growth instabilities induced by elasticity in a vicinal surface
TRANSCRIPT
Growth Instabilities Induced by Elasticity in a Vicinal
Surface
Christophe Duport, Paolo Politi, Jacques Villain
To cite this version:
Christophe Duport, Paolo Politi, Jacques Villain. Growth Instabilities Induced by Elastic-ity in a Vicinal Surface. Journal de Physique I, EDP Sciences, 1995, 5 (10), pp.1317-1350.<10.1051/jp1:1995200>. <jpa-00247139>
HAL Id: jpa-00247139
https://hal.archives-ouvertes.fr/jpa-00247139
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J. Phys. I £Yance 5 (1995) 1317-1350 oCToBER1995, PAGE1317
Classification
Physics Abstracts
61.50Cj 68.55Bd 62.20Dc
Growth Instabilities Induced by Elasticity in a Vicinal Surface
Christophe Duport(~), Paolo Politi(~) and Jacques Villain(~)
(~) CEA, Département de Recherche Fondamentalesur
la Matière Condensée, SPSMS/MDN,
38054 Grenoble Cedex 9, France
(~) CEA, Département de Recherche Fondamentalesur
la Matière Condensée, SPMM/MP,38054 Grenoble Cedex 9, France
(Received 31 May 1995, accepted 26 June 1995)
Abstract. We study the eifect of elasticity on the step-bunching mstability m heteroepi-taxial growth
on avicinal substrate. Three main eifects
aretaken into account: the Schwoebel
barrier and the step-step and the step-adatom elastic interactions. The firstone is always sta-
bilizing, the second one generally destabilizing and the last one, stabilizing or Dot according to
the sign of the misfit with the substrate. At very low values of the flux F, Orly the step-stepinteraction is
effective: the stabilization is provided by the broken bond mechanism and the
destabilization by the misfit mechanism. At increasing F, the stabilization is governed by the
Schwoebel eifect and only the misfit mechanism needs to be taken into account. A stabilityphase diagram, taking into account aise
apossible instability for island nucleation, is drawn in
the plane (F, T) % (flux,temperature):a
stable region appears, at intermediate values of F.
1. Introduction
A non-hydrostatic stress may give rise to morphological instabilities, when applied to a solid
with a surface. This is because, when the thickness h of the stressed solid is high enough, the
elastic strain energy (proportional to h) overcomes the surface tension (independent of h) and
a plane surface may be no more energetically favoured. It is important to point out that this
process is governed by the mass transport, which can be provided by surface diffusion or by a
partide reservoir(a gas or a liquid) in contact with the sohd.
In the past, starting from the pioneering work of Asaro and Tiller [ii, several authors (Grin-feld [2], Srolovitz [3], Nozières [4]) have studied the stability of a plane interface. The main
result of a linear analysis is the existence of a critical wavelength Ào, beyond which the in-
terface is unstable with respect to a sinusoidal perturbation of arbitrary amplitude (it will be
referred to the following as ATG instability). The existence of Ào is due to the capillaritywhich stabilizes the flat surface with respect to short wavelength deformations. The gravity
or the van der Waals interactions may provide a similar stabilization for excitations of longwavelength [4], inducing a threshold ac in the non-hydrostatic stress a for the rising of the
mstability. As a = ac, the perturbation of wavevector k=
kc < 1/Ào becomes unstable.
© Les Editions de Physique 1995
1318 JOURNAL DE PHYSIQUE I N°10
For the comprehension of the later stages in the evolution of the morphology of the film,
a non-linear analysis is necessary. Nozières [4] has performed a non-linear energy calculation,taking the gravity into account. Such a calculation shows that at the bifurcation point (k
=
kc, a =oc), the first non-linearity lowers the free energy (negative sign of the quartic term in
the interface displacement), so that the instability is subcritical: the analogous of a first order
transition.
Numerical studies have been performed by Yang and Srolovitz [5] and by Kassner and Mis-
bah [6]. They find that, after the above-mentioned instability, there is the formation of deepcracklike grooves which become sharper and sharper as they deepen. According to these
authors, this could connect the ATG instability with the fracture process, even if the two phe-
nomena have completely different characters, because in the latter there is no mass transportand the fracture propagates by breaking the chemical bonds.
Expenmentally, there are different examples of instabilities driven by the strain energy.Thiel et ai and Torii and Balibar [7] have applied an external stress to a
~He crystal and
have shown that above a threshold, grooves form on the surface. In most cases, the stress
is due to the presence of a substrate whose lattice constant does not match the adsorbate
(heteroepitaxy) In films of pure germanium on silicon, D.J. Eaglesham and M. Cerullo [8] have
reported a dislocation-free Stranski-Krastanov growth, with an initially commensurate, layerby layer growth followed by the formation of bubbles. A. Ponchet et ai. [9] have observed
spatial modulations in strained GaInASP multilayers, which allow partial elastic relaxations.
Also, Berréhar et ai. [10] have studied the stress relaxationm single-crystal films of polymerized
polydiacetilene, m epitaxy with their monomer substrate. Polymerization is obtained throughirradiation with low-energy electrons and induces a uniaxial stress. Above a critical thickness,
the authors report a wrinkle surface deformation, but their interpretation as an ATG instability
may be questionable, because of the slow diffusion process of such big molecules.
Snyder et ai. [l ii have stressed the importance of kinetic effects on the instability, by showinghow the surface diffusion determines the growth mode of heteroepitaxial strained layers: thus
a reduced kinetics in films of InxGai-xAs on GaAs gives rise to highly strained pseudomorphicfilms.
In an epitaxial film, the relaxation of trie misfit stram may also occur via misfit dislocations,which eventually lead towards an incommensurate structure. However, only if the relaxation
is total the ATG instability should be forbidden. On the other hand, we can also observe
dislocations after the instability. In experiments on strained Sii-xGex layers on Si, Pidduck
et ai. [12] have observed an mitially defect-free Stranski-Krastanov growth, followed by the
formation of dislocations; these authors also speculate on a possible subsequent increase mthe
modulation period.The instability has a
technological interest, too. In fact the growth in strained structure may
grue rise to the self-organized formation of semiconductor microstructures such as quantum dots
and wires. Moison et ai. [13] have shown that the deposition of InAs on GaAs, after a first
two-dimensional growth proceeds by the formation of single-crystal, regularly arrayed dots, of
quantum size. Similar microstructures have been obtained by Nôtzel et ai. [14], during the
growth of stramed InGaAs/AlGaAs multilayers on gallium arsenide substrates.
With regard to the rising of an mstability, an important feature is the orientation of the
surface. In the case of a non-singular one, a continuum approximation is correct and the
dassical picture of the ATG instability emerges. On the other hand, in the case ofa high
symmetry orientation, the surface tension is non-analytic and Tersoff and LeGoues [15] have
shown in this case the existence of an activation barrer.
Usually, the mstability is studied as a problem of thermodynamical equilibrium. Kinetic
effects have been taken into account by Srolovitz [3] and by Spencer et ai. [16], but their
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1319
Q~ ~, i ,j~~ ~ ~
D«f~W
D'p+ j~titi
~,,p-
Fig. 1. In this Figure, a growing stepped surface is shown: there is a flux F of atoms pet second
and pet lattice site; alter the landing of the adatomon the surface, it moves with a diffusion coefficient
D; close to a step, the adatom has a probability 1' (1')pet unit time to stick to the step from above
(below). Analogously, the probabilities of detachment from the step to the upper and lower tenace are
~, + ~,,f and~ ,
respectively. In this Figure,we have also depicted the diiferent types of instability
to which a stepped surface may be subjected: step-bunching, island formation, step meandering.
approximations make the analysis valid for a non-singular surface, close to the equilibrium.However, Duport et ai. [iii have studied the case of a vicinal surface and they have shown that
the discrete character of the lattice may be very important, since the adatom-step interaction
is otherwise neglected, and that when the adsorbate grows, the surface kinetic may stabilize
the growth process via the Schwoebel effect. Also, if neither growth nor evaporation is present,the vicinal surface has a behaviour similar to that of a non-singular surface: an instability
appears and no activation barrier is present.The main result of their paper is to show the existence of a new possible instability in
MBE growth; it is an elastic instability other than the ATG instability, since trie underlyingmechanism is different and it appears or does not appear, according to the sign of the misfit
ôa. Let us now describe very briefly the growing process for a vicinal surface and let us explainhow such a mechanism works.
A vicinal surface does not grow layer by layer, but through step-flow, with all the stepsmoving at the same velocity. After an adatom has reached the surface, it diffuses on the
terrace until it sticks to a step (see Fig. l). The growth velocity of each step is therefore
determined by the rate of attachment of the adatoms on it and the stability of the step-flowgrowth (uniform velocity of all the steps) will depend on which ledge the adatoms prefer to
stick to. In the presence of a uniform flow, the number of atoms reaching a given terrace is
proportional to the terrace area; so, if the sticking to the upper step is favoured, a larger terrace
(larger than the neighbouring ones) becomes smaller, and viceversa: the step flow growth is
stable. On the contrary, if sticking to the lower step is favoured, a large terrace becomes largerand larger and a step bunching instability arises.
How does the adatom choose the step to stick to? During the diffusion process, it feels
the step-adatom interaction: the steps produce an elastic deformation proportional to the
misfit ôa, which interacts with the deformation induced by the adatom ~ia the so-called broken
bond mechanism, and whose strength is proportional to a force dipole moment m. Thus, the
resulting interaction may be stabilizing (if the adatom is pushed towards the upper step)or
destabilizing (in the opposite case), according to the sign of m and ôa.
1320 JOURNAL DE PHYSIQUE I N°10
However, as pointed out in [iii, the asynlmetry mthe probabilities of attachaient, from
above and front below, to a step (Schwoebel effect) is a possible source of stabilization of trie
growth process.In reference [iii it was assumed that the sticking of an adatom to a step is an irreversible
process. As a consequence, the adatonl is not able to reach thermal equilibrium and minimize
the step-step interaction, which therefore results to be ineffective. In this paper we will take
the step-step interaction into account, andwe will be able to interpolate between our previous
analysis and the works of Srolovitz [3] and Spencer et ai. [16]. As in reference [iii and in the
current literature, the thickness of the substrate will be assumed to be infinite.
The paper is organized as follows: in Section 2 we study the possible origins of elastic
deformations on a stepped surface and we give the expressions for the adatom-step and step-
step interactions. In Section 3 we introduce the Schwoebel effect, while in Section 4 we show
the destabilizing character of the step-step interaction. Then, we perform a stability analysisof the step-flow growth: in Section 5 we wnte the equations of motion for each step, and in
Section 6 we linearize and solve them. Only the main formulae will be given, while ail the
details of the calculation will be deferred to the Appendices. Section 7 is devoted to the studyof the limiting case of low flux, this will allow
us to compare our results with those of Spenceret ai. [16]. Finally in Section 8 we will discuss the hypotheses underlying our stability analysisand in Section 9 we will summarize the main points we have addressed in the paper.
2. Elastic Interaction between Adatom and Steps
The elastic interaction between an adatom ion a high symmetry surface) and the substrate may
have two different origins. The first one is a deformation of the surface, because of the different
lattice constants of adsorbate and substrate; ii is called "misfit mechanism" and occurs only in
heteroepitaxy. In the second case, the deformation originates from the dipole of forces which
the adatom exerts on the other atoms; it is called "broken bond mechanism" and is effective
even in homoepitaxy. Let us focus on the latter mechanism.
It is described in Figure 2a in the case of central, pairwise interactions between nearest and
next-nearest neighbours. The adatom exerts forces fR on the other atoms, the location of
which is designated by R. Ai eqmlibrium the total force is zero, but the dipole tensor
mn~a~=
£ Rn f( il
R
has non-zero diagonal elements. If the x and y axes are chosen parallel to the surface, symmetryimposes
: mxx = m~~ = m. Having introduced the atomic areaa~, the force dipole moment
mn~ results m having the dimension of a force divided by a length that, is the dimension of a
surface stress.
At a long distance r of an adatom, the components ~xx, u~~ of the strain tensor are pro-
portional [18-20] to~~
~~~~~~~ '~~, wherea is the Poisson coefficient. The part of this
r
expression, which is proportional to mzz is often ignored [18, 21]. Quadrupolar and higherorder effects will be neglected.
The broken bond effect is not additive, in that the number of broken bonds of a pair of
adatoms is not twice as much as for a single crystal. Accordingly, the stress exerted by a largeterrace is localized near the ledge (Fig. 2b), so that the elastic effect of a terrace is that of a
distribution of force dipoles along its edge [22].The second mechanism of elastic interaction is the so-called "misfit mechanism", and ii
results from the fact that the adatoms would like to have an interatomic distance different
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1321
a
~,.
2 f'2
1.i i'i(1 .~
o o o
o o o o o o
b
e e © o--
@ @ 3 j tl
e tl 16 e @ e @ e
Fig. 2. We have represented the broken bond mechanism in the case of pairwise interaction between
nearest and next-nearest neighbour atoms. a) For an adatom. We bave also mdicated the moduli of
the forces. b) In thecase of a tenace, where the forces
arelocalized near the tenace edge. In bath
cases we have considered only the forces due to the presence of an adatom or a step, Dot of the surface;
m the latter case the forces disappear after surface relaxation.
from that of the substrate. In opposition to the previous one, this effect is additive, I.e.
proportional to the size and the thickness of the terrace. However, if the substrate is infinitelydeep, the stress exerted by complete layers does non need to be taken into account, smce the
size of these layers is fixed by the substrate.
On the contrary, if the adsorbate has a modulated surface of variable height h(x,y), ii is
equivalent to an adsorbate with a plane surface on which the substrate acts as an external
surface stress given by (see Appendix Al):
~[~ "
))lôxn+ôw)à@hl~,iÎ) 12)
where E is the Young modulus, q has the dimension of a surface stress and ~~) is the relativea
difference (or misfit) between the lattice constant of the substrate (asb) and that of the free
adsorbate (aad)~~
=
~~~ ~~~.a asb
In the case of heteroepitaxy, addressed in this paper, the misfit mechanism (being additive)dominates the elastic effect of a large terrace (at least far from ils edge), while it is ineffective for
an isolated adatom, which interacts via the broken bond mechanism. The elastic deformations
give rise to an effective interaction between defects: the adatom, through the broken bond
effect, feels the deformation produced by the terrace through the misfit mechanism. Ii is the
form of the potential, felt by the adatom, which determines the stabilizing or destabilizingcharacter of such an interaction.
1322 JOURNAL DE PHYSIQUE I N°10
f,
j~,
fl++O
,©w
,i
,-*
(a)'
'
,
,
i
~i
©w
(bl'
i
i
i
-Én/2~ ÎÉn/2 $
~n+1
n-W teoeace
sdbstrate
Fig. 3. At the bottom,a
one-dimensional array of straight steps and the labelling used in the text.
Above, the potential felt by the adatom because of the interaction with the steps and the Schwoebel
barrer: bath the cases ofa
stabilizing (a) and destabilizing (b) step-adatom interaction, areshown.
The Asaro-Tiller-Grinfeld instability derives from a step-step interaction due only to the
misfit mechanism: therefore the potential is proportional to (ôa)~ and the instability takes
place for any sign of ôa. On the contrary, the instability which will be addressed in the present
paper depends on a step-adatom interaction where both the mechanisms (misfit and broken
bond mechanism) coule into play; since m and ôa may have identical or different signs, the
elastic interaction can be destabilizing (in the former case)or stabilizing (in the latter case).
Now, it is important to evaluate the elastic energy Unix) for an adatom on the n'th terrace,
wherex is defined in Figure 3. This can be done under the assumption [18, 21-23] that
adatoms and steps can be represented as force dipoles acting on a plane surface. For straightsteps parallel to the y axis, the only nonvanishing element of the force dipole tensor, which
results from the misfit mechanism, is qxx la. Since the elastic interaction energy between two
force dipoles is minus the product of one force dipole limes the local deformation mduced by
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1323
the second one, after integration on the plane of the terrace, in the limit » a we obtain [20]
Unix)=
Ûf l~
~ ~ ~(3)
k=0~lp=0 ~n+p + X Î ~lp=0 ~"-P
~ Î
with
U=
j( ~ 'j a~ôajjl a)m amzz) j4)
The term proportional to mzz had been neglected in reference [iii.In the same approximation, the elastic energy of an atom incorporated into the n'th step
can be seen as the variation of the step-step interaction when an atom is added to the n'th
step, that is to say,
En "
if~ ~
~(5)~ ~
În+p
~jÎn-1-pP"o p=0
with
~ ~Î ~~~~~~~ ~~~
3. Trie Schwoebel Effect
As originally pointed out by Ehrlich [24], even in a free diffusion process ii-e- without adatom-
step interaction) adatoms have no reason to go with the same probability to the lower and to
the upper ledge. This effect will be taken into account by introducing two different coefficients:
D" la is the probability per unit time for an adatom near an upward step to stick to that edge,D' la is the same probability for a downward step. Several tl~eoretical studies [25, 26] confirm
the existence of an energy barrier in the proximity of the upper ledge of a step (see Fig. 3), that
is to say, D' < D" and the adatom prefers to go to the upper ledge. Schwoebel [27] was the
first to notice that, as a consequence of this, the growth of a parallel array of steps is stabilized
(Schwoebel effect).It will be useful, in the following, to define the quantity L
= a(§ + )). This length
measures the difficulty, for the adatom, to stick to a step. We shall assume that §ci 1 and
§ < 1, so that L cia§.
4. Elastic Interaction between Steps
In the previous section it has been darified how the adatom, during its diffusion process,
interacts with the steps. Because of this interaction, it goes preferably to the upper or lower
ledge according to the sign of (m mzza/(1- a))ôa: the destabilizing case (lower ledge)corresponds to a positive value. When ii is near a downward step, the adatom feels the
Schwoebel barrier, which reduces its probability of attachment.
In the Introduction, it was said that the ATG instability is a consequence of step-stepinteraction. Indeed, once the adatom is incorporated, this interaction may come into play, but
it is effective only if the adatom can escape from the steps and eventually attach to the other
ones in order to reduce the step-step energy. Duport et ai. [iii made the assumption that the
incorporation of the adatoms at the step was irreversible, so that the ATG instability was not
1324 JOURNAL DE PHYSIQUE I N°10
f i
,
,
fÎ
i
©j ,3à~
Î~
l
,
i
i
i
X~
Fig. 4. Asm Figure 3a, but
Dowalso the destabilizing step-step interaction is shown.
induded in their treatment. On the contrary, in trie present work, atom detachment from stepswill be taken into account.
It will first be checked, by a simplified argument taking only interactions between nearest
neighbour steps into account, that the step-step interaction (5) destabilizes a regular array of
steps, in agreement with the general ATG instability. The exact proof (taking ail terraces into
account) results from the calculations of the next sections.
If an atom is transferred from the (n+ll'th to the n'th step, the energy variation is
ôE= en en+1 =
<~ ) )l17)
n n n-
The atom transfer is energetically favorable if ôE < 0, 1-e- if in is larger than tn+i and in-i (seeFig. 4). Since the transfer lengthens the n'th terrace, wide terraces become even wider. This
means that atom exchange between terraces is destabilizing. Even so, the previous argumentis vahd only if equilibrium can be reached, that is to say if the incorporation of the adatoms
at the step is reversible within a reasonable time before new adatoms come from the beam.
The probabilities of detachment of an atom per unit lime and per unit length of the stepfrom the n'th step to the upper and lower terrace will be called D'p$la and D"pj_i la,respectively. The quantities p$ and pj are close to the equilibrium density of adatoms po,
with a small correction due to elasticity. The probability of detachment D'p$ lais related
to the probability of attachment D'la of an adatom (supposed to be close to the step) bythe detailed balance principle. This principle states that the ratio PS of the two probabilities
is proportional to the exponential of the difference between the energy of an adatom at the
respective positions, divided by kBT. This energy may be written as a quantity independentof elasticity, plus a correction. This correction is equal to (5) for the incorporated atom, and
to U (~") (see Eq. (3)) for the atom close to the step.2
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1325
Accordingly, we can write:
vivo exp à lu
iii nii~a)
vn vo exp à luii n+ii
i~b)
with1là=
kBT.
5. Equations of Motion for trie Steps and trie Terraces
Let us now investigate the linear stability of a regular array of straight, parallel steps. In ouf
one-dimensional mortel, each terrace is characterized only by its length in= xn xn+i, whose
time dependence is a function of the length of ail trie terraces:
f~) =F lll~l) 19)
We shall linearize this equation near the stationary solution in=
1, corresponding to the
step-flow growth.Within each terrace, we can write a conservation equation for the adatom density pn(x, t)
~pn(x,t)
=
-)Jn(x,t) + F (10)
where Jn is the current of adatoms and F is the beam intensity. The evaporation rate has been
neglected. It will be assumed that the time variation of pn is very slow with respect to trie
adatom motion, so that the conservation law takes trie stationary form
)Jn(x,t) + F=
0 iii)
and trie time dependence of pn and Jn will be ignored in the following.The integration of iii) is straightforward
Jn(x) Fx=
An (12)
where An is an integration constant, depending on trie tenace. If F=
0, the current is uniform
on each terrace, and ii is equal to AnTrie relation between trie density pn and trie current Jn is provided by trie Einstein-Fokker-
Planck equation [28]
~"~~~ ~ ÎÎÎ~ ÎIÎÎ ÎÎ
~~~~
where D is the diffusion constant and Unix) is the energy of an adatom on the n'th terrace,
given by (3).From Equations (12) and (13), the following expression for the adatom density pn ix) is found
pn(x)= pn
(-j)exp (fl
Un (-j) nix)]
j /~ dY(An + FY) exP (fl[Un(Y) Un(x)Î) (14)
1326 JOURNAL DE PHYSIQUE I N°10
where the middle of each terrace is taken as the origin for the terrace itself.
Now the boundary conditions at x =
+~"are necessary. They are obtained by writing
2that the sticking rate is, on the one hand, equal to the absolute value of the current, and on
the other hand proportional to the difference between the local adatom density and the stepequilibrium adatom density:
Jn))
=
~' pn(~")
$j(lsa)
tl 2
Jn (~))
"
~~" pn(~
))vii (lsb)
tl
Jointly with (12), we have
~' ~
iPn (il
Ptl -An + Fi i16a)
~" f~ f~Î ~"
2~" ~" ~ ~
2~~~~~
Finally, to study the stability of the regular array of steps, we need the quantity
()=
lxn xn+i) (ii)
Since it results that
12 t~"~~~~" ~Î~ ~"~~ ~Î ~
~Ît~"~~~~~ ~"~~ ~~Î~ ~" Î~
with the aid of Equation (12), the required expression is attained
(/=
)lin-i Én+1) + 2A~ A~+i A~-1 j18)
6. Linear Stability Analysis of trie Step-Flow Growth
In the spirit of a linear stability analysis, we will consider small deviations from the stationarysolution in(t) =1, so that we can consider a single Fourier mode:
Énjt)=
+ (jt) cosjKn #jt)j eii + ôÉnjt) j19)
and we will seek the equation governing the amplitude ((t).The quantities Am entering equation (18) are obtained starting from equation (14) (see
Appendix B), and depend on all the lengths (in), so that we can define AS and ôAn throughthe relation
An ((in))=
An((1)) + ôAn + AS + ôAn (20)
where AS is independent of n, and ôAn is a very complicated expression
ôAn=
£ ~~"ôim (21)
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1327
In ail trie following treatment, we assume that the elastic energies are very small compared
to the thermal energy:~,
< kBT=
~. Within ibis limit, ôAn takes tl~e forma fl
ôAn=
-flpoD~~"~
)"~~ ( ~ ~~ ~f~ ôta ~) ~~~U( (~) ôta
+ ii + L) ii + L)
fÎ< Yul lv)du pF IL w)~~
i+ flF 2
~ôin + ~ IL 1)
~U~ ~ ôin
ii + L) ii + L)
pF Î iflôUn 1- +) IL fl) ôUn 16)] 26 iÎ~ vUniv)du~
2 t + L
pff aDflôUn 1-ft) + IL fl) ôUn 16) + à fÎ~ Univ)du
2 ~D" ~(t + L)~
~~~~
where ôX, as m equation (20), is trie part of X proportional to (. Trie different quantifiesappearing in (22) are written below and derived in trie Appendix:
à/~
yUn(y)dy=
Ù1
+ in ôtn +
f~k (tn+k + in-k] (23)
% 2 tl~~~
with~
~~( ~k
~ 2(kÎ
1)~~
~ Î~~~~
ô/
Unlv)du=
) £ (~ )) loin+k ôIn-k) 125)~Î li
ôUn))
=
j £ Ch (ôta-k ôta+k-1) (26a)Î~
ôUn (-))=
£ Ch (ôta+k ôtn-k+1) (26b)Î~
withm
Ch=
~m
(27)
~~~P
ô~n ô~n+1" ~~
~~~~~~~
2 É~~ ~~~"~~
~~~"
~~~"~~ ~ ~~~~~ ~~ ~~~~
k=1
U( (~) =
~(29)
a
/Î~ ~)yU~(y)dy
=2Ut 1 j in cx (30)
-) l~
1328 JOURNAL DE PHYSIQUE I N°10
with~
cx =
£ ((k + in (1+ ii (31)
~=~2 k
With trie previous expressions, trie evolution equation (18) for trie terraces array, for weak
deviations ôAn is written as
~ ("=
((ôta-i ôta+1) + 26An ôAn+i ôAn-1 (32)
Insertion of (23-31) into (22), allows us to put ôAn in trie form
ôAn=
R( cos(Kn #) + Q( sin(Kn #) (33)
where R and Q are t-dependent, Substituting such an expression into (32) and using trie
definition (19) of ôta, we find
12 Î~" ~~~ "~~ Î~ ~°~~~~ ~~
+4Q( sin~ ~sin(Kn #) + 2F( sin(K) sin(Kn #) (34)
2
By comparison with trie equation
(in=
~~cas(Kn #) +
(jsin(Kn #) (35)
a t dt t
trie following relation is established
(~ =4R(sin~ ))
(36)a t
where we can identify trie relaxation frequency a~Q(K), defined by (~ =Q(K)(,
asa t
Q(K)=
4R(K) sin~ 1( (37)
The step-flow growth will become stable if Q(K), 1-e- R(K), is negative for any value of K.
Then, the next step is just to calculate R(K), which is done in Appendix E. It is worth notingthat the stability condition R(K) < 0 can be formally put into the form
Re1[ ÎÎÎ e~~~"~~~ < ° 138)
which generalizes the condition~l<
0, found in reference [iii.
In Appendix E, R(K) is found to be equal to
R(K)=
Ro Rif
~k cas(Kk) + R2 sm(~) f
Ch sinK
(k (39)
~~~2
~=~2
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1329
with RD, Ri, R2 given respectively by:
Ro=
-1(L
SI ~ i Î(1+ 2flFU Il Ill + fIFU l~l Il
j~ + ij iL~
2(L
t) (Lfl (fl)~j~~~
t +
Î ~ ~~~ait + L)3 ~~~~~
Ri=
2flFÙ) (40b)
~E1+,~
pp~ D iLt + 4 Lfl (fl)~
~~x 1 a
~~~~~ ii + L) t2~ ~~~~
(t
+ L)2 ~~~~~
Expression (39) has two extrema for K=
0 and K= x. Analytical expansions for small K
and numerical tests led us to condude that trie first one is a minimum, the second one is a
max1nlum and that no other extremum exists. The stability condition becomes R(x) < 0, that
is to say
~~~~ ~° ~~ ~ ~~~~~ ~ ~~
~
(2k l)2 ~ ~ ~~~~
After having evaluated the two sunlmations, we obtain the final form (~ is the Euler constant)
Ro + )Ri (~ in ))+ ~R2 < 0 (42)
Two limiting cases are worthy of mention: the case of terraces very large with respect to L
and the case of weak flux.
Limitt>L>a
In this limit, equation (42) becomes
~~ > flÙ~2
In 2 +~ ~~ ~~~ ~ ~~
+XED~ ~' (aôa)~ ~~( (43)
~~ ~ ~' ~~
The term on the left derives from the Scl~woebel effect, while tl~e terms on tl~e rigl~t correspond,the first one, to the step-adatom interaction and, tl~e second one, to tl~e step-step interaction.
The first one is proportional to Ù and tl~erefore it may be stabilizing (Ù < 0) or destabilizing(Û > 0), according to the sign of the product (m mzza fil a))ôa.
~2The stability condition (12) of reference [iii is only recovered iii » and if the term of
astep-step interaction is neglected. If so, (43) is rewritten as
> fll~~
ôa(i + a) in 2 144)
Tl~is is the condition found in reference iii], apart from a factor 2 in 2 (instead of 2 as daimed
in iii]).
1330 JOURNAL DE PHYSIQUE I N°10
Limit of weak flux and large L
The limit of weak flux will be discussed extensively in the next section, with reference to the
paper of Spencer et ai. [16]. For the moment, let us observe that the condition of low flux is
written as
~ ~~Î
~~~~
In this limit, we can retain (see Eq. (E.6)) only the term of step-step interaction independentof F, and the Schwoebel term of zero order in fl. Then the stability condition takes the form
~ , ~Ft~L~
(46)2xEDfi (~~~) fil'° ~Î + L
Indeed, a further condition on step-adatom interaction must be satisfied to obtain (46):flÙ < L~ lit + L). Since we have already supposed flÙ < a, the previous condition is always
fulfilled unless t > L~ la, in which case we fait within the limit of large terraces.
Ii is noteworthy that in the limit of strong Schwoebel effect (L » 1) we recover the same
stability condition (46), because in both cases the step-adatom interaction is negligible.
7. Limit of Low Flux
In this section we want to compare our results with the theory of Spencer, Voorhees and
Davis (16] (SVD in the following), which is valid for a nonsingular surface close to equilibrium.Indeed, SVD have assumed a flux so low that only the step-step interaction needs to be taken
into account.
In the previous section, we have already said that in this limit only two terrils can be retained:
the step-step interaction, which is independent of F, and the term coming from the existence of
a Schwoebel barrier, which is of zero order m fl. The stability criterion takes the form (46), but
it is not directly comparable with equation (3) of SVD. So, let us go back to our equation (36),which -in the case of low flux- takes the form
ÎÎ ~~~~~~ ~Î~lÎ
lé ÎL)2 ~ (Î~ÎÎÎ2 ( ~~ ~~~~Î)
~~~
~~~ÎÎ ~Î~~~~
The summation can be done explicitly
C° ~ ~j~ ~ ~) m sin ~f~
~ ~2( ~~ ~~~2
~~~2
§ j~2~4 8
~~~~
=i p=
so as to obtain
J dt~~ ~~~
2 2 l~ + L)~~
l~ + L)~~~~~
2
Î~~~~
In the approximation K < 1, if we introduce the 'true' wavevector k=
~,the previous
trelation can be written as
~ÎÎ~~~
Î (tÎÎ)2 ~ ÎÎÎ~Î~~Î ~~~~
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1331
The second term on the right (proportional to k~) corresponds to the term of elastic interaction
in equation (3) of SVD, if tl~e l~ypotl~esis of equal elastic constants for adsorbate and substrate
is done. Actually, the only difference is the presence in our equation (50) of the prefactoril ii + L), which takes into account how the step-step interaction is modified by the Schwoebel
effect.
The comparison of the stabilizing terms dearly shows their dioEerent origin: SVD consider a
nonsingular surface and the stabilization is provided by the capillarity, proportional to k~. On
the contrary, we consider a stepped surface, whose total area does not change during the growth,and trie stabilization derives from trie Schwoebel effect, wl~ich contributes to equation (50)with a term proportional to k2. While the capillarity stabilizes against fluctuations of short
wavelengths, the Schwoebel barrier stabilizes the region of long wavelengths, and the instabilityarises for
~ ~ ~~'~~ ~~ÎÎ~~~ Î ~
~~~~
However, since ouf stabilizing term derives from a kinetic mecl~anism (trie Schwoebel effect)it cancels m tl~e hmit F
=0. TO fill tl~e gap between our results and tl~ose of SVD, we must
reconsider tl~e nature of tl~e step-step interaction: in fact, we bave supposed tl~at tl~e interaction
of order il /t2) between tl~e dipole forces located at tl~e ledges of tl~e steps and coming from tl~e
broken bond mecl~anism, were negligible witl~ respect to tl~e interaction of order (In t), comingfrom tl~e misfit mecl~anism. In otl~er words, we have neglected a stabilizing interaction, wl~icl~
corresponds -for a nonsingular surface- to the capillarity, and which is important at equilibrium(F
=0).
In trie following, we will remove such a hypothesis and the flux F will be supposed to be
so low that the stabilization provided by the Schwoebel effect can be neglected. Then, if in
analogy to m we define f as tl~e sum of tl~e dipole forces acting at tl~e edge of tl~e steps and
coming from tl~e broken bond mecl~anism, equation (47) can be written as
ÎÎ ~~ "~~ (Î~ (ÎÎÎ) ~~~~ ~~~+~ ~~~~
with
ô~(=
ô~n ~~~ j~~~~~~ f~ ~
~rk=o
lLp=o În+p)3 lLp=o in-i-p)3
~~~ ~ '~a~ fôa
f~
~
~(53)
~k=o
(Lp=oÉn+p)~ (Lp=oÉn-i-p)~
By analogy to ~n, ~( is the total variation of the step-step interaction when an adatom is
incorporated into the n'th step. In equation (53), the three quantities on the right correspond
to the interaction by misfit mechanism, to the interaction by broken bond mechanism and to
tl~e step-step interaction, respectively if botl~ the mechanisms are effective, one for eacl~ step.Alter a manipulation similar to tl~at performed in Appendices C and E, at order k~
we find
ÎÎ ~~~ÎÎ ~ ~~ Î~~ ~~~ Î~Î~~~~~~~~~
-~~~ ~'~a~£fôa (~ ln(kt))~j
(54)~r 2
Let us now suppose that one of the two mechanisnls prevails, so that the third term is alwaysnegligible. In the two opposite l1nlits, we obtain:
1332 JOURNAL DE PHYSIQUE I N°10
Interaction ~ia misfit mechanism: Et~~» f
a
In this limit we simply have equation (47) ai F=
0, and there is instability for ail the
physical wavelengths > t.
Interaction ~ia broken bond mechanism: Et~~< f
a
In this case, equation (54) beconles:
ÎÎ ~~ÎÎ
Î"~~~~
ÎÎÎ~~~~~~~~Î
~~~~
and an instability arises for > Àmin, with
So, we have recovered the result of SVD, in the case of a vicinal surface. Their result is valid
in the limit of non too large terraces, otherwise the step-step interaction ~ia misfit mechanism
cannot be neglected. Trie validity conditionEt~~
< f corresponds to Àm;n > t, 1-e- triea
stability region corresponds to a physical domain of trie spectrum.
8. Discussion
The present section will be mainly devoted to the discussion of the hypotheses underlying our
stability analysis. Some of them have already been stated at the beginning of the paper.First of all, we have supposed (see the Introduction) tl~at tl~ere are no misfit dislocations,
wl~icl~ is experimentally verified in some systems [8,12]. Tl~is is correct if trie misfit (ôa la) and
tl~e tl~ickness h of tl~e adsorbate are not too large. For the heteroepitaxy of semiconductors
for example Gexsii-x/Si), a misfit (ôa la)cf
1% -corresponding to x ci 0.25- can be considered
as a typical value.
For this value J-C- Bean et ai. [29] found a critical thickness hc C~ 100a. Such results, and
the following tl~eoretical interpretation [30] bave been criticized because tl~ey concem a non
equilibrium case [31].Anyl~ow, tl~e system we bave been discussing is actually eut of equilibrium and trie mechani-
cal equilibrium theory of Matthews and Blakeslee [32] gives a critical thickness hc of trie same
order for a misfit (ôala)ci
1%. We condude tl~at, for a significant range of values of tl~e
tl~ickness h of tl~e adsorbate, tl~el~ypotl~esis of no misfit dislocations is reasonable.
Moreover, tl~e temperature T is supposed to be l~igl~er tl~an tl~e elastic interactions (seeSection 6):
T »~
# a~m(1 + a) (~~) (57)a a
A typical value of tl~e misfit is ôa lacf
1%; aise, in Appendix A2 we give a rough valuation of m
for a triangular lattice: a~m# 0.l7Wcah witl~ Wcah m 35000 K. Tl~en, the previous condition
is written T » 60 K. Since typical values of T for MBE growth are around 400-800 K, the
above approximation is correct.
In addition to tl~e previous ones, there are some hypotheses concerning other kinds of insta-
bilities which may occur dunng the growth of a stepped surface.
N°la GROWTH INSTABILITIES INDUCED BY ELASTICITY 1333
First of ail, a serious limitation of our model is that it is one-dimensional and therefore
applies only to straight steps. In reality, steps form meanders because of thermal fluctuations
or stochastic fluctuations resulting from diffusion. Worse than that, even if deterministic
equations are accepted, wl~icl~ generalize tl~ose of Section 5 in tl~e two-dimensional case wl~en
elastic effects are neglected, straight steps tutu eut to be unstable [33, 34]. Even worse, a non-
linear study suggests that tl~ere is no steady state, but a cl~aotic state [35] unless a suiliciently
strong anisotropy stabilizes a steady state [36]. However Johnson et ai. [37] have shown that
the Schwoebel effect does prevent step-bunching in step-flow growth on a two d1nlensional
surface. Presumably this happens in heteroepitaxy toc, when elastic effects are taken into
account. Tl~us, ouf treatment can be expected to be qualitatively correct, if £ represents a
local average of the distance between steps at a given time.
Finally, let us discuss the possibility of islands formation on the terraces, which has been
disregarded up to now. Generally speaking, this is correct if tl~e flux F is non too higl~ and
trie tenace width t not too large, so that the average lifetime of an adatom is short and
consequently the probability of island nudeation, low. A rough valuation of such probabilityand of the consequent instability threshold can be given.
We shall assume a 'minimal model' for the island formation: an adatonl diffuses on the
terrace until it sticks to a step or it encounters another adatom, forming a stable pair which
can neither decay nor diffuse. The addressing quantity is trie diffusion length tD, tl~at is to
say, the average distance travelled by the adatomm
the absence of steps, before the creation
of a stable pair. In this min1nlal model, tD equals the average distance between nudeation
centers and the average size of the islands just before their coalescence. We shall assume that
the step-flow growth is stable against island formation if £ < tD (37]. The order of magnitudeof tD [38, 39] is
tD Cf
~~~ ~~~
(58)F
and the stability condition is easily found to be: F < Da~/t6.In Figure 5 we show tl~e regions of stability in the plane (F, T), at fixed t. We can observe that
at a given temperature the step-flow growth is destabilized by step-bunching at low flux and byisland formation at high flux. At high temperatures no stability domain is predicted. However,
our l~ypotl~esis tl~at a pair of adatoms is stable is not correct at l~igh temperatures: actually,this is correct only at low temperatures while during the growth of T a stable nudeation center
is made of three, four or more adatoms. This fact makes island nudeation more diilicult, lifts
up the upper curve and stabilizes the step-flow growth.Quantitatively, finis means that tD is no longer tD oc
F~~/6 Tl~e correct exponent is now
given [39] by ~ =i*/(2i* + 4) where (1* +1) is trie number of adatoms forming
astable
nudeation center: so, ~ =l/6 for1*
=1 (tl~e dimer is stable) and ~ =
l/4 for1*=
2 (tl~etrimer is stable). Unfortunately, trie proportionality constant does not depend only on D, but
also on all trie average lifetimes (r2, r3,.,
rj~ of tl~e unstable nudeation centers.
Another comment on trie validity of trie previous calculation is appropriate. Tl~e stabilitycondition derives from a comparison between a diffusion length iD, valid for a high symmetry
surface, and the terrace length of a stepped surface. If there is no Schwoebel effect, this is
correct because when the adatom feels the step, it is automatically incorporated. Instead, in
the presence of a Schwoebel barrier the adatom con be reflected, so that the time r necessary
to be incorporated increases and the probability of forming a stable pair increases as well. In
the former case(no Schwoebel effect),
r is the time necessary to reach the nearest step, that
is to say (see Fig. 3) r m (1/2 (x()~ ID and if we average over x we find f ml~/12. In
the latter case, if we consider the limit ofan infinite Schwoebel barrier, the adatoms can be
iowNaL ce nmsiquB i -r s, w ta ocroAm iws s2
1334 JOURNAL DE PHYSIQUE I N°10
F (1/S)
/
/
/
/
0 1
~
/
/350
Fig. 5. Stability phase diagram for a stepped surface with 1= 30a and amisfit ôala
=1%. Tl~e
values of D, po, L, m, Eare given in
the text. A step-bunching mstability arises in the region below
the full line andan
instability for island nucleation in the region above the dashed line.
incorporated only at tl~e upper ledge and r is smaller than il /2 + x)~ ID,so that f < i~ /3. We
condude tl~at tl~e presence of a Scl~woebel barrer does not modify qualitatively Figure 5. A
more quantitative analysis of tl~e stability against island nudeation will be tl~e object of future
work.
9. Conclusions
We bave studied the elasticity-driven step-bunching instability of a vicinal surface. The surface
has been modeled as a one-dimensional array of straight steps; the growth occurs ma step-flow,
I.e. tl~rough sticking of the adatoms to tl~e steps.
Tl~e step-flow growtl~ is stable if tl~e adatom prefers to stick to tl~e upper ledge. Wl~icl~
ledge is indeed tl~e favoured one depends on kinetic and tl~ermodynamical effects, tl~e latter
being effective only if adatoms are allowed to detacl~ from steps after incorporation, so tl~at
equilibrium can be attained. Kinetics effects are tl~e Scl~woebel effect and tl~e step-adatominteraction: tl~e first one is stabilizing, wl~ile tl~e second one may be stabilizing or not, according
to the sign of the misfit (ôa la). Tl~is fact gives rise to a new possible mstability in MBE, which
has been discussed in reference iii]. Here we have also taken into account thermodynamicaleffects -that is to say the step-step interaction- and we have generalized the step-adatominteraction, previously restricted to nearest neigl~bours: tl~e latter extension bas no important
consequence on tl~e stability condition.
In heteroepitaxy, it is generally correct to restrict tl~e step-step interaction to tl~e misfit
mecl~anism: tl~e latter is destabilizing and it prevails on tl~e otl~er possible destabilizing terms
(the step-adatom interaction) within the limits of strong Schwoebel effect (L » t)or weak flux.
However, if F is too low or even zero, the stabilization provided by the Schwoebel barrier is no
longer effective and the step-step interaction ~ia broken bond mechanism must be invoked; such
an interaction can be considered, for a stepped surface, the analogous to the surface energy(capillarity).
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1335
If stability with respect to island nucleation is considered, we find that at a given tempera-
ture, tl~e step-flow growtl~ is destabilized by step-buncl~ing at low flux and by island formation
at high flux, while it is stable at intermediate values.
Acknowledgments
P. Politi acknowledges financial support from the EEC program Human Capital and Mobility,under Contract ERBCHBICT941629.
Appendix A
Proof of Equation (2)
Let us consider an adsorbate, grown on a substrate with the same elastic constants.
The elastic free energy in the substrate can be written as
Fsub= /~ d~T L Qil~npir)~~~ir) jA.i)
BU, oP~i
where the integral is on tl~e volume of tl~e substrate, (cx, fl,~,() are the coordinates x,y,z,the quantity ~(r) is the strain tensor at point r and Q is the elastic tensor. In the case of an
isotropic solid, it is given by [40]:
titi=
vlôn~ôpi + ôniôp~) + Àônpô~t lA.2)
wl~ere and p are tl~e Lamé coefficients.
Tl~e substrate will be assumed to be infinite in tl~e -z direction. Tl~erefore, tl~e strain
vanisl~es for very large values of -z. Thus it is natural to count the strain from an originsuch that the substrate strain is zero. The substrate actually introduces a strain b into the
adsorbate. However, the strain in the adsorbate will be counted from such an origin that it is
zero if the adsorbate is in registry with the substrate. Then the free energy in the adsorbate
film is
Fad=
/ d3r £ ail inp jr) + bnplk~i jr) + b~~l lA.3)~~d nP~f
where id is the volume of the adsorbate. In the absence of any interaction with the sub-
strate, the adsorbate strain (with respect to the substrate) would take the value ~ =-b which
minimizes (A.3).Assuming that the axes x and y parallel to the surface are equivalent, the components bnp
along these axes are obviously
bxx= b~~ =
-~~(A.4a)
a
The other nonvanisl~ing component of the tensor b is easily deduced from elasticity theory,namely (for
an isotropic solid)
2À ôa 2a ôajA_4b)~~~
+ 2~t a 1 a a
wherea = ~
~) is trie Poisson coefficient.
1336 JOURNAL DE PHYSIQUE I N°10
For a fixed number Nad of adsorbed atoms, the term of second order in b, in (A.3), which is
proportional to Nad, is a constant and will be omitted. Formula (A.3) can therefore be written
as
Fad= /~ d~r L Qil~opir)~~~ir) L inp /~ d~mnpir) iA.5)
~d nP~t n,P=x,y ~d
where
4np =
fllilib~i lA.6)
~f
plays the role of an externat volume stress. The stress tensor qnp is independent of the surface
shape, and therefore ils symmetry properties may be derived assuming a plane surface. Since
relaxation is free in thez
direction, the components qnz are zero at equilibrium. On the other
hand, symmetry implies qx~=
0 and qxx=
q~~. In particular, for an isotropic solid, from
(A.6) and (A.4a-A.4b)we obtain
~°~#
~~ )(ôxn + ôm)ôn~ (A.7)
a a a
For an infinitely thick substrate, the average strain on the adsorbate perpendicular to z
vanishes. Therefore, the contribution of the completely filled layers to the second integral at
the right hand side of (A.5) vanishes toc. The integral can thus be limited to the incompletelyfilled layers.
~n,p(x, y, z) may be assumed to be mdependent ofz in the uncompleted layers, so the second
volume integral in (A.5) reduces to a surface integral, where the number h(x, y) of incompletelylayers has to be introduced. If we define the external surface stress
Qn,~ "40,~hlX, Y)> (A.8)
equation (2) is recovered.
Value of trie Dipole Moment m
Let us calculate m for a triangular lattice (bidimensional model). We shall assume that the
adatom interacts with the surface through pair interactions V(r) with its nearest (nn) and
next-nearest (nnn) neighbours (see Fig. 2).In the bulk, the condition of minimum energy is wntten as
)[NV(ri) + NV(r2)1
=0 (A.9)
ri
where N=
6 is the number of (nn) and (nnn), and ri,r2 "
Viriare the distances between
(nn) and (nnn).So, the relation between fi (nn)-force and f2(nnn)-force is
fi"
Vi f2 (A.10)
At the surface, a relaxation phenomenon occurs: the interatomic distances and the forcesf(, f()
are modified. We shall assume, for the moment, that the distances are non changed, so
the equilibrium condition for the adatom is wntten as
Vif(+ 2 f(
=0 (A.Il)
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1337
Now, because of its simplicity, a Lennard-Jones interaction will be assumed:
V(r)=
l§(~° ~~
(~°~j
(A.12)r r
but the resulting value for the dipole moment is expected to be typical also for other, some
realistic potentials.If the nnn-force is supposed to be unchanged at the surface
fi=
f2, iA.13)
the following relation between f) and fi can be established
fÎ=
)fi lA.14)
Accordingly, the diagonal elements of the dipole tensor (1) are easily calculated:
lf[_3vS~> 7 f2jA.isa)~xx
a 2 2
~2Via
~zz ~ ~Î~~ÎÎ Î
~~ ~~~~
The interatomic distance a results nearly equal to a m2~/6ro, and the force f2 can be deter-
mined
~~Î
~d~
Î'~~'~~~
Since the cohesion energy per atom Wcah nearly equals fi Via) (, it results
a~(mxx(= a
~fi
'~
~Vo ~'0.l7Wcah IA-lia)
2Vi 54
a~(mzz(=
a~f2
CÎ
~C~ 0.07Wcah (A.17b)
If the distance between the adatom and the substrate is allowed to change and a numerical
calculation is performed, the above results are nearly unmodified. Also, a calculation on a
tridimensional bcc lattice gives values of the same order of magnitude: a~(mxx( ci 0.l6Wcahand a~(mzz( ci 0.24Wcah.
Appendix B
In the following Appendix we will show the details of the calculation in order to obtain the
equations discussed in Section 6. The organization of the Appendices is the followmg: in Ap-pendix B we obtain the general expression for ôAn (Eq. (22)); in Appendix C we calculate
exphcitly the different quantities appearing môAn, while Appendix D is devoted to the calcu-
lation of some summations; finally, in Appendix E we give R(K) (Eq. (39)) and the stabilitycondition m the different limits.
Let us start deriving An from equation (14), by evaluating the adatom density pn(x) for
~ =
~"and tl~en elimmating pn
~"and pn
-~"by means of (16a) and (16b). Tl~e
2 2 2
1338 JOURNAL DE PHYSIQUE I N°10
result is:
jt~ [gePUnl-%) ePUnl%)j +D' [p~ePUnl-+) pjePUn16)] Fj' fi
yePUn<y)dy~ ~ ~ ~it
" ePUn16)+ jePUnl-%)
+ § j f epu~(~)dy" -it
(B.l)and after substitution of p) tl~rough equations (8), we have
~
Î~" (Î~'~~~~~'~
~~~~~~~j ~ ~~f~~~~~~~ ~~~~Î
~Î'f~iÎ~~~~~~~~iÎ
~ ~~~~l+~~ &~~~~l~+~ ~ 5 fly ~~~"~~~dY
~~ ~~
In the hypothesis that the elastic energies are much smaller than kBT=
~, at the first orderfl
in fl we obtain:
Fj J-i +&& g~~j_ ç)_~~jç)~
~
~ " ~ ~
~ ~"à~i~$
~Î'PO(~n ~n+1) W fÎ~ ilUnlil)dit~ tn§+1+$
i~ (~~) (~~" (~~Î~ ~ ~n (~~ ~
~ fÎ@"(iÎ)d~j
~, ~, ~(B.3)
(~"~~i~W)
We can now calculate the variation ôAn, resulting from a small sinusoidal deformation of
tl~e regular array of steps (see Eqs. (19-21)). Tl~e result is tl~e following:
~~~ ~~ÎÎ~ÎÎ +~i~~
Î (~~) ~ Îà (~~Î 1~ ~Î (ÎÎ~j
~~+~, ~, nim +1+ p
i~ (~~) ~ ~Î~ (~~Î Î~ ~Î (~~j
Ù f~( iÎ~Î(iÎ)~~D'
~~ n(fD' ~ ~ D'j aD
aD D"
~/ (~~~" ÎÎ~ ~~" (~~j ~~ flfiÎ~"~Î~~~iÎ
~ i§ +1+ $
1 Î~' ~) Î~'~Î~~ ~ ~Î
~~ ~~ f~) ~Î(~)~ilj
~~~ n(1§ +1+ $)
~/ ~~~) (~~~" (~ÎÎ~ ~ ~~" (~Î~ ~
~~ fÎ~ n(Y)dYj
(Éà +1+$)~
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1339
+ 2
~ ~~ ~ ~~ ~~ ~~ ~ ~~ ~~ ~~
~~~ ~~~~~~~~ ~~ôin (BA)
ji D' ~ i ~ D' aDaD D"
Tl~e first term is tl~e only one taken into account by Srolovitz [3] and by Spencer et ai. [16].Trie next two terms are finite at fl
=0 for a finite Schwoebel effect, stabilizing the growtl~
at high temperatures. Anlong the last terms, only a few have been considered by Duport et
ai. [iii.Using the parity of the function Uj(x)
=-U((-x), introducing the quantity L defined in
Section 3 and after a rearrangement of the different terms, equation (22) is found
ôAn= -flpoD~~"~
~~"~~ ~ ~ ~~ ~f~ ôin~~ ~~
~
Uj (~) ô£n+ L 2 ii + L) 2 (1+ L) 2
j~
+flF~~' ~~"~~~ôin +
~~(L 1)
~~ ~~~~ U( ôinii + L) 2 (1+ L) 2
pF1lflôUn 1-+) IL fl) ôUn 16)] 26J$ vuniy)dy
~2 1+ L
préj~aD ~j
flôUn 1-+) + IL fl) ôUn 16) + ôJÎ~ Univ)du~~
2 D" (1+L)~
Appendix C
Computation of trie Integrals Involving yU and U
The potential U~(x) (see Eq. (3)) has been defined as:
U~(x)=
Ù£ l~~ ~ ~
(3)Î~
£~~~ £~+p + x y £~~~ in-p x g
It follows that,/~ ~
yUn(y)dy=
Ù 2Io +f(Ik
+ I-k) (C.l)~~t+~
k=1
where a cut-off q has been introduced. We put q = a in ID and q =0 otherwise, that is to say
~/ ~ l/dl/ / ~ l/dl/
~-b L(=o Én+p Ît + v -% L(=o Én+p h
v
for k # 0 and
j/ ~
~ YdY / ~~ YdY
o i 1-%+~ Î+Y -%+~ Î~Y
Ail the integrals have the general form
/_~_ i~ ~ ~~~
=in vo In
~° ~
%+~ vo + y vo @ + q
1340 JOURNAL DE PHYSIQUE I N°10
For vo =
~"and q = a we obtain
2
~ ~in
(in~" à ~~ i
while for vo "
~ in+p~"
and q =0 we have
~2
P=
Ik in in+Pil
in lllii Iii[
The linearized variations are given by
ôlo"
("~(" In (~) (C.2a)
a
and,fork#0
ôlk=
ôin (j~ ôin+p ~) ln (~)
(k + ()~~[°~)~~ ~"~/~"~~
~=o
2(kÎ1)
~~ ~ Î~~" ~
( ~~"~~ Îk ~ 2(kÎ1)
~~~ Î
~~'~~~
Using the explicit expressions of ôlkwe derive
ô/ ~ yUn(y)dy
=26Io +
£(ôlk+ ôI-k)
~~~~ Îi
=
ôin-ôinIn(~)+£(Î-In~~~)ôin£l
Î~ k+1 k
~
( ~~~~"~~~ ~"~~~j k
~ 2(kÎ
1)~~
~ Î~~'~~
Defining the quantity
~~( ~k
~ 2(kÎ1) ~~~ Î ~
~~~~
we can finally wnte
à l'~ ~
yUn(y)dy=
U~i
+ ln )j ôin +
f~k[ôin+k + ôin-kÎ (23)
@+~ 2 a~=i
Let us now proceed with calculating the integral of U.
l'~ ~
Un (l/)dy#
Ùf /~ ~
~ ~ ~ ~
dy%+~
k=0Î~+~ £p=0 ~"+P ~ Î £p=0 ~"~P ~ Î
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1341
Since the k=
0 term does not contribute (the corresponding integrand is an odd function)and for the other the cut-oOE q can be omitted, the integral takes the form
l'~~~ ~~~~~~~ ~
1
~'~£~=0
n+~+ ~Î £~=0
"~~
~Î
~~
W j~k~
f~~~ ~jk~
f~_~=
U £ In )" In )"k=1
~lp=1 ~n+P £p=1 ~n-p
and its variation can be wntten
/_~~ ~"~~~~~ ~Îk ÎÎÎÎ~
~~ i~~~~~~ Îk ÎÎÎi ~~
~~ ÎÎ~ ~
At the end, we have
à/ ~
~
Un(y)dy=
) ~j (~ j) (ôin+k ôin-k) (25)~~
+~
Î~k +
Computation of U on trie Steps
From the definition (3) it follows that
~"Il
~l ~
o
lutin+p iii1-p+
)i~.~~
and~
Un11
+~l
-
U
~
~j~1[~~+ ~
~j[i~~) iC.5)
For an array of terraces of equal length, we have
~o (É) _~o (_É) ~If f 1
~l 1
" 2 " 2 (k + 1)£ ki £ a
=
~(29)
In equation (C.4),we can omit q because in the only term (k
=0) where the cutooE enters,
it appears as a constant and therefore it is needless for tl~e variation. Thus we obtain
~~in
~/ ôÉn ~f ~ÎÎ=o ôÉn+p ~ÎÎ=i ôÉn-p
" 2 12~_~
(k + 1)212 k212
Tl~e first quantity on tl~e rigl~t l~and side is exactly tl~e k=
0 term of tl~e summation, so that
we can rewrite
ôu (~") =_Ù
f ~Î~"Î ~~"+P Î=1~~"~P)" ~ ~2f2 ~2f2
k=1
1342 JOURNAL DE PHYSIQUE I N°10
Introducing tl~e quantities~
Ck=
~ (27)
~~~P
the previous relation becomes
ôUn (~" =
jj £ Ch (ôin-k ôin+k-1) (26a)2 Î~
A similar calculation for equation (C.5) produces
ôUn (-~" =
-j £ Ch (à/n+k ôin-k+1) (26b)2
Î~
Computation of ô~n ô~n+i
Starting from equation (5)
~°Il~n"é~ j~)jk ~k
k=o p=o n+p p=o n-1-p
we find
and therefore
ô~n ô~n+1 j2 ~~ ~~~"~~ ~~~~
~~~~~~ ~ ~~~~~ ~~ ~~~~
Integrals Involving U: Uniform Case
From the definition (3) of U, we immediately recover that
Il~Unlv)du
=° 13)
-1
On the other hand
/~ ~~~~~~~~ ~ Î~ll
+ + QÉÎ +
Él/
~~
~f Il ~ Î+QÉ Î+QÉ~
~=o-j Y+Î+QÉ Î+QÉ-Y ~
= Ù12i-iIn~+f(2i-2(~+qi)In~~~j)a
~=i
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1343
and finally
/ yU((y)dy=
2Ùi il In~ j
(30)) a
where
cx =
£(k + ln (1+ )) ii (31)
Î~2
The value of o will be given in the following paragraph (Eq. (D.9) ), but it is already possibleto check the convergence of the summation, since for k > 1,
~~
~~~
~~
~12k2
Appendix D
Calculation of ~p and cx
~p has been defined in equation (24) as
'fP
É~~ ~ 2(kÎ1) ~~
~ ~Ù ~~ ~~
~ Îp 2(sÎ1) ~~'~~
k=p q=P
The connection with the Euler constant ~ is apparent, in that
~i " ~~
#0.07721 (D.2a)
~2 = 'ii In 2 + +=
0.02035 (D.2b)
and so on.
We are interested in the behaviour of ~p for large p. Let us define the function R(p, s) such
tl~at
~p =lim R(p, s). (D.3)
s-m
Tl~en,
~ ~~~Î~ ~Î~Îq 2sÎ2 ~~~Î ~Î~~p 2sÎ2
f Il dz f 1 Ii dz j° dz 1
~~-jq+~
q~o
p+~~ _is+1+~~2p2s+2
We can develop tl~e integrands for small (~) and (~p s
5 ' 2 5 ~ 2
R CÎ~ /~ + ~~ dz ~
+ /~ ~j + ~~ dz
~~~ ~~-)
~~~ ~~ 0 ~ ~
~ /~i ~~
~~~
p2s 2
1344 JOURNAL DE PHYSIQUE I N°10
R ce
f /~ ~~dx+ /~ (-~j+~~)dx+ /~ (-~j+~~)dx+)-~~
~=~~~-j o -j s s s s +
~ "
~j~ 1q3 12~
12~
12" /~
113 12~
12~
12
~(~.~~
F~lla~~Y' ~'~~ ~~~
~P ~ @#
~~ ~~ ~~~ ~~~ ~~~~ ~~~ ~~~mation £$i ~P
(~~ ~ ((~21Q+1)~à~~~Ù~
~ s~ffl((~21Q+1)~à~~~Ù~
which can be rewritten, by interchanging p and q:
~~P = ~'fllj
~ ~ ~jq+ i)
+ fi 'n ~
q
li ~i ~i ~
- ~'iu~
((i ~iq1i~ in ~ i iD.5a)
~~ÎJ~ ~
Î
~~~
~~ q~~~ ~~ ~~~
q=1 q=1
If we rewrite the last term on the right as the logarithm of a product, we easily obtain
~~~ ~~ô
~ ~Î
~~ ~~ Î~~~
P"1 q#Î
If the Stirling formula is used
SI=
ris + 1) ci(s +1)~+)e~~~~@, (D.6)
we have
'~~
~Îl~ Îj~qll ~~~Î~~~~i~
=lim lj In(s +1) ~j l + ln(2x)5~C°
2
Î~Q+1 2
=
ln(2x)~i (j ID.ia)
or equivalently~~j
~p + ~i #
In(/ù)=
0.169 (D.7b)
p=1~
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1345
The coefficient cx can now be deduced from (D.5a), since
~~Îf$~ ~
Q~~~
q=
~~P + ~Îi$ ~
~jq + i)
~i ~ li ~i
Then the summation (31) becomes
° "im
~
~~" Î~~~
~ ~~~
lj =ji~
f 1 q + 1
~~"q=1
2~~
q 2j j~ ~
and from (D.l) for p =
°Î~ j~ ~~ ~~'~~
p=
Finally
°
~~~~~~ ~~à ~~~~~ ~~~~
and~
)m +£
~p = cx(D.10)
p=1
Appendix E
We have seen in equation (37) that the interesting quantity, in order to establish if the growthis stable or unstable, is given by R(K), defined in equation (33) as the part of ôAn in phasewith the shift ôin
Let us call (ôAn)a,b,c... the dioEerent terms of ôAn and Ra,b,c... the corresponding quantitiesin R(K). Then, the second term in (22) gives immediately the contribution
~/ ~ ~ m)Rb
"
$' (E.la)2 (1+ L)
The first terril in (22), which is equal to
(ôAn)a=
-flpoD~~" ~~"~~
using the equation (28) becomes
(ôAn)a= <(~°
( j fCk (ôin+k-i ôin-k ôin+k + ôin+i-k)
+k=1
and R will gain the quantity
Ra= ê(~°
( j £ Ck (2 cos(K(k 1)) 2 cas(Kk))+
~
~~ÎÎÎ ~~Î~ ~~~~ ~~~ ~~~ÎÎ
~~ ~~'~~~
JOURNAL DB FHYSIQUBI-T. s, N° la OCmBER 1995 53
1346 JOURNAL DE PHYSIQUE I N°10
Let us now consider the terms in ôAn which depend on trie integral of yU(y):
fÎ< Yul lv)dy à j)~ yu~jy)dy(ôAn id
=flF 2
~ôin flF 2
ii + L) 1 + L
or, using equations (23) and (30)
(ôAn)d"
2ÙflFt~ ~~~~~( °ôtnlé + L)
ÙfIF ~~~ ~ ~~ ~ ~~" ~ ~~~ ~~~~~"~~ ~ ~~"~~~
which can be rewritten as
iôAn)à-
uàf liiiii~ ôin +i'i Ill ôin i~iiii~~n ~~i
~kii~ii + ~nk~l
(E.lc)The corresponding quantity Rd results
Rd=
ÙfIF ~~~~~
ôta +~~~
~~Î ôta ~~~ ~ ~~ ~~" ~ ~~~ ~~ ~°~~~~~(E.ld)
ii + L) ii + L) Î + L t + L
Let us now tum to the term in (22) involving trie integral of U(y)
~
~~~"~~ ~Î~ ~~Î~ ~)~~ ~ÎÎ~~~~
If equation (25) is used, (ôAn je becomes
(ôAn)e=
-~) (~)~ L)Ù~~~
~~~ ~ ~~~~ ~~" ~~
lé + L)
Since (ôta+k ôt~-k) ocsin(Kn ii, trie corresponding contribution to R is zero
Re=
0. (E.le)
The other terms in (22) can be rewritten in trie forai:
(ôAn)c=
~~ ~~
~
U( (~) ôt~ +~~
IL t)~~
~~~~U( (~) ôt~
2 (t+L) 2 2 (t+L) 2
flft aD ôUn )) flft(aD
~ D' j ôUn ))2 D' t + L 2 D' D" (t + L)~
flft aD ôUn (- ))~
flft (aDj~
(D"j ôUn (- ))
2 D" 1 + L 2 D" D' (1 + Lj2
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1347
which can be manipulated using (26a,26b,29) up to obtain
i~An)c=
flF~ ~~~ + 21L ii ILJ ig~ j
~ ii + L)~ôi~
+ i + ~ ~
~~' l~ + L)~ P( ~k lôÉn-k ôin+~_~
~ ÎÎ~ ~~iÎ L)Î"Î ( ~~ ~~~"~~~~ ~~"~~~
The insertion of (19) gives for (ôAn)c:
Ù £L2 + 2(L 1) (Lfl fl j
~~~"~~ ~~a
ii + L)3~°~~~~ ~~
~Î~ ÎiÎ~ÎÎ21 ~~ ~~~ Î~~ Î) Î
~~~ ~~ Î)
+~~~ ~~ ~ ~ ~~~ f
Ch sinK
(n + Ii sin K (kD" Ii + L)~ É~~~~
2 2
and for Rc
Ù iL2 + 2(L £) (Lfl (fl) j
~ ~~a
ii + L)3
~~~~ ~ ~iÎ
L)2 ~~ ~ (~~ ~~~
Î~~~ ~
~~~'~~~
All the contributions Ra,.,
Rd can be sunlmed up and the result put in the form:
R(K)=
Ro Rif
~k cos(Kk) + R2 sin (~ fCh sin
K(k (39)
~=i2
~=i
2
where Ro, Ri, R2 have already been given in equations (40).The two double sums in (39) can be simplified by the following procedure. Let us consider
the quantity
m
Sk=
~ f(Pl
p=k
where, in our case f(p)=
for Sk=
Ch and f(p)=
+ ln~ ~
~] for Sk" ~k.
P 2P 21P +1) PThen, we have to calculate the real and imaginary parts of F(K):
ce
~jjçj s ~z(Kk-~)~jk
k=1
1348 JOURNAL DE PHYSIQUE I N°10
ce cejjj~ fj )~~(Kk-~)Îl
k=lp=k
m p
=
£flPl£ei<Kk-~)p=1 k=1
ce siu (éÉ)j~ f j ~ ~[(p+1) § -~j
_~
~sin ~
For exanlple, for Sk"
Ck and #=
§we obtain
~
~~ "~ ~~~Î Î~I~ ~
~~'~~
and equation (39) can be rewritten as
n~ n~~j~2 j)
R(K)=
Ro Ri ~~k cas(Kk) + R2 ~
~
~ (E.3)
k=1 p=1~
As mentioned below (40), the stability condition is given by R(x) < 0
~ ~
~~~~ ~° ~~[~~~~~~~
+ ~~ [12k i)2 < ° ~~~~
y~2The second sum is equal to [41]. For trie first one, some manipulations are necessary:
8
m ~
~ ~~~~~"
i
~iP~~~n~ i~l~i -
~
fi~p i~
~~Ùl~~l.3.5.7..(2s-1)~ 2~r
/Ù ~~ ~ÎÎÎÎÎ~~ j/Ù ~~ ~ÎÎÎÎÎ~~ ~~~~~Î
=lim
2sIn 2 + In ~~~~~~ n(2s)j ~
5-ce (2s)2~+1 2 2
= (In)
~) =-0.0628 (El)
The stability condition (41) can be written as
Ro + Ri (~ ln1) +ÎR2
< 0, (42)
that is to say
F Dj~ (ô)~l cx
LIn ())2
~D" ii + L)2
~ ~~~~~ii + L)2
~ ~~~ii + L)2
N°10 GROWTH INSTABILITIES INDUCED BY ELASTICITY 1349
iL2 + 2(L -1) (Lfl (fl)~j1 ~~~~~
ail + L)3~~~i
+ L~~
2
~~~~~
~~~~~~Î /Î)
~ ~~Î~ ~~ ~
~ÎÎ~)~~~~~
~ ~ ~~'~~
If both 1, L »$
the previous formula becomes
Îii Î~)2 ~ ~~~~~ (Î+ 12 ~ ~~~ ~Î )~
~ ~~~ a(ÎÎ~)3
~~~~i
ÎL
~~ ~ ~~~ ~~~~~~2 /Îi
~~~Î~
ii /L)2 ~ ~ ~~'~~
In the limit £ » L, (E.6) reads
((~ + 2ÙflF °
+ fIFÙ ~~(j + fIFÙ
$
-fIFÙ ln 1+ XED
~ ~ (aôa)~ ~(° +fIFÙ~ (
< 0 (E.7)
Since the last term is negligible with respect to the first one, and it results 2(1 a) in~
=
22 in 2, equation (43) is recovered.
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