1994 the influence of dimerization on the stability of ge hutclusters on si(001)
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ELSEVIER Surface Science 317 (1994) 5864
The influence of dimerization on the stability of Gehutclusters
on Si( 001)
F. Tuinstra, P.M.L.O. Scholte *, W.I. Rijnders, A.J. van den Berg
Department of Applied Physics, Solid State Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, Netherlands
Received 16 June 1993; accepted for publication 16 May 1994
Abstract
The epitaxial growth of Ge on Si(OO1)initially proceeds twodimensional. After a few monolayers, a large number
of threedimensional Ge nanocrystals are formed with well defined, highly anisotropic shapes and bounded by (105)
facets. These facets are unstable in the morphology of macroscopic crystals. Apparently a different set of parameters
governs the crystal stability, size, shape and orientation of nanocrystals. By applying elastic continuum theory we
calculate the stability of Ge nanocrystals on Si(OOl), bounded by different facets. We show that (105)faceted
nanocrystals are indeed the most stable. The model identifies some of the principal parameters which control the
stability of nanocrystals: the strain due to the lattice mismatch between substrate and nanocrystal, the size of the
nanocrystal, and the surface energy (or the reconstruction) of the substrate and of the facets of the nanocrystal.
1. Introduction with well defined, highly anisotropic shapes
(aspect ratios up to 1: 81, and bounded by (105)
With the introduction of the scanning tunnel facets. Upon annealing above 800 K, the hutclus
ing microscopy @TM) in the field of crystal ters are replaced by much larger crystals with
growth, the initial stages of nucleation and mor (113) facets; the {105} facets having disappeared
phology of nanocrystals have become accessible [3]. The appearance of (10.5) facets is surprising
to experimental investigation [1,2]. During the since according to the morphology of macro
epitaxial growth of Ge on Si(OO1) nanocrystals scopic crystals, 1105) facets are not stable. At the
emerge with a morphology that is unexplained by same time the appearance of {113} facets after
the morphology of macroscopic crystals 1351. annealing is understood by macroscopic morphol
The epitaxial growth of Ge on Si(OO1) initially ogy 161.
proceeds twodimensional. After a few monolay Apparently a different set of parameters gov
ers, a large number of threedimensional Ge erns the stability, size, shape and orientation of
nanocrystals (the socalled hutclusters) are formed nanocrystals.
The object of this paper is to present a model
for the morphology of the hutclusters, that identi
* Corresponding author. Fax: +31 15 783251. Email: fies some of the principal parameters which con
scholte@duttncb.tn.tudelft.nl. trol the stability of nanocrystals: the strain due to
00396028/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved
SSDZ 00396028(94)00315Z
2.
the lattice mismatch, the size of the nanocrystal, continuously rotated around the [OOl] axis at 0.2
and the chemical surface energy of the substrate revolutions/s. RHEED images were taken at an
and of the nanocrystal. acceleration voltage of 10 kV and an angle of
By applying elastic continuum theory we calcu incidence of 2.1& 0.2”. A CCDcamera was read
late the stability of the Ge nan~~stals on SKOOl), out about one hundred times per samplerevolu
bounded by different facets. We show that {lOS} tion with a video framegrabber. The stored
faceted nanocrystals are indeed the most stable. RHEED patterns were combined by a computer
Recent work of Tromp and Tersoff corroborates program into a single pattern which has the same
our model by identifying the size and strain as the features as a pattern obtained by low energy
parameters that determine the shape of Ag clus electron diffraction (LEED). The resulting dia
ters on Si@Ol) 171. gram displays the projected intensity of the
This paper consists of two sections. In the first diffraction rods onto the U&O) plane in recipro
section of the paper a modified RHEED (reflec cal space.
tion high energy electron diffraction) technique is To illustrate the feasibility of the method, the
introduced. We have used this technique to es result for a clean Si(OO1)surface is shown in Fig.
tablish the presence of hutclusters during the 1. The horizontal and vertical directions in this
growth of Ge on Si, under the circ~stances as figure coincide with the [llO] and [liO] directions,
described by MO et al. [3]. The principal advan respectively. Clearly the diffraction pattern of the
tage of this technique is that it gives a projection 2 X 1 reconstructed surface can be discerned. The
of the reciprocal space along all azimuthal direc splitting of some spots in Fig. 1 is an artefact of
tions, while conventional RHEED uses one az the method and is due to irregularities in the
imuthal direction only. rotation speed of the sample during the recording
In the main part of the paper a theoretical of the RHEED images.
model for the morphology of hutclusters is pre In Fig. 2a the transformed RHEED pattern is
sented. First we will consider the respective con shown of a Si(OO1)surface with 9 monolayers Ge
tributions from the bulk and the surface to the
total energy of a hutcluster separately. In the
final section we will discuss how the corrobora
tive effect of all contributions stabilizes clusters
bounded by (105) facets.
2. Experimental observation of the hutclusters
The observation with STM of the evolution of
the morphology during molecular beam epitaxy
(MBE) growth, seems as yet beyond reach. Yet,
diffraction techniques like RHEED and GIXD
(grazing incidence Xray diffraction) can be em
ployed without much interference with the depo
sition process. In particular the low incidence of
the beam favors substantially the diffracted inten
sity of protrusions on the substrate, over the
contribution of the smooth crystalline substrate.
To observe the appearance of the hutclusters,
RHEED diagrams have been recorded during the Fig. 1. Generated “LEED image” of clean Si(OO1) surface.
deposition of Ge on Si(tOl) at 460°C. The deposi The image has been constructed from approximately 60
tion rate was set at 1 A/s, and the sample was RHEED images.
3.
60 F. Tuinstra et al. /Surface Science 317 (1994) 5864
grown onto it. In addition to the well pronounced function of the hutclusters appears as broad fea
spots of the Ge bulk structure and those of the tures.
2 x 1 surface reconstruction, a raster of weak Therefore the continuous and narrow appear
lines along the [lOO] and [OlO] direction is ob ance of the lines in Fig. 2a can be attributed to
served (Fig. 2b). protruding structures which have a large exten
The streaky radial pattern in the background sion in the [loo] and small dimension in the [OlO]
of Fig. 2 is due to fluctuations of the background direction, and the other way around. According
intensity of the RHEED patterns. In general to the diffraction pattern the ratio between length
artefacts in the transformed image have a radial and width of these nanostructures is at least 100;
nature due to the fact that the sample is rotated the width cannot be more than a few atomic
around an axis perpendicular to the image. This distances. MO and Lagally observed an aspect
is corroborated by computer simulations in which ratio of up to 1: 8. Due to the finite penetration
we calculated the effect of experimental errors on depth of the RHEED electrons in Ge, the elec
the transformed image, such as wobbling of the trons skim only the tops of the protruding hut
wafer during the rotation and a nonconstant clusters. Consequently the aspect ratio observed
rotation speed. We conclude that the weak lines with RHEED is much larger than that observed
in Fig. 2b, originate from a surface effect and with STM.
cannot be attributed to an artefact of the trans Also, the projection along the [OOl] direction
formation method. that is used to transform the RHEED images
In the kinematic approximation, the diffrac gives an overestimation of the aspect ratio, since
tion spots from the crystalline structure of the diffraction rods from the vicinal surfaces that
nanocrystals are convoluted with the Fourier bound the hutcluster are not parallel to [OOll.
transform of their shape. The transform of a long The weak lines appear in the transformed pat
narrow form is a thin flat slab, the normal of tern after growth of 6 monolayers of Ge. MO and
which is parallel to the long dimension of the Lagally report the hutclusters to nucleate after
crystallite. Thus in reciprocal space the shape 34 monolayers [3]. In order to be able to observe

[l 1 OJ
Fig. 2. (a) “LEED image” of 9 monolayers Ge on Si(oO1) constructed from 100 RHEED images. (b) Schematic clarification of Fig.
2a, showing the diffraction spots from the crystalline surface and the lines due to the appearance of the hutclusters.
4.
F. Tuinstra et al. /Surface Science 317 (I 994) 5864 61
them after transformation of the RHEED im of Ge on a Si substrate (Fig. 3). The strip is
ages, enough hutclusters have to be present. bounded in the y and the zdirection, respec
Therefore the appearance of the weak lines after tively parallel and perpendicular to the substrate.
6 monolayers corresponds very well with the ap The complete strain tensor can be deduced
pearance of the hutclusters as reported by MO straightforwardly from the symmetry of the Ge
and Lagally. strip. The misfit of the epitaxial Ge adlayer intro
Aumann et al. have calculated the RHEED duces a compressive strain E of 4.2% at the
pattern for hutclusters with a fixed azimuth of the interface between substrate and adlayer. At the
incident beam along the [lOO] and [llO] directions SiGe interface the strain is uniaxial because of
[8]. In our experiment we find the same patterns, the equivalence of the [RIO] principal axes. Along
if we fix the azimuth of the incident beam. There the long axis the strain cannot relax, since the
fore we conclude that the raster of weak lines in strip is thought to be (infinitely) long. In the
Fig. 2b, is due to the appearance of the hutclus ydirection, however, a gradual elastic relaxation
ters. over the height of the strip is possible. As a
consequence the strain is anisotropic in the bulk
of the cluster. Since the strip can freely expand
3. The morphology of hutclusters along the [OOll direction, the stress along the
zdirection is zero.
The morphology of a hutcluster is character
From these considerations we find up to first
ized by a high aspect ratio and the orientation
order in the strain E at the interface:
along the [loo] and [OlO] directions. In the next
sections a hutcluster is modelled by a long, nar C,*
&i =E, ,,=e;, Es= $&i+E*),
row cluster bounded by 4 facets. A Cartesian 11
coordinate system is attached to the cluster with Y
the xaxis parallel to the long axis of the cluster, h
the yaxis parallel to the short axis, and the zaxis where h is the height of the strip, and Cij are the
perpendicular to the substrate (see Fig. 3). The elastic stiffness constants with respect to the sym
facets bounding the long ends of the cluster are metry axes xyz of the strip. Now the elastic
taken to be parallel to the yz plane, the other energy can be calculated straightforwardly:
two facets are parallel to the xdirection. The
azimuthal orientation of the hutcluster is given by E=/ dV $ ~C;E~&~  AC,[ ( e1  Q)’
Cluster id
the angle 4 between the xaxis and the [lo01
direction of the substrate.
We assume the hutcluster to be coherent with I
&6” cos2+sin24
i
.
the substrate for all azimuthal orientations, i.e.
for all values of 4. This means that the principal
axes of the crystallographic structure of the hut
cluster may not be parallel to the xyzaxes that
describe the symmetry of the morphology.
To calculate the total energy of a hutcluster,
the energy is split into a term related to the bulk
of the cluster, and a term related to the surface
of the cluster. We will consider both contribu
tions separately in the next sections.
3.1. Elastic misfit energy
Fig. 3. Infinitely long Ge strip on a Si substrate. The strip is
The first stage in the nucleation of a hutcluster stressed at the interface with the Si substrate and relaxed at
can be approximated by an (infinitely) long strip the top.
5.
62 F. Tuinstra et al. /Surface Science 317 (1994) 5864
AC, is the anisotropy constant of the elastic (001) terraces, separated by equivalent monos
tensor of rank 4. For Ge we find [9]: teps. In a similar fashion it is also possible to
construct the competing {lln} facets from (001)
AC, = Cfi  CfZ  2C& =  53.2 GPa.
terraces. In this case the terraces are separated
Ct. are the (tabulated) elastic stiffness constants by monosteps and double steps. For the {lln}
of Ge with respect to its principal axes. Since the facets the monosteps are inequivalent, since they
Ge cluster is assumed to be coherent with the are alternatingly parallel and perpendicular to
substrate, the symmetry axes of the Si coincide the dimer rows of the (001) terrace. The steps
with the crystallographic principal axes of the and terraces on the (10n) vicinal facets on the
cluster. contrary, are all equivalent; all of them are mak
The first term in the elastic energy is indepen ing angles of 45” with the dimer rows on (001).
dent of the azimuthal orientation of the Ge strip The atoms on the (001) terraces try to mini
on the Si substrate. The second term, however, mize the number of unsaturated dangling bonds.
does depend on &Jand is minimal if 4 = 0, i.e. if One way to do that is by the formation of dimers.
the strip is oriented along the principal axis of the Roberts and Needs have calculated the energy
Si substrate and maximal if the strip is oriented gain due to the 2 x 1 reconstruction of SXOOZ)to
along the Ill01 or [liOl direction. be approximately 2.08 eV per asymmetric dimer
bond [lo]. This value compares reasonably well
3.2. Chemical surface energy with the strength of a single covalent SiSi bond:
1.8 eV. Therefore we estimate the lowering of the
The second contribution to the internal energy chemical surface energy by the Gedimer bonds
that is considered, is the chemical surface energy to be approximately equal to the strength of a
of the (10n) and (lln} facets. GeGe bond in bulk Ge: 1.6 eV.
{lOn} and {lln) facets can be considered as The chemical surface energy is calculated by
vicinal (001) planes, especially if IZ is not too counting the number of dangling bonds that are
small. In Fig. 4 this is illustrated for a (104) and a left, making sure that if possible the dimers are
(1051 plane. It can be seen that they consist out of formed on the terraces. (See Fig. 4.) It is assumed
(1051 {lo41
Fig. 4. Side view and top view of (104) and (105) vicinal planes. Possible dimers are indicated by connecting the atoms. The height
of the atoms is indicated by their color: the lighter the atom, the deeper it lays.
6.
F. Tu~~r~ ei al. /Surface Science 317 (1994) 5864 63
that only those atoms dimerize that have at least surface energy of the SK000 interface that is
two dangling bonds. In doing so we neglect re covered by the cluster.
bonding at the bottom of steps. Rebonding com In Fig. 5 the excess surface energy is shown of
pensates a number of dangling bonds that could a 100 nm long cluster of 10’ atoms. The cluster is
not be removed by the dinner reconstruction of bounded by {10nf or {lln) facets, that make an
the facets. But the bonds at the rebonded steps angle fi with the substrate. Generally a hutclus
are highly strained and therefore rather weak. ter is much longer than it is wide. Therefore in
Chadi calculated the energy gain due to rebond the calculation the contribution from the small
ing to be 0.16 eV/bond [ll]. Although this can facets at the end of the long axis has been ne
not be neglected compared to the energy of a glected. In Fig. 5 the excess surface energy has
dangling bond ( 0.8 eV1, the only effect is that been normalized on the total number of atoms in
the differences in surface energy of different the cluster. The figure shows a remarkable peri
facets are smoothened somewhat. odicity, due to the interference of the dimeriza
Depending on the width of a terrace, not all tion and the terrace structure of the facets. The
atoms can dimerize, as is illustrated by the (104) periodic&y of n = 4 of the {lOn) line is related to
plane in Fig. 4. In the case of the {lo51 facet an the atomic arrangement in the diamond struc
efficient dimer reconstruction is possible, leaving ture, the basis of which consists of two atoms, one
only 1 dangling bond per atom on each terrace. shifted over (0.25, 0.25, 0.25) with respect to the
On the {104} facet atoms are left with 2 dangling other.
bonds, making the surface energy of this facet
slightly higher than of (1051.
To be able to quantify the effect of the mor 4. discussion and conclusion
phology on the surface energy of a cluster, an
excess surface energy has been defined. The ex As a Ge adlayer grows on the Si(OO1) sub
cess surface energy is calculated by subtracting strate, the elastic misfit energy stored in the
from the total surface energy of a cluster, the adlayer increases. The adlayer may release this
elastic energy partly by developing steps. Conse
quently if the elastic energy raises above a certain
level, the layer starts to facet. This facetting in
creases the surface area, and consequently the
total chemical surface energy increases. The layer
finds an optimum by developing a facet that is as
steep as possible, i.e. with as many steps as possi
ble, while at the same time has a lowest possible
chemical surface energy. From Fig. 4 it can be
seen that both (105) and {113) fulfil these criteria.
a
The azimuthal orientation of the hutcluster is
determined by the elastic contributions to the
total energy of the cluster. Tersoff and Tromp
showed that highly anisotropic Ag islands are
stabilized by the stress induced by the lattice
mismatch between Ag and Si. By the same effect
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 a highly anisotropic Ge island will nucleate dur
aa
ing the first stages of epitaxial growth of Ge on
Fig. 5. Excess surface energy of a 100 nm long cluster of 10’ Si(OO1). If we consider the elastic energy that is
atoms versus the angle B of the facets with the substrate. The
stored in the bulk of a long Ge strip, we find that
total energy has been divided by the number of atoms. Solid
squares represent the clusters with {lOn} facets (tg 6 = l/n), it is oriented along the principal axes of the Si
open squares the clusters with {lln) facets (tg 9 = .,‘2/n). substrate. The energy difference between the ori
7.
64 F. Tuinstra et al. /Surface Science 317 (1994) 5864
entations along the [lOO] and [llO] directions is been observed recently during the surfactant
approximately 1.0 meV/atom. Including the third mediated growth of Ge on Si(ll1) [12].
order term in the elastic energy raises the energy
difference up to 1.4 meV/atom. Although this
energy difference seems small, a 4 nm by 100 nm Acknowledgments
Ge strip of 4 monolayers high contains lo4 atoms
The authors gratefully acknowledge mr. K.
already, making the energy difference between
Werner and mr. 0. Schannen for the opportunity
the two orientations 14 eV.
to use the MBE system of the Delft Institute of
We conclude that the elastic stress due to the
Microelectronics and Submicron technology (DI
dimerization and the misfit favors the nucleation
MES) for the RHEED experiments.
of anisotropic islands along the principal axis of
the Si substrate. When the growth proceeds, small
crystals nucleate on top of the strips. The elastic References
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876.
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[4] U. Kiihler, 0. Jusko, B. Miiller, M. Hornvon Hoegen
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171 J. Tersoff and R.M. Tromp, Phys. Rev. Lett. 70 (1993)
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