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: kct@tf.uni-kiel.de (K. Tillmann).

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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.

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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.

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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.

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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

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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

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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.

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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.

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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

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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.

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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.

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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

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