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ELSEVIER Surface Science 317 (1994) 58-64

The influence of dimerization on the stability of Ge-hutclusters 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 two-dimensional. After a few monolayers, a large number of three-dimensional 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 the introduction of the scanning tunnel- ing microscopy @TM) in the field of crystal growth, the initial stages of nucleation and mor- phology of nanocrystals have become accessible to experimental investigation [1,2]. During the epitaxial growth of Ge on Si(OO1) nanocrystals emerge with a morphology that is unexplained by the morphology of macroscopic crystals 13-51.

The epitaxial growth of Ge on Si(OO1) initially proceeds two-dimensional. After a few monolay- ers, a large number of three-dimensional Ge nanocrystals (the so-called hutclusters) are formed

* Corresponding author. Fax: +31 15 783251. E-mail: scholte@duttncb.tn.tudelft.nl.

with well defined, highly anisotropic shapes (aspect ratios up to 1: 81, and bounded by (105) facets. Upon annealing above 800 K, the hutclus- ters are replaced by much larger crystals with (113) facets; the {105} facets having disappeared [3]. The appearance of (10.5) facets is surprising since according to the morphology of macro- scopic crystals, 1105) facets are not stable. At the same time the appearance of {113} facets after annealing is understood by macroscopic morphol-

ogy 161. Apparently a different set of parameters gov-

erns the stability, size, shape and orientation of nanocrystals.

The object of this paper is to present a model for the morphology of the hutclusters, that identi- fies some of the principal parameters which con- trol the stability of nanocrystals: the strain due to

0039-6028/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved

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the lattice mismatch, the size of the nanocrystal, and the chemical surface energy of the substrate and of the nanocrystal.

By applying elastic continuum theory we calcu- late the stability of the Ge nan~~stals on SKOOl), bounded by different facets. We show that {lOS}- faceted nanocrystals are indeed the most stable. Recent work of Tromp and Tersoff corroborates our model by identifying the size and strain as the parameters that determine the shape of Ag clus- ters on Si@Ol) 171.

This paper consists of two sections. In the first section of the paper a modified RHEED (reflec- tion high energy electron diffraction) technique is introduced. We have used this technique to es- tablish the presence of hutclusters during the growth of Ge on Si, under the circ~stances as described by MO et al. [3]. The principal advan- tage of this technique is that it gives a projection of the reciprocal space along all azimuthal direc- tions, while conventional RHEED uses one az- imuthal direction only.

In the main part of the paper a theoretical model for the morphology of hutclusters is pre- sented. First we will consider the respective con- 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 X-ray 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 deposition of Ge on Si(tOl) at 460°C. The deposi- tion rate was set at 1 A/s, and the sample was

continuously rotated around the [OOl] axis at 0.2 revolutions/s. RHEED images were taken at an acceleration voltage of 10 kV and an angle of incidence of 2.1& 0.2”. A CCD-camera was read out about one hundred times per sample-revolu- tion with a video frame-grabber. The stored RHEED patterns were combined by a computer program into a single pattern which has the same features as a pattern obtained by low energy electron diffraction (LEED). The resulting dia- gram displays the projected intensity of the diffraction rods onto the U&O) plane in recipro- cal space.

To illustrate the feasibility of the method, the result for a clean Si(OO1) surface is shown in Fig. 1. The horizontal and vertical directions in this figure coincide with the [llO] and [liO] directions, respectively. Clearly the diffraction pattern of the 2 X 1 reconstructed surface can be discerned. The splitting of some spots in Fig. 1 is an artefact of the method and is due to irregularities in the rotation speed of the sample during the recording of the RHEED images.

In Fig. 2a the transformed RHEED pattern is shown of a Si(OO1) surface with 9 monolayers Ge

Fig. 1. Generated “LEED image” of clean Si(OO1) surface. The image has been constructed from approximately 60 RHEED images.

60 F. Tuinstra et al. /Surface Science 317 (1994) 58-64

grown onto it. In addition to the well pronounced spots of the Ge bulk structure and those of the 2 x 1 surface reconstruction, a raster of weak lines along the [lOO] and [OlO] direction is ob- served (Fig. 2b).

The streaky radial pattern in the background of Fig. 2 is due to fluctuations of the background intensity of the RHEED patterns. In general artefacts in the transformed image have a radial nature due to the fact that the sample is rotated around an axis perpendicular to the image. This is corroborated by computer simulations in which we calculated the effect of experimental errors on the transformed image, such as wobbling of the wafer during the rotation and a non-constant rotation speed. We conclude that the weak lines in Fig. 2b, originate from a surface effect and cannot be attributed to an artefact of the trans- formation method.

In the kinematic approximation, the diffrac- tion spots from the crystalline structure of nanocrystals are convoluted with the Fourier transform of their shape. The transform of a long narrow form is a thin flat slab, the normal of which is parallel to the long dimension of the crystallite. Thus in reciprocal space the shape

function of the hutclusters appears as broad fea- tures.

Therefore the continuous and narrow appear- ance of the lines in Fig. 2a can be attributed to protruding structures which have a large exten- sion in the [loo] and small dimension in the [OlO] direction, and the other way around. According to the diffraction pattern the ratio between length and width of these nanostructures is at least 100; the width cannot be more than a few atomic distances. MO and Lagally observed an aspect ratio of up to 1: 8. Due to the finite penetration depth of the RHEED electrons in Ge, the elec- trons skim only the tops of the protruding hut- clusters. Consequently the aspect ratio observed with RHEED is much larger than that observed with STM.

Also, the projection along the [OOl] direction that is used to transform the RHEED images gives an overestimation of the aspect ratio, since the diffraction rods from the vicinal surfaces that bound the hutcluster are not parallel to [OOll.

The weak lines appear in the transformed pat- tern after growth of 6 monolayers of Ge. MO and Lagally report the hutclusters to nucleate after 3-4 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.

F. Tuinstra et al. /Surface Science 317 (I 994) 58-64 61

them after transformation of the RHEED im- ages, enough hutclusters have to be present. Therefore the appearance of the weak lines after 6 monolayers corresponds very well with the ap- pearance of the hutclusters as reported by MO and Lagally.

Aumann et al. have calculated the RHEED pattern for hutclusters with a fixed azimuth of the incident beam along the [lOO] and [llO] directions [8]. In our experiment we find the same patterns, if we fix the azimuth of the incident beam. There- fore we conclude that the raster of weak lines in Fig. 2b, is due to the appearance of the hutclus- ters.

3. The morphology of hutclusters

The morphology of a hutcluster is character- ized by a high aspect ratio and the orientation along the [loo] and [OlO] directions. In the next sections a hutcluster is modelled by a long, nar- row cluster bounded by 4 facets. A Cartesian coordinate system is attached to the cluster with the x-axis parallel to the long axis of the cluster, the y-axis parallel to the short axis, and the z-axis perpendicular to the substrate (see Fig. 3). The facets bounding the long ends of the cluster are taken to be parallel to the y-z plane, the other two facets are parallel to the x-direction. The azimuthal orientation of the hutcluster is given by the angle 4 between the x-axis and the [lo01 direction of the substrate.

We assume the hutcluster to be coherent with 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 xyz-axes 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

The first stage in the nucleation of a hutcluster can be approximated by an (infinitely) long strip

of Ge on a Si substrate (Fig. 3). The strip is bounded in the y- and the z-direction, respec- tively parallel and perpendicular to the substrate.

The complete strain tensor can be deduced straightforwardly from the symmetry of the Ge strip. The misfit of the epitaxial Ge adlayer intro- duces a compressive strain E of 4.2% at the interface between substrate and adlayer. At the Si-Ge interface the strain is uniaxial because of the equivalence of the [RIO] principal axes. Along the long axis the strain cannot relax, since the strip is thought to be (infinitely) long. In the y-direction, however, a gradual elastic relaxation 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 along the [OOll direction, the stress along the z-direction is zero.

From these considerations we find up to first order in the strain E at the interface:

C,* &i =E, ,,=e;, Es= -$&i+E*),

11

Y

h where h is the height of the strip, and Cij are the elastic stiffness constants with respect to the sym- metry axes xyz of the strip. Now the elastic energy can be calculated straightforwardly:

E=/ dV Cluster

$ ~C;E~&~ - AC,[ ( e1 - Q)’ id

-&6” I cos2+sin24 . i

Fig. 3. Infinitely long Ge strip on a Si substrate. The strip is stressed at the interface with the Si substrate and relaxed at the top.

62 F. Tuinstra et al. /Surface Science 317 (1994) 58-64

AC, is the anisotropy constant of the elastic tensor of rank 4. For Ge we find [9]:

AC, = Cfi - CfZ - 2C& = - 53.2 GPa.

Ct. are the (tabulated) elastic stiffness constants of Ge with respect to its principal axes. Since the Ge cluster is assumed to be coherent with the substrate, the symmetry axes of the Si coincide with the crystallographic principal axes of the cluster.

The first term in the elastic energy is indepen- dent of the azimuthal orientation of the Ge strip on the Si substrate. The second term, however, does depend on &J and is minimal if 4 = 0, i.e. if the strip is oriented along the principal axis of the Si substrate and maximal if the strip is oriented along the Ill01 or [liOl direction.

3.2. Chemical surface energy

The second contribution to the internal energy that is considered, is the chemical surface energy of the (10n) and (lln} facets.

{lOn} and {lln) facets can be considered as vicinal (001) planes, especially if IZ is not too small. In Fig. 4 this is illustrated for a (104) and a (1051 plane. It can be seen that they consist out of

(001) terraces, separated by equivalent monos- teps. In a similar fashion it is also possible to construct the competing {lln} facets from (001) terraces. In this case the terraces are separated by monosteps and double steps. For the {lln} facets the monosteps are inequivalent, since they are alternatingly parallel and perpendicular to the dimer rows of the (001) terrace. The steps and terraces on the (10n) vicinal facets on the contrary, are all equivalent; all of them are mak- ing angles of 45” with the dimer rows on (001).

The atoms on the (001) terraces try to mini- mize the number of unsaturated dangling bonds. One way to do that is by the formation of dimers. Roberts and Needs have calculated the energy gain due to the 2 x 1 reconstruction of SXOOZ) to be approximately 2.08 eV per asymmetric dimer bond [lo]. This value compares reasonably well with the strength of a single covalent Si-Si bond: 1.8 eV. Therefore we estimate the lowering of the chemical surface energy by the Ge-dimer bonds to be approximately equal to the strength of a Ge-Ge bond in bulk Ge: 1.6 eV.

The chemical surface energy is calculated by counting the number of dangling bonds that are left, making sure that if possible the dimers are 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.

F. Tu~~r~ ei al. /Surface Science 317 (1994) 58-64 63

that only those atoms dimerize that have at least two dangling bonds. In doing so we neglect re- bonding at the bottom of steps. Rebonding com- pensates a number of dangling bonds that could not be removed by the dinner reconstruction of the facets. But the bonds at the rebonded steps are highly strained and therefore rather weak. Chadi calculated the energy gain due to rebond- ing to be 0.16 eV/bond [ll]. Although this can- not be neglected compared to the energy of a dangling bond (- 0.8 eV1, the only effect is that the differences in surface energy of different facets are smoothened somewhat.

Depending on the width of a terrace, not all atoms can dimerize, as is illustrated by the (104) plane in Fig. 4. In the case of the {lo51 facet an efficient dimer reconstruction is possible, leaving only 1 dangling bond per atom on each terrace. On the {104} facet atoms are left with 2 dangling bonds, making the surface energy of this facet slightly higher than of (1051.

To be able to quantify the effect of the mor- phology on the surface energy of a cluster, an excess surface energy has been defined. The ex- cess surface energy is calculated by subtracting from the total surface energy of a cluster, the

a

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

aa

Fig. 5. Excess surface energy of a 100 nm long cluster of 10’ atoms versus the angle B of the facets with the substrate. The total energy has been divided by the number of atoms. Solid squares represent the clusters with {lOn} facets (tg 6 = l/n), open squares the clusters with {lln) facets (tg 9 = .,‘2/n).

surface energy of the SK000 interface that is covered by the cluster.

In Fig. 5 the excess surface energy is shown of a 100 nm long cluster of 10’ atoms. The cluster is bounded by { 10nf or {lln) facets, that make an angle fi with the substrate. Generally a hutclus- ter is much longer than it is wide. Therefore in the calculation the contribution from the small facets at the end of the long axis has been ne- glected. In Fig. 5 the excess surface energy has been normalized on the total number of atoms in the cluster. The figure shows a remarkable peri- odicity, due to the interference of the dimeriza- tion and the terrace structure of the facets. The periodic&y of n = 4 of the {lOn) line is related to the atomic arrangement in the diamond struc- ture, the basis of which consists of two atoms, one shifted over (0.25, 0.25, 0.25) with respect to the other.

4. discussion and conclusion

As a Ge adlayer grows on the Si(OO1) sub- strate, the elastic misfit energy stored in 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.

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 a highly anisotropic Ge island will nucleate dur- ing the first stages of epitaxial growth of Ge on Si(OO1). If we consider the elastic energy that is stored in the bulk of a long Ge strip, we find that it is oriented along the principal axes of the Si substrate. The energy difference between the ori-

64 F. Tuinstra et al. /Surface Science 317 (1994) 58-64

entations along the [lOO] and [llO] directions is approximately 1.0 meV/atom. Including the third 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 Ge strip of 4 monolayers high contains lo4 atoms already, making the energy difference between the two orientations 14 eV.

We conclude that the elastic stress due to the dimerization and the misfit favors the nucleation 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 relaxation propagates to the growing nucleus, forcing it to be elongated along the strip. Adding successively a few layers the islands start to de- velop facets in order to allow further elastic re- laxation.

The trade-off between the step structure of the vicinal (10n) facets, and their dimer recon- struction stabilizes the (105) facets, because of the efficient removal of dangling bonds that is possible on these facets.

It must be stressed that this model proposes that the strain in the hutcluster is relieved par- tially only. The strain at the base of the cluster stabilizes the orientation of the long axis of the clusters, while the strain is relaxed elastically at the top of the clusters. If growth proceeds fur- ther, the clusters starts to coalesce and they relax plastically by introducing dislocations at the inter- face. The anisotropy in the elastic energy lessens and it becomes energetically favorable to form { 113) facets. Plastic relaxation of Ge clusters has

been observed recently during the surfactant- mediated growth of Ge on Si(ll1) [12].

Acknowledgments

The authors gratefully acknowledge mr. K. Werner and mr. 0. Schannen for the opportunity to use the MBE system of the Delft Institute of Microelectronics and Submicron technology (DI- MES) for the RHEED experiments.

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