Growth Instabilities Induced by Elasticity in a Vicinal Surface

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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 Elasticity in a Vic- inal Surface. Journal de Physique I, EDP Sciences, 1995, 5 (10), pp.1317-1350. �10.1051/jp1:1995200�.

�jpa-00247139�

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 Fondamentale _sur la Matière Condensée,

SPSMS/MDN,

38054 Grenoble Cedex 9, France

(~) CEA, Département de Recherche Fondamentale _sur 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 a} vicinal substrate. Three main eifects _are taken into account: the Schwoebel barrier and the step-step and the step-adatom elastic interactions. The first _one 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-step

interaction _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 stability phase diagram, taking into account aise _a possible 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

capillarity

which 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

long wavelength

[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

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

deep

cracklike _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} transport

and 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,

layer by 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 relaxation} m

single-crystal

films of

polymerized polydiacetilene,

_{m epitaxy} with their _monomer substrate.

Polymerization

_is obtained

through

irradiation 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 showing

how 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

pseudomorphic

films.

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 _m the 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 of _a

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}

Q~ ~

, i ,

j~

~ ~ ~

«f~

D

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 adatom

on 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

underlying

mechanism 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

explain

how such _a mechanism works.

A vicinal surface does not grow

layer by layer,

but

through step-flow,

with all the steps

moving

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-flow growth (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

larger

and

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 _m the

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, ^and _we 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 analysis

of 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

study

of the

limiting

_case of low

flux,

this will allow _{us to compare our} results with those of

Spencer

et ai. [16].

Finally

in Section 8 _we will discuss the

hypotheses underlying

_our

stability analysis

and 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,

symmetry

imposes

_{: mxx = m~~}

= m.

Having

introduced the atomic _area

a~,

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

~~ ~~~~~~~ '~~,

_where

a is the Poisson coefficient. The part ^of this

r

expression, which is

proportional

to mzz is often

ignored [18, 21]. Quadrupolar

and

higher

order 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

large

terrace 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 the _case of _a tenace, ^{where the} forces

are localized _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

infinitely deep,

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 relative difference

(or misfit)

between the lattice _constant of the substrate

(asb)

anda 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

destabilizing

character 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 of _a stabilizing

(a)

and

destabilizing (b)

step-adatom interaction, _are shown.

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, where _{x 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

straight

steps 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)

₌

^Û f 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-p

P"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 _ci

a§.

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-step interaction.

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 ,

à~3 Î~

l

,

i

i

i

X~

Fig. 4. As _m Figure 3a, but _Dow also the destabilizing step-step interaction is shown.

induded in their treatment. On the contrary, ⁱⁿ trie present

work,

atom detachment from steps will 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

(see Fig. 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 argument

is 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 step from 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$ la

_is related to the

probability

of attachment

D'la

of _an adatom

(supposed

to be close to the

step) by

the 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

independent

of

elasticity, plus

a correction. This correction is

equal

to

(5)

for the

incorporated

atom, and to U

(~")

₂

^{(see Eq. (3))}

^{for the atom close to the step.}

Accordingly,

_{we can} write:

vi

_{vo exp}

à lu iii ⁿⁱ _i~a)

vn

_{vo exp}

à lu ii 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 An

Trie 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

2

that 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 step

equilibrium

adatom

density:

Jn

))

=

~'

pn

(~") ^{$j (lsa)}

tl 2

Jn

(~ ))

"

~~" pn

(~ ))

vii ^(lsb)

tl

Jointly

with

(12),

_we have

~' ~

i

Pn (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 stationary solution

in(t) =1,

_so that _{we can} consider _a

single

Fourier mode:

Énjt)

= +

(jt) cosjKn #jt)j

_e

ii

₊

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

through

the 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 form}

a

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

aD

flôUn 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

quantifies appearing

in

(22)

_are written below and derived in trie

Appendix:

à

/~ _yUn(y)dy

=

Ù 1

+ in

ôtn

₊

f

~k

(tn+k

+

in-k] (23)

%

^{2 tl}

~~~

with

~

~~

( ^~k

_~

2(k

Î

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~

with

m

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 #)

₊

(j _{sin(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)(,

^as

a 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

noting

that 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 Ri f

~k

cas(Kk)

+ R2 sm

(~) f

Ch

sin

K

(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 i Lt ₊ 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.

~2

The

stability

condition

(12)

of reference

[iii

is

only

recovered iii _» and if the term of

a

step-step interaction is

neglected.

If _so,

(43)

is rewritten _as

>

fl l~~

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

independent

of

F,

and the Schwoebel term of _zero order in

fl.

Then the

stability

condition takes the form

~ , ~ Ft~L~

2xEDfi (~~~) fil'°

~

(46)

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

t

relation _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

prefactor il 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

instability

arises 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), coming

from 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

~ ~

~r

k=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

always

negligible.

In the two

opposite l1nlits,

_we ^obtain: