Send reproduction fee to Elsevier Science S.A. · with numerical finite element simulations, ......
Transcript of Send reproduction fee to Elsevier Science S.A. · with numerical finite element simulations, ......
Critical dimensions for the formation of interfacial mis®t dislocations ofIn0.6Ga0.4As islands on GaAs(001)
K. Tillmanna,*, A. FoÈrsterb
aCentre for Microanalysis, University of Kiel, D-24143 Kiel, GermanybInstitute for Thin Film and Ion Beam Technology, Research Centre JuÈlich Ltd., D-52425 JuÈlich, Germany
Received 26 September 1999; received in revised form 24 November 1999; accepted 11 February 2000
Abstract
The critical dimensions for the introduction of interfacial mis®t dislocations into epitaxially grown In0.6Ga0.4As islands on GaAs(001)
substrates are investigated by high-resolution transmission electron microscopy. By the combination of the continuum theory of elasticity
with numerical ®nite element simulations, equations are derived in order to calculate a phase diagram separating the coherent from the
incoherent phase in dependence on the island size. It is demonstrated that, if the elastic relaxation of the laterally limited epilayers is taken
into account in a quantitative manner, the derived expressions for the coherent-to-incoherent transition are in excellent agreement with the
experimental observations. When geometrical and material parameters are modi®ed correspondingly the model calculations may also be
applied to the analysis of other heteroepitaxial systems for which dislocation formation is not kinetically limited. q 2000 Elsevier Science
S.A. All rights reserved.
Keywords: Transmission electron microscopy; Molecular beam epitaxy; Nanostructures; Computer simulations
1. Introduction
Self-organization and spontaneous formation of lattice
mismatched nanostructures are synonyms which are used
to describe methods for preparing nanoscale structures
with tailored electronic properties by direct epitaxial
growth. Especially self-assembled island structures grown
by advanced deposition techniques such as molecular beam
epitaxy (MBE) are presently of interest as they are known to
exhibit the carrier con®nement properties of quantum dots
[1±3]. In this respect, a key issue is that initially planar
epitaxial layers under stress can lower their elastic strain
energy not only by the formation of interfacial mis®t dislo-
cations but also by an elastic relaxation mechanism due to
the formation of three-dimensional islands [4]. Indeed,
highly-mismatched InxGa12xAs layers (x $ 0:25) epitaxi-
ally grown on GaAs(001) substrates follow the Stranski±
Krastanow [5] growth mode as has been shown in a number
of previous analyses [6±8]. Under such conditions growth
starts with the formation of a two-dimensional wetting layer
until a certain nominal layer thickness hc is reached which is
in the order of one to three monolayers depending on the
indium content x chosen [9,10]. Above hc, a morphological
transition occurs and three-dimensional islands nucleate on
top of the wetting layer which remain free of dislocations in
their earliest stages of formation. These islands partially
relieve their strain energy by an expansion of the InxGa12xAs
lattice close to the free surfaces of the islands and by the
distortion of the GaAs substrate beneath the islands
[4,11,12]. Finally the initially coherent islands evolve into
larger islands by ripening or coalescence, in which plastic
strain relaxation takes place by the introduction of interfa-
cial mis®t dislocations [13±16].
In the present context, a crucial point with regard to the
technological utilization of self-assembled InxGa12xAs
island structures as quantum dots is the accurate determina-
tion of critical geometrical dimensions below which islands
will remain free of dislocations. This issue of giving quan-
titative information on the coherent-to-incoherent transition
is of less interest to take stock in the formation of disloca-
tions but of special practical interest since experimental
analyses are often carried out by techniques which reveal
information on the layer geometry but not on structural
defects of the epilayer, e.g. atomic force microscopy
(AFM), scanning tunneling microscopy (STM) or scanning
electron microscopy (SEM). Through pioneering work [17±
20] it is well established that equilibrium models may be
Thin Solid Films 368 (2000) 93±104
0040-6090/00/$ - see front matter q 2000 Elsevier Science S.A. All rights reserved.
PII: S
0040-6090(00)00858-0
www.elsevier.com/locate/tsf
* Corresponding author. Present address: Institute for Solid State
Research, Research Centre JuÈlich Ltd., D-52425 JuÈlich, Germany. Tel.:
149-431-775-72-508; fax: 149-431-775-72-503.
E-mail address: [email protected] (K. Tillmann).
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
applied to describe the evolution of the dislocation density
as a function of the island size when assuming pure tetra-
gonal lattice distortions of the epilayer [15,21]. However,
not surprisingly the model predictions on critical island
sizes for the formation of the ®rst mis®t dislocation barley
agree with experimental measurements since models
neglect the possibility of the elastic strain relaxation.
Hence, model calculations usually yield a signi®cant under-
estimate of critical island sizes by a factor of two to three
compared with experimental results obtained, e.g. by trans-
mission electron microscopy (TEM).
In the present study analytical expression for critical
island dimensions are derived based upon thermodynamic
equilibrium theory which precisely take into account the
island geometry and the corresponding amount of the elastic
strain relief by the application of ®nite element method
(FEM) calculations. Experimental results obtained from
the analysis of In0.6Ga0.4As/GaAs(001) heterostructures,
characterized by a nominal lattice mismatch of f �12 asub=alay � 0:04121 with asub � 0:56533 nm and alay �0:58963 nm denoting the bulk lattice parameters of the
GaAs substrate and the In0.6Ga0.4As layer, respectively, are
compared with the model calculations. It is demonstrated
that the model predictions on the coherent-to-incoherent
transition are in excellent agreement with the experimental
results gained by high-resolution transmission electron
microscopy (HRTEM) when considering the elastic strain
relief in a quantitative manner by an effectively reduced
lattice mismatch. The model calculations may also be
applied to the analysis of other heteroepitaxial systems
when geometrical and material parameters are modi®ed
correspondingly.
2. Experimental
The heteroepitaxial growth of the In0.6Ga0.4As layers was
performed by MBE on buffered GaAs(001) substrates
slightly misorientated by 28 towards the crystallographic
[010] direction. A constant growth rate of 0.2 mm/h and a
temperature of TS � 5208C were chosen with nominal layer
thickness ranging between 6 and 12 monolayers. As shown
in a previous analysis [22] growth under such conditions
results in the formation of coherent as well as incoherent
islands characterized by a quite narrow size distribution for
both types of islands. Electron-transparent specimens were
prepared by mechanical standard procedures followed by
Ar1 ion milling at 4 keV with liquid-nitrogen cooling
until perforation and, subsequently, at 2 keV to minimize
the thickness of an amorphous surface layer induced by the
ion milling process. The layer morphology and defect struc-
ture were analyzed by conventional TEM of plan-view and
cross-sectional samples using a PHILIPS CM30 electron
microscope as well as by recording high-resolution micro-
graphs taken with a JEOL 4000EX electron microscope
characterized by a Scherzer resolution of 0.17 nm. To calcu-
late the strain distribution of elastically relaxed In0.6Ga0.4As
island structures on GaAs(001) three-dimensional FEM
simulations were carried out on an IBM RS/6000-R50
machine by application of the ABAQUS FEM code [23].
3. Layer morphology and defect structure
Fig. 1 shows a bright-®eld TEM image of an In0.6Ga0.4As/
GaAs heterostructure characterized by a nominal layer
thickness of 12 monolayers. The plan-view micrograph
has been taken under [001] zone axis conditions, i.e. with
the electron beam oriented parallel to the growth direction
of the sample. Two basic types of island structures may be
distinguished, smaller coherent and larger incoherent
islands.
A characteristic cross-shaped image contrast resulting
from the local misorientation of lattice planes is associated
with the smaller islands. These bright-dark contrast slopes
must not be misinterpreted as a direct image of a potential
square based island geometry. Indeed, recent image contrast
simulations for bright-®eld images oriented along the [001]
zone-axis demonstrate that the observed 4-fold symmetry of
the image contrast rather re¯ects the symmetry of the cubic
lattice than the geometry of the islands and circular based
structures will show a similar contrast behaviour as a matter
of principle [24]. Moreover, atomic force and scanning elec-
tron microscopic images obtained from the samples reveal
an approximate circular base island geometry [25] which is
in agreement with AFM analyses of island structures grown
under comparable conditions [26±28].
For the majority of the larger islands the image contrast is
dominated by a network composed of two orthogonal sets of
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±10494
Fig. 1. Plan-view micrograph of an In0.6Ga0.4As layer with a nominal thick-
ness of 12 monolayers grown on a GaAs substrate taken along the [001]
zone axis of the sample. mo ire fringes along the [110] and the [1Å10]
directions indicate that larger islands are plastically relaxed by the forma-
tion of mis®t dislocations. The bright-dark contrast slopes forming a cross-
shaped pattern associated with the smaller islands result from local crystal
misorientations due to the elastic relaxation of lattice mismatch induced
elastic strains.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
moire fringes along the [110] and the [1Å10] directions indi-
cating that the critical island sizes have been exceeded and
mis®t dislocations have been generated. Using the ~g £ ~b � 0
extinction criterion under weak-beam imaging conditions
[29] most of the dislocations are found to be of Lomer
type with Burgers vectors ~b � alay=2k110l oriented perpen-
dicular to both the dislocation line and the (001) interface
normal. However, having a closer look at the defect struc-
ture by analyzing high-resolution micrographs it is found
that mismatch accommodation by these pure edge disloca-
tions, located solely in the islands interior, takes place only
in those islands containing a larger number of defects [22].
Contrastingly, single 608 dislocations with burgers vectors
of type ~b � alay=2k110l are observed exclusively in the near-edge regions of those islands which are characterized by
only one or two dislocations as demonstrated by the lattice
image shown in Fig. 2. Similar experimental observations
have recently been published for pure InAs islands grown by
metalorganic vapor phase epitaxy on GaAs substrates [21].
It is reasonable to assume that these 608 slip dislocations
will have nucleated, perhaps as surface glide loops, at the
free surfaces close to the island edges, i.e. the regions of
highest stress, and will have moved on {111} slip planes to
the heterointerface. At the later stages of growth sessile
Lomer dislocations will have formed by one or another of
the commonly suggested mechanisms implying the combi-
nation of two 608 dislocations with appropriate oriented
Burgers vectors [30±32]. However, with regard to the calcu-
lation of critical islands sizes the matter of consequence is
that the ®rst dislocations nucleated close to the island edges
are of 608 type.
Generally, the determination of critical island dimensions
may be performed by both, the measurement of the maxi-
mum sizes of coherent islands as well as the measurement of
the minimum sizes of incoherent islands. In the following
both complementary techniques will be used for the analysis
of the coherent-to-incoherent transformation geometries.
Fig. 6 later shows a diagram separating the coherent (®lled
dots) from the incoherent (®lled triangles) phase dependent
the island radius r measured at the island basements and the
inverse aspect ratio 1=p � 2r=h with h denoting the island
height. Data result from the analysis of a large number of
cross-sectional high-resolution micrographs. Special care
has been taken in the selection of images suitable for the
quantitative determination of critical island dimensions.
High-resolution transmission microscopy is a two-dimen-
sional projection technique which requires rather thin speci-
mens with a typical thickness along the direction of the
electron beam in the range between 5 and 50 nm. Hence,
there is a certain probability for cutting through an island
not including the island centre and recorded islands may
appear smaller than they have actually been before the
preparation of the specimens.
On the one hand, this potential source of error does not
represent a severe problem in the determination of the maxi-
mum coherent island size when measurements are
performed on a large number of islands. Although some
of the coherent islands will be arti®cially reduced in size
there is also a fair chance of ®nding non-carved islands
which increase with the number of islands investigated.
Therefore the largest coherent island sizes found will indeed
represent a suggestive estimate for the maximum dimen-
sions of coherent structures.
On the other hand, preparatory artifacts must be expelled
when experimentally determining the minimum sizes of
incoherent islands. In this case the imaging of truncated
island regions may result in a signi®cant underestimation
of the critical island dimensions. Therefore, during the
analysis only those incoherent islands have been considered,
for which the image contrast motifs con®rmed a sample
thickness t larger than the island radius r which ensures
that the actual island dimensions are correctly measured.
For this purpose the defoci Df of individual micrographs
have been measured from diffractogram analyses of the
central parts of the In0.6Ga0.4As islands and the correspond-
ing specimen thicknesses t have been determined by
comparison with simulated images calculated by the appli-
cation of the macTempas package [33]. By this means a
systematic under-evaluation of incoherent island dimen-
sions resulting in an underestimation of critical island
sizes may be ruled out. In principle, a more accurate experi-
mental determination also of the coherent island dimensions
would require one to choose the same exclusion procedure
for both types of islands. However, since this procedure is
quite time consuming and no relevant information is lost
with regard to the coherent-to-incoherent transformation
when additionally considering some coherent islands arti®-
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±104 95
Fig. 2. High-resolution micrograph of a plastically relaxed In0.6Ga0.4As
island on a GaAs substrate taken along the [110] direction. The positions
of interfacial mis®t dislocation are marked by the white arrows. The defect
on the right is identi®ed as a 608 dislocation with a closing failure vector of
type ~b� alay/2k101l as can be seen by performing a Burgers circuit around
the dislocation core in the magni®ed clipping. The defect on the left repre-
sents an incipiency in the formation of sessile Lomer dislocations with one
608 dislocation already located at the heterointerface and with a second one,
being split up in two partials bounding an intrinsic stacking fault, only later
to merge with the ®rst one to a perfect edge dislocation.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
cially reduced in size, corresponding analyses have been
omitted in this study.
Looking at Fig. 6 the criterion used for the selection of the
incoherent islands yields a clear separation between the
coherent and the incoherent phase, i.e. there is no distinct
intermixing between the ®lled dots (coherent) and the ®lled
triangles (incoherent) in different areas of the plot. The plot
demonstrates a broad scattering of the geometrical dimen-
sions for both types of islands which allows one to draw a
clear separation between both phases. The measured island
size distribution itself will be determined by different para-
meters and effects. The data result from samples character-
ized by different nominal layer thicknesses while all other
growth parameters have been retained during the deposition
of the layers. Hence larger island sizes may be intuitionally
assigned to the samples characterized by the larger nominal
layer thicknesses. Moreover, incoherent islands will vary in
size most probably due to ripening and coalescence during
the epitaxial growth. This explanation is corroborated by the
plan-view micrograph shown Fig. 1 which demonstrates a
wide range of lateral extensions of the incoherent islands at
a ®xed nominal layer thickness. In addition to these expla-
nations, the rather broad distribution of the coherent island
sizes measured from the cross-sectional micrographs may
be due to preparatory artifacts since plan-view images
substantiate a rather narrow size distribution at least of the
lateral island extensions, cf. Fig. 1. Additionally, the scatter-
ing of the island aspect ratios at a ®xed radius may be
attributed to kinetic effects preventing individual islands
from attaining their equilibrium shape during the epitaxial
growth.
4. Model calculations
In the following a mixed analytical and numerical
approach striving for the quantitative determination of criti-
cal island sizes is presented assuming a conical island shape.
Other possible island geometries frequently observed in
other experiments will be discussed later. Calculations are
based upon an energy equilibrium approach which implies
that the formation of dislocations is not suppressed by
kinetic reasons. This assumption is well founded when
clearly distinguishing between the limitations in the growth
kinetics with regard to the evolution of the equilibrium
surface morphology and the potential kinetic limitations
with respect to the nucleation of dislocations. As already
mentioned, under the chosen growth conditions the scatter-
ing in the island size distribution may indeed result from an
insuf®cient adatommobility preventing individual islands to
reach their equilibrium shape at a ®xed nominal layer thick-
ness. However, this does not affect the nucleation of mis®t
dislocations, e.g. by the formation of surface glide loops
which is assumed to be the dominant mechanism for their
formation. From previous investigations focusing on the
calculation of conditions for the spontaneous formation
and the subsequent glide of half loops within two-dimen-
sional InxGa12xAs layers on GaAs(001) it is known that the
system may be treated as an equilibrium system for an
indium content of x $ 0:6 and a growth temperature of
TS $ 5008C [34,35]. Additionally, recent analyses reveal a
signi®cant lowering of the activation energy for the forma-
tion of surface half loops close to the edges of three-dimen-
sional islands compared to two-dimensional layers [21,36].
This ®nding results from a superposition of the lattice
mismatch induced stresses and local stresses by the bending
of the lattice planes due to the overall elastic relaxation of
the strain energy. Hence, for the epitaxial system investi-
gated in this study a thermodynamic equilibrium model may
be applied for the calculation of critical island dimensions.
(It is worth mentioning that critical dimensions of Ge
islands on Si(001), i.e. a system characterized by a lattice
mismatch comparable to the In0.6Ga0.4As/GaAs(001) system
investigated here, are found to be distinctly larger even
when deposited under comparable growth conditions [37±
39]. This discrepancy results from a much larger degree of
metastability in the GexSi12x/Si(001) system while the
InxGa12xAs/GaAs(001) system behaves much like an equi-
librium system.) For the calculations isotropic elastic mate-
rial properties are assumed since the rather complex
analytical expressions associated with the elastic strain
®elds of dislocations taking elastic anisotropy into account
[40,41] suggest this simpli®cation for the sake of clarity.
Fig. 3 gives a sketch of de®nitions of geometrical para-
meters frequently used in the following.
4.1. Critical island sizes for the formation of mis®tdislocations
From an energetic point of view the condition for the
nucleation of the ®rst mis®t dislocations at a given island
size is that the strain energy gain DEela associated with the
plastic reduction of the nominal lattice mismatch f has to
balance the energy DEdis associated with the strain ®eld of
the mis®t dislocation itself, i.e.
DEela 1 DEdis # 0 �1�Assuming isotropic elastic material properties the elastic
strain energy density r coh of a biaxially strained coherent
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±10496
Fig. 3. Plan-view (left) and cross-sectional (right) representation of the
island and defect geometry for illustration of parameters frequently used
in this study, cf. text.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
epilayer on a (001) oriented substrate is given by
rcoh �Glay
12 nlay12110 1 2nlay11101 �110 1 12�110
� ��2�
rcoh � 2Glayf2 11 nlay12 nlay
�3�
with Glay and n lay denoting the shear modulus and the Pois-
son ratio of the epilayer, respectively, and 1110 � 1 �110 � 2f
denoting the [110] and the [1Å10] in-plane components of the
elastic strain tensor. Upon formation of the ®rst interfacial
mis®t dislocation with, say, a dislocation line ~u oriented
along the [110] direction the value of 1110 � 2f remains
unchanged and the orthogonal strain component is reduced
to 1 �110 � 2f 1 beff =2r with beff denoting the absolute value
of the [1Å10] component of the Burgers vector. Hence, the
strain energy density r incoh associated with a plastically
relaxed island containing only one dislocation is given by
rincoh � 2Glayf2 11 nlay12 nlay
12beff2rf
1b2eff
8r2f 2�11 nlay�
" #�4�
and the total strain energy reduction due to the formation of
the ®rst mis®t dislocation amounts to
DEela � �rcoh 2 rincoh�V �5�
DEela � 2Glayf2 11 nlay12 nlay
beff2rf
2b2eff
8r2f 2�11 nlay�
" #V �6�
with
V � p
3hr2 � 2p
3pr3 �7�
representing the volume of the conical island.
In general, the strain energy E per unit length L associated
with the strain ®eld of an interfacial mis®t dislocation is
given by [42]
E
L� 1
2p
GsubGlay
Gsub1Glay
j ~b j2 cos2f1sin2f
12 nlay
" #
� ln R= j ~b j� �
1 1h i
�8�with Gsub and f denoting the shear module of the substrate
and the angle inclined between the Burgers vector ~b and the
dislocation line ~u, respectively, as well as R representing an
outer cut-off radius of the dislocation strain ®eld. If, at ®rst,
no further assumptions on the in plane position of the mis®t
dislocation along the [1Å10] direction are made, a suitable
choice for the length of the heterointerfacial dislocation
segment will be to choose this length equal to the average
island basement extension when slicing the island at an
arbitrary position. Hence, the length of the heterointerfacial
dislocation segment may be calculated according to
l � 2Zr
0dx
���������r2 2 x2
psZr
0dx � p=2r �9�
which results in a total energy associated with the elastic
strain ®eld of the dislocation given by
DEdis � lE
L�10�
Strictly speaking, a dislocation line extension l smaller
than the diameter 2r of a circular based island implies that
margining regions of the island are still strained in full
compression. By analyzing the volume of these chamfered
ends sketched grey in Fig. 3 it is found that the fractional
volume of these regions amounts only to
2��12 �p=4�3�2 �3=p� ��p=4��������������12 p2=16
p1 arcsinp=4�
�12 �p=4�2�� < 0:007
and may hence be reasonably neglected.
Unlike the case of the two-dimensional layer growth,
where the cut-off radius R of the dislocation strain ®eld is
usually assessed either by the layer thickness or by the
average mutual half dislocation distance [35], the island
geometry and the position of the mis®t dislocation itself
must be considered when choosing adequate R values in
the case of three-dimensional island structures. Taking
into account that iso-contours of the local strain energy
density of a mixed screw and edge dislocation of 608 typeare characterized by `butter¯y-like' symmetry with a maxi-
mum of values inside the heterointerfacial plane and a mini-
mum perpendicular thereto when slicing the material along
the dislocation line [42], an appropriate choice will be to
asses the cut-off radius by the geometric mean
R � ���������h�r1�r2
pof both the dislocation core distance h(r1) towards the
islands free surface along the [001] direction and the
distance r2 � r 2 r1 towards the nearest island edge along
the [1Å10] direction. As can be seen from simple geometrical
considerations with l de®ned according to Eq. (9) these
distances are given by
r1 � �12 d�r �11�
r2 � dr �12�
h�r1� � dh �13�with the parameter d de®ned as
d � 12�������������12 p2=16
ph i�14�
and resulting in a cut-off radius amounting to
R � rd���2p
p �15�At a ®rst glance this de®nition of the cut-off radius, which
is primarily determined by the r1/r2 ratio, may seem to be a
quite arbitrary choice, but it is not with respect to the experi-
mental observations. Eqs. (11)±(14) mean that r1=r2 �1=d2 1 � 1:625 is assumed for the calculation of the cut-
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±104 97
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
off radius. Measurements on the micrograph shown in Fig.
2, however, con®rm r1/r2 ratios of 1.564 and 1.857 for the
left-hand side and the right-hand side defect of the island.
Therefore, the de®nition of the cut-off radius according to
Eq. (15) is also reasonably motivated from an experimental
point of view.
By combination of Eqs. (1)±(15) an expression according
to Eq. (16) for the critical island radius r � rc;cone is
obtained above which the incorporation of the ®rst mis®t
dislocation into the island becomes energetically favorable.
rc;cone�p� � 2
beff f£(
b2eff8�11 nlay� 1
"cos2f1
sin2f
12 nlay
#
£ ln rc;cone���2p
p �12�������������12 p2=16
p�= j ~b j
� �1 1
h i
£"G�l;V� 12 nlay
11 nlay
Gsub j ~b j 2Gsub 1 Glay
#)(16)
In this equation
G�l;V� � 1
4p
l
Vr 2 � 3
16pp�17�
represents a geometry factor which only depends on the
length of the interfacial dislocation segment l and the island
volume V. Numerical G(l ,V) values have to be modi®ed if
island geometries different from the conical shape consid-
ered here are assumed.
Eq. (16) represents an implicit expression to be evaluated
numerically. The equation gives information on the critical
island radius depending on the aspect ratio p, the defect
structure (~b,beff,f ) and material properties ( f,Gsub,Glay,n lay).
In Fig. 6 corresponding model predictions on the critical
island radius rc,cone are given by the lower solid curve as a
function of the inverse aspect ratio 1/p using material para-
meters summarized in Table 1 for In0.6Ga0.4As epilayers on
GaAs substrates. According to the preceding experimental
analyses plastic mis®t relaxation by a 608 dislocation char-
acterized by ~b � a lay=2k011l, beff � alay=��8
pand f � 608 is
assumed. For the experimentally relevant range of aspect
ratios 1=2 $ p $ 1=10 the calculations yield critical island
radii in the range between 5 and 14 nm, i.e. Eq. (16) predicts
coherent islands only for those island radii and aspect ratios
represented by the darker grey shaded area in Fig. 6. Experi-
mental results, however, demonstrate the existence of much
larger coherent islands as sketched by the ®lled dots in the
®gure.
4.2. Elastic relaxation of coherent islands
In the derivation of Eq. (16) it has so far been assumed
that coherent islands are homogeneously and fully strained
in biaxial compression, i.e. 1110 � 1 �110 � 2f , resulting in a
strain energy density according to Eq. (3). Compared to a
biaxially strained layer, e.g. an in®nitely extended two-
dimensional epilayer, three-dimensional islands are elasti-
cally relaxed at least in part, i.e. their elastic strain energy is
reduced due to a lateral bending of lattice planes close to the
islands free surfaces. Hence, the formation of mis®t disloca-
tions will take place at larger island sizes compared to the
predictions according to Eq. (16) when the possibility of the
elastic strain relief is taken into account.
In order to consider the impact of the elastic strain relaxa-
tion on the quantitative values of critical island sizes, three-
dimensional ®nite element simulations, taking fully into
account the island geometry and the elastic material proper-
ties according to Table 1, are applied to calculate the inho-
mogeneous elastic strain ®elds associated with coherent
In0.6Ga0.4As island structures on GaAs substrates. The
heterostructure geometry is meshed with three-dimensional
20-node cuboid ®nite elements. For simplicity the existence
of the completely strained ultrathin two-dimensional
wetting layer is neglected since it will have only a negligible
in¯uence on the strain distribution of the three-dimensional
islands. Based upon the experimental results, conical islands
are assumed which allow that only pie-like sectors of the
heterostructures have to be modeled thus saving central
processing unit time. By reasons of symmetry this strategy
implies to set appropriate bounding conditions during
processing, e.g. by ®xing all nodes on the free side faces
of the model against displacements in the respective direc-
tion parallel to the face normal and by ®xing the central part
of either the island an the underlying substrate against radial
displacements. Additionally, all nodes at the bottom of the
model are ®xed against displacements along the [001] direc-
tion to avoid rigid body motion during processing. The
accuracy of the FEM algorithm was checked by the modi-
®cation of boundary conditions and the treatment of border-
line test cases for which analytical solutions are available,
e.g. biaxially strained epilayers, for which perfect coinci-
dence between numerical and analytical results were found.
The contour plot in Fig. 4 shows the radial in-plane strain
component 1r � 1110 � 1 �110 as obtained from the simula-
tions for an island characterized by an aspect ratio p � 1=5.
As can be seen from the contour legend only the island
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±10498
Table 1
Numerical data on the lattice properties and the elastic constants of GaAs
and InAs used for the calculations in this study [43]. The values for
In06Ga04As follow from a linear interpolation between the corresponding
binary alloy data. Shear modules G and Poisson ratios n represent isotro-
pically averaged values calculated as G� (c11 2 c12)/51 3c44/5 and
n � (c11 1 4c12 2 2c44)/(4c11 1 6c12 1 2c44) according to [44]
Material GaAs In0.6Ga0.4As InAs
a (nm) 0.56533 0.58963 0.60583
f 0.04121 0.06686
c11 (GPa) 118.10 97.21 83.29
c12 (GPa) 53.20 48.44 45.36
c44 (GPa) 59.40 47.51 39.59
G (GPa) 48.62 38.21 31.24
n 0.23297 0.25583 0.27107
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
basement is compressively strained in the order of the lattice
mismatch (grey and black dotted contours with
1r < 2f � 20:04121) while upper parts of the island are
nearly completely unstrained (darker grey contours with
1r < 0). Moreover, below the central part of the island the
underlying substrate is slightly strained in dilatation (white
dotted contours with 1r $ 0) and the strain ®eld close to the
island edge becomes rather complicated actually represent-
ing a numerical singularity. However, the matter of conse-
quence is that the elastic relaxation reduces the nominal
lattice mismatch f to an effective mismatch feff according to
feff � fF�p� �18�where the correction function F (p) is given by the geome-
trical mean value of the radial strain component 1 r of all
island-related nodes i1{1¼n} of the ®nite element mesh,
i.e. by
F�p� � ���������������������������F1�p�¼F i�p�¼Fn�p�n
p �19�with
F i�p� � 1r;i�p�=f �20�By carrying out FEM simulations assuming different
island aspect ratios a discrete set of F(p) values represented
by the black dots in Fig. 5 is obtained. For experimentally
relevant aspect ratios 1=10 # p # 1=2 these data are well
approximated by an exponential function
F�p� � ������������������12 exp�2kp�p �21�
which is given by the solid curve in the ®gure. For conical
In0.6Ga0.4As islands on GaAs the decay parameter deduced
from the ®t amounts to k � 0:087. For p ! 0 Eqs. (18) and
(21) yield an effective mismatch feff�p ! 0� � f , thus repre-
senting the borderline case of a homogeneously strained
two-dimensional epilayer. For aspect ratios 1=10 # p #1=2 Fig. 5 demonstrates a lowering of elastic strains in the
order of 50% compared to the nominal mismatch of the
heterostructure. A comparable reduction of the nominal
lattice mismatch as well as a similar functional behaviour
of feff�p� / F�p� has been reported in a number of previous
analyses based upon either analytical [9,45,46] or numerical
[47±50] approaches.
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±104 99
Fig. 4. Contour representation of the radial in-plane strain component 1 r(r,z) of a conical In0.6Ga0.4As island on a GaAs substrate as gained by ®nite element
calculations. Strain values are given by the contour legend and displacements are magni®ed by a factor of ten for visualization purposes. The contour
representation reveals that only the island basement is strained in full compression in the order of the lattice mismatch (grey and black dotted contours with
1 r <2f� 0.04121) while the upper parts are nearly completely elastically relaxed (darker grey contours with 1 r < 0).
Fig. 5. Correction function F(p) according to Eq. (21) for a conical
In0.6Ga0.4As island on GaAs in dependence on the aspect ratio. For experi-
mentally relevant values 0:1 # p # 0:5 the calculated data (X) gained by
®nite element simulations may well be adjusted to a function F�p� ��������������������12 exp�2k=p�p
with k � 0.087 while different decay parameters kaccording to Table 2 have to be used when considering different island
geometries.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
4.3. Coherent-to-incoherent transformation
The impact of the elastic strain relaxation induced lower-
ing of the strain energy on the critical island size may now
be considered by applying the replacement rule
f ! feff � fF�p� �22�in Eq. (16), i.e. by replacing the nominal lattice mismatch
with the effective mis®t. Since the dominant contributions in
Eq. (16) indicate, that the critical island radius in a ®rst order
approximation scales as rc;cone / 1=f with the lattice
mismatch, in round terms a gemination of rc,cone values is
expected when assuming an effective mismatch being
halved compared to the nominal one. However, precise
numerical evaluation of Eq. (16) considering Eq. (22)
results in a functional behaviour of the critical island radius
rc,cone(1/p) given by the upper solid curve in Fig. 6. Due to an
increased elastic relaxation with increasing aspect ratios the
rc,cone values are disproportionately enlarged for rather small
1/p values compared to the assumption of a biaxially
strained epilayer. For instance, calculations demonstrate
an approximate 3-fold increase of the rc,cone value for 1=p �1=2 when the elastic relaxation is taken into account as can
be seen from Fig. 6. Not shown in Fig. 6, the upper and the
lower curve will merge at quite large 1/p values because of a
vanishing elastic relaxation for rather small aspect ratios.
4.4. Comparison with experimental results
The calculated rc,cone(1/p) curves in Fig. 6 in a manner
represent a phase boundary separating the incoherent
phase above the curves from the coherent phase below
(darker or lighter grey shaded areas dependent on the
assumption of a biaxially strained or an elastically relaxed
epilayer) in dependence on the island radius and the inverse
aspect ratio. As can be seen from Fig. 6, neglecting the
elastic relaxation results in a pronounced underestimation
of the critical island radii while its consideration yields an
excellent agreement between calculations and experimental
data obtained from the high-resolution micrographs.
Despite this obvious agreement between the theoretical
predictions and the experimental results some potential
sources of accidentally benevolent conjunctures with
respect to the measured island sizes and shapes of incoher-
ent islands need to be discussed. From real time growth
observations focusing on Ge islands on Si(001) it is
known that the island aspect ratios decrease upon the forma-
tion of mis®t dislocations via mass transport from the island
tips towards the island edges [51]. On the one hand this
behaviour does not in¯uence the predictions on rc,cone(1/p)
since the re-arrangement of deposited material takes place
only after the nucleation of the ®rst dislocation, i.e. the
elastic strain distribution associated with the geometry of
the coherent island is relevant for the calculation of critical
island dimensions in contrast to the geometry the incoherent
island adopts posterior. On the other hand the incoherent
island dimensions measured from the cross-sectional micro-
graphs may not represent the coherent-to-incoherent trans-
formation geometries but those of potentially modi®ed
incoherent islands at later stages of growth which may be
characterized by decreased aspect ratios. However, this
potential experimental artifact may be excluded in all like-
lihood because of the following two reasons.
Up to now a signi®cant mass transport within islands due
to the formation of dislocations has only been observed at
extremely small growth rates and rather high growth
temperatures [51] as well as upon the epitaxial overgrowth
of the islands with a capping layer [52] and may hence be
completely suppressed under the growth conditions chosen
in this study. In addition, if the rather broad distribution of
the coherent island sizes should result from kinetic limita-
tions preventing the islands from adopting their equilibrium
shape via mass transport as discussed above, the same argu-
ment makes a good case against potential changes of the
shape of incoherent islands for which mis®t strain driven
shape alterations will be further reduced due to an already
plastically decreased strain energy.
Even if any kind of mass transport is considered the
following worst case approximation yields that correspond-
ingly adopted experimental data are still consistent with the
model calculations. When assuming that an incoherent
island close to the coherent-to-incoherent phase boundary,
say, those characterized by the measured geometrical para-
meters rc;cone;f � 18 nm and 1=pf � 7:8 in Fig. 6, has under-
gone a reduction of the aspect ratio by a factor of two after
the nucleation of the ®rst mis®t dislocation, which is much
more than what is expected from the in situ experiments
[51], the conservation of the total island volume would
require an initial island radius of
rc;cone;i �������pf =pi
3p
rc;cone;i �����1=2
3p
rc;cone;i � 14:3 nm
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±104100
Fig. 6. Calculated critical island radius rc,cone in dependence on the inverse
aspect ratio 1/p assuming conical In0.6Ga0.4As islands on GaAs. The lower
solid curve results from the assumption of a biaxially strained epilayer,
while the upper curve takes elastic relaxation of the islands into account.
They grey-shaded areas below either curve represent the total range of
geometrical parameters for which coherent islands are predicted. The
experimental data on the geometry of coherent (®lled dots) and incoherent
(®lled triangles) islands gained by the analysis of HRTEM images only
coincide with the calculations when the elastic strain relief is considered.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
at an initial aspect ratio of pi � 2pf � 1=3:9. Having a look
at the calculated phase boundary shown in Fig. 6 this initial
value rc,cone,i(1/pi) would still be in fair agreement with the
model calculations. Similar considerations may be applied
for all other geometrical parameters of incoherent islands
shown in the plot.
4.5. Comparison with force calculations
In addition to the energy equilibrium approach presented
in this study Johnson and Freund have recently given a
criterion for the nucleation of dislocations in laterally
limited island structures [36]. Their approach is based on
balancing the image force on a dislocation close to an island
edge with the mismatch induced force promoting the incor-
poration of the dislocation at the heterointerface. Although
there exist some common features in the results obtained by
both approaches, e.g. an increase of the critical island radius
with a decreasing aspect ratio, the models may not easily be
compared in a quantitative manner. This problem arises
from the fact that Johnson and Freund assume mis®t accom-
modation by pure edge dislocations within a two-dimen-
sional circular arc shaped layer geometry, actually not
representing a quantum dot but rather a quantum wire struc-
ture hence characterized by a different elastic strain distri-
bution.
Moreover, a dislocation core distance r2 �j ~b j (denotedd in [36]) towards the island edge is assumed. However,
presuming a lattice mismatch of f� 0.04 their calculation
yields a critical island radius in the order of rc,JF < 200 j ~b jwhich is nearly independent from the aspect ratio of the
island for 1/p# 4. Although in contradiction to the experi-
mental results gained from the cross-sectional micrographs
mis®t accommodation by, say, a pure edge dislocation of
Lomer type with ~b � alay=2k110l and beff � alay=��2
pmay be
discussed in order to compare the predications of both
models. In this case a critical island radius of rc,JF < 83
nm would be expected from the equilibrium of forces calcu-
lation. Contrastingly, Eq. (16) predicts a critical island
radius of rc;cone�1=p � 4� � 29 nm when assuming mis®t
accommodation by pure edge dislocations and keeping all
other parameters ®xed.
Apart from the different layer morphologies assumed in
both approaches, this obvious discrepancy will be primarily
due to the difference in r2 values chosen in the models.
Indeed, as can be seen from Eq. (17) in [36], an increase
in the dislocation core distance towards the island edge
being only a few multiples of the amount of the Burgers
vector will also result in a decrease of rc,JF values due to
moderated image forces on the mis®t dislocation. A corre-
sponding modi®cation of r2 values, unfortunately not
numerically evaluated in [36], would also be in agreement
with the experimental results presented in this study and in
other investigations [21] demonstrating that dislocation
formation actually takes place at a distance of ®ve up to
ten monolayers away from the island edges.
4.6. Modi®ed islands geometries
Heteroepitaxial growth under deposition conditions
different from those chosen in this study may result in differ-
ent layer morphologies. Indeed, island geometries such as
pyramidal with different facets [1], truncated pyramidal
[15,21,53] and lens shaped or spherical cap like [24,54±
56] have been reported in a number of previous analyses
focusing on highly lattice mismatched InxGa12xAs/GaAs
heterostructures.
Only minor modi®cations to some of the previous equa-
tions will be necessary when focusing on islands with
geometries and compositions different from those consid-
ered up to now. Having a look at Eq. (16), in those cases
modi®cations of the island geometry factors G(l ,V) and thecut-off radii R of the dislocation strain ®eld will be neces-
sary. Moreover, since island shapes characterized by differ-
ent volume distributions may allow a more or less effective
elastic mismatch reduction at a ®xed aspect ratio compared
to the cones, the decay parameters k according to Eq. (21)
must be re-calculated using a different mesh geometry when
performing ®nite element simulations
When calculating the elastic strain distribution of square-
based islands elongated, say, along crystallographic k110ldirections at least quarter islands parts must be modeled
instead of the pie-like sectors suf®cient for the circular
based islands. Details on corresponding procedures have
been published in an earlier analysis [48]. In those cases
the determination of the correction function F (p) according
to Eq. (19). implies that individual contributions F i(p) are
calculated as F i�p� � ������������1110;i1 �110;i
p=f instead of the expres-
sion given in Eq. (20) with 1 110,i and 1 1Å10,i denoting the in-
plane components of the elastic strain tensor at each node of
the ®nite element mesh.
To simplify matters, Table 2 gives a summary of required
displacement rules to preceding equations and parameters
when assuming spherical cap like, pyramidal, truncated
pyramidal and cuboid island shapes instead of the conical
geometry. For island shapes characterized by a square base-
ment, i.e. pyramids, truncated pyramids and cuboids, the
variable r refers to the half island basement extensions in
these cases, thus 2r representing the island lengths. These
islands are assumed to be elongated along the k110l direc-tions thus yielding increased lengths l � 2r of the hetero-
interfacial dislocation segments compared to the circular
based island geometries characterized by l � p=2r. For
better comparability the in-plane dislocation core distances
r2 � r 2 r1 towards the island free edges are assumed to be
identical with the conical model for all island geometries
considered. Despite these identical distances r2 the cut-off
radii R of the dislocation strain ®elds, again given by the
geometrical mean values of r2 and the island heights h(r1)
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±104 101
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
above the dislocation cores, will depend on the island
geometries as can be seen from the table. Re-evaluated
decay parameters k demonstrate that island geometries as
cones or pyramids characterized by rather small volume
fractions close to the island tips yield the largest k values
hence representing the geometries allowing the smallest
amount of elastic strain relief at a ®xed aspect ratio.
Fig. 7 shows the functional behaviour of correspondingly
re-evaluated critical island radii rc,shape in dependence on the
1/p ratio assuming different island geometries. Values are
plotted normalized with respect to those values rc,cone(1/p)
already calculated for conical islands. For the numerical
treatment R, G and k variables and parameters according
to Table 2 have been chosen.
As can be seen from Fig. 7, pyramidal islands character-
ized by identical cut-off radii R, geometry factors G and
decay-parameters k as the conical islands will also be char-
acterized by identical critical island radii, i.e.
rc;shape=rc;cone � 1 for all 1/p values. For spherical cap like
islands rc,shape values are decreased by a factor of rc,shape/
rc,cone < 0.8 but being slightly dependent on the aspect
ratio. This decrease is mainly due to increased island
volumes compared to the cones resulting in an enhanced
elastic strain energy at ®xed heterointerfacial dislocation
segment lengths, thus yielding decreased critical island
radii despite a slightly more effective elastic strain relief.
A further decrease in critical dimensions is anticipated
with some of the square based island geometries with rc,shapevalues decreasing all the more as the total island volume,
which is at a maximum for the cuboid geometry, increases at
a ®xed aspect ratio. By geometrical reasons, the curves for
the truncated pyramids terminate at 1=p � 2=tan 25:248 �4:24 and at 1=p � 2=tan 54:738 � 1:41 when assuming
{113} and {111} faceted islands, respectively, since at
these inverse aspect ratios the truncated pyramids actually
represent pyramids. However, with decreasing 1/p values a
steep increase in the rc,shape/rc,cone ratios is predicted for the
truncated pyramids as well as the cuboid islands, which
results from two effects. On the one hand, these geometries
allow a more effective elastic strain relief compared to the
cones, thus postponing the formation of the ®rst mis®t dislo-
cation towards larger islands sizes more effectively than
conical islands at small 1/p values. On the other hand the
chosen de®nition for the cut-off radius of the dislocations
elastic strain ®eld implies a disproportionate increase in R
values for small 1/p ratios, since h(r1) values will become
quite large especially for the truncated pyramid and the
cuboid geometries. Hence, for these island geometries E/L
values according to Eq. (8) represent upper estimates at
small 1/p values which implies that the calculated rc,shape/
rc,cone ratios also represent maximal estimations.
4.7. Modi®ed layer compositions
When focusing on InxGa12xAs islands on GaAs substrates
characterized by an indium content x different from x � 0:6
the numerical values of the critical island dimensions will
also be affected. As can be seen from Eq. (16) the critical
island radius scales as rc,cone / 1/f with the nominal lattice
mismatch f when neglecting that the absolute length of the
Burgers vector and that the elastic constants will also be
in¯uenced by a decreasing or an increasing indium content.
Hence, critical island radii smaller than those plotted in Fig.
6 are expected for epilayers with indium contents x $ 0:6,
e.g. for the frequently investigated InAs/GaAs system. In
those cases a precise numerical treatment would require a
complete re-calculation of rc,cone(1/p) curves taking modi®ed
material parameters into account. However, the decay para-
meters k given in Table 2, which result from the analyses of
In0.6Ga0.4As/GaAs heterostructures, may still be used, at
least for indium contents 0:5 # x # 1:0. In those cases re-
calculated ®nite element data using a ®xed mesh geometry
but correspondingly modi®ed elastic material parameters
demonstrated only minor deviations j Dk j# 0:001 of the
decay parameters compared to those values summarized in
the table for a number of island geometries.
4.8. Further experimental evidence
In order to check the applicability of the model calcula-
tions for island shapes different from the conical geometry
already discussed in full details, calculated rc,shape(1/p)
values have also been compared with earlier microscopic
results on the coherent-to-incoherent transition for islands
of various shapes and compositions. For instance, Al-Jassim
et al. [13] reported on the largest radius of a coherent sphe-
rical cap like InAs island on GaAs amounting to rc;exp � 7:7
nm at a speci®c aspect ratio of pexp � 1/3.84. For this system
and island geometry the model calculations predict
rc;shape�pexp� � 7:3 nm, which is in fair agreement with the
experimental observation. Moreover, as shown by Guha et
al. [14] spherical cap like In0.5Ga0.5As islands on GaAs
remain free of dislocations up to a critical radius of rc;exp �12:5 nm at an aspect ratio of pexp � 1=3:13 and indeed calcu-
lations yield rc;shape�pexp� � 13:0 nm which is again in
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±104102
Fig. 7. Functional behaviour of the critical island radius rc,shape in depen-
dence on the inverse aspect ratio 1/p assuming a variety of conceivable
island geometries. Discrete rc,shape(1/p) values (X) are plotted normalized
with respect to the critical radii rc,cone(1/p) calculated for conical islands, cf.
the upper curve in Fig. 6, while the solid curves result from an interpolation
between these data.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
accordance with the experimental measurements. In a recent
investigation Wagner [21] analyzed the formation of 608dislocations close to the edges of approximately {111}
faceted truncated InAs pyramids on GaAs substrates. In
this study the smallest incoherent island observed by
cross-sectional HRTEM is characterized by a critical island
radius of rc;exp � 8:2 nm at an aspect ratio of pexp � 1=1:86,
cf. Fig. 14 in [21]. Model calculations for this system predict
a critical island radius of rc;shape�pexp� � 7:8 nm, once more
demonstrating the applicability of the model to a variety of
island geometries and compositions. Thus, in whole, the
comparisons between individual calculated data and experi-
mental measurements con®rm that the model predictions on
critical island radii rc,shape(pexp) fairly agree with experimen-
tal results rc,exp(1/pexp) with a margin of error
j rc;exp=rc;shape 2 1 j# 0:06.
5. Conclusions
Critical island sizes for the formation of mis®t disloca-
tions into highly lattice mismatched In0.6Ga0.4As islands
epitaxially grown on GaAs(011) have been determined by
high-resolution transmission electron microscopy on cross-
sectional samples. Experimental analyses reveal that the
islands are of conical shape and that the ®rst dislocations
nucleated close to the island edges are of 608 type under thechosen growth conditions. The critical island radii are found
to be in the order of 15 nm but being slightly dependent on
the aspect ratio of the islands. Based upon equilibrium
theory of elasticity analytical expressions have been derived
in order to calculate a quantitative phase diagram separating
the coherent from the incoherent phase dependent on the
island radius and the aspect ratio. When the possibility of
an elastic strain relief of coherent islands is considered in
the framework of an effectively reduced lattice mismatch,
which has been numerically evaluated by the application of
®nite element simulations, model calculations on rc,cone(1/p)
are in excellent agreement with the experimental results.
Calculations demonstrate that neglecting the elastic relaxa-
tion mechanism results in an underestimate of critical island
sizes by a factor of two to three. Rather simple modi®cations
to geometrical and material parameters also yield a fair
agreement between experimental data extracted from
various earlier analyses and the model calculations for a
variety of different island shapes and compositions.
References
[1] M. Grundmann, O. Stier, D. Bimberg, Phys. Rev. B 52 (1995) 11969.
[2] A.P. Alivisatos, Science 271 (1996) 933.
[3] M.S. Miller, Jpn. J. Appl. Phys. 36 (1997) 4123.
[4] D.J. Eaglesham, M. Cerullo, Phys. Rev. Lett. 64 (1990) 1943.
[5] I.N. Stranski, L. Nrastanow, Akad. Wiss. Wien Math.-Naturwiss. K1
IIb 146 (1939) 797.
[6] A. Marti-Ceschin, J. Massies, J. Cryst. Growth 114 (1991) 693.
[7] P. Chen, Q. Xie, A. Madhukar, L. Chen, A. Konkar, J. Vac. Sci.
Technol. B 12 (1994) 2568.
[8] A. Trampert, E. Tournie, K.H. Ploog, Phys. Status Solidi (a) 145
(1994) 481.
[9] C. Ratsch, A. Zangwill, Surf. Sci. 293 (1993) 123.
[10] C.W. Snyder, J.F. Mans®eld, B.G. Orr, Phys. Rev. B 46 (1992) 9551.
[11] F. Glas, C. Guille, P. Henoc, F. Houzay, Inst. Phys. Conf. Ser. 87
(1987) 71.
[12] S. Guha, A. Madhukar, R. Kapre, K.C. Rajkumar, in: C.V. Thompson,
J.Y. Tsao, D.J. Srolovitz (Eds.), Evolution of Thin-Film and Surface
Microstructure, Pittsburgh, USA, November 26±December 1, 1990,
202, MRS Symp. Proc, 1991, p. 519.
[13] M.M. Al-Jassim, J.P. Goral, P. Sheldon, K.M. Jones, in: D.N. Sadana,
L.E. Eastman, R. Dupuis (Eds.), Advances in Materials, Processing
and Devices in III±V Compound Semiconductors, Pittsburgh, USA,
November 28±December 2, 1988. MRS Symp. Proc., 144 (1989) 183.
[14] S. Guha, A. Madhukar, K.C. Rajkumar, Appl. Phys. Lett. 57 (1990)
2110.
[15] M. Lentzen, D. Gerthsen, A. FoÈrster, K. Urban, Appl. Phys. Lett. 60
(1992) 74.
[16] A. Trampert, E. Tournie, K.H. Ploog, J. Cryst. Growth 146 (1995)
368.
[17] J.H. van der Merwe, J. Appl. Phys. 34 (1963) 123.
[18] W.A. Jesser, D. Kuhlmann-Wilsdorf, Phys. Status Solidi 19 (1967)
95.
[19] R. Vincent, Philos. Mag. 19 (1969) 1127.
[20] J.W. Matthews, Surf. Sci. 31 (1972) 241.
[21] G. Wagner, Cryst. Res. Technol. 33 (1998) 681.
[22] K. Tillmann, D. Gerthsen, P. Pfundstein, A. FoÈrster, K. Urban, J.
Appl. Phys. 78 (1995) 3824.
[23] Hibbit, Karlsson & Sorensen Inc, ABAQUS 5.8 User's Manual, HKS,
Pawtucket RI, 1998.
[24] X.Z. Liao, J. Zou, X.F. Duan, D.J.H. Cockayne, R. Leon, C. Lobo,
Phys. Rev. B 58 (1998) R4235.
[25] V. Zaporojtchenko, Department of Materials Science, University of
Niel, Kiel, F.R.G., private communication.
[26] D. Leonard, K. Pond, P.M. Petroff, Phys. Rev. B 50 (1994) 11687.
[27] G.S. Solomon, J.A. Trezza, J.S. Harris, Appl. Phys. Lett. 66 (1995)
3161.
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±104 103
Table 2
Displacement rules to variables and parameters when assuming islands of various shapes instead of the conical geometry investigated in this study. In case of
the truncated pyramids a represents the angle inclined between the island facet normal vector and the substrate normal vector amounting to a113 � 25:248 and
a111 � 54:78 when considering square based {113} and {111} faceted islands elongated along crystallographic (110) directions, respectively
Geometry variable Cone! Spherical cap Pyramid Truncated pyramid Cuboid
V, Eq. (7) 2/3ppr3 ! (11 4/3p2)ppr3 8/3pr3 4/3tana[12 (12 2p/tana)3]r3 8pr3
l , Eq. (9) p /2r! p /2r 2r 2r 2r
R, Eq. (15) rd����2p
p ! r����2p
p ���������������������p2=�16�11 4p2��p
rd����2p
prd
������tana
pr
��d
p ����2p
pG, Eq. (17) 3/(16pp)! 3/(8(31 4p2)pp) 3/(16pp) 3=�8ptana�12 �12 2ptana� 3�� 1/(16pp)
k , Eq. (21) 0.087! 0.082 0.087 0:086 ja�a113
¼ 0:079 ja�a111
0.073
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
[28] R. Leon, T.J. Senden, K. Yong, C. Jagadish, A. Clark, Phys. Rev. Lett.
78 (1997) 4942.
[29] D.J.H. Cockayne, J.L.F. Ray, M.J. Whelan, Philos. Mag. 20 (1969)
1265.
[30] K.H. Chang, P.K. Bhattacharya, R. Gibala, J. Appl. Phys. 66 (1989)
2993.
[31] K.P. Nvam, D.M. Maher, C.J. Humphreys, J. Mater. Res. 5 (1990)
1900.
[32] T.J. Gosling, J. Appl. Phys. 74 (1993) 5415.
[33] M.A. O'Keefe, R. Nilaas, Scanning Microscopy Supplement 2 (1988)
225.
[34] J.W. Matthews, A.E. Blakeslee, S. Mader, Thin Solid Films 33 (1976)
253.
[35] E.A. Fitzgerald, Mater. Sci. Rep. 7 (1991) 87.
[36] H.T. Johnson, L.B. Freund, J. Appl. Phys. 81 (1997) 6081.
[37] D.J. Eaglesham, F.C. Unterwald, D.C. Jacobson, Phys. Rev. Lett. 70
(1993) 966.
[38] D. Dutartre, P. Warren, F. Chollet, F. Gisbert, M. Berenguer, I. Berbe-
zier, J. Cryst. Growth 142 (1994) 78.
[39] D.J. Eaglesham, R. Hull, Mater. Sci. Eng. B 30 (1995) 197.
[40] A.N. Stroh, Philos. Mag. 3 (1958) 625.
[41] T.Y. Zhang, J. Appl. Phys. 78 (1995) 4948.
[42] J.P. Hirth, J. Lothe, Theory of Dislocations, Wiley, New York, 1982.
[43] K.-H. Hellwge (Ed.), Numerical Data and Functional Relationships in
Science and Technology New Series, Vol. 17, Springer, Berlin, 1982.
[44] T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff
Publishers, The Hague, 1982.
[45] S. Luryi, E. Suhir, Appl. Phys. Lett. 49 (1986) 140.
[46] R. Kern, P. MuÈller, Surf. Sci. 392 (1997) 103.
[47] D. Vanderbilt, L.K. Wickham, in: C.V. Thompson, J.Y. Tsao, D.J.
Srolovitz (Eds.), Evolution of Thin-Film and Surface Microstructure,
Pittsburgh, USA, 26 November±1 December, 1990. MRS Symp.
Proc., 202 (1991) 555.
[48] K. Tillmann, A. Thust, M. Lentzen, P. Swiatek, A. FoÈrster, K. Urban,
Philos. Mag. Lett. 74 (1996) 309.
[49] P. Van Miegham, S.C. Jam, J. Nijs, R. Van Overstraeten, J. Appl.
Phys. 75 (1994) 666.
[50] S. Christiansen, M. Albrecht, H.P. Strunk, P.O. Hansson, E. Bauser,
Appl. Phys. Lett. 66 (1995) 574.
[51] K.K. LeGoues, M.C. Reuter, J. Tersoff, M. Hammar, R.M. Tromp,
Phys. Rev. Lett. 73 (1994) 300.
[52] X.W. Lin, Z. Liliental-Weber, J. Washburn, E.R. Weber, A. Sasaki,
A. Wakahara, Y. Nabetani, J. Vac. Sci. Technol. B 12 (1994)
2562.
[53] K. Georgsson, N. Carisson, L. Samuelson, W. Seifert, L.R. Wallen-
berg, Appl. Phys. Lett. 67 (1995) 2981.
[54] G.S.. Soloman, J.A. Trezza, J.S. Harris, Appl. Phys. Lett. 66 (1995)
991.
[55] G. Medeiros-Ribeiro, K.H. Schmidt, D. Leonard, Y.M. Cheng, P.M.
Petroff, in: E.D. Jones, A. Mascarenhas, P. Petroff (Eds.), Optoelec-
tronic Materials: Ordering, Composition Modulation and Self-
Assembled Structures, Boston, MA, November 28±30, 1995. MRS
Symp. Proc., 417 (1996) 221.
[56] C. Lobo, R. Leon, J. Appl. Phys. 83 (1998) 4168.
K. Tillmann, A. FoÈrster / Thin Solid Films 368 (2000) 93±104104
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.
Send
repr
oduc
tion
fee
to E
lsev
ier S
cien
ce S
.A.