Different Regimes in the Ehrlich-Schwoebel Instability

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Different Regimes in the Ehrlich-Schwoebel Instability

Paolo Politi

To cite this version:

Paolo Politi. Different Regimes in the Ehrlich-Schwoebel Instability. Journal de Physique I, EDP

Sciences, 1997, 7 (6), pp.797-806. �10.1051/jp1:1997193�. �jpa-00247363�

J.

Phj.s.

I IYance 7

(1997)

797-806 JUNE

1997,

PAGE 797

Dilllerent Regimes in the Ehrlich-Schwoebel Instability

Paolo Politi

(*)

CEA-Grenoble, Ddpartement

de Recherche Fondamentale _sur la MatiAre

Condens4e, SPMM/MP,

38054 Grenoble Cedex 9, France

Centre de Recherche _sur les Trks Basses Tempdratures,

CNRS,

BP 166, 38042 Grenoble Cedex 9, France

(Received

20 December 1996,

accepted

20

February 1997)

PACS.81.10.Aj Theory

and models of

crystal growth; physics

of

crystal growth,

crystal

morphology

and orientation

PACS.68.35.Fx

Diffusion;

interface formation

PACS.05.40.+j

Fluctuation

phenomena,

random processes, and Brownian motion

Abstract. A one-dimensional

high-symmetry growing

^surface in presence of

step-edge

bar-

riers is studied

numerically

and

analytically, through

_a discrete

/continuous

model which

neglects

thermal detachment from steps. The

morphology

of the film _at different times

and/or

different sizes of the

sample

is

analyzed

in the overall range of

possible step-edge

barriers: for _a small barrier, we have _a strong

up-down

asymmetry ^of the

interface,

and _a

coarsening

_process

with _an increasing size of mounds takes

place;

at

high

barriers _no

coarsening

exists, and for infinite barriers the

up-down

symmetry is

asymptotically

recovered. The transition between the two

regimes

_occurs when the so-called Schwoebel

length

is of order of the diffusion

length.

1. Introduction

The stable

growth

mode of _a

high-symmetry

surface in Molecular Beam

Epitaxy

is

layer-by- layer growth: newly

fallen adatoms diffuse _on the surface till

they

meet _a

"trap",

which may be another

adatom,

or the

step

^of a

growing

island.

Afterwards,

islands coalesce

by leading

to the

completion

of the

layer,

and the _process starts

again. Nevertheless,

adatoms

approaching

a

step

^{from above} may be hindered from

descending it,

because of

step-edge

barriers

(Ehrlich-

Schwoebel effect

ill).

This _{causes on one} side _a

higher

adatom

density

and therefore _a

higher probability

of

nucleation,

and _on the other side _an

up-hill

current [2]

(also

called diffusion

bias)

both effects _determine the formation of mounds.

If the main relaxation mechanism of the surface is surface diffusion and vacancies

/overhangs

can be

neglected,

the surface evolves

by preserving

the volume _so that the

following Langevin- type equation

can be written for the local

height z(x, t):

btz

=

-b~j

+ F ₊ ~.

(l.1)

(*)Present

address:

Dipartimento

di

Fisica,

Universith di

Firenze, L-go

E-Fermi 2, 50125

Firenze, Italy (e-mail: politipflfi.infn.it)

©

Les

#ditions

_de

Physique

1997

o o o o o F

~ D' TOP

~°

vIcwAL BOTTOM

Fig.

1. Profile of the one-dimensional

high

symmetry

growing

surface. F is the flux of

particles (per

unit time and unit

surface). Top,

vicinal and bottom terraces are indicated, D and D'

are the

sticking coefficients,

from above

(D')

and below

(D).

Their asymmetry defines the Schwoebel

length (see

the

text),

Here and

throughout

the paper, the lattice constant is put

equal

to _one. F is the flux of atoms:

it will be

incorporated

in _z

(z

~ z

Ft). Finally,

_~ is the

noise,

which may have different

origins (shot noise,

diffusion

noise,

nucleation

noise).

A most used

expression

for

j

is

[3-5]:

j

₌

j~(mj

+

Km"jxj

_{m -}

o~zjx,tj. ji.2j

The first term will be called Schwoebel _current and it is due to the above mentioned diffusion bias. It

depends

_on the Schwoebel

length,

which is _{a measure} of the asymmetry ^{in the}

sticking

coefficients to _a step

(see Fig. Ii,

from above

(D')

and below

(D):

is

=

~) _{l) (1.3)}

Its actual form may have [4] or not

[5,

6] zeros at finite

slopes

_m

= +m*. In the former _case, the

growth equation (I.I)

is

equivalent

to the Cahn-Hilliard

equation,

where _m

plays

the role of order

parameter:

o~m

=

oj(£)+o~~ (1.41

F =

/

_dx

(((b~mi~

⁺

^{(mij U'(mi}

⁼

^{-Js(mi (isi}

So,

the surface

undergoes

a

spinodal decomposition

_process, with the formation of

regions

of size

L,

where the

slope

is

alternatively

in the two minima _m

= +m* of the

potential U(m).

These

regions

are

separated by

domain walls and their

typical

size increases in time

according

to _a power law: L

+~ t". For _a conserved order

parameter,

n =

1/3

_[7] if ~ is due ^to

shot-noise,

while L increases

logarithmically

if ~ is absent [8]. ^In 2+1

dimensions,

as

pointed

out

by Siegert

and Plischke

[4],

the

growth equation

has _{no exact}

counterpart

^{in the} Cahn-Hilliard

equation

(one

_reason is that _m satisfies the extra-condition: T7 Am ₌

0),

but the evolution is very

similar and numerical solutions of the

corresponding Langevin equation give

_[4] n =

1/4.

The

previous

scenario is

expected

to be valid if _a

strong

term of the form

j

₌

Km"(x)

is

present

in the current, also at the

largest slopes

allowed

by

the

dynamics.

Such term _was introduced

by

Mullins [9] ^to

explain

the relaxation of _a

grooved

surface via surface diffusion. _Its

microscopic origin

is

thermal

detachment from

steps,

and its form

simply

translates _a current

deriving

from _a chemical

potential proportional

to the surface curvature.

In the present

article,

_we will

study

the

opposite

^limit of _a far from

equilibrium

system, when thermal detachment _can be

neglected (as

found

experimentally

in Fe

/Fe _[10]

at not too

high temperatures),

and other terms appear ⁱⁿ the current.

N°6 EHRLICH-SCHWOEBEL INSTABILITY: DIFFERENT REGIMES 799

A first

important

^{issue is} the

following:

the current

11.2) changes sign

if _{z ~ -z,}

meaning

that the

resulting morphology

is

up-down symmetrical.

This

property

^is not

always

satis-

fied,

neither in

experiments ill,12],

_nor in simulations

[6,13,14]. So,

_a term of the form

jc

₌

b~A(m~),

whose

justification

_was

given

in reference

[6],

^{will be added: it has the}

simplest

form which breaks the

up-down symmetry, by keeping

the reflection

symmetry (x

~

-xi.

Even if thermal detachment is

neglected,

other mechanisms

[6,15]

_may contribute to _a Mullins-like term in the current: for

example,

random nucleation and diffusion noise. In the limit of

large terraces,

random nucleation

prevails [15],

^and

only

this _one will be retained.

However,

since nucleations

depend

_on the

length

of the terrace

(see below),

this _means that K is indeed _a function of the

slope:

K

=

Kim),

which vanishes if nucleation events are very

rare. This is _one of the main _reasons

why

the

resulting morphologies

in _a far from

equilibrium

system

differ _a lot from the

quasi equilibrium

_case.

2. The Model

The

discrete/continuous

model which will be used

throughout

the _{paper was} introduced

by

§lkinani

^{and Villain}

[16].

^{In this}

model,

a one-dimensional

high-symmetry growing

surface

"contains"

steps,

but not adatoms, which _are treated in _a continuous way via _a diffusion

equation,

whose

boundary

conditions at

steps give step

velocities.

The diffusion

equation

is

integrated by

_an

expression giving

the

probability

of nucleation

Pit)

_per unit time _{on a}

given

terrace of width t. Let _us

briefly

recall its form.

The

probability Pit)

_may be determined _as the rate

(Ff~)

of

incoming

atoms _{on a} terrace of dimension d

id

=

1, 2)

times the

probability

_p for each adatom _to meet another _one

(before sticking

to a

step).

_p is

approximately given by

the number of distinct sites

visited'by

the

adatom (+~

f~,

if

logarithmic

corrections _are

neglected)

times the

(average) density (pi

of adatoms.

Finally:

P(f)

_m

^{Ff~ f~} (pi. (2.I)

We want to stress that

(pi depends

_on

is

and _on the-

type

of terrace.

By characterizing

_a

terrace

according

to the number of

ascending

and

descending steps (see Fig. Ii,

we will

speak

of

top

terraces

(no ascending steps),

vicinal terraces

(both types

of

steps)

and bottom terraces

(no descending steps).

Adatom

density

is not influenced

by step-edge

barriers on a bottom terrace, whilst _{on a}

top

terrace

(pi

_goes to

infinity

when

is

~ cc, and this

strongly

^increases

the

probability

of nucleation

P(f) (see

the

Appendix).

For _a weak Schwoebel effect _{we can}

neglect

the

dependence

of

(pi

_on

is

and the solution of the diffusion

equation gives (pi

_rs

^Ff~ ID,

_so that

~n2~2(d+ij

P(f)

_m

(2.2)

We define nucleation

length

the

typical

linear size of _a terrace

just

^before a nucleation process takes

place,

_or in other words _as the

typical

size of _a terrace for which the

probability

of nucleation

during

the

deposition

of _one

layer (t

= I

IF)

is of order _one:

P(f) IF

_m I. For

a bottom

terrace,

or

equivalently

in the

submonolayer regime,

we will _use

alternatively

the _{more common} term of diffusion

length (fd). So,

from

equation (2.2)

_we obtain

D

+

id

"

~-) (2.3)

F

This well-known result

[17]

^{is valid in the}

hypothesis

that the nucleation of two adatoms is _an irreversible _process

(not

_too

high temperature)

and that islands _are not fractal

(not

too

low

temperature).

More details and the

explicit expressions

for the nucleation

lengths

^of _a

top

(f)),

_a bottom

iii

_e

id)

and _a vicinal

(f$)

_{terrace are}

_given

in

Appendix.

In this paper, the

following general expression

^for the surface current will be used

[18]:

" ~~ ~ ~~ ~ ~~ ^~

211

+

ts/td ~~~/)(i

+

imifd) -b~ ~j~

()~~~j)

⁺

K(m)m"(xi _12.41

It is

noteworthy

that

jc

does _not

depend

_on

is (see

also Refs.

[19,20]);

on the other

side,

the actual

dependence

of K _on

is

and _m is not

exactly

known. This

equation

will be

exploited

in the

following

sections _to accompany the numerical simulations of the model described above.

3. Aim of the

Paper

The aim of the

present

work is to

study

the different

morphologies

of _a I+I dimensional

growing

surface which is

strongly

out of

equilibrium.

In this

limit,

it has been shown in reference [6]

that

deep

_{crevaces appear} at

sufficiently long

time

/length

scales.

The scenario

given

in reference [6] ^is

representative

of _a low

is regime ifs

_<

id

In this _case two critical

wavelengths exist,

_a lower _one

(~)~~)

and _{an upper one}

(~[~P).

^For scales smaller than ~(~~ ^the flat surface is

stable,

but after _a

typical

time t* mounds

develop

with _a size L

(=

_~(~~) which increases in time via _a

coarsening

_process. These mounds _are not

(up-down)

symmetric

and _are

separated by angular points.

When L

=

~[~P coarsening stops

and

deep

_crevaces form in between. It is

noteworthy

that the existence of the

coarsening

_{process may} be

explained

in _terms of the

stability properties

of

stationary mounds,

which exist in the interval ~(~~ < L _<

~[~P.

^mounds are stable with respect ^to

height

fluctuations

(which

_means that

they keep

their form

during coarsening)

and unstable with respect to width fluctuations

(which

_means that _a

large

mound "eats" _a smaller

one).

When L ~ ~(~~ the

height

of mounds goes ^to zero,

meaning

that the flat surface is sta- ble at

sufficiently

small

scales,

while for L >

~[~P

mounds become unstable _even for

height

fluctuations.

The _case of infinite Schwoebel effect has been

studied,

for _a far from

equilibrium

system,

by Krug [19].

^He

asymptotically

finds

up-down symmetric configurations,

which he calls

"wedding

cakes". He also find their

analytical profile,

which is

given by

the inverse of the Error function.

In the

following

_we will

study

the transition between the _two

regimes, by focusing

_on the

following questions: ii

^{is the} scenario found in reference

[19]

^for

is

= cc valid also for

sufficiently large is? iii

how does the transition between the two

regimes

occur?

iii) why

at infinite

is

is the

up-down

_symmetry

asymptotically

recovered?

4. Results and Discussion

In the

following

_we will call the behaviours described in reference _[6]

(is

«

fd)

and in refer-

ence

[19] its

=

cc), respectively

_as

"small-ts"

and

"large-ts" regimes.

^{To understand} how _a transition between them _can occur, let _us focus _on the _upper critical

wavelength.

First of

all,

we will show that

~[~P

is _a

decreasing

function of

is.

If _we

adopt

_a

purely

deterministic

point

of

view, according

to which nucleation takes

place

on a terrace

only

when its width

equals

its nucleation

length,

_we have _a much

simpler

model

N°6 EHRLICH-SCHWOEBEL INSTABILITY: DIFFERENT REGIMES 801

c

d

Fig.

2.

- Typical orphologies

of the

surface

at

ifferent ilues of the Schwoebel length

(is

_{lid =} 0.I (al,1 (b),100 (c), _cc

(d)).

It is

shown the

from the

small is

large

is one.

Vertical and

horizontal units

are _arbitrary. For the morphology

(d), t

of eposited

yers),

which results long to recover

the

_up-down ^ymmetry. (The morphol-

ogy

which is unable to

explain

_~)~~

(whose origin

is

just

random

nucleation),

but which _can

explain qualitatively

well

~[~P

^{In the limit}

(m(

> I

lid,

the current

(2.4)

writes:

~ ~~ ~ ~~

2(1+ (m[is)[m[

^{~ 4m3 ox ~~'~~}

The

equation j

₌ 0 _can be solved if the

"starting" slope

_mo is known. In _a deterministic model _mo is

nothing

but the inverse of the nucleation

length

of _a

top

terrace

(the only

_one

where nucleation takes

place).

This way, we can writ,e down the

expression

of

~[~P.

~

[i) _(is))

^~

i~ (is)

~c~~(det

"

~i ⁺

~

~

(4.2)

where _we have stressed the

is dependence

of

I).

As detailed in

Appendix, I)

_goes to _zero when

is

~ cc, so that

~[~P

^is a

decreasing

function

offs

and

~[~P(cc)

₌ 0.

Anyway,

the

previous

statement may be

misleading,

since the minimal

meaningful length

on a

high symmetry

surface is the diffusion

length id-

In other

words,

the minimal distance between _crevaces is set

by

the nucleations in the very first

layers,

which

give

rise to islands at an average distance

id (see,

for

example,

Ref.

[10]).

Therefore

~(~P

can not be smaller than this distance:

~[~P(is)

>

id

~

is

<

I(~P (4.3)

By using

^the deterministic formula

(4.2)

it is found

I(~P

=

0.lsid

and

I)(I(~P)

=

0.8id.

If

is

_>

_I(~P

the flat surface is

already

unstable at the "minimal" scale

id-

This

morphology (see Fig. 2c)

is rather similar to the _one found

by Krug [19] (see Fig. 2d)

in the _{extreme case}

is

= cc, but while at finite

is

_an

up-down asymmetry

^{is maintained} at all

times,

for

is

= cc

the

symmetry

^is

asymptotically

recovered. At this

regard,

_a first trivial observation is that

I)

keeps

finite for finite

is

and it vanishes

only

for

is

= cc. This

implies

that the

top

of the

profile naturally

differs from the bottom. On the other

hand,

the

comprehension

of the recovered

up-down symmetry

at

large

times for

is

= cc

requires

_{a more}

deep analysis.

Inspection

of

equation (4.I)

shows that both

js

^and

jc keep

^finite for

is

= cc

(indeed, jc

does not

depend

_on

is). Anyway,

for

is

>

I(~P

no

stationary configuration

exists and therefore there is _no need for

js

and

jc

to

compensate.

Let _us evaluate the ratio between the curvature term and the Schwoebel term, ^{in the} surface

current

(the slope

_m will be

supposed

to be

positive)

) 1~2 ~ilts

^~

~l'

~~'~~

For _a

strong

^Schwoebel

effect,

the ratio

(I /mts)

is smaller than _one and

j~

^m'

b~js _btz[~~=~

~

^~ _{2m2 F F} ^~~'~~

So,

the condition

jc

«

is

is satisfied if the rate of

growth (btz)

of the

surface,

due to the Schwoebel current, is much smaller than the _rate

(F)

of

incoming

atoms.

(Let

_us remind that the "real"

height

of the surface is

z(x, ii

+

Ft.)

We _can check the

previous

condition

by using

the solution for

jc

e 0

[19].

^{The evolution}

equation btz

= Fz"

/2(z')~

is

separable

and the differential

equation

_can be solved: the

profile

is

given by

the inverse of the Error function and it is

easily proved

that

is

^" 2Ft ~~'~~

Since the

height z(x, t)

increases _as

/,

_{in the limit t ~}

cc the curvature term

(jc)

is

negligible

with

respect

to the Schwoebel _term

(js).

This _means that the

up-down symmetry

^is

recovered,

but

only asymptotically.

After

having

discussed the

region is

>

I(~P,

let _us turn to the

region is

<

I(~P

^In

general

terms, the lower critical

wavelength

_~(~~ is determined

by

the linear

stability analysis

of the flat surface. In this _case

jc

(+~

b~m~

at small

slopes m)

is irrelevant and

j

has the form:

j

₌

jj(o)

_{m +}

Km"jxj j4.7j

where

j((0)

₌

Fisid/(2(1+ is lid))

The _exact

expression

of K is not

known,

but K

surely

does not increase with

is.

For small

ifs lid),

dimensional

analysis suggests

the form K

=

Kofi(,

Ko being

a numerical constant [6].

The solution of the linear

equation btz

=

-b~j gives

the

following expression

^for ~)~~.

~~ ~

~~fi~

_~~

i ~[

~

ll'

~~'~~

So,

_~(~~ too is _a

decreasing

function

offs. Furthermore,

the

limiting

value does not

depend

too much _on the actual form of K: if

K(is

=

ccl

=

0,

then ~(~~

(is

=

cc)

=

0; conversely,

^if K has

a weak

dependence

_on

is (K

_m

Kofi(),

then ~(~~

(is

=

ccl

=

2~@Gid. _{By comparison}

with the numerical results at small

is,

where ~(~~ ^{is well} defined

(see Fig. 3),

the determined value of

Ko gives

rise to

~(~~(ts

=

ccl

_ct

0.8/1.2 id-

Since

lengths

smaller than

id

_are

meaningless,

there is not _a

big

difference between the _{two cases}

(constant

K and

decreasing K).

Now,

^{what does}

happen

to ~(~~, ^when

is

is

smaller,

but of the _same order of

I(~P?

Since

by

definition ~(~~ ^<

~[~P,

a second critical value I(~~ ^is

expected,

but two

possibilities

exist:

~(~~ =

~[~P (Fig. 3a)

_{or ~[~~}

=

id (Fig. 3b),

when

is

_{= I(~~}

(The

two

possibilities

coincide if

tint

= fsup

s s

N°6 EHRLICH-SCHWOEBEL INSTABILITY: DIFFERENT REGIMES 803

L L

SUp ~Up

,nf inf

~C ~C

is is

o o

a) bj

Fig.

3. Schematic forms of the two

possible "phase diagrams".

The

coarsening region

is in between

the two _curves. Note that i~ varies in [0,

cc],

while L in

lid, cc].

The evolution of the

surface,

in the interval I[~~ <

is

<

I(~P,

differ in the two _cases. In the former _one, at

increasing

L

(or

the time

t),

_{we pass}

directly

from _a stable flat surface to _an

instability

with formation of

deep

_crevaces. In the latter _case, mounds _appear with _a

wavelength

~

=

id,

_we have

coarsening,

^{and afterwards} the upper

instability

takes

place.

In other

words,

the flat surface is _never stable

(on

scales

larger

than

id).

Which scenario

really

_occurs is not evident. In

fact,

for

is

m I(~~ ^the

coarsening region

if exists! is very narrow and mounds have _a

typical

size which is

only slightly larger

than

id

This _means that fluctuations _{are very}

important,

because nucleation events take

place exactly

on a

length

scale

id

A second

difficulty

is that ~(~~ ^and

~[~P

can not be located with extreme

precision. ~[~P

can be determined for

example by changing

the value of L

(small

size of the

sample)

and

by checking

when _a mound is _{no more} stable.

Nevertheless,

if L _m

id stochasticity

in nucleation makes the interface thickness

/hz(=

_zmax

zmm)

neither constant in time

(meaning

_a

stationary mound)

_nor

constantly increasing

in time

(meaning

_an

instability) rather,

/hz shows _an intermittent

behaviour,

with formation and

healing

of

deep

_crevaces.

The location of ~(~~ ^is even more difficult: _{some reasons were}

explained

in reference

[6],

^but here _we will add _{a more}

pertinent

comment. A continuum local

equation

results from _a

spatial

average on a distance of order

id,

_so the notion of lower critical

wavelength

which results from _a linear

stability analysis

of the current

(4.7)

breaks down if ~(~~ ct

id-

This

happens just

when

is

_ci

_I(~~,

therefore the difficulties in

drawing

the

phase diagram (see Fig. 4)

in the

region is

_m

_I(~~,

L _m

id-

In

Figure 2,

^different

morphologies,

for different values of

is

_are shown: for

is lid

= 0.I

(Fig. 2a)

_we have mounds which would be

stationary

in absence of

coarsening,

and whose _cur- vature is

always negative.

All the other structures

(Figs. 2b,

_c,

d)

_are unstable

configurations,

which _are also characterized

by

the

appearing

of

pieces

^of curve with _a

positive

curvature. The size of the flat

region,

on the top,

equals I),

_{which decreases at}

increasing is.

For

is

₌ cc this size

vanishes,

but the

up-down symmetry

_appears

only

at

large

times.

In

Figure 4,

it is shown the

"phase diagram"

of the one-dimensional

surface,

in the

plane its lid, Llid).

The variable L _may have two different

meanings:

for _a small

sample size,

^L is the

sample size;

^for a very

large sample,

L is the

typical

size of the

emerging

structure.

Because of the

dynamics,

this size increases in

time,

_so L _{may even} be

thought

_as the "time".

ioo

t§

O L io

I

_~

O

.

f

x

O Ox

4i WO x

. .. 4i x

X x

~o.oi io.oo

ls

Fig.

4. "Phase

diagram"

of the one-dimensional surface. Both is and L _are

given

^{in units} of the diffusion

length

id- The full rhombuses _mean that the flat surface is

stable;

the circles that

stationary

mounds _are present; the _crosses that _no

stationary configuration

exists and

deep

_crevaces form. Other details _are

given

in the _text. For _a

given

value of

is,

if _{a cross} is marked for L

= L~ then other crosses are present for all the L > L~.

Analogously,

^if _a rhombus is marked for L

= Lo then other rhombuses are present for L _< Lo.

The lower critical

wavelength

is located between full rhombuses and

circles,

and the _{upper one} between circles and _crosses. When two different

symbols

_are

superimposed,

it _means that it has not been

possible

to discriminate between them: for

example,

that it has not been

possible

to decide if the flat surface is stable _or

not,

at the

corresponding lengthscale.

This _occurs when

is

_{ct I(~~} From the

figure,

it is

possible

to locate

t(~P

^{in the} range

2/4 id

5. Conclusions

We have studied _a one-dimensional

high-symmetry surface,

whose

growth

in presence of

step-edge

barriers is unstable

(Ehrlich-Schwoebel instability).

If the flux is not too

high

and thermal detachment from steps ^is

effective,

current

(1.2)

should be relevant and the

picture given

in

previous

works

[3,4] applies.

In this _case, the

up-down _asymmetry

is weak and the actual value of the Schwoebel

length

should not influence

qualitatively

the

resulting morphology; nonetheless,

it determines how

long

the

crystal

surface

keeps

flat.

In this paper we have studied the

opposite

limit of _a far from

equilibrium

^surface. Even if thermal detachment is

absent,

_a Mullins-like term in the surface current is induced

by

random

nucleation,

so the linear

description

^of the surface evolution is the _{same as} in the

quasi-equilibrium

_case.

When

slope increases,

_a

strong up-down asymmetry

^{in the surface}

profile

_appears: it is _a consequence of the nonlinear symmetry

breaking

term, ^{but also of the}

inability

of random nucleation to heal the surface at

large slopes. Furthermore,

the

morphology

^of the

growing crystal

at variance with the

quasi-equilibrium

_case

depends

_on the value of the Schwoebel

length,

^{and for}

is

= cc, the

up-down _symmetry

is

asymptotically

recovered.

N°6 EHRLICH-SCHWOEBEL INSTABILITY: DIFFERENT REGIMES 805

The main features of the

present

model _are I) the

neglect

of thermal

detachment,

and

it)

the

vanishing

of the lattice constant _a

(continuum limit).

The former determines _{an upper} critical

wavelength,

the latter that

slope

_{can go} to

infinity.

In reference _{[6] we} have

argued

that

~[~P

should survive _even if

ii

is

relaxed,

if detachment is weak.

Anyway,

it will be

important

to

study

more

carefully

this

point,

_{so as} to

investigate

the transition between the

quasi-equilibrium

_case and the far from

equilibrium

_one, treated here. We want also to stress that the existence of

~[~P

^does not

depend

_on

iii

_: in

fact,

the maximal

slope

at the

angular points, just

^before the formation of

deep

_crevaces, is weaker than I

la. Crystalline

effects _come into

play

after _crevaces have

appeared,

and _{m ci} I

la.

Acknowledgments

I _am indebted to J.

Krug

and J. Villain for _a lot of

stimulating

and

interesting

discussions.

D. Wolf is

gratefully acknowledged

for his kind invitation to the

University

of

Duisburg,

where

this _{paper was}

prepared.

I

acknowledge

financial

support

^{from the EEC} _program Human

Capital

and

Mobility,

under Contract ERBCHBICT941629.

Appendix

Nucleation

Lengths

The nucleation

length in

of _a terrace is determined

by

the condition

(see Eq. (2.I)):

P(in) 'j

"

f]~(Pl

_" 1

~~'~~

where

(pi depends

_on the

type

^of terrace.

The diffusion

equation

F +

Dp"(x)

= 0 is solved in the interval _x E

(-i12,i12),

with the

following boundary

conditions

(x

₌

+i12):

p'(x)

=

p(x) descending

_step

is p(x)

= 0

ascending

step.

So,

if

is

= 0 _we need

p(x)

_{[step "}

0,

because adatoms _are

automatically incorporated.

For

is

= cc, we have

p'(x)

_[step

" 0.

It is found that

(capital

letters

B, T,

V refer

respectively

to _a

Bottom,

a

Top

and _a Vicinal

terrace):

PB(Xl

_"

~

~~)

X~l

~

PTIXI

"

~)+PB(X)

~~ ~~~

2D~~~ _t) ~)

^{~ ~~~~~'}

It is manifest that pT ~ cc if

is

~ cc, whilst pv

keeps

finite.

Finally,

from

(A.I)

_we obtain the

following expressions lequations

for the nucleation

lengths,

in I+I dimensions:

~

l2Dj~/~

~~ ~~

F

ltl)~

₊

fits ltl)~ tl

" o

(t$)~

₊

4is(I$)~ t((I$

₊

is)

= 0.

For

is

_»

id, I)

_ci

id (id /6is)~/~

and

t)

_ci

_id Il.

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R.L.,

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experimental

results is in order here.

Experimental techniques

like

Scanning Tunneling Microscopy,

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profile

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growing film,

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^{It is similar to the} one

proposed

in reference

[6], except

^that now

is

is written in _a form valid for _any value

offs,

and not

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for small

is.

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