Growth roughness and instabilities due to the Schwoebel effect : a one-dimensional model

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Growth roughness and instabilities due to the Schwoebel effect : a one-dimensional model

Ilya Elkinani, Jacques Villain

To cite this version:

Ilya Elkinani, Jacques Villain. Growth roughness and instabilities due to the Schwoebel ef- fect : a one-dimensional model. Journal de Physique I, EDP Sciences, 1994, 4 (6), pp.949-973.

�10.1051/jp1:1994238�. �jpa-00246957�

Classification Phy.n<..i Ahsn.a<.li

68.95 05.20D

Growth roughness and instabilities due to the Schwoebel

effect

_{: a}

one.dimensional model

Ilya

Elkinani and

Jacques

Villain

CEA, Département ^{de Recherche} Fondamentale _sur la Matière Condensée, SPSMS. MDN, CENG, 17 _rue des Martyrs, F-38054 Grenoble Cedex 9, France

(Re<.eii'e(13 Jamiaiy 1994. a<.<.epled /4 Fehriiari 1994)

Abstract, A very simple, one-dimensional model (« Zeno model ») of

crystal

growth by

molecular beam epitaxy ^{is studied} numerically. The essentiel feature of the Zeno model _is that ii takes into account the asymetry ^{of the} sticking coefficient of adatoms _to steps (Schwoebel effect),

i-e-, the

diffusing

_atoms stick preferably to the Upper ledge. In contrast with other, more

microscopic

descriptions, the Zeno model takes the diffusion of adatoms into account through a

determiniUic diffusion

equation,

so that the computing ^lime is

greatly

reduced and _a

systematic

investigation of the effect of the different parameters is

possible. Deep

cracks form _even for _a weak Schwoebel effect, but they form after _a time which is very long if the Schwoebel effect _is very weak. In certain cases, the roughness increases proportionally to time, _in agreement ^with expenments on silicon and with other calculations. In the absence of Schwoebel effect, surface defects _are healed dunng

growth.

1. The Schwoebel effect and its consequences.

In the

growth

^of a

crystal by

Molecular Beam

Epitaxy (MBE),

the

freshly deposited

atoms

(adatom.ç) which diffuse _on the surface (as shown

by Fig.

l _in the _case of _a

high

symmetry

Fig.

l. Schematic representation ^{of the MBE} growth of _a

high

symmetry ^surface.

Deposited

adatoms diffuse to a step or nucleate _{a new tenace.}

surface)

do net stick with the _same

probabihty

to the

edge

of Upper and lower _terraces. The importance ^{of this}

peculiarity

_was first

emphasized by

Schwoebel

[1]

who showed that this mechanism

(hereafter

called Schw,oevel

effect)

_{generates an}

instability

in the

evaporation

of _a vicinal surface. More

recently,

several authors

[2-10] pointed

ont other

important

_conse-

quences of the Schwoebel effect. The present ^{work is much related} to that of

Eaglesham

and Gilmer

[8]

and of

Zhang

et ai.

[9].

Before

going further,

_it is of interest to recall the mechanism of the Schwoebel effect.

Generally,

adatoms stick

preferentially

to upper terraces, because there is _a

potential

barrer which makes it difficult for them to go down _{to a} lower _terrace

(Fig. 2a).

The _reason is

basically

that, in order to go downstairs, _an adatom _must either

push

another one

(Fig. 2b)

or

go

through

an uncomfortable

position (Fig. 2c)

where _it has _too few

neighbours

and _too

high

an

energy. The

exchange

mechanism of

figure

2b

(analysed by

Hansen _et ai.

Il1]

_in the _case of

ordinary

diffusion _{on a}

high

symmetry

surface)

is often believed _to be _more favourable in

typical

menais than that of

figure

2c,

investigated by

Bourdin et ai.

il 2]. Experimentally,

the Schwoebel barrier has been evaluated to be about 2 000 K for

W(110) Il 3].

a

b

c

~> ^D

d

Fig. ^{2. The Schwoebel} effect. a) Potential _seen by _an adatom, b) The

exchange

mechanism suggested

by

Hansen _et ai. _in the _case of adatoms diffusion without steps. cl Mechanism treated by Bourdin _et ai.

d) Probability of

jump

_per unit time.

The

subject

to be addressed in the present ^{work is the}

growth

^of a surface of

high

symmetry below _ils

roughening

transition. It _is clear

[4]

that, for such _a surface, a strong ^Schwoebel effect

produces

an

instability.

Indeed, on any growing terrace, adatoms land from the beam and, since

they

_{cannot go} downstairs,

they

are

obliged

to nucleate a new terrace _on which

another terrace will be nucleated, etc., so that the surface

is covered

by

towers rather than flan.

Even _a weak Schwoebel effect may be

expected

to

produce

an

instabihty.

Indeed, it

implies

that the current of

diffusmg

adatoms has the _same direction _as the

height gradient

j

=

K Vz

(1.1)

with _a positive coefficient K. Under certain

fairly

realistic circumstances

(implying

_in

particular

the absence of

evaporation),

the local

height z(r)

satisfies _a conservation

equation,

à=/ôt

=

div

j,

^which has to be modified because of the effect of the beam. Thus the

corresponding equation

is

~~

=

Fa div

j (1.2)

where F is the beam

intensity. Comparing (1.1)

and

(1.2),

_one obtains

É=Fa-KV~z.

(1.3)

A Fourier component _=j ^of

z(r)

satisfies, if k _# 0, the equation

art

,

=

KL- ri

Since K is

positive,

there is _an

instability. However,

this

argument

^is not

fully convincing

because _a nonlinear _term

proportional

_to

(grad z)~

should be added to

(1.3). Moreover,

the above arguments say

nothing

about the actual

shape

of the surface.

The best theoretical information about the consequences of the Schoebel effect _on the

shape

of _a

growing

surface has been obtained

by

Monte carlo simulations

[5,

8,

10].

Experimentally,

direct evidence of the Schwoebel effect has been obtained

by

^{field ion}

microscopy [2].

The

instability

observed

by Eaglesham

et ai.

Il 4]

in the

growth

of the

(001)

surface of silicon has been attnbuted _to the Schwoebel effect

[8].

2, The Zeno model,

In the present ^work we propose an

oversimplified

mortel

which,

_we believe, will

usefully complement

the Monte Carlo work. For _reasons which will become clear in the

following,

this model will be called Zeno model, from the

philosopher

born _in the lucanian

city

of Eleas in the sth century ^{B,C. This model} was

already

described in _a

preliminary

communication

Il 5].

This model is, to a great extent, deterministic. In

particular,

the fluctuations of the beam intensity ^{F in} _space ^and time, as well _as the fluctuations of the diffusion current around the average value

(1.1),

will be

neglected.

These

approximations

_are

justified

if the number of

partides

ⁱⁿ _some charactenstic _time is

large enough,

_more

precisely

if the

typical

_{terrace size}

Î

_{is much}

larger

than the atomic distance.

In

comparison

with Monte Carlo

simulations,

such _a deterministic model has the

advantage

(in addition to its

simplicity)

to isolate the contribution of the different factors which can be

responsible

for kinetic

roughness.

Beam fluctuations and current fluctuations _are mdeed such factors [4], and it _is of interest _to

study

a model

deprived

^of such fluctuations, _in order to

study

the other _causes of

roughness

more

precisely.

One of these _causes is the Schwoebel effect,

although

the _{term «}

instability

_{» may} be _more

appropriate

than _«

roughness

_» to

designate

_a deterministic process. However, _m

expenmental

data

(microscope

_image or diffraction

spectrum)

^the distinction _{is not} easy ^to make, _since stochastic

phenomena

_are

always

important.

A pi-tort, there _may be another _source of

roughness, namely

the nucleation of _{new terraces} of

atomic thickness which will form the _{successive atomic}

layers.

The

place

and time where

nucleation _{occurs are} stochastic _{to a}

large

extent. Can

roughness

_anse from this

stochasticity

?

The _{answer can}

hardly

be obtained from _an

analytic

argument ^because

long

time,

oscillatmg

current correlations result from nucleation

(Fig. 3).

An important ^{result of the} present work will be that nucleation

stochasticity

atone cannot

produce roughness,

at least in _one dimension.

~ /

ll~~s

it'

ô~ -

/

Fig. 3. After nucleation of _an island (ai the _current in the vicinity ^of the nucleus _remains directed

toward the nucleu; dunng _a time

z and then in the opposite ^direction 16) during _a time T'. _z and T' depend _on the distance _to the nucleus, but _are both of order (though smaller than)

1/(Fa ).

Another feature of the Zeno model is that _it is

« + »-dimensional, _i-e- the surface is _one dimensional. Of _course the real world is

(2

₊ 1) dimensional. A

(1

₊

1)-dimensional

world may appear as an

unacceptable ~implification.

However, it is such _a

huge simplification

that it is a necessary first step,

just

_as mean field is a nece~sary first step in the

study

^of

phase

transitions. Indeed the

dynamics

of _a two-dimensional surface

depends

on many parameters, in

particular

the diffusion _constants of _atoms

along

a step. ^{It is of interest} to

investigate

a model with few parameters.

We have used _two versions of the model the continiious version and the disc.rete _version.

Bath versions _are discrete in the direction _z

perpendicular

to the

high

symmetry ^{surface. In the} contimiou.ç _version, there is _{no atomic} distance _in the

high

symmetry direction.i

parallel

to the average

surface,

so that _terraces may be

mfinitely

small. The

advantage

of this approximation

is to reduce the number of parameters.

Actually,

it will be _seen that the _continuum model

depends only

on a

surgie

parameter, ^which characterizes the

strength

^{of the Schwoebel effect.}

A discrete version, where _a finite interatomic distance is introduced, will also be used. It will be descnbed in section 6. In the present section the continuous Zeno model will be defined.

Any

state of the surface is charactenzed

by

the positions ^{of the} steps. The absciwa of the n-th

step ^{from the left will be called} i,,. We need to

specify

two

points:

ai the

velocity

v,, of each step ; b) the nucleation

probability

_{p~,, on} each _terrace. The former quantity can

easily

be deduced from the dassical

theory,

it is shown in

Appendix

A that the absolute value of the

velocity

_r,, of _a step separating an upper tenace of width

Î,,

and _a lower terrace of width Î,( is

r,, =

f (f,,

_{+ g}

(fi _{(2. i)}

where the

expressions

of

f

and g will be

given

below, _assuming the upper terrace to be _on the

left hand side of the step and the lower terrace to be _on the

right

hand side, _so that

v~, ^is positive. The opposite case will

easily

be obtained

by

_symmetry.

The forms of

f

and _g

depend

on the type ^of terrace, which may ^{be either} a « top terrace _» a

« bottom terrace _» or a « vicinal _{terrace »}

(Fig. 4).

T

B

Fig.

4. T is _a top tenace, B is _a bottom terrace. The other _{tenaces are}

« vicinal

»

For _a top terrace, g does not exist

(since

that terrace cannot be the lower

terrace)

and

f

is given on both

sides,

if _a is the _atomic

distance, by f(Î)

=

FaÎ/2, independently

of the

Schwoebel effect. The rote of the Schwoebel effect _is

just

to increase the nucleation

probability.

For _a bottom terrace,

f

does not exist

(since

that terrace cannot be the topmost

terrace)

^and

g is given on both sides

by g(Î)

=

FaÎ/2, independently

of the Schwoebel effect.

For _a vicinal terrace, it is shown _in

Appendix

A that

aF ail

j

(2.2)

f(~)"i

ii ~m ~f+fi

'

~z

and

g(Î)=aFÎ-Î ^{_aFÎ Îj}

~ i _~

~'

2

~

~

Î

₊

Îj ^(2.3)

where _{a is} the interatomic

distance,

F is the beam intensity per ^unit time and unit

length,

^and

ii

is _a

length

which characterizes the Schwoebel effect. It tutus out to be

given by

ii

=

~~Î (2.4)

where

Dla~

_{is the}

probability

_per unit _time for _an adatom _at abscissa _{x to}

jump

to a

position

at abscissa (x + 1) or (x

1)

if this

jump

does not reqwre a

change

of

height

for the adatom.

D'la~

is the

probabihty

_per unit time for _an adatom _to

jump

to a

neighbouring

site when this

implies

_going to a lower _terrace

(Fig: 2d).

D is the diffusion _constant of _an adatom _on the

ideally

flat surface.

After

having

_given the formulae for the step

velocities,

we have _to calculate the nucleation

probability

_per unit time _on a terrace. Its detailed form

depends

on the features of the nucleation process. For instance, if we assume that _two adatoms which meet form _a pair which

is stable and immobile, then the nucleation

probability

_per unit time on a top terrace is found

(App. A)

to be

~~~~~~~ ÎÎÎ ^~

~

~~

~~'~~

The nucleation

probability

_{on a} bottom _or vicinal _{terrace is} given m the

Appendix. They

_are

not essential _{since most} of nucleation _events take

place

_on top terraces.

An important quantity is the

typical

size of _a top terrace when nudeation takes

place.

It _is

argued

in the

Appendix

that this _size

corresponds

to a value of the above

probabihty

of the

JOURNAL DE PHYSIQUE T 4 N'6 )UNE 1994

~~

order of Fa

F~ Î) Îj

fi

^{~ ~}

_Î~

^~~

^~~'~~

We shall call

f~o

^{the value of}

Î~

m the absence of Schwoebel effect. Thus

FÎ)o

m 12 Da

(2.7)

or

ij~i~

₊ 6

ii)

=

ij~ (2.8)

STEP _{MOTION IN} THE CASE OF MULTIPLE STEPS. The presence of step ^{bunches ares} net

modify

the nucleation

probability

nor the characteristic

lengths,

but does

modify

the step

velocity.

A bunch of _p steps may ^be considered _{as a} facet _on which adatoms diffuse

freely.

The

possibibty

of nucleation _on such _a facet will be

neglected.

An adatom which reaches _a step bunch from above will be assumed _to

glide instantaneously

to the bottom of the bunch

(Fig. 5).

This

approximation

is _not

satisfactory

for

large

bunches.

---l _---

a

b

c

d

Fig.

5. Advance of _a multiple step.

Consider _a bunch of _p steps separating an upper terrace of width f and _a lower _terrace of

width

Î'.

_As

before,

_we call

f(Î)dt

the number of atoms

sticking

^from above _{in time}

dt and g

(Î')

dt the number of atoms

sticking

from below.

f(Î

_{and g}

(Î')

_are still given

by (2.2)

and

(2.3)

for vicinal terraces, and _are

equal

to

FaÎ/2

in other _cases. The atoms commg from

above and from below distribute themselves in different ways : an atom

coming

from above will stick above the

previous

_one and _{an atom}

coming

from the _same level will stick aside the

previous one. Therefore the

f (Î )

dt atoms

coming

^from above will

displace

the bunch

position by

the _amount

v ~

dt

=

f

_~~ ~~

(2.9a)

while the

g(Î')

dt _atoms commg from the _same level

(Fig. 5b)

will

displace

the lower step ^of the bunch atone

by

the _amount

v~~

^dt =

g(Î')

dt

(2.9b)

Thus,

the bunch tends to separate into an isolated step at the bottom and a bunch of

~/J

1)

steps. ^{But this} separation ^will

really

take

place (Fig. 5c) only

if that bunch does _not

catch the bottom step

again

due _to the

f(Î)dt

_atoms

coming

from

above,

therefore if

vj ~ ^<

v~~

^or

f(Î)

< ~p

1) g(Î').

In the opposite case

(Fig. 5d),

the step ^{bunch does} not dissociate and _{moves at} the

velocity

~

f(f)

₊

g(f')

(2.io)

p

3. Growth without Schwoebel effect.

3.1 NO SCHWOEBEL EFFECT, SURFACE INITIALLY FLAT. We have first

performed

simu-

lations without Schwoebel

effect, starting

from _a flat surface. The

only

random process was the nudeation of _new terraces, with the

probability given

in the

Appendix.

A

typical profile

_is shown _m

figure

6. The surface thickness is

essentially

reduced _{to two or} three levels

(Fig. 7a),

but _a few

higher bumps (or deeper holes) occasionnally

_appear. If the time _unit is the

layer filhng time,

the

largest

terraces have the

height

t at mteger ^time t, and there _are small,

freshly

nucleated tenaces at

height (t

₊

1),

and small terraces at

height (Î 1)

which are on the verge of

disappearing.

^On the other

hand,

if the _time is

half-integer,

ail _terraces have

heights (Î

₊

_1/2)

and

(Î _1/2)

and _are

approximately equal.

2002

2001

2000 z

1999

199s

~~~~.0

5.0 10.0 15.0 20.0 25.0

x

Fig.

6.

Typical

surface

shape

at integer coverage in the absence of Schwoebel effect for _an

mitially

flat surface. The vertical direction _is the height in monolayers _; the horizontal direction _is the distance, using Î~~ as length unit.

1200 500

iooo

800

N(ZJ ₆₀₀ N(1)

400

200 100

0 0

1997 1998 1999 2000 2001 2002 2003 Ô-Ô 0.4 0.8 12 1.6 2.0 2.4 2.8 3.2

z i

a) b)

Fig.

7. Growth of _an

mitially

flat surface without Schwoebel effect. a)

Histogram showing

the proportion of the surface _at

heights

1, and ₊ at integer time t (there are aise two tenaces at height

( + 2 and _{one at1} 2 which _cannot be _{seen on} the histogram), b) Terrace size distribution after filling 10 000 layers without Schwoebel effect.

The conclusion _is that random nucleation does _not lead _to

roughness.

The _size distribution of the _terraces at an integer time is shown _in

figure

7b.

3.2 HEALING OF DEFECTS FOR SAWTOOTH PROFILES OF SHORT WAVELENGTH. lt _is weÎÎ

known

expenmentally

that defects such _as

bumps

_or

valleys

_are healed

by

MBE

growth.

Our mortel does

reproduce

this property. ^For

simplicity,

_we restncted _our

study

to sawtooth

profiles.

We have studied

numerically

the

growth

of _a

crystal

limited

by

a surface

exhibiting

_a

periodic

succession of

straight

facets

(Fig. 8).

We have

imposed

a determmistic nucleation rule,

namely,

a new terrace is nucleated _at the _centre of each top terrace when _{its size} reaches the value

Î~

₌ Î~~. ^Nucleation on terraces other than top terraces never occurs.

Figure

8 shows

1=0 toe20

t=40 -t=80

120

100

~---

---;

80

Z ₆₀ .~

,

40 .'

20

0

0.00 0.20 0.40 0.60 0.80 1.00 1.20

x

Fig.

8.

Healing

of _a saw-tooth

profile

without Schwoebel effect. Only one penod is

represented

at increasing limes (0, 20, 40 and 60 layers). Nucleation _is determmistic.

the successive _states of the

profile. They correspond

to a

period

L

=

1.2 Î~~ ^and to an initial

height h(0)

=

100.

Eventually,

a flat surface

corresponding

to

figure

6 is reached. In the intermediate _states

(Fig. 8)

the surface _consists of flat _{terraces at its} top, ^which are bounded

by

a steep

edge

made of several

coalescing

atomic steps ^and

separated by

cracks. We will _now try

to understand this _structure in the _case when the initial _terrace width _is shorter than

Î~.

This _is the

experimentally interesting

_case, because

Î~

is

generally

_very

large

in MBE.

The variation of the interface thickness

(height

difference between topmost ^{and lowest}

point)

is

displayed

for _a sawtooth

profile

in

figure

9a in _a

typical

case. The initial decrease

corresponds

to the

flattening

of the top ^{of the}

profile,

when the width Î~ ^{of the} topmost terrace is smaller than Î~~. ^{Then there is} no nucleation _at the top and, _since bottom _{terraces are}

annihilated,

the thickness decreases. When

Î~

^{become of the order of}

Î~o,

^the thickness decreases but very

slowly.

120

so ioo

80 60

Ah Ah 60

40

40

20

0 0

0.0 20.0 40.0 60.0 80.0 100.0 0.0 8000.0 16000.0 24000.0 32000.0

Time Tlme

a) b)

Fig.

9. Variation of the

height

difference (« interface thickness ») between the

highest

and the lowest point ^{of the surface} as a function of time. There _{is no} Schwoebel effect. In _case (a), the

period

^is

L=1.2i~~

and the initial height difference is 100 atomic layers. In _case (b), the penod is

~0 i~~ ^{and the} initial height difference is 100. The _time unit _is (1/Fa).

The evolution shown in

figure

8 _is

easily

understood if the initial tenace width

is much shorter than

i~.

Indeed

(cf. Fig. loi

the topmost tenace of each

period readily

takes _a width

i~ w1. Therefore, according

to

(2.2)

^and

(2.3),

^and because

ii

=

0 without Schwoebel

effect,

the topmost steps ^have a

velocity

of order

Fai~/2,

which is much

langer

than the

velocity Fai12

_{of the other steps. Therefore, the topmost steps coalesce with the next steps} and form _a bunch of _two steps. ^{This bunch will then}

proceed

with _a reduced

velocity.

The reduction factor _is

approximately

2 because the progression of the topmost steps is

mainly

due to atoms

falling

^from the upper tenaces, which _{are m} identical

quantity

with _or without

bunching,

and this

quantity

must be shared between two

steps.

^{Thus additional}

bunching

will

follow,

and _a

bunch of 3 steps, ⁴ steps, ^and so on, will form. Thus the

shape

of

figure

8 emerges.

The

healmg

time of _a sawtooth

profile

with

period

2 L

~ 2

i~

_is

easily

calculated. Indeed there _{is no} nucleation until the surface _is healed. Therefore the

healmg

time r~~~j ^is

just

the time

necessary to fill each

valley.

For _a sawtooth

profile,

the

valleys

have _a

height

h

=

Lli

_and

an

area Lh/2

=

L~/21,

_and

are filled after _a time

~~~~~

2

Îa~

^~~

1

_ a

Fal _-

_ i

i

-

j

i b

i i i i i i i i i

C

Fig.

10. Growth of the top ^{ofthe surface. Ifthe width}

i~

of the topmost tenace is much

larger

^than the width f of the lower _terraces, the upper steps form bunches.

We have aise studied the _case L

~

i~.

It tums eut to be

surprisingly compbcated.

^The

profile

shows a step ^bunch near the top as

before,

but the _new feature _is that _waves

develop

below the bunch

(Fig.

ii

).

The

wavelength

increases with time, _so that the number of

bumps

decrease before

eventually vanishing.

As _a consequence, the surface thickness has also _an

oscillating

character

(Fig. 9b).

It decreases

abruptly

when a

bump disappears

from the

profile.

350 600

(a) (b)

330

310 560

Z

290 540

270 520

(c) (d)

s30 ioso

sio ioôo

z

790 1040

770 1020

750

0.00 4.00 8.00 12.00 16.00 u.00 4.00 8.00 12.00 16.00 20.00

x x

Fig. Il- Profile observed at successive limes (a) t =250, b) t=500, c) t=750 and d)

t 000) when heahng _a sawtooth

profile.

The period ^is equal to 20 f~~. ^There is no Schwoebel effect and the nucleation process is determmistic. The units are the _{same as m}

figure

6.

We have net been able to undestand the

origm

of this

instability.

We first

thought

of _a

possible

artefact of time

discretization,

but _we have checked that the time step was short

enough

to avoid any artefact.

Possibly

the

nonlinearity responsible

for the

instability

^arises from collisions between steps, but _we have _not studied these collisions.

Healmg

of cracks has aise been

reproduced

in the _case of _a weak Schwoebel effect,

again

usmg a determmistic nucleation mechanism.

However,

these results _{are not} reliable because it will be _seen in the _next sections that in the _case of tandem

nucleation,

cracks may be

produced

even m the _case of _a very small Schwoebel effect.

4. Evolution of _a surface with Schwoebel effect.

4.1 CRACKS. In this section _we report simulations with Schwoebel

effect, starting

from _an

initially

flat surface. Random nucleation _is assumed. The

strength

of the Schwoebel effect is characterized, as said above,

by

the _{ratio ~}

=

ijli~~,

_note that it is convenient _to

distinguish i~,

the actual characteristic

length

of nucleation _on the topmost tenaces, and i~~ given

by (2.7).

i~

and

i~~

are related

by (2.8).

With _a strong Schwoebel effect

(say,

for _{~ ~} l cracks

develop

_very

early. They

do not heal within the time of _ouf

calculation,

_so that their

depth

^is

essentially equal

to Fat. This feature is

a consequence of the continuum

approximation.

The

density

of cracks _cannot be

langer

than

lli~,

and seems to converge to a value of this order when _~ increases.

Actually,

_even with _a

fairly

weak Schwoebel

effect,

cracks also appear very

rapidly.

Figure

12 shows _a

typical profile

obtained for

ijli~~

= 0.2. The cracks look similar to those _m

a strong ^{Schwoebel effect}

they

become very

deep

^{while their} width increases very

slowly.

However

they

_appear

progressively.

For instance, if _~

=

0.20,

the first cracks

typically

_appear

after

filling

20

layers (in

a

sample

of _size 100

i~~).

^Then the cracks'

depth

_grows

nearly proportional

to time

(the

coefficient of

proportionality

is atomic size of

depth

_pet

deposited layer).

The

deep

cracks observed in

figure

12a _are

unphysical

because their width is smaller than

the interatomic distance.

However,

the other features

(density

of cracks, crack

shape,

roughness)

_are the _{same as} for _a

physical

model with _a finite _atomic distance

(Fig. 12b).

We shall _now try to understand the evolution of cracks. if the crack

profile

_may be

approximated by

an

analytic

function _z

(x)

of the _« horizontal _» coordinate

(parallel

to the

high

symmetry

orientation),

the evolution _can be treated

by

the

theory

of Franck

[16].

The bottom and the top ^{of the crack} require

special

attention because, as will be

argued,

the

profile

there _is

700 700

600

soo soo

400 400

Z Z

300 300

200

ioo ioo

o 0

0.00 20.00 40.00 60.00 80.00 100.00 0.00 20.00 40.00 60.00 80.00 100.00

X X

a) b)

Fig.

12.

Typical profile

at successive times for

ijli~~

=

0.20 in the Zeno model with random nucleation. a) Continuum limit. b) Fimte atomic distance, _a

=

0.001f~~.

net

analytic.

In the next subsection _we consider _a

deep

crack, ^without _caring about how _it formed.

4.2 EvoLuTioN OF THE BOTTOM OF A CRACK. We consider a crack with _a very

sharp

bottom. It _is clear that the

profile

is _not

analytic

at the bottom. A

sharp

bottom

implies

that the

tenace widths _are small,

i~

«

ii.

This

implies

that the Schwoebel effect for these _{tenaces can}

be considered _as very strong. ^This argument ^is

only

valid in the _continuum

approximation,

and has _not

necessarily

_a counterpart ⁱⁿ a reabstic

problem.

Assummg

the Schwoebel effect to be

infinite,

the

equations

of evolution for _a crack

having

_a symmetry axis at ~r =

0 _are

(Fig.

13a)

where

x,,(t)

denotes the

position

of the _n th step at time _t.

à 'Z

x ~

~

~

2

i~

~

---~----/---_

à

x~ z

x

b

1---$

Fig. 13. a) Labelhng the steps in the bottom of _a deep crack (a) and at the top ^{of the} profile (b).

4.3 THE _{TRACK OF} ZENO OF ELEAS.

Equation (4, la)

^is

independent

of the other _ones and its solution _is

xi

(t

_{= A"1(0 exp}

(-

Fat

(4.2)

This

apparently

^{harmless result}

implies

that xi

(t)

_never vanishes.

During

the nominal

layer filling

time, which is il

(Fa

),

only

a fraction

(1

île

)

of the lowest

layer

is filled. After _{a time}

n/(Fa),

which is

nominally

the time _to fill _n

layers,

a fraction

xi

~

=xj(0)e~" (4.3)

Fa

is still unfilled. This is reminiscent of Zeno's

paradox according

to which Achilles will _never catch the turtle if the latter has _some initial

advantage,

_say km. Indeed when Achilles will have _run that kilometer, the turtle will advance, _say one meter. Then, m the time necessary for Achilles to mn that meter, the turtle will _{mn one} millimeter. At the n'th step ^{of the} argument, the turtle will still have _an

advantage equal

tu

d~ ₌ ^{10~ ~~ kilometers}

(4.4)

The

similarity

between

(4.4)

and

(4.3) justifies

the

terminology

« Zeno

instability

_» for the effect described above. There _are of _course differences between Zeno's

original paradox

and its modem _version

applicable

_to MBE. First of

ail,

_ouf derivation and _ouf result _are

strictly

correct within the continuum model. Zeno's argument was

obviously

_wrong. As fat _{as we}

understand, it

just

reflects the

ignorance

of

Algebra

^{which affected} the ancient Greeks. It should be remembered that Zeno's time _was very

bright

for literature, but the great ^{time of} Greek Science was two centuries later On the other

hand,

Zeno's

theory

_was formulated in _a

dispute against

the

disciples

of

Pythagoras. Presumably [17, 18],

the latter may have known the _sum of the

geometrical

series with

general

term

(4.4) although they

were not able _to

formulate the calculation _{m a}

convincmg language.

The

simple algebraic language

which is

nowadays taught

in ouf schools _was not invented before the end of the sixteenth century

[19].

The

general

solution of

(4,1)

is

easily

seen to be

x,,

(t )

=

~jj' ^~~ [~ ^Y

x,,

~

(0

_exp Fat

(4.

5

Thus,

if xi, x~,

...,

x~

^{are known at time}

0, they

_are also known at a later time t.

Dur numerical simulations _on the model described in section 2 confirm this result _: if the Schwoebel effect is strong

enough,

cracks form their

depth

increases

proportionally

to lime, and

they

^do not heal

during

the simulation.

Then,

how _{can one} understand that, for appropnate

values of the Schwoebel effect, the crack

ultimately

heals after _an initial increase ? To _answer,

one should worry about the

edges

of the crack.

4.4 EVOLUTION OF THE CRACK EDGES. While it is easy, as seen m the previous section, to

study

the evolution of the bottom of _a

crack,

it is _more difficult _to

figure

ont the evolution of _a crack

edge.

A

priori,

two

simple

schemes may be

expected.

The two

possibilities

_can be best

described in the case of _a

periodic profile

with deterministic nucleation. In contrast with the previous

section,

it is convenient tu label the steps ^{from the} top ^{of the}

profile (Fig. 13b).

A first

possibility

is

that,

near its

edge,

the

profile

_goes to a well-defined hmit. In other

words,

if _one considers for instance

only

the times t where _{a new tenace is}

nucleated,

_one has for any given n

Lifrl

1,,(t

₌ 1~°

(4.6)

Co

Of _course, in the continuum model addressed

here,

the crack

depth,

and therefore the

maximum value of _n goes to

infinity

with time. Therefore the bottom of _a crack _cannot go ^to a

limit.

The other

possibility

is that the

freshly

nucleated steps merge into _a step bunch, as occurred without Schwoebel effect when

healing

a sawtooth

profile. Then,

the

position

_u

(t)

of the step brunch _on the left of _a maximum _{is an}

increasing

function of

time,

_except if _a bunch appears above _a bunch _an event which has never been observed. Let h

(t

be the

height

of the

bunch, H(t)

the crack

depth

and L the

period. H(t)

is counted from the top ^{of the}

bunches,

_so that

h(t) ~H(t).

The numerical calculation suggests ^{that the nucleation} rate, as well _as the increase of H, is

approximately uniform,

_so that

H(t)~F't

where F' may be different from F. The

quantity

of matter which faits mto the crack pet unit time is also

approximately uniform,

and therefore it is

bigger

than

F"i~,

where F" is a

number. This _{matter is} to be distributed among the h steps ^{of the} bunch, _so that the width

(L/2 u)

of the crack decreases

by

the _amount F"/h in _a time unit. Therefore

du

F"i~ F"i~ F"i~

à

_h ^~ H

~F't

Integration yields

F "

i~

"~~ ~

F'

~~

~~'~~

This expression goes to

infinity

with time _t. This

implies

that the crack width

L/2

u(t)

vanishes after _some time. Thus the cracks should heal.

The two scenarios described above

(crack reaching

_a

stationary shape

and crack

heabng)

are

the

simplest possible

_ones. In the absence of Schwoebel

effect,

and also _m weak Schwoebel

effect, preexisting

cracks _seem to heal

according

to the second _scenario.

Figure

14

gives

an

idea of the evolution of _a crack for _a

fairly

_strong Schwoebel

effect, iili~

= 0.5. As shown _m

figure

lsa, the

slope

at the crack

edge regularly

_{increases as}

,à.

_{This is}

consistent with the second of the _scenarios described above. The crack will

presumably

described above. The crack will

presumably

heal after _a very

long

time,

although

this has _not been observed in the present calculation.

5. Weak Schwoebel effect.

5.1 EVOLUTION OF THE PROFILE wiTH TIME.

-Figure16

shows the

profile

obtained for

ijli~~

= 0.05 at successive times from _an

mitially

flat

profile, using

the velocities defined

by

(~.l-3),

and the nucleation

probability (2.5).

As _in the _case of _a strong ^{Schwoebel effect} studied in the previous section, there _are

deep cracks,

visible _m

figures

16c _{to e.}

However,

there _are

significant differences,

_m

particular

_:

i)

For coverages lower than 1000

layers,

the

profile

is

just

a little

rougher

than without Schwoebel effect. There _{is no}

striking

difference between the

profile shapes

observed after

deposit

of 500 and 1000

layers.

ii) Healmg

of cracks _is

occasionnally

observed. For instance, the

deep

crack observed _near

1= 80 in

figure

16c _{is no}

longer

observed _{as a}

deep

crack in

figure

16d.

However,

_a small

2050

(a) (b)

iooo

9so 19so

z 900 1900

sso isso

soo

(c) (d)

3000

2950 7950

Z 2900

2850

2800 7800

2750 7750

0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20

x x

Fig.

14. Profile of _a crack _at coverages 1000 (a), 2000 (b), 3000 (c) and 8000(d) for

f~/f~~ ₌ ^{0.5 in the continuous model.} Nucleation _is deterrninistic.

iooo.o

100.0

-,-__--

' '

-~

-,-

~

i

~

- .

iooo.o

Time Time

a) b)

Fig. 15.

increase as _functions of t'~

depression

_is observed _at the _same

place,

and is

presumably responsible

for the crack which reappears later _at the _same

place.

iii)

For coverages lower than 2000

layers,

the bottom of ail cracks _moves

upward,

_m

contradiction with the argument ^{à la Zeno} _given m section 4.2. As _{seen m} section

4.4,

this can

be understood _{as an} effect of the crack

edges. Figure

17 shows the interface width for _a

sample

99s

~ 990

485 _?Si

48Ù

~~j ~~j 98Ù

iioo

z

(c) (ii

iioo

145Ù

14ÙÙ Z

13so

1300

(ti (o

Ù.ÙÙ lÙ,ÙÙ 4Ù,ÙÙ lÙ,ÙÙ 8Ù.ÙÙ Ù.ÙÙ lÙ.ÙÙ 4Ù,ÙÙ lÙ,ÙÙ 8Ù.lÙ IÙÙ.ÙÙ

x x

Fig.

^16. Profile at coverages 500 (a) ¹⁰⁰⁰ (b) 15 000 (c) 2 000 (d) 2 500 (e) and 5 000 (fl monolayers for

fj/f~~

= 0.05. Note that the ordinate~ have different scaies in the different pictures. The

units are the _{same as m}

figure

6.

of _size 100 i~~ ^for a

particular

run. It shows

straight

_pieces of

slope

terminated

by

an

abrupt

decrease. This decrease

corresponds

to the

healmg

of the

deepest

crack.

At

longer

times, the lifetime of the cracks becomes much

longer.

For _instance, at coverages between 3 300 and 5 000, the interface width is _a linear function of time with

slope (Fig.

17).

This _means that the

deepest

crack dues not heal

during

this time.

5.2 ROUGHNESS. The

roughness

at time t _is defined _as

IL

^{L 1/2}

ôh(t)=-

_ΰ d£r

dV[h(V,t)-h(,1,t)l~) ^(5.1)

o o

where the

angular

^{brackets denote} an average on many runs. In

practice,

we have made

only

one run. The results _are shown _m

figure

18. It is of interest _to compare this

roughness

^{with the}

experimental

observation of

Eaglesham

et ai.

[14, 8]

that the

roughness

mcreases

linearly

^with

time until _a critical coverage of _a few hundreds of

monolayers,

^{when the material becomes}

3000 200

2500

150 2000

" isoo " ioo

iooo

so soo

0

~~~~~ ~ 20000.0 3°°~°'°

0

~ 2000,0 3000.0 4000.0 5°°°'° ~'~

Tlme

0.0 1000.

~~

a) b)

Fig. 17. Interface thickness _{as a} function of time _: a) contmuum model with

ij/f~~

= 0.05 _; b) discrete model with f~/f~~ 0.03, and

a/f~~

O.COI. The _{units are} the _{same as in}

figure

9.

30.0 10.0

25.0

8.0

)

20.0

(

6.0

i~ 15.0

~

Î )

_4.0

5.0 2.0

0.0 0.0

0.0 10000.0 20000.0 30000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0

Time Time

a) b)

25.0

20.0

(

_6.0

(

lS.0

Z z

Îi Î~

4.0 1 _10.0

2.0 5.0

0.0

0.0 400.0 800.0 1200.0 1600.o 0.o 1000.0 2000.0 3000.0 4000.0 5000.0

Time Time

c) d)

Fig.

18.

Roughness

_{vs. time} for

fj/f~~

0.03 (a), 0.05 (b), O.l (c) and 0.2 (b). Curves (b) and (c) correspond to the _continuum model, and _curves (a) and (d) correspond to an atomic distance _a, with a/f~~ 0.005. The coordinates _are the

same as m figure ^9.