Growth Instabilities Induced by Elasticity in a Vicinal Surface

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>

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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.


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


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.


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


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


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


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)


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)


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~


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


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]).


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 ~~~~


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:


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


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).


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.


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)


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


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+$)~


+ 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


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


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


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


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


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~


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


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


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


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


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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