Damping of Growth Oscillations in Molecular Beam Epitaxy: A Renormalization Group Approach

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Damping of Growth Oscillations in Molecular Beam Epitaxy: A Renormalization Group Approach

Martin Rost, Joachim Krug

To cite this version:

Martin Rost, Joachim Krug. Damping of Growth Oscillations in Molecular Beam Epitaxy: A Renor- malization Group Approach. Journal de Physique I, EDP Sciences, 1997, 7 (12), pp.1627-1638.

�10.1051/jp1:1997105�. �jpa-00247476�

DanJping of Growth OsciUations in Molecular Beam Epitaxy:

A Renormalization Group Approach

Martin Rost

(*)

^and

^Joachim Krug

Fachbereich

Physik,

Universitht GH Essen, 45117

Essen, Germany

(Received

28

July

1997,

accepted

28

August 1997)

PACS.81.10.Aj Theory

and models of

crystal growth; physics

of

crystal growth, crystal morphology

and orientation

PACS.68.35.Rh Phase transitions and critical

phenomena

PACS.64.60.Ht

Dynamic

critical

phenomena

Abstract. The conserved Sine-Gordon equation ^with nonconserved shot noise is used to

model homoepitaxial crystal growth. With

increasing

_coverage the renormalized pinning _po-

tential changes from strong to weak. This is interpreted _as a transition from

layer-by-layer

to

rough growth.

The associated

length

and time scales _are identified, and found to agree with recent

scaling

arguments. ^A

heuristically

postulated nonlinear term

i7~(i7h)~

is created under

renormalization.

1. Introduction

In Molecular Beam

Epitaxy (MBE)

it is

possible

to control the amount of

deposited

matter

through

oscillations of the surface

roughness,

^which indicate that the surface grows

layer by layer [1-3j.

A

simple picture represents layer-by-layer growth

on a

high symmetry

^surface as

follows: After

deposition

out of the beam atoms diffuse _on the surface until

they

either meet other

diffusing

atoms to form _a stable

island,

or

get incorporated

at the

edge

of _a

previously

nucleated island. If most atoms

deposited

_on

top

of _an island _are assumed to be

incorporated

into its

edge by performing

a downward

hop (which implies

that the

suppression

of

interlayer transport by

Ehrlich-Schwoebel-barriers [4j ^is

negligible),

little nucleation _occurs in the second

crystal layer

before the first

layer

is

completed. Consequently

the surface width at

layer completion

is

nearly

_zero, after

having

_gone

through

_a maximum at half

filling

of the first

layer.

As

long

_as the

layer-by-layer growth

mode

persists,

the surface

morphology

exhibits oscillations with _a

period given by

the

monolayer completion

time.

In

general

this scenario is

only transient,

and the oscillations _are

damped.

A

variety

of mechanisms contribute to the

damping:

The surface _may be

slightly

miscut

[2, 5j,

or the average beam

intensity

_may ^be

inhomogeneous [3j. However,

_even in the absence of such

(experimentally unavoidable) imperfections,

the stochastic beam fluctuations _are sufficient to

destroy

the

temporal

coherence of

spatially separated regions

on the surface. Provided the beam noise is the sole

damping mechanism,

it _was

recently

shown that the critical coverage 9

(*)

Author for correspondence

(e-mail: marost@theo-phys.uni-essen.de.)

@

Les

#ditions

_de

_Physique

₁₉₉₇

at which the oscillations

disappear

scales with the ratio of the surface diffusion _constant

Ds

to the

deposition

flux F _as [6, 7j

1

~

jDs /F)&

₁₁

with _an exponent

~

'~4~~d'

^~~~

where d denotes the surface

dimensionality

d

= 2 for real

surfaces)

and the

exponent

_~y characterizes the

dependence

of the diffusion

length (or typical

island

size) iD

_on

Ds/F [8j,

iD

~

lDs/Fi~.

₁₃₁

Equation ii

_may therefore be rewritten _as

j

~

f4d/(4-d)

~

Id j~j

where

j

~

~4/(4-d) (5)

D

is the coherence

length,

_an estimate of the size of

coherently oscillating regions [7j.

The

theory

of reference [7j ^is based _{on a}

phenomenological

stochastic continuum

equation

for the

growing

surface. It is not clear _a

priori

that such _a continuum

description

would be able _to capture the

phenomenon

of

growth oscillations,

which is

distinctly

_a lattice effect. As _a first

step

towards _{a more}

complete treatment,

in the

present

work _we therefore

perturbatively

include the lattice structure

by analyzing

the

driven,

conserved Sine-Gordon

equation

3th

=

-KA~h AA(Vh)~

VA sin

~~~

+ F + _i~

(6)

a i

for the surface

height h(x, t).

In this

equation

the constant F denotes the average

deposition flux,

while the noise

i~(x,t)

models its "shot noise" fluctuations. The noise is assumed to be

Gaussian with _{mean zero} and correlator

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

= 2D

bit t')dlx x').

₁₇₎

To motivate the

systematic

terms _on the

right

hand side of

equation (6),

we note that it can

be written in the form of _a

continuity equation

3th+V.J=F+~ (8)

reflecting

the absence of

desorption

and defect formation under

typical

MBE conditions

[9-1Ii.

The current is

given by

Fick's

law,

J

=

-Vp,

where the

quantity p(x, t)

= -KAh

A(Vh)~

V sin

~~~

(9)

al

can be

interpreted

_as the

spatially varying

part of a coarse

grained

adatom

density

_{[9j or}

chemical

potential ill].

The first term

incorporates

_a

generalized

Gibbs-Thomson effect

[12j, according

to which the

density

is enhanced _near maxima and decreased _near minima of the

surface,

while the second term reflects the

dependence

of the

density

on the local

vicinality

[7,

II,13j.

The third _term models the lattice

potential:

Adatoms _are

preferably

driven to

places

where

they

_can be

incorporated

such that the surface remains _at

integer multiples

of the vertical lattice constant al-

Only

the lowest harmonic of the

periodic

surface

potential

is

kept,

since

components

^of

periodicity

_al

/2,

_al

/3,.

_are less relevant

(see

below and

[14j).

It is _necessary to

distinguish

the vertical _al and horizontal

ajj)

^lattice

constants,

^since

they play

_very different roles in the renormalization _group calculation.

The

analysis

of reference [7] ^was based _on

equation (6)

with V ₌ 0. Here _we show

that, by explicitly including

the lattice

potential,

the characteristic time and

length

scales

(1,5)

_emerge

naturally

in the renormalization group

(RG)

flow

equation

of the

potential strength

V. More-

over the

(Vh)~ nonlinearity

in

(9)

is _seen to be

generated

under

renormalization,

thus

relieving

us from the task of

postulating

its

microscopic origin;

_{we may} set

= 0

microscopically.

A similar scenario is valid for the nonconserved Sine-Gordon model

[15].

These results _are ob- tained

by applying

the Nozibres-Gallet RG scheme

[14j

to

equation (6).

An RG

analysis

of

(6)

was

previously presented by Tang

and Nattermann

[16j,

however these authors considered _sep-

arately

the _cases V

=

0, ~

0 and V

~ 0,

₌ 0 and thus _were not able to address the

generation

of from the lattice

potential;

in

addition,

_our

analysis

includes

explicitly

the flat initial condition of the

surface,

which is essential for

describing

transient behavior.

Since the

interpretation

of the RG results

depends crucially

_on

relating

the

"mesoscopic"

coefficient K to the

microscopic length

scale

iD,

^the next section will address this issue within the framework of the linear

equation IV

= =

0).

It turns out that the _mere existence of

a vertical lattice constant _al is sufficient for the nontrivial time and

length

scales 11,

5)

to emerge from the continuum

theory,

_even if this scale has _no

dynamical

effect

(I.e.,

V

=

0) [17j.

The full

problem

with V

~

0 is treated in the

following

sections. After

briefly recalling

the RG

scheme,

the renormalization of the conserved Sine-Gordon

equation

in 2+1 dimensions is carried out in Section 3. The RG-flow

equations

for the

parameters

in

equation (6)

are

interpreted

in Section 4, and the extension _to

general

dimensionalities and

general

relaxation mechanisms is

briefly

addressed. Some conclusions _are

given

in Section 5.

2.

Length

Scales in the Linear

Theory

On the most basic

level,

the

growing

surface

morphology

evolves in response ^to the

competi-

tion between

disordering

beam fluctuations and

smoothening

surface diffusion. The

simplest

continuum

theory

that

incorporates

^{both effects} is the linearization of

(6),

3th

=

-KA~h

_{+ F + y/.}

(10)

As _was mentioned in Section

I,

the coefficient K arises from _an

expansion

of the local adatom

density

in the surface curvature Ah. Under

near-equilibrium conditions,

it would therefore be

expected

to be

given by

the

product

of the surface stiffness and the adatom

mobility [12,18j.

However,

far from

equilibrium

other processes may contribute

to,

and in fact dominate K

[9,19j.

In

particular,

it has been

suggested _[20]

that random island nucleation

produces

_a contribution

K+~

Fit, ill)

but the

underlying microscopic

mechanism is not known. In the

following

we show how this relation follows from _a

simple reinterpretation

of

(10)

in the presence of _a finite vertical lattice

constant al

The

straightforward

solution of

(10) ii Ii

shows

that, starting

from _a flat substrate at time t ₌

0,

^after time t surface correlations have

developed

_up to a scale

ijt)

~e

jKt)1/4 j12)

and the surface width grows as

Wit)

_~

iD/K)~/~fit)~ _i13)

in dimensionalities d < 4, where

(

=

~ ~

(14)

2

is the

roughness exponent

^{of the linear}

equation ill].

Together

with the average

growth

rate F the presence of the vertical lattice constant induces

a fundamental time

scale,

the

monolayer completion

time

~~L = ai

IF. jis)

Setting

t ₌ TML in

(12)

_we obtain a

corresponding

lateral

length

scale

((TML),

the scale _on which lateral structure has

developed

after

deposition

of _one

monolayer. Clearly

it is very natural to

identify

this scale with the diffusion

length iD,

and hence the coefficient K in

(12)

can be identified _as

K _m

it _/TML

= a

j~fi[ (16)

in accordance with

ill).

The correlation

length (12)

can then be

expressed

in terms of the coverage 6

=

Ftlal

_as

j(t)

_m

tD9"~ ₍₁₇₎

To _see how the

scaling

laws 11,

5)

_can be obtained

along

similar

lines,

note first that for shot noise the noise

strength

D is

proportional

to the beam

intensity F,

D _m

ala(F. (18)

Thus

(13)

takes the form

W _{i al}

(aj liD)~~~9~~~~~/~ (19)

If _{one now}

postulates, plausibly

_[7], that the lattice effects

disappear

when the

roughne8s

due to

long wavelength

fluctuations becomes

comparable

to al, the characteristic coverage 9 _can be defined

through W(9)

_{1 al} and is

given precisely by equations (1, 2).

These considerations _may be viewed _{as a} zeroth order assessment of lattice

effects,

to be

justified by

the

systematic

calculation

provided

in the remainder of the paper.

They easily

extended to

general

linear

equations

with _a

dynamical exponent

_z

ill],

in which _{case one} finds [7j

d = ~y

~~

(20)

z d

forz>d.

3. Renormalization

Group Analysis

3.I. THE RENORMALIZATION SCHEME _OF

NOzIkRES

_AND GALLET. Because of the struc-

ture of

equation (6)

_we use an

approach

which is suitable for

general

forms of the

nonlinearity.

It _was introduced

by

Nozibres and Gallet for the

dynamical

renormalization of the Sine-Gordon

equation

to obtain the

roughening

transition

[14].

^{A detailed}

presentation

_can be found in their

work,

which _we recall

briefly.

Consider _a

Langevin equation

3th

= £h ₊

fit(h)

_{+ ~}

(21)

with _a linear

part £h,

_a nonlinear _term

fit(h)

and Gaussian noise

y/ with _{mean zero} and

correlator

(y/(k,t)y/(k',t'))

=

2(k(~"Dd(t t')d(k

+

k')9((k( A).

The cutoff A m

llajj

is

introduced to model the lateral lattice structure which does not allow for fluctuations _on scales smaller than the horizontal lattice _constant ajj. ^{In the}

following _ajj

will

only

_appear in the cutoff

A,

hence _{we can}

disregard

the distinction between ajj ^{and al and set} al " a.

Two

types

^of noise can be considered: Either volume

conserving noise,

which

corresponds

to the _{case p}

= I

("diffusion

noise"

[6]),

or

nonconserving

"shot" noise p = 0. Here _we focus

on the nonconserved

contribution,

_p

= 0, which

always

dominates _on scales

larger

than the diffusion

length _{iD (6, 7].}

A

study

of the conserved _case has been

presented

in reference

[21].

Renormalization of

equation (21)

^is

performed

in the

following

_way:

. We average over the short _wave

components

dy/ of the noise. In

k-space

_{dy/ is nonzero}

only

^for modes k with

ii dl)A

_{< (k(} < A. Define the

averaged

_{or coarse}

grained

field h _e

(h)&~.

.

Equation (21)

is

split

in two; one for the _coarse

grained

field

3th

= £h +

(fit(h

₊

_dh))&~

₊

_ij

and _a second _one for the difference dh _e h h. One _now takes _an

approximation

of

(fit(h+dh))b~

in terms of the

(hopefully all)

relevant

operators appearing

in

equation (21).

For this _one calculates dh

(respectively

its correlation

functions) perturbatively

in the

nonlinearity fit.

Since

fit

is not of _a

simple polynomial form,

_a

Rayleigh-Schrbdinger expansion

is used the

only

feasible _one, albeit

poorly

controlled.

.

Time,

lateral and vertical

length

_are rescaled with different exponents: x ~

(l dl)x,

h _~

ii ( dl)h

and t ~

(l

_z

dl)t.

This _{causes a}

corresponding rescaling

of the terms in

(21) yielding

the RG flow

equations

for £ and

fit.

The detailed

application

of these

steps

to

equation (6)

is the

subject

of the next. section.

3.2. APPLICATION _{To THE} CONSERVED SINE-GORDON

EQUATION.

Consider

equation (6)

in _a frame

moving

with the average

growth speed

F

3th

=

-KA~h

_{V A sin} ^~~

lh Ft)

⁺ i~.

(22)

a

Epitaxial growth

starts on an

atomically

flat

surface,

_so at time t=0 the initial

configuration

is hm0. Fluctuations _are caused

by

the noise at later times. This will

play

_an

important

role in the

interpretation

of _our results.

We

neglect

the

perturbative

contribution of the

-AA(Vh)~-term.

In the Introduction it

was

argued

to reflect the effect of the

dependence

of the

macroscopic

adatom

density

on the local surface tilt. Therefore _{on a}

microsopic

level in the continuum

equation

it is absent. It is

generated

under renormalization

by

the lattice

potential

and the

driving

force F to order

V~.

Although

it is a relevant operator at the linear fixed

point,

_we expect it to be

negligible

_on

small and intermediate

scales,

_as

long

_as the lattice

potential

contributes to renormalization of K and A. The _same is observed for the

(Vh)~-nonlinearity

_in the nonconserved Sine-Gordon

equation [15].

Averaging

_{over di~} leads to two

coupled equations,

where _we

expand

to lowest order in the infinitesimal

quantity

^dh

3th

= -K

A~h

_VA

sin b

(h Ft) ^{1 ~~ dh~))}

⁺

⁽

a a

3tdh

= -K

A~dh

VA

CDs

^~~

(h Ft)

^~~

^hj

^{+ dy/.}

⁽²³⁾

a a

The second

equation

of

(23)

is solved

by

_a

Rayleigh-Schr6dinger perturbation

ansatz

dh~°~

ix, t)

=

d~z' ~ dt' G(x x',

_t

t') _dy/(x/, t') (24)

dh~~~

ix, t)

₌ ^~~~

/ _{d~z' /~ dt' G(x x',}

t

t')

a o

x

A'

_cos ^~~

h(x', t') Ft'

dh~°~

ix', t')

a

In

k-space

the linear propagator ^{G is}

given by exp( -Kit t') (k(~).

^{In the}

sequel

_we will _use the notation

k',

_{i~', when the}

_argument

_{is the}

integration

variable

(x', t')

and

unprimed symbols

for

quantities

at

ix, t).

The

corresponding

derivatives

(Laplace operators)

_are denoted A and A' to mark this difference. For the convolution

integral f d~z' fj

_{dt' the shorthand}

_f

_{is used.}

Insertion of

(dh~)

=

(dh~°~~)

+

2(dh~°~dhl~~)

+

O(V~)

in the first

equation

of

(23) generates

the corrective terms for h to order

V~

(dh~)

=

(dh~°~~)

~~~

/(A'G)

^{cos ~~}

_(h' Ft') ^{C, (25)}

a a

where C _e

(dh~°~dh~°~')

^{is the}

unperturbed

correlator. Its Fourier transform is

given by

~-K(t-t'j)k)~

_~

/jj~j~j4) ¹ _~-2Kt]k]~j

correction

~~

In

eneral

_nsion

is

replaced

by

unit

sphere.

To handle

the

-~~ ) A sin ~~ h-Ft cos ~~

h'-Ft'

~

a a

e

^plit

the

It wouldcreate

higher

armonicsof the ^ttice

which are

less^relevant (see Eq. ₍₂₆₎ )

than the fundamental.

In

the mainingxpression we

split

into _terms with h

and Ft

- _~~~)~ A / [sin

~~ Fit'

_-

t)

cos ~~ (h - h') _(A'G)

a

a

+ _cos ~~

a a

erms

integrating

over d2(

we

get

~~~)~

_[A(Vh)~)

)sin

(~~F(t'-

t)j

a _a

- ~~)~

a

Up

to _{now we} have

expanded

the

averaged nonlinearity (fit(h))&~

to order

V~

_in

projections

onto the relevant

operators.

The last

step

consists in

rescaling

_space and time. To examine the behavior close to the linear fixed

point V=0,

A= 0 _we choose its

scaling exponents,

z = 4 and

( =2-d/2

=1 in two dimensions

(see (14)). Accordingly

the

growth

rate F and the lattice constants al and

ajj

^are rescaled _as

dF/dl

=

(z ()F, dal /dl

₌

(al

and da

/dl

= -a

3. 3. FLow EQUATIONS. The RG flow will be examined in terms of dimensionless

quantities,

in which the effect of the trivial

rescaling

has been eliminated. As

elementary length

scales _we

use the vertical and horizontal lattice constants _al and ajj, ^{and the basic time scale is}

given by

the

monolayer completion

time _TML

" al

IF.

Thus time is measured

through

the coverage

9

=

t/TML,

and the coefficients

K,

V and

appearing

in

(6)

are

replaced by

the

expressions

x~ _%

(~j

"

liD/~il)~

~ll v U

=

j

j~2

~ ^l

~~4

Due to the conserved form of the

equation

of motion

(6),

the noise

strength

D is not _renor- malized

[10,16],

and need _not be considered further.

Using

the

scaling parameter

~ eexp 41, ^the flow

equations

then take the form

ii)

4~

(~

₌

^{-4x~ ~} _~l e~~~°&) ^U,

~ x~

Iii) ~~)

"

il~ _glx~/~,91'

128) (iii) 4~()

=

il~ _flJ~/~,°).

The functions

f

and _g

depend

_on the

integrals

in

equation (27)

and are

given

in the

appendix.

Equations (28)

_are the central result of this _paper. The

following

sections are devoted to the discussion of their

physical

content.

4.

Interpretation

of the RG Results

As in the _case of the

roughening

transition

[14],

^the renormalization of U determines the relevance of the lattice _on

large

scales. In the present ^{situation the} lattice

potential always decreises

_under renormalization

(Eq. (28(I))). Indeed,

since in the absence of the lattice

potential

the

roughness exponent (14) (

> 0 in two

dimensions,

the lattice becomes irrelevant

asymptotically;

_a

roughening

transition is

possible only

if

(

=

0,

such _as for

equation (6) subject

to conserved noise

[21].

Nevertheless _on finite

length

_or time scales U remains

finite,

and its

dependence

_on 9 and _{~ may} be used to describe the transition from

(lattice-dominated) layer-by-layer growth

to

rough,

continuous

growth.

^It can be shown that for small U the

amplitude

of the surface width oscillations is

proportional

to

U;

thus the value of U _{on a}

given

time _or

length

scale is _a direct _measure of the observable

signatures

^of

layer-by-layer growth.

In the

following

_{we assume} that the rate of

particle deposition

is small

compared

to the diffusion rate, which is true for

typical

MBE conditions and

implies

that the dimensionless

stiffness

parameter

^Xl» I. The renormalization of Xl due to the lattice

potential,

_as

expressed by equation (28(it) ),

_can then be

disregarded,

and Xl becomes _a constant. This

decouples

the flow

equation

for the lattice

potential

U and allows for _a

straightforward solution,

which _can be

used to extract the

damping

time and the coherence

length

I. In Section 4.3 the

generation

of the

A(Vh)~ nonlinearity,

_as described

by equation (28(iii) ),

will be discussed.

4. I. DAMPING TIME. The

asymptotic

~ ~

cc)

value of the lattice

potential

U is obtained from

equation (28(I))

_as

UC~ =

e~~~°lUo (29)

with

~~~~

~~ _~~~

~ ~~~~~~ _'~

~

_~~~~

in the relevant

regime

Xl9 » 1. Thus

writing 1(9)

=

/~

_the characteristic _coverage is obtained _as

~

(7

~

~~~~~)(~~'

~~~~

in

agreement

with

(4)

for d

= 2.

Moreover,

since in this _case the surface width of the linear

theory

is

proportional

to

(9/#)"~ _(see

Sect.

2),

we see that the

decay

of the lattice

potential

is of the form

Uco/U0

_"

~XPl~Cw~19)1 132)

where C _> 0 is _a constant. This behavior has been found to describe the

decay

of oscillation

amplitudes

in

layer-by-layer growth

in numerical

simulations,

and _can be derived

assuming

a

discrete

probability

distribution of the

heights taking

at each

possible height

the value of the

corresponding

continuous Gaussian distribution

[22].

It is instructive to extend these results to other dimensionalities and linear relaxation mech- anisms with _a

general dynamic _exponent

_z. For

general

_z, the natural

scaling

variable is

~ =

expzl.

Then the

only qualitative change

of the

expressions

discussed above is that the

algebraic part

of the

integrand

in

(30)

becomes

~~(/~~~

_{instead of}

~~"~

_To

see

this,

note that the

only quantity changing

under

rescaling

in the

time-independent part

of the correction

dV/V

in

equation (26)

is the vertical lattice

spacing

_{al +~}

~~(/~ _Thus,

_for

(

< 0 the

integral (30)

_converges even when 9 _{~ cc,}

implying

that the surface remains smooth. For

(

_> 0 the

integral

becomes

1(9)

+~

Xl~~(Xl9)~~/~

+~

W~, (33)

showing

that

(32)

remains valid in the

general

_case.

Writing 1(9)

=

lo /9)~(/~,

the characteristic coverage is of the order

j

~

~z/2(-1 ~d/(z-d) j~~)

,

where in the last

step

the

scaling

relation _z

= d +

2(

for linear

growth equations ii Ii

has been used. This becomes

equivalent

to

(20) by noting

that

(16)

is

replaced by

K _m

ii /TML

^for

general

z.

4.2. COHERENCE LENGTH. For 9 _{~ cc} the solution of the flow

equation (28(I)

at finite _~ is

U(~)

=

exp[-(2x~ /Xl)(vi I)]Uo. (35)

If the renormalization is

stopped

at a lateral scale

Ljj

=

~"~ajj,

the

remaining

value of the lattice

potential

_may therefore be

written,

for

Ljj lajj

»

I,

_as

~~~ ~~~

~

~~"~~~ (36)

with

~ 12

I =

ajj

/$

+~

W, ₍₃₇₎

2~T ~ll

in

agreement

^{with the}

expression (5)

for the coherence

length.

If the

system

is smaller than I the lattice

potential

remains

relevant,

the surface remains smooth and

growth

oscillations will be

present

^{for all times} [7].

As in the

previous section,

these

considqrations

can be

generalized

to

arbitrary

z and d.

Then

(36)

becomes

U(Ljj)/Uo

_"

exp[-(Ljj li)~(]

with

j

~

~i/(2()

=

~i/(z-d)

~

ji~ lajj)~/~~~~~ (38)

in accordance with

(20)

and reference [7].

4.3. GENERATION _{OF THE} CONSERVED KPZ NONLINEARITY. We _now focus _on the flow

equation (28 (iii ),

^{which describes the}

generation

of the nonlinear term of the conserved Kardar-

Parisi-Zhang [23j (CKPZ) equation, A(Vh)~, through

the

interplay

of the lattice

potential

V, the

growth

rate F and the effective stiffness K. We consider the

stationary regime

9 _{~ cc,} and

again

assume that Xl is

large,

_so that its renormalization _can be

neglected. Setting

^L ₌ ^A ₌ 0 at the

microscopic scale,

the solution of the flow

equation (28(iii)

then reads

L(~)

=

) _/~

dz

z~~/~ exp[-(4x~ _/Xl) (fi I)] f(K/x), (39)

where the solution

(35)

for the flow of U has been used. For _{~ ~ cc,}

equation (39)

tends to a

finite

limit,

which for Xl _{~ cc} has the

simple

form

LC~ ₌

2x~Uj. (40)

The numerical solution of the full set

(28)

of

coupled

flow

equations

^shows that the

limiting

value

(40)

is attained for Xl >

10~, corresponding

to a diffusion

length ID

=

100ajj (See Fig. I).

In the range

10~

_< J~ <

10~

the nonlinear

coupling

LC~ is

po8itive

and increa8es with

increasing

Xl, while for smaller values of Xl it is

negative.

In _terms of the bare

parameters

of the

original equation,

the relation

(40) implies that,

_on

large

scales and for

large

diffusion

lengths,

the CKPZ coefficient is

positive

and of the form

v2

m °

(41)

Fa(

It is instructive to compare this to _a heuristic estimate of A, ^based on Burton-Cabrera-Frank

(BCF) theory [7, iii. According

to BCF

[24j,

the adatom

density

_on a terrace can be

computed by solving

a

steady

state diffusion

equation

with sinks at the surface steps. ^The

density

decreases with

decreasing

_step distance _or

increasing tilt,

and becomes

independent

of the tilt when the

step

^distance is of the order of the diffusion

length ID

The coefficient of the

leading

order

expansion (9)

around _a

singular

surface is then

positive (~)

^and

given by

ABCF

*

~~~ (42)

a

(~ For _a vicinal surface growing in the step flow mode _{one can} show that A < 0, _see [13].

02 Lm

o 15

o i

o 05

o

-o.05

lo~ lo~ lo~ lo~ io~ lo~ lo~°

Fig. 1. Numerical solution of

L(~

_~

cc)

_as defined in equation

(39)

using the full flow equations

(28).

The potential

strength

is Uo ₌ 0.1; for

large

values of K >10~ the value

27r~U/ (40)

is attained.

To

identify equations (41, 42)

we would need _to

require

that the bare

pinning potential

_Vo

depends

on the diffusion

length

_{as Vo *}

Fi[,

and hence the dimensionless

potential strength

is

Uo m

(iDlajj)~

=

fi

_»

_{1, (43)}

which is

clearly

inconsistent with _our

perturbative

treatment of the

potential. Thus,

the

expressions (41, 42)

_are not

equivalent,

but rather

correspond

to different

limiting

situations:

Our calculation is _an

expansion

for small Uo and fixed

(large)

Xl, while the BCF

picture

assumes

perfect crystal planes,

which would be

formally represented by taking Uo

~ cc at

fixed Xl.

Writing

LC~

=

LC~(Uo,

Xl) we have shown that LC~ =

U] #(Xl)

for

Uo

~

0,

where

#

is _an

increasing

^fimction of Xl, and the BCF argument ^{indicates that} LC~ _rw Xl for Uo ~ cc.

Clearly

the latter

regime

^is not accessible

by

_our method. Nevertheless it is remarkable that the CKPZ coefficient _emerges from the RG calculation with the correct

sign

^{and the} correct

qualitative dependence

_on the diffusion

length.

5. Conclusions

In this work _we have derived renormalization group flow

equations

for the conserved Sine- Gordon

Equation

with nonconserved shot noise.

They

_were used _to model the _crossover in

homoepitaxy

from

layer-by-layer growth

to

rough growth.

The _crossover can be

quantitatively

characterized

by

_a characteristic

layer

coherence

length I

and _an associated _coverage

9,

which

is a measure of the number of

growth

oscillations that can be observed under

optimal growth

conditions

(that is,

in the absence of miscut or beam

inhomogeneity).

The

dependence

of these

length

and time scales _on

growth parameters

is in agreement ^with dimensional

analysis

and numerical simulations [7].

Our

approach

also

predicts

the presence of _a nonlinear term of the form

A(Vh)~

_on

large scales,

which _was

suggested previously

_on heuristic

grounds [9,10]. By

power

counting

^{it is} seen

to be relevant in the

long

time limit and it has nontrivial effects _on the

scaling

behavior

[25j.

Two extensions of this work _seem to be

possible

within the conserved Sine-Gordon ansatz:

First,

renormalization of _a tilted surface

(as performed

for the

original

Sine-Gordon model in

Ref. [14] should

clarify

the influence of _a small miscut _on the

damping

of

growth

oscillations [2].

Second,

and _more

ambitiously,

the

implementation

of _an Ehrlich-Schwoebel-effect _{[4] may}

provide

_a

systematic approach

to

computing

the surface current induced

by step edge

barriers

[9, II, 26j,

^and thus contribute to

understanding

the transition from

layer-by-layer growth

to a

coarsening

mound

morphology ill, 27].

Acknowledgments

We thank H. Kallabis for

helpful

discussions. This work _was

supported by

Deutsche

Forschungs- gemeinschaft

within SFB 237

Unordnung

und grosse Fluktuationen.

Appendix

A

Scaling

Functions

The functions used in the flow

equations (28i)

and

(28ii)

are

1_

~-2W/~

91~/~ ₉₎

-

87r~

ii~/~~~

+ yr~i~

~/K I-i~/~)~ ^47r~i~/~)~

⁺

^7r~)

+ B

cos(27r9)

+ A

sin(27r9) e~~"/~

i

~-2~0/~

fi~/~ ₉₎

- 167r~

ii~/~~~

+ yr~i~

7r li~/~)~ ^87r~i~/~)~

⁺

^r~)

+ -A

cos(27r9)

₊ B

sin(27r9) ~~~%~l

with the

polynomials

A =

4x6~(Xl/~)~

+

lx 8x~6~)(Xl/~)~

+8x~6(Xl/~)~

+

(8x~ 4x~6~)(Xl/~)~

+

8x~6Xl/~ ^x~

B

=

-46~(Xl/~)~

₊

46(Xl/~)~

+

ii 8x~6~)(Xl/~)~

+16x~6(Xl/~)~

+

4(x~ x~6~)(Xl/~)~

+

12x~6(Xl/~)~ 5x~Xl/~.

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