Metastability of Step Flow Growth in 1+1 Dimensions

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Metastability of Step Flow Growth in 1+1 Dimensions

Joachim Krug, Martin Schimschak

To cite this version:

Joachim Krug, Martin Schimschak. Metastability of Step Flow Growth in 1+1 Dimensions. Journal

de Physique I, EDP Sciences, 1995, 5 (8), pp.1065-1086. �10.1051/jp1:1995177�. �jpa-00247115�

Metastability of Step Flow Growth in 1+1 Dhnensions

Joachim

Krug

and Martin Scuimscuak

IFF, Theorie II, Forschungszentrum Jülich, 52425 Jülich, Germany

(Jleceived

3 ApriJ 1995j received in final form 10 ApriJ 1995j accepted 19 ApriJ

1995)

Abstract. We introduce _a "minimal" model of crystal growth in 1+1 dimensions, which mcludes random deposition, surface diffusion of singly bonded adatoms, and perfect step edge barriers ta completely _suppress interlayer transport. We show tuat tue stable step ^flow regime

predicted by Burton-Cabrera-Frank type ^tueories is destabilized by Island formation. Trie tran, sition to _an a8ymptotic Po1880n-like growth mode, _m wuicu the Surface width _grows indefinitely

as trie square ^root of trie number of layers, _occurs after _a transition time _T

mJ

f~~(D/F)~/~,

where £ is the 8tep 8pacing of trie vicinal surface, D is the surface diffusion constant and F is the deposition rate. Trie Poisson regime is preceded by _an intermediate scaling regime _in which the surface width _grow8 linearly with trie number of layer8, _as lias been reported in recent

experiment8. Trie relation to trie Cahn,Eilliard theory for thermodynarnically metastable state8 18 outlined. Stable 8tep flow18 po8sible _in the limit D

IF

- co. This _case is solved exactly, and

the _terrace length8 _are 8hown to have _a Po1880n d18tribution.

1. Introduction: The Schwoebel Elfect

Thirty

_{years ago} Schwoebel and

Shipsey iii pointed

out that step

edge

barriers which control trie _mass transport between

loyers

_con bave drastic elfects _on trie

stability

of _a

growing

or

evaporating crystal

surface. Trie basic mechanism

operating

in trie _case of _a vicinal surface

is illustrated in

Figure

1. Under

growth

conditions reduced

interlayer

transport

implies

that trie main contribution _to trie motion of _an

advancing step

is due _to trie atoms

arriving

from trie lower _terrace. If this _terrace

temporarily

becomes

larger

thon _average, trie step

velocity

increases and trie average ^terrace size is restored, thus

stabilizing

the

uniformly spaced

_step

train. On _a

singular

surface without

preexisting

steps the _saine elfect _cari lead to _a

growth instability,

_{as was} first

predicted by

Vîllain [2]. ^Such instabilities bave been

reported

in several

recent

experirnents [3,4].

For _{a more}

quantitative

treatment, it is useful to introduce trie concept of _an inclination,

dependent, growth-induced

surface current [2,

si.

We

explain

trie basic idea

by

_a

simple

calcu, lation in trie framework of trie Burton-Cabrera-Frank

(BCF) theory

of vicinal surfaces

[6-8].

Consider _a step train with uniform terrace

length

^£. Trie

microscopic

_processes indicated in

Figure

1

irnply

that trie one-dimensional concentration

profile n(x)

of isolated adatoms between

© Le8 Editions de Phy81que 1995

o

~

r, ^~

~ @

E

o i

x

Fig. l. Schematic of _a vicinal cry8tal Surface according to BOF theory. Trie lower graph indicate8 the energy land8cape re8pon8ible for trie Schwoebel elfect.

two steps

satisfies,

in

steady

state [8],

Dn" _{+ F}

aDn~

= 0

ii.i)

where D is the surface diffusion constant, ^F is the

deposition

rate, and _a is _a capture

elliciency characterizing

the formation of

(immobile) dirners;

additional terras

describing

the eflect of step motion _on the adatom concentration I?i ^{and the}

deposition

onto nearest

neighbor

sites of adatoms [8] can be

neglected

in the _range of _{terrace sizes} of interest here [9]. ^The step

edge

barriers enter

through

the

boundary

conditions at trie

descending ix

=

0)

and

ascending ix

= £) steps

iii,

n'(0)

=

n(0) Il-, n'(1)

=

-n(1)/1+, (1.2)

where

l~

=

D/r~

_are capture

lengths

^related to the

incorporation

rates r~ for _{atoms ap-}

proaching

trie step from below

(+)

_or above

i-i-

Once trie adatom

density profile

bas been

computed

^from

Il.li

and

(1.2),

the _net surface _current J is obtained

by averaging

trie local diffusion current

-Dn'(x)

_over trie terrace,

J =

-lD/É)lnlÉ) n1°)1. Il.3)

In

general,

the current is

governed by

the

interplay

of trie

four length scales1, £-, £+,

and the

diffusion length(1)

ÉD =

lDlaf)~/~ _l14)

beyond

which the diffusion term _in

equation Il.i)

_can be

neglected;

for1» ÉD trie adatom

density

is limited

by

island

formation,

and

n(x)

saturates at nD =

(FlaD)~/~

=

Fl[ ID. Iiài

Here _we

simplify by assuming perfect

step

edge

barriers

jr-

₌

0)

and instantaneous _incorpora- tion from the lower terrace

(r+

=

ccl,

_so that

11.2)

^is

replaced by n'(0)

=

n(1)

= 0.

Rescaling

(~) ^Th18 expre8810n for the difiu8ion length 18 not quantitatively correct, becau8e

we bave neglected the capture of adatom8 by181and edge8. In tact iD _'~

(D/Fj~/~

_{on a} two-dimen810nai Surface, and fD

'~

ID /F)~/~

in _one dimen810n, though for other reason8; 8ee

[loi

and reference8 therein. For _our purpo8es the es8ential feature of trie 181and formation term _in

I.1)

₁₈ that it limit8 trie adatom den81ty

on large terraces.

-z

-io-5 0 5 io

~@ ^~

~w o

$

W(u)

z

-0.5 1

~

-io

u

-1

-4

In/1

Fig. 2. Dimensionle88 Surface current _as a function of Surface inclination fD

If.

Trie _upper in8et 8how8 trie "free _energy den8ity"

V(u)

obtained by integrating the

(negative)

_current function, while the lower in8et depict8 trie potential

W(u)

=

-V(u) J(m)u

that appears in trie mechanical analogue

m Section 4; the dotted fine indicates trie _energy of trie partiale following trie "bounce" trajectory.

n

by

_nD and _x

by

ÉD,

equation(1.1)

is then made

dimensionless,

and the _current

(1.3)

_can be written _as [4]

J =

FlDv7(1/lD). Il.6)

The function

q7(o)

can be

expressed

in terms of _an

elliptic integral(~)

and is shown in

Figure

^2.

Its asymptotics

q7(o

_-

0)

m

n/2, q7(o

-

ccl

_m

1la Il.ii

is

easily

^{inferred from}

equation il.1).

In

Figure

2 _we have in fact

plotted

^the current _{as a} function of trie inverse terrace

size,

which is

proportional

to the surface inclination _m

=

a/1,

where _a is trie lattice

spacing.

This is convenient for

discussing

trie

stability

of trie surface [Si. ^On a coarse

grained

scale

(much larger

than trie _terrace

size1)

trie surface

dynamics

_can be described

by

_a continuum

equation

of conservation type [2],

ôth

₊

ôxJ(ôxh)

₌

F, (1.8)

where

h(x, t)

represents trie surface

profile

at time t and J is the surface current evaluated at trie local

slope ôxh,

_1-e-,

setting a/1

₌

ôxh

in

equation Il.Gi. Expanding

around _a solution of

(~) The _inver8e of ~k(a) e oqJ(cv) ^{8at18fie8 the} bound8

(21b)~~~~

lniil

+

~b)/il

_~b)i 1 _OE 1 12~b/il ~b~)i~~~

fixed

slope

_{m, as}

h(x, t)

_{= mx} + Ft +

é(x, t),

the deviation _é is found _to

satisfy

_a diffusion

equation

ôté

=

vôjé _il.9)

with diffusion coefficient _v

=

-J'(m).

Trie

stability

condition is

J'(m)

_{< O,} and

comparing

to

Figure

2 _{we see} that surfaces _are stable

(unstable)

if1 _< ÉD

Il

_{> ÉD}

).

Trie relation 1m ÉD defines trie transition between step flow and island formation _m

epitaxial growth

_[8]. We bave thus arrived at trie well known conclusion that trie Schwoebel eflect stabilizes trie step ^flow

growth

mode but is

destabilizing

in trie island formation

regime.

These arguments

disregard

two

important

_aspects of actual

epitaxial growth. First,

real surfaces _are two dimensional and real steps can therefore meander in trie transverse direction.

In 2+1 dimensions trie second term in

equation (1.8)

becomes trie

divergence

of _{a current} vector, which _can be

written,

under trie

assumption

of

in-plane isotropy,

_as

là,12]

J(Vh)

=

F£Dv°(alô~(Vh(~~)(Vh(~~Vh. ^Iiiùi

Repeating

the hnear

stability analysis

around _a surface tilted in trie

x-direction, h(x,y, t)

=

mx + Ft +

e(x,

_y,

t),

_{we now} obtain

ôté

=

ujjôje

+

viô(é Iiiii

with coefficients [9,

iii

vjj ⁼

jF£D/m~)ç7'ja/£Dm),

_vi

"

-IFÉD/m)ç7ja/ÉDmj. jl.12j

While trie

stability parallel

to trie tilt follows trie pattern

already discussed,

trie transverse coef- ficient _vi is

always negative,

and

consequently

trie surface is

always

unstable

against

transverse

fluctuations(~).

This _{is a} manifestation of trie step

meandenng instability

described

by

Bales and

Zangwill

[13]. ^It is

clearly

^related to trie

specific

form of the function

q7 obtained from trie BCF

theory,

which _can be

justified only

for vicinal

surfaces; truly

stable orientations _are pos- sible if _more

complicated

^functions _are used which

include,

_e-g-, additional

symmetry-related

zeroes [5,

12].

Returning

to the

simpler

world of1+1

dimensions,

_we remark that _a second

important

feature

missing

in the BCF calculation _are

fluctuations.

Elkinani and Villain [14]

recently investigated

the combined eflects of step

edge

barriers and _a

specific

kind of fluctuations due to the randomness of island nucleation events "nucleation noise"

),

in a rnodel of _a

singular,

_one-

dimensional surface.

Here,

_we

investigate

_a model which incorporates ail relevant fluctuation

sources

ils] deposition

_noise, surface diffusion noise and nucleation noise and ask how trie

fluctuations affect trie stabilization of one-dimensional vicinal surfaces

predicted by

trie BCF

theory.

Dur central conclusion is that one-dimensional step flow _is destabilized

by fluctuations,

but in _a rather

interesting

_way: for

large

values of ÉD

Il

_we find _an extended metastable step flow

regime.

In trie _next section _we introduce the model and describe trie main numerical and

analytic

results. In Sections 3 and 4 _we present some evidence in favor of _our view that trie metastable behavior _is generic and not due to trie rather drastic

simplifications

inherent in

our model. Section 3 contains _a brief discussion of related behavior _{in a more} realistic

growth

model

[16],

^{while Section 4}

places

the

phenornenon

into the framework of the Cahn-Hilliard continuum

theory

for the

decay

^of metastable states

iii].

Conclusions _are olfered in Section 5. Some exact results

pertaining

to trie D

IF

_{- cc} limit of _our model _are

relegated

_{to an}

Appendix.

(~) ^This point _was ^{overlooked in references} [2] ^and [Si.

1" "1

k (. _~

S. à ₄

)

; é_j '

~ '~ '.

~ f ~

Fig. 3. The 8tochastic Schwoebel model. The _move8 indicated by cro88ed _{arrow8 are} forbidden.

2. The Stochastic Schwoebel Model

We represent ^{the surface}

by

_a set of

integer height

^variables hi defined _{on a} one-dimensional lattice of L sites. An _average tilt _m is

imposed through

helical

boundary

conditions

hi+L

₌ hi + mL.

(2.i)

The average terrace size is thus 1

=

1/m.

The initial condition is _a

regular

step train

h~ =

intj/j j2.2)

prepared

at time t

= 0.

Deposition (hi

_- hi +

i)

_occurs at rate F at

randomly

selected sites.

Singly

bonded atoms attempt ^diffusion

hops

to nearest

neighbor

sites at rate D. Due to an

infinitely high

barrier these

hops

_are restricted to remain within the _same atomic

layer;

in

particular,

_an atom

deposited

_on top of another

singly

bonded atom is not allowed to

hop

down.

Dimers and

larger

islands _are also immobile and stable. Trie allowed _{moves are} summarized in

Figure

3.

2.1. LIMITING CASES. Trie

Orly

_parameters in trie model _are trie surface inclination _m

=

i

Ii

and the diffusion

length

ÉD, whicu is controlled

by

the ratio D

IF; throughout

the _paper

we wiil _use tue number of

deposited layers

_{as our} time

scale,

such that

elfectively

F

= i.

By

virtue of _our somewuat restrictive

assumptions,

tue model

incorporates

two

simple limiting

cases.

First,

for _m

= 0

("singular" surface)

trie

complete

absence of

interlayer

transport ^allows

one to prove [18] ^{that trie}

heights

bave _a Poisson distribution witu _mean Ft. This

implies,

in

particular,

that the surface widtu W is

given by

W~lt)

=

((hi(t) lhilt)))~)

=

lhilt))

= Ft

12.3)

independent

of D

IF

and

independent

of trie systern size

L,

_{1-e-, tue} widtu behaves _as in random

deposition

_[19]. Tue actual surface

morphology

in this

regime

consists of _{an array} of steep

hills,

with _a spacing determined

by

trie spacing between nucleation _events in trie first few

monolayers

(Fig. 4).

This

length

scale _can be

extracted,

_e-g-, from trie first

peak

of trie

height

dilfierence

a)

¹⁰

t=2.4 ML t=3992 ML 4200

8

4100 6

k ₄₀₀₀

4

3900

~ 3800

0 60 120 180 240 0 60 120 180 240

~

i

~

t=2.4 ML t=3992 ML

~~~~

8

4100 6

k ₄₀₀₀

4

3900

~ 3800

0 60 120 180 240 0 60 120 180 240

Fig. 4. Depo81t8 grown on a flat 8ub8trate with variou8 ratio8

D/F;

a)

D/F

= 5 _x 10~ _and b)

Dl

F ₌ 5 _x 105. _The

8ame 8y8tem8 are 8hown iii _an early Stage

(left)

and in _a later Stage

(right).

correlation function

Gjr,tj

"

((hz(t) hz+r(t) +Tl~i)> 12.4)

and it

corresponds precisely

to the diffusion

length

ÉD of interest in studies of

submonolayer

epitaxy. The data shown in

Figure

⁵ confirm the

prediction

_[10]

ÉD ~-

ID /F)~/~ (2.5)

in one dimension. On scales _{r <} ÉD ^the correlation function

(2.4)

increases

linearly, Gir, t)

_m

G(1, tir,

hence

G(1, t)

_measures trie

slope

^{of the hillsides} in

Figure

4. Since G becomes of the

order of the width W at _{r m} ÉD this

slope

increases _as

Gji, t)

_~-

iii jftji/2 _j2.6j

The

morphology

of

singular

surfaces _con also be understood within trie continuum

theory

described in Section 1. At _m

= 0 trie diffusion coefficient in equation

(1.9)

is _v

=

-Fi[.

If _we start from _a

slightly

disordered initial condition, then surface fluctuations with

spatial

wavenumber _grow at rate

w(q)

=

Fl[q~.

In trie absence of

stabilizing

surface tension elfiects

(see

Section

4)

^{the dominant wavenumber is determined}

by

^the structure of trie first

~ 20

é

10

~ m=0 _a

m=1/2 _o

m=1 ~

(D/F)°'~~°'°~

1000 _{10000 100000}

D/F

Fig. 5. Diffusion length iD for _vanous surface tilts extracted from trie tower density _pT at fate limes

(between

t ₌ 10~ ML for _m

= 0 and t ₌ 10~ ML for _m

= 1, for _a system of size L

=

10080).

Trie

tower density _is defined in Section 2.2; in the asymptotic Poisson regime _pT

=

lliD.

The predictiou

£D _+~

ID IF)

^~" is verified for ail _m. Since the relaxation of _pT towards trie asymptotic Poisson value is very slow for _m # 0, the diffusion lengths for _m

=

1/2

and 1 _are overestimated. Nevertheless there is _a

slight variation with trie surface tilt. Alternatively £D _con be determined from trie first minimum of the correlation function

Gir, t)

at fate times which corresponds to

£D/2

and yields, within the statistical

errors, the _same result.

monolayer.

Thus the relevant mode _{is q}

r~

1liD,

which _grows ai rate _{ùJ r~} F

independent

of

iD1 consequently

the hills form

already during

the

deposition

of the first few

monolayers. Clearly

under these conditions the continuum

approach

is _non very useful.

Simple analytic predictions

_are also available for vicinal surfaces with

D/F

= cc. In this

limit every

deposited partide

is

immediately

transferred to the

ascending

_step, and step ^flow

growth

is enforced

"by

hard". The elfect of fluctuations _con be obtained

by adding

a Gaussian shot noise

term(~) i~(x,t)

with

l~l

= °,

l~lx,t)~lx',t'))

=

Fôlx x')ôlt t') _12.7)

to the linearized continuum

equation Iig),

which becomes then trie one-dimensional Edwards- Wilkinson

(EW)

equation [2, 20,

21].

^For tD _{- cc} at fixed t the surface current

(1.6)

reduces to

J(m)

=

F/2m, (2.8)

and the diffusion coefficient in

ii-g)

is

given by

v =

-J'(m)

=

F/2m~. _(2.9)

(~) Other types of fluctuations [15] are irrelevant for trie large scale behavior.

100000 _2.5

2

10000 _1.5

1000

(~(t)

_{~Î) î'~}

~ 0.5

(0.566i0.003) _m .''

_

0

Ù 0 2 3 4

~~~ ^m

~Î

,,'

, ^m=Ù o

,,'

^m=1/5+

,,'

^m=1/20

lÙ ,J m=1 _~

,,'

m=2 a

,,'

m=4 _.

,~ t

, ij~

/ _~t

J' ,'

,

~

,'

l 10 100 1000 10000 100000

tmÀi

Fig. 6. Time evolution of the squared surface width W~ _for

vanous surface tilts

IL

= 10080, single

rua).

The solid fine of slope corresponds to the trie random deposition behavior in trie Poisson

regime, trie dashed fine of slope

1/2

to the Edwards-Wilkinson scaling _m trie step flow regime, and trie dashed fine of slope 2 shows that trie width increases linearly _in the intermediate regime. Inset:

Amplitude of

(W~

W,~) in the step flow regime where W, denotes trie intrinsic width. Trie slope of the solid fine matches the analytical prediction ~~~'~ _ts 0.564.

Solving

the EW equation _one finds that [21]

w2(1)

=

j(Fi)1/2 ^(~,io)

for times 1 < Fi < m~L~. In the

following

_{we compare} these

predictions

to simulations carried out at finite

(but large)

values of tD

Ii.

Some further results for D

IF

= cc con be

round in the

Appendix.

2.2. BREAKDOWN OF STEP FLow.

Figure

6 shows the surface width _{as a} function of time, for D

IF

₌ 5 _x 10~

(corresponding

_to

tD

*

40)

^and various tilts _m. For _m > 0 _we observe

an initial step ^flow

regime,

_more extensive for

langer

_m

(smaller

terrace sizes

1),

in which W~

r~

t~/~

_with

a

prefactor given by equation (2.10) (inset).

The step flow

regime

terminales ai a well-defined transition lime t2,

beyond

which the width increases much _more

rapidly;

ai _a

second charactenstic time T the

asymptotic

Poisson behavior

(2.3)

is reached. In the transition

region

t2 < t < T W _grows

faster

thon the

purely

ramdam

t~/~ behavior,

_as W

r~

t~ with

fl

m 1.

This is

interesting

in _view of several recent experiments where values of

fl

close to

unity

have been observed, and _a relation to the Schwoebel elfect has been

conjectured

_[22].

The microscopic

origin

of the

instability

is illustrated in

Figure

7. Around t ₌ t2 well- defined features appear on the surface which _are charactenzed

by

_a

large

tilt opposite to the average surface inclination. As the

growth proceeds

_more and _more of these structures appear,

iooo

800

600

400

20D

0 60 120 180 24D

Fig. 7. Surface evolution of _a deposit _{grown on a} tilted surface with _m

=

1/3

and D

IF

= 5 _X 10~.

leading asymptotically

to _a

hill-and-valley morphology

similar to thon of the Poisson

regime (see Fig. 4).

It appears therefore thon the

instability

is associated with

large

^ramdam deviations

from the average

inclination,

which _are net

captured by

the lineanzed fluctuation

theory.

L. PT +

100 G(1 ) _~

J _O

~~' t~~~,,"""'

+ ,,'

~~J ,,"

ç~ 1Ù

il

,""

~ + ~ ~ ,"

~ _++H+

m ++ W

p +++

~

W WW

Q5

+W ++ + +

~Q _",

cL

Q ".

O '-_ f

".

Ù-1 "-

".

".

"_

10 100 1000 10000 100000

t imLi

Fig. 8. Time evolution of the number of towers L _{* pT,} the average step height

G(1,t)

and trie

surface current J _{on a} vicinal surface with _m

= 1 and

D/F

= 5 _X 10~

IL

₌ 10080, single rua). Trie step height is measured relative to the average tilt

(see

Eq.

(2.4)).

Other

quantifies

_con be used to monitor the

instability.

In

Figure

8 _we show the surface current

J,

the average step

height G(1, t),

and the

density

of _towers

(defined

_as froc

standing

columns of unit width and

height

>

2)

_as functions of time. In the step flow

regime

^the current is

given by

the

analytic expression (2.8),

but ai t _{m T a}

rapid decay

sets in,

approximately

described

by

J

+~

Ill-

The step

height G(1, t)

is constant

(of

order

m)

in the step ^flow

regime

but _starts

growing

_as

(2.6)

around t

= T.

Finally,

^trie tower

density

_pT is _seen to increase

linearly

with time within the step flow

regime

and _saturate ai t m T; this last observation

provides

the basis for the

analysis

that _we present next.

2.3. ANALYTICAL ESTIMATES.

Retuming

to

Figure

7, _we note that each of trie localized features contains

exactly

one tower at its

highest point.

In the Poisson

regime

the distance between _towers is

given by

the diffusion

length tD,

hence _pT saturates ai

tô~.

It is therefore useful to introduce the number

of

towers pet

diffusion length p(t)

=

tDpT(t).

To descnbe the lime evolution of this quantity, _we make _a number of

simplifying assumptions.

First,

_{we assume} thon the diffusion

length tD (defined

from the correlation function

(2.4)

in the

asymptotic

Poisson

regime)

is

independent

of the surface inclination _m, and of the order

ID /F)~R _numencally

_we detect _a weak

dependence

_{on m,} but the relation

(2.5)

refrains valid

(see Fig. 5). Second,

_{we assume} that _every tower that is formed _{on one} of the terraces survives and

rapidly

_grows into _one of the features shown in

Figure

7; this

neglects

trie

possibility

that

a small tower is

caught

by _a moving step and trie step flow mode is

temporarily

restored,

but it _seems to be _a reasonable approximation _in view of the sudden

morphological

evolution

evident in

Figure

7. Third, we assume that the adatom

density

in the step ^flow

regime

_con be

With these caveats in

mind,

_{we con} write down _an evolution

equation

for the number of

towers per diffusion

length.

In the step ^flow

regime

_a tower is formed whenever _an atom _is

deposited

_on top ^of a mobile adatom. The

probability

for this to

happer

is

given by

^{the adatom}

density (2. Il).

Therefore

dp/dt

₌

(F/D)t~tD Il p) (2.12)

where the factor

il p)

accounts for the tact that at time t

only

_a fraction

il p)

of the surface is still

growing

in the step ^{flow mode. The solution of}

(2.12)

is

pli)

= 1-

expi-t/T) j2.13)

with the characteristic lime

T r~

(D/F)t]~t~~

r~

m~(D/F)~H (2.14)

Ai limes of order _T most of the surface bas

completed

tue transition to tue Poisson

growth

mode.

Tue various

quantifies

discussed in Section 2.2 _{con now} be obtained

by treating

tue surface _as

a

superposition

of _two

phases,

_one with

weight

_p that follows tue asymptotic Poisson behavior and _one with

weight

1- p thon is described

by

the step ^flow

predictions.

For the width this

yields

w~lt)

= pi +

Ii p) )t~/~ ^12.isl

~

(here

the

prefactors

have been included _in order to _ensure the _correct

limiting

behaviors for small and

large t).

The two terms in equation

(2.15)

become

comparable

ai _a time

t2 _~

(mT)~/~

r~

m~(D/F)~/~

_{< T,}

(2.16)

associated with tue initial point of

departure

from step ^{flow behavior} in

Figure

6.

Moreover,

in the transition

region

t2 < t _{< T}

(which

_{con encompass many} orders of

magnitude

for

large

D

IF)

trie width increases

linearly

with

lime,

^W~ _r~

t~/T,

_in agreement ^with

Figure

6. We _note in

passing

^that t2 is also tue time at which tue Edwards-Wilkinson correlation

length (A14)

becomes of the order of the diffusion

length

tD.

The transition limes for other

quantifies

_may differ from both

(2.14)

and

(2.16).

For

example,

for the step

height

the

superposition

ansatz reads

G(1, t)

_m

ptô~t~/~

+

(l p)m, (2.17)

compare ^to

(2.6),

and the two terms become of similar

magnitude

ai the lime to +~

(mtDT)~/~

+~

m~(D/F)~/~ (2.18)

which is intermediate between t2 ^and T. In

fact,

_a ^whole _sequence ^of transition times and associated

roughness

_exponents describe the behavior of the

height

fluctuation moments defined

by

[23]

lwolt))~

~

([ht Ftl~)

'~

pt~/~

₊

Ii pl'~~~/~t~/~ 12'19)

(the

standard width defined

by (2.3) corresponds

to

W2).

For Wq ^the

instability

sets in at _a lime

tq,

^{and in} the intermediate

regime

_t~ < t < _T the

growth

law is Wq r~

t~q,

with

tq r~

m~(D/F)~/~~+~)

and flq =

1/2 +1/q. (2.20)

This type ^{of behavior is} sometimes referred _{to as}

multiscaling [23, 24].

Finally,

_we

give

_a

rough exploration

for the behavior of the surface _current J shown in

Figure

8. In the Poisson

regime

^{the surface} consists

entirely

of _very steep

regions

where almost

ail terraces have unit width. Diffusion _{moves are}

possible only

when _a fluctuation

brings

two terraces to the _saine level. Since the average

slope

increases _as

t~/~ (see Eq. (2.6)),

the

probability

for such _a "vertical collision" decreases _as

Ill;

this _accounts for the

decay

of J in

Figure

8.

2.4. THEORY VERsus SIMULATIONS. In the above

analysis

_we stated the existence of _two

types ^of crossover limes, _one

ii,

₌

0,1, 2..) depending

_on trie

quantity

under

consideration,

which describes trie _crossover oui of trie step flow

regime,

and _a second

jr)

_common to ail quantifies, ^{which describes the} onset of the

asymptotic

Poisson

regime;

unless _{t~ m T an} intermediate _regime exists for _t~ < t < T.

Although

we are Dot able to resolve ail of these

crossover times, our numerical data support trie

approach

to describe the _crossover behavior _as

a

superposition

of two

phases weighted by

the normalized tower

density p(t). Figure

9 shows

scaling plots

of W~ _obtained

by superimposing

data for fixed

slope

m and different values of D

IF, (a)

for the first _crossover around t ₌ t2, and

(b)

for the second _crossover ai t _{= T.} From

equation (2.15)

_one expects the

scaling

forms

W~

"

lD/F)~~~fillD/F)~~~~t) _12.21)

for t ~3 t21 ^a~~

w2

=

iD/F)3/4/~iiD/F)~~/~t)

_~~'~~~

for t m T, in excellent accord with the numerical data.

In order to resolve the _{crossovers even} for the smaller ratios

D/F

_we have used _a rather

large

slope m = 1 in these

plots.

Since there is _an

underlying

intrinsic

roughness

due to lattice effects

[25],

^{the choice of such} a

large

_m has aise the

advantage

that this intrinsic contribution

is

negligible.

The intrinsic width has to be taken into account,

however,

in order _to

verify

the

slope dependence

^{of the} crossover times. For fixed

D/F

the

superposition

ansatz

(2.15) yields

a

scaling

form

W~

=

m~g(t/m~), (2.23)

however the best data

collapse

_was achieved when

rescaling

lime with _a

slightly larger

_power

of _{m, as} m~.~

(Fig. 10).

The _sonne

applies

to the

analysis

of the step

height G(1,t).

For the current J the best overall data

collapse required

_a

rescaling

of time with

m~.~~(D/F)°.~

(Fig. Il).

In this _{case a}

comparison

with the

superposition

ansatz is limited

by

the tact thon the behavior of the current in the Poisson

regime

is net known

(apart

from the

qualitative

argument

given

ai the end of the

preceding section).

The

discrepancy

in the

slope dependence

of the _crossover times _carnet be attributed to terrace _size fluctuations: In the

Appendix

ii is shown that the variance of the

stationary

terrace

length

distribution in the step flow

regime equals

^i~ + t, hence

replacing

(1)~

by (i~)

in

(2. Il)

would lead _{to a crossover} to a linear

dependence

_{on m} for

large

_m

(small t). Instead,

_we

believe thon the weak

point

in _our arguments ^is trie

assumption

thon trie diffusion

length iD

used _as the normalization factor _in the definition of

p(t)

is

independent

of the _average surface

slope,

whereas in fact ii increases with

increasing

m

(compare

to

Figure 5).

However since _we

have _no

insight

into the

ongin

^of this

elfect,

_we chose non to include _it in the

analysis.

>~

%

#

°Îk1

o-1

01

b)

D/F=5e2 ... ...

F=5e3

.---

10 D/F=5e5 x

1

~

(

°Î~ _~'~~

coi

.oooi ""'~"'

0.0001

0.001

t

lmlUl

x

~~~~~~

(x/K)~~~

ioooo

iooo

CQ$

É

_IÙÙ

~Ît~

io

i

o-i

o-i i io ioo i ooo ioooo i ooooo

t imLj i m~.~

Fig. 10. Scahng plot of W~ _{at fixed} D IF and _vanous slopes _m

(D/F

= 5 _X 10~, _m

=

ils,.

,

4, L =

10080),

for comparison with trie scaling form

(2.23).

i

o.oi

o.coi

1e-05 o.oooi o.coi o.oi o-i i io ioo

t imLj (m2.33(Difj°.7j

Fig. Il. Scahng plot of trie surface current. The plot certains data for D

IF

= 5X10~,5X10~,5X10~

with _m

=1/5, 1/2,

1, 2, 4 and

D/F

₌ 5 X10~ with _m

=1/8,

1/5,

1/4, 1/3,

1/2, 1, 2, 3, 4

IL

=10080).

;."

Q _;."

(

100 "'

/,

/ /

10 "'

/ ,' / / ,' /

10 100 1000 10000

t jmLj

Fig. 12. Time evolution of W~ _{for various surface tilts in the realistic model with} isotropic surface diffusion [16]

(D/F

₌ 200, ^E ₌ 3, L ₌ 120, averaged _{over up} to 1000

ruas).

Compare to Figure 6.

3. Realistic Models

Ai this point the reader _{may worry} thon trie

phenomena

described in the previous section _are artifacts of the

cornplete suppression

of

interlayer

transport in our rnodel. We will instead argue

thon,

while _some of tue detailed estimates in Section 2.3 may indeed

depend

_on this

simplification,

the

metastability

scenario itself is

generic

for

(1+1)-dimensional growth

in the presence of step

edge

barriers. Dur argument ^{consists of} a numerical and _an

analytic

part. ^Here

we demonstrate that the

phenornenology

of trie stochastic Schwoebel rnodel _con be recovered within _a rather realistic rnodel for

epitaxial growth

_[16],

provided

the step

edge

barners _are made strong

enough.

In the

following

section _we show how the concept of

metastability

arises

naturally

from _an

analogy

with

phase

separation.

Figure

12

depicts

simulation results obtained with _a

growth

model [16] ^{thon contains non}

only interlayer

transport, but in fact realizes _an

isotropic

surface diffusion

algorithm

^which

allows for the formation of

overhangs

and bulk vacancies; step

edge

barriers _are

implemented by suppressing

'around the corner' _moves

along

the

arclength

of the one-dimensional

surface,

1-e- _moves to next nearest

neighbor

sites _on the square

lattice, by

_a factor

exp(-E).

For strong

barriers, E

= 3, the behavior of the width is _seen to be

closely analogous

to that shown in

Figure

6. The main difference lies in the laie lime behavior t > T. Instead of _an unlimited Poisson

regime

the realistic model shows _a

proliferation

of bulk defects and _an associated transition to

amorphous growth govemed by

the

Kardar-Parisi-Zhang equation

[16]

(Fig. 13).

10uBNAL DEPBY81QuEL T.5.W8. AuGu8T 1995 43

6oo

400

'

t ~Î

'

'

Il

t'

'

,

I ' '

Fig. 13. Deposit _{grown on a} flat substrate _usmg the model with isotropic surface diffusion

(D/F

₌

200, E ₌ 3). The figure shows the fate stage of growth, 1-e-, the topmost 700 loyers after the deposition of _a total of 20000 monolayers. Compare to Figure ^4.

4. Continuum

Theory

of

Metastability

The

analogy

between

growth-induced

surface instabilities and

phase separation

has been

pointed

oui

by

^{several authors}

(5,9, II,12, 26]. Adding

_a fourth order denvative term

-~ô]h

to account for

capillarity-driven

surface diffusion [27], ^{the continuum}

equation (1.8)

takes the form of _a

Ginzburg-Landau equation

for the conserved "order

parameter" u(x, t)

=

ô~h(x, t),

ôtu ₌

-à(J(u) «à]u

=

à((ôF/ôu) (4.1)

where F is the "free

energy"

functional

J~ =

/

_dz

jj~/2)ja~u)2

+

vjujx,i))j j4.2)

and the local "free energy

density"

V is determined

by

the inclination

dependent

current,

vju)

=

/" _{dwJjw). j4.3)}

The

capillarity

term is absent in the stochastic Schwoebel model because there is _no

interlayer

transport.

Physically,

the coefficient _« is the

product

of the surface stilfness and the adatom

rnobility

_[28]; it is therefore

proportional

to the collective surface diffusion constant, ^which however need _net be

simply

related to the "tracer" diffusion coefficient D of

single

adatoms used _so for _{m our} discussion [29].

The presence of the surface stiffness introduces _a further

length

scale

t~ +~

lK/F)~/~ 14.4)

scription

is

inapplicable,

_compare to Section _2.1. Since

t~

r~

F~~H

but

iD

~

F~~/6

_{in two}

dimensions

(loi,

the condition t~ > tD is

always

fulfilled at small

deposition rates(~).

The value of _« is

directly

accessible to _mass transport experiments (27, 29,

30].

A

typical

order of

magnitude

is (30] ~ m

1(pm)~

_per

hour,

which

yields t~

_m

1000À

_ai

a

deposition

rate of1 MLS~~.

The free _energy

density

V

corresponding

to the BCF current

(1.6)

is shown in the upper inset to

Figure

^2. It has _a

single

_{extremum, a} maximum ai u = 0, but _no stable minima.

Consequently

ail values _{m =}

lu)

of the average inclination _are either unstable

(if V"(m)

₌

-J'(m)

< 0, 1-e-, _m <

1/td)

_or metastable

(if V"(m)

_>

0).

In the metastable

regime

trie

homogeneous

state is

expected

to

decay through

the formation of _a critical nucleus (31]. ^Here

we outline the calculation of the nucleus and the associated free _energy barrier within the classical Cahn-Hilliard

theory Ii?i.

^While this

theory

is non without

conceptual problems (32],

ii is sulficient _to illustrate _some

pertinent

features.

The strategy ^{of Cahn and Hilliard} (17] ^{is based} on trie idea that trie critical nudeus is _a saddle point of the free energy functional

(4.2). Adding

_a

Lagrangian multiplier

to enforce

the conservation law

f dx(u(x) m)

= 0, the Euler

equation ôF/ôu

= 0 reads

~Q

^"

^~'(U)

^{+ "}

^~J(il)

^{+ À.}

^(4.6)

For

large

systems the violation of conservation due to the nucleus _can be

neglected,

and the

Lagrange multiplier

_is set to

=

J(m)

to _ensure that _~

= m far away from the nucleus. The Euler equation

(4.6)

_can then be viewed _as the equation of motion for _a partiale coordinate _~

traveling

^{in lime} x and

subject

_{to a}

potential W(~)

₌

-V(~) J(m)(~ m).

The critical nucleus

corresponds

to _a "bounce"

trajectory

_in ^which the

partiale

starts ai rest ai _{~ =} m, rolls downhill past ~ = 0, turns around ai a

negative

value m* _< 0 determined

by W(m)

=

W(m*)

and returns to _{~ = m}

(see Fig. 2).

Thus _a first conclusion _is that the

decay

of the metastable _{regime requires} the formation of _regions of

negatiue slope

m* < 0, in

qualitative

accord with

Figure

^7. A detailed

analysis(6)

shows that m*

= -0.278465 _m for

iDm

» 1.

In order _to estimate trie time scale for trie

decay

of the metastable state we need to compute

the _excess free energy of trie

nucleus,

_{i e.} the

integral

àJ~

= 2

/

_dz

_{jvj~~jx)) vjm)j}

=

w /~

_d~

_jwjm) wj~)ji/2, _j4.7)

where _~b

lx)

denotes trie bounce solution and

"energy

conservation" for trie classical mechanics

problem (4.6)

has been used. From the

general

form

Il.6)

of the current function it follows that

AF

=

Àjiô~Ç(miD), _(4.8)

(~) ^In the expenment of Orme et ai. [3] ^it was indeed observed that the initial wavelength of trie instability exceeded the diffusion length by _a factor of 15; _see the discussion _in (9].

(~) More precisely, _o

= -m*

/m

is the solution of 1 + _o + In _{a =} 0.