On the theoretical side, two general approaches
are used for the calculation of the elastic strain connected with self-assembled islands, namely,
atomistic and continuum elasticity simulations [41] . In the former approach, two- and many-
atom potentials are used for the calculation of the elastic energy stored in the sample; the strain
is obtained by minimizing this energy. In the continuum elasticity calculations the actual struc-
ture is replaced by an elastic continuum and the strain distribution is obtained again by minimiz-
ing the elastic energy or by solving the elasticity equilibrium equation. The relevant parameters
within this approach, besides the elastic constants, are the misfi t strains:
ε
a
aa
a
AlN GaN
GaN
0 024.
ε
c
cc
c
AlN GaN
GaN
0 039.
(7.1)
The lattice constants of wurtzite GaN and AlN can be found in Table 7.2 . Note that here the lat-
tice mismatch is more moderate than in the case of InAs/GaAs dots, where it is around 0.07.
Both modelling strategies, atomistic and continuum, have been applied to every kind of pseu-
domorphic structures, and they appear to give the same qualitative features and very similar
quantitative results in many cases [42] . In the case of III-N wurtzite QDs, the atomistic valence-
force-fi eld method has been used to obtain the strain distribution in [43–46] . On the other hand,
continuum elasticity models with various degrees of sophistication have also been applied to
those nanostructures [47–51] .
Table 7.2 Material parameters of bulk wurtzite
GaN and AlN used to calculate the strain [52] .
GaN AlN
a (Å) at T 300 K
3.189 3.112
c (Å) at T 300 K
5.185 4.982
C
11
(GPa) 390 396
C
33
(GPa) 398 373
C
12
(GPa) 145 137
C
13
(GPa) 106 108
C
44
(GPa) 105 116
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Optical and Vibrational Properties of Self-assembled GaN Quantum Dots 237
In order to illustrate the main features of the strain distribution of GaN/AlN(0001) QDs, we
present here model calculations for a simplifi ed situation, in which linear elasticity is assumed
and the dot is considered as a misfi tting inclusion in an otherwise infi nite matrix. The model is
based on the Eshelby’s method of inclusions [53] generalized to account for the transversely
isotropic elastic symmetry of the wurtzite crystal structure [54] . The calculations are similar to
those presented in [47] , except for the fact that a single inclusion in an infi nite matrix is consid-
ered here, whereas a free-standing periodic array of dots is analysed in [47] . In order to avoid
time-consuming three-dimensional calculations, the reported pyramidal shape of the dots (see
section 7.2.1) was modelled with a truncated cone shape, with base radius R (diameter D 2 R ),
height h and an angle between the base and lateral surface equal to 30°. Unless otherwise stated,
this shape will be adopted for the remainder of the calculations shown in this work. The differ-
ence between the results for both shapes, pyramid and cone, is negligible except in the surface
of the dot [47] . One important parameter for the specifi cation of the dot geometry is the aspect
ratio, r h / D . The results for a QD of typical dimensions are shown in Fig. 7.6 .
6
4
0
0
0
0
2
10 10886644220
0
0
0
2
4
ρ (nm)
ρ (nm)
z (nm)
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
ε
zz
ε
⊥
3.0
2.5
2.0
1.5
1.0
0.5
0.5
1.0
1.5
2.0
2.5
3.0
0
2
Figure 7.6 Contour plots of the strain components ε
zz
and ε
⬜
( ε
xx
ε
yy
)/2 (in %) in a GaN QD with dimensions
R 8.3 nm and h 3.6 nm (i.e. r 0.2), laying on a WL of thickness d 0.5 nm. For the sake of simplicity, equal
elastic constants (those of AlN, see Table 7.2 ) have been assumed throughout the whole structure. The dashed contour
lines correspond to zero strain and the rest of the lines correspond to successive increments of 0.25%.
On account of its relative fl atness, one approximation frequently used for the state of deforma-
tion of the QDs is to assume that it is equivalent to that of a GaN QW grown pseudomorphically
on AlN (0001). This so-called biaxial strain is given by:
εεε
xx yy a
εε
zz xx
C
C
2
13
33
,
,
.
GaN
GaN
(7.2)
The elastic constants of GaN and AlN can also be found in Table 7.2 . The results in Fig. 7.6 reveal
the limitations of the above simplifi ed approximation. Due to the presence of the lateral surfaces,
the strain in the GaN/AlN QD deviates considerably from the biaxial case. The general features
are the following:
(i) The strain is not homogeneous inside the dot, in contrast with the biaxial approxima-
tion. In general, the whole volume of the dot is compressed in the in-plane directions,
i.e.
ε
⬜
( ε
xx
ε
yy
)/2 0. On the contrary, as a consequence of the barrier constraint
through the lateral surfaces, as one moves from the base to the top of the dot the strain
ε
zz
passes from tensile to compressive. In the case of Fig. 7.6 , the average values at the QD
base ( z 0 ) are ε
⬜
2.12% (less negative than the misfi t strain ε
a
2.4%) and
ε
zz
0.75% (much less than that predicted by the biaxial approximation 1.3%),
and their ratio ε
zz
/ ε
⬜
0.35 is clearly away from the biaxial value ( 0.53). This different
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238 Handbook of Self-assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
behaviour accounts for the partial relaxation of the stored elastic energy associated with
the dot formation.
(ii) The strain is not zero in the barrier: in the regions above and below the dot, the AlN
matrix is expanded in the plane and compressed along the z -direction. The deformation
presents a maximum at the bottom and top surfaces and slowly decays with the distance.
(iii) The shear components of the strain (not shown in Fig. 7.6 ) are non-vanishing, though
they are strongly localized to the perimeter of the bottom and top surfaces of the dot (in
general, to the edges joining surfaces with different orientation), where they can reach
rather large values.
It is to be noted, however, that the results shown in Fig. 7.6 change qualitatively when the geo-
metrical parameters, most notably the QD aspect ratio r , are varied. Thus, for higher values of r
(e.g. 0.25), ε
zz
can become compressive through most of the dot volume, in radical contrast to
the biaxial approximation, whereas for smaller values (e.g. r 0.15), the biaxial approximation
gives reasonable results. Fortunately, it is usually found that the aspect ratio of the dots within a
given sample is approximately constant. For the sake of simplicity, some authors assume an iso-
tropic elastic approximation in the study of the strain in wurtzite-type QDs [51, 55] . It must be
warned that, while the general trends shown in Fig. 7.6 can be emulated with such approxima-
tion, it is impossible to reproduce accurately the quantitative values.
It is also instructive to comment about the modifi cations in the strain when going from an iso-
lated dot to a series of stacked dots. These results are relevant for the interpretation of experiments
performed on multiple dot layer samples, as will be seen in sections 7.3 and 7.4. In Fig. 7.7 we
present the results of the on-axis ( ρ 0 ) strain for various stacks of varying number of QD layers N .
1.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
0.5
1.0
1.5
2.0
2.5
60 40 20 20 40 600
1.0
0.5
0.0
z (nm)
ε
zz
(10
2
)
ε
⊥
(10
2
)
(a)
(b)
Figure 7.7 Strain components ε
⬜
(a) and ε
zz
(b) along the axis of a stack of N 1, 3, 7, 11 and 25 dots. The
grey colour of the curves goes from darker to lighter as N increases. The dots lie on a WL of thickness d 0 . 5 n m .
The sizes of the dots have been taken following the experimental results of [37] , i.e. R 17.5 nm and h 3.8 nm
(therefore, r 0.11), with an interdot spacer thickness of 8.6 nm.
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