Scaling behavior of driven solid-on-solid models with diffusion

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Scaling behavior of driven solid-on-solid models with diffusion

Martin Siegert, Michael Plischke

To cite this version:

Martin Siegert, Michael Plischke. Scaling behavior of driven solid-on-solid models with diffusion.

Journal de Physique I, EDP Sciences, 1993, 3 (6), pp.1371-1376. �10.1051/jp1:1993185�. �jpa-00246800�

Classification

Physics

Abstracts

61.50C 05.70C 68.22

Scaling behavior of driven solid-on-solid models with diffusion

Manin

Siegert

and Michael Plischke

Department

of

Physics,

Simon Fraser

University, Bumaby,

British Columbia, Canada VSA lS6

(Received J February 1993, accepted ⁹ February J993)

Abstract, We investigate a solid-on-solid model in which the interface of the

deposit

relaxes

through

_a surface-diffusion process controlled

by

a local energy function and obeys ^detailed balance. We find that in the steady state the

dynamic

structure factor of the interface

obeys scaling

with exponents which,

remarkably,

are

equal

to those of the Edwards-Wilkinson model for sedimentation. Thus the interface fluctuations in _two dimensions

diverge only logarithmically

^and

are therefore much smaller than those of all

previously proposed

models. We discuss the relevance of these results _to molecular beam

epitaxy.

Currently

there is considerable interest in the

modelling

of

growth

_processes such _as molecular beam

epitaxy (MBE)

in which the surface of the

growing

film relaxes

primarily through

diffusion of the

deposited particles.

This class of

growth

_process

gives

rise to _a much richer _« kinetic

phase diagram

» than is found for the _case of relaxation

through evaporation.

The latter processes seem to be well-described

by

the

Kardar-Parisi-Zhang (KPZ) equation [I],

which leads to the well-known

scaling

of the width of the interface

f(L, t) L' f(t/L~)

where

(at

least for one-dimensional

substrates)

the exponents ( and _{z can} take on

only

two values

corresponding

_to the two fixed

points

of the KPZ

equation.

In _contrast to this

relatively simple situation, growth

models which do not allow

evaporation

of

particles (as

_seems to be the _case in MBE

growth

at low

temperatures) generally

lead _to very

large

fluctuations of the interface

[2-4],

often with _an

exponent

( _m

I,

and to instabilities such _as the spontaneous ^{formation of} a

pyramidal

structure

[4].

Vfhether _{or not} either of these effects is relevant to

experiment

^remains

to be understood.

Most of the discrete models studied _to date _use ad-hoc rules _to model diffusion of

newly deposited particles.

In the Wolf-Villain model

[2]

the _most

recently deposited particle

_moves to

the _nearest

neighbor

site with the

strongest binding.

The rest of the

deposit

is inert.

Similarly,

in the work of Das Sarma and Tamborenea

[3]

and Yan

[5] particles

_are

only

allowed to _move a

limited number of

steps and, again, only

the _most

recently deposited particle

diffuses. The model of Kessler _et al.

[6]

allows all

particles

that _are not

fully

coordinated to move, but

only

to sites with _a

larger

coordination number. In _our

approach [4]

_we have

adopted

the

point

of view that at least the entire

exposed

surface of the

growing

film _must be allowed _to

participate

in the relaxation process so

that,

in the absence of _an

incoming flux,

the correct

equilibrium

state is obtained. We have considered _a solid-on-solid model

(no

voids _or

overhangs)

and have

1372 JOURNAL DE PHYSIQUE I N° 6

assumed that diffusion

along

the surface is driven

by

a local energy function. In

particular,

_we

have used the

following

Hamiltonian _:

H

=

J

£

_{[h~ h~ [~}

(l)

<>.1>

where the _sum extends _over nearest

neighbor

sites of the

underlying

substrate and where _n is _an

integer

that _we take to be

1,

2 _or 4. Particles _are

deposited randomly and,

between

deposition

events, the interface relaxes

by

the

hopping

of

particles

in the

exposed

parts ^{of the}

cluster, I-e-, particles hop

^{from the} top ^of a

given

column, _say I, to the top ^of a nearest

neighbor

column

Q)

at a rate

given by

W

=

r[exp(p AH(h,

-

h,

I, _{h~ - h~} + I

))

+

1]~

where AH is the

energy

change

due to this _move and _r is _an

attempt frequency.

This process

obeys

detailed balance and results in the correct

equilibrium

state of model

(I)

if the

incoming

flux is _{set to}

zero. It should be noted that the

equilibrium phase diagram

of model

(I)

does _not

depend

qualitatively

on the parameter n

whereas,

as we discuss below, the

dynamics

is very sensitive to this

microscopic

detail.

In _our

previous

work

[4]

we have

investigated

model

(I)

^with n =

2 and _n

=

4 in _one dimension. We found that the _n

=

2 model

displays qualitatively

the _same behavior _as other

previously investigated models, I-e-, scaling

of the width

f(L, t)

and the associated

height- height

correlation function. The

exponent

f is

roughly 1.2, indicating

that this model is

unphysical

in the _sense that when fluctuations _are of this

magnitude overhangs

and voids will

certainly play

a role and _must be included in the

description

of the

growing

cluster.

Surprisingly,

for the _n

=

4 model the

growth

_process is unstable and the

deposit

evolves toward _a

steady

state with _a broken symmetry,

I-e-,

_a

pyramid-

_or

sandpile-like shape

of the

interface.

Superimposed

_on this ordered state are the usual

long-wavelength

fluctuations which scale with exponents

indistinguishable,

for _our system

sizes,

from those of the _n

=

2 model.

This

instability

exists also for two-dimensional substrates and will form the

subject

of _a separate ^detailed

study [7].

In this article _we report our results for model

(I)

with _n

= I for _one- and two-dimensional substrates.

Although

identical _to the other models in _terms of its

equilibrium phase diagram,

the _n

= I model is

quite

different when driven

by

_a non-zero flux of

incoming particles.

Our main conclusions _are the

following

_:

(ii

the exponents y and _z

obey

the

scaling

relation

[2]

Y = z with _y

=

2 in both _one and _two dimensions _at all temperatures ;

(it)

because the exponent y which characterizes the

divergence of

the

steady-state

structure

factor

at small k is related _to the

surface roughening

_exponent (

through

(

=

(y d)/2,

the width

ofthe interface diverges only logarithmically for

two-dimensional

substrates,

in contrast to all

previously proposed

models

;

(iii)

the

instability

_seen in the _n

= 4 model is absent in the _n

= I model

(iv)

at low

enough

temperatures the width

of

the

inte~fiace of

the _n ;

=

I model is small

enough

that the

growth

_process is

essentially equivalent

to

layer-by-layer growth,

_even

for extremely

thick

films.

We _now

proceed

_to document and discuss these results.

We have carried _out

computer

simulations for model

ill

with _one- and two-dimensional

(square lattice)

substrates. Particles _are

deposited

at random and between

depisition

_events the film relaxes

through

the diffusion process described above. We have used

periodic boundary

conditions and have

investigated

this model for _a range

of-temperatures

both above and below the

roughening

temperature

[8]

in _two dimensions. The average

deposition

rate was set at one tenth of the diffusion

attempt-frequency-

We have also carried _{out a} few simulations for

a

deposition

rate of _one fifth of the diffusion

attempt-frequency

and found _no

change

in behavior. We characterized the

steady

state of _our

deposits primarily through

the behavior of the

height-height

correlation function and its Fourier transform. We define the function

4l(k, t)

to be

4l

(k, t)

₌ lim

(h(k,

t +

r) h(-

k, _r

)) (2)

where

h(k, t)

₌

L~~~ £h(r, ^t)e'~~,

^the sum extends _over the sites of the d-dimensional substrate and h

(r,

t

)

is the

height

measured in the

co-moving

frame. For t

=

0 this function

[9]

is the

steady-state

structure factor

S(k) and,

for small

k,

has the

scaling

form

4l(k, t)

=

k~ ^Y C (k~

t).

In

figures

I and 2 _we show the function

S(k)

in _one and two dimensions. In the simulations the entire surface _was

updated (deposition

or

diffusion)

20 L~ times in order _to reach the

steady

~e

- 0~ _T~0

~

'

tJ~

~

T~J/2k~

0

o L=512

D L=256

O L=128

o.oi

o.ooi o.oi o-i o.5

k/1c

Fig.

I.- The

steady-state

structure factor S(k) for the _n =1 model in _one dimension for two

temperatures. ^{Here k} ₌ 2 «j/L with j

= 1, 2, The data for T

= J/2 k~ have been scaled by a factor of 1/20 in order to attain

separation

of the _two

plots.

Statistical _{errors are} less than I fG. The

straight

lines _are fits to the form S(k)

=

Ak~~

o~

o L=128

_10~

~

° Lm64

$

T=2J/k

~ ~ ~~

1000

~ ~ L~16

T=o i o o

io

i

o.oi o-i o.5

Fig.

2. The

steady-state

structure factor S(k) for the _n

=

I model in two dimensions at T

= 0 and _{at a}

temperature ^{above the}

equilibrium roughening

temperature. ^In order to reduce statistical errors we have taken S(k) ₌ 0.5 [S(k~, k~) ⁺

S(k~,

k~Ii for k

= (k~ +

()"~

with k~ = 2

«j/L,

j ₌ 1, 2, The

straight

lines _are fits _to the form S(k

= Ak~^~

Except

for the smallest k for L

= 128, statistical errors are less than I fb.

1374 JOURNAL DE

PHYSIQUE

I N° 6

state. Data was then collected for another 20 L~

updates

_per

sample. Except

^for the

largest

system

(128

_x 128 _at T

=

0,

35

samples)

_we have data from several hundred

samples.

In the

figures

_{we see,} for small

enough k,

_an

asymptotic power-law

^{behavior with} an exponent _very close to y = 2. This is most evident at T

= 0 where the

asymptotic

behavior is well established

in both _one and _two dimensions for the smallest values of k for substrates _as small _as

L

= 128. We note that in two dimensions the value T

=

2J/kB

is

considerably

above the

roughening

temperature

[8]

and it _seems that this

equilibrium

transition is not reflected in the

non-equilibrium steady

state of the _n

=

I model.

Figures

3 and 4 show the function C (k~ t

=

4l

(k, t)/S(k)

for several values of k in _one dimension and for _two different substrate sizes and several values of k in _two dimensions. It is clear that the choice _z

= 2

provides

_an excellent

collapse

of the data to a universal _{curve over a}

+ L=256, k

0.8 ^x L~256, k~

-

O Lm256, k~

~$ ~

& L~256, k~

i~

^~ o L~256, k~

~i

^'

~i _0.4

i~

0.2

o

O 2 3 4 5

(kin)?

t

Fig.

3. The scaling ^{function C} (k~ t)in the T

=

o steady state of the one-dimensional _n

= I model for

a substrate of length L

=

256 for the five smallest k-vectors,

plotted

_{as a} function of the scaled variable k~ t. In this

figure

and in

figure

4, time is measured in number of monolayers deposited.

o L=64, k

_ a L=64, ^~

~$

0.8

o L=64, k~

it

x L=64, k~

~i

+ L=64,

k(

fli

°.6

a L=32, k

e '

0,4

0.2

0 DA 0.8 1.2 1.6 2

(k/1c)~

t

Fig.

4. The

scaling

function C (k~ t) for two-dimensional substrates of size 32 _x 32 and 64 _x 64 for several of the smallest values of k at T

=

o.

substantial range of the function C. We

conclude,

that _z

= 2.0 _± 0. I

independent

of substrate

dimensionality and,

_as

well,

that _z and _y

obey

the

scaling

relation

[2]

_z

= y. We

emphasize

that these results _are

substantially

different from those found for all

previously proposed

models

[2-6]

^which have either

yielded

values of _z and _y much closer to 4 than _to 2 _or the exponents ^of the KPZ

equation.

We _now discuss _our results in the context of _a continuum

Langevin equation

which

should,

in

principle,

be

capable

of

describing

the

scaling

behavior of this class of models.

Since the diffusive

part

^{of the}

growth

_process is the _{same as} for

equilibrium dynamics,

_we

expect

^{that the}

appropriate Langevin equation

in the

comoving

frame of reference will be of the form

[4,10]

3~h(r, ti

= v~ Ah

(r, t)

+

Ail

₊

(Vhi~i"~ _j ~~j)~i

+ 1~

(r, ti (31

where

J

^{is the}

Laplace-Beltrami

operator ^{and F}

[Vh

the free energy functional of the surface.

The noise function

7~

(r,

t represents fluctuations in the

incoming

flux and is nonconservative.

The coefficient _v~ of the

Laplacian

term is

normally expected

to be _zero. However, Villain I I

], starting

from _a

microscopic picture

of

step-motion, argued

that v~ can be

negative

if there is «diffusion bias _»

(Schwoebel

^effect

[12]).

Such diffusion bias, I-e-, the

repulsion

of

particles

from the

edge

of a downward

step,

^{exists in the} n ₌ 2 and _n

=

4 models but not in the

n ₌ I model.

Beginning

with the

Langevin equation (3),

Golubovic and Karunasiri

[10]

showed

that, quite generally,

a

negative

_v~ will be

generated by

renormalization.

However,

our

previous

results

[4]

for the _n

= 2 and _n

= 4 models and the results of the

present

^work are

consistent with the

following

somewhat different

picture.

^In the _case of the _n

=

4 model _we

believe that v~ < 0 and that the

pyramid instability

_can be

explained by

this fact. On the other

hand,

the _n

=

2 model does _not show this

instability

and, as the

exponent

z is close _to

4,

_{we are} forced to conclude that the

leading

derivative _on the

right

hand side of

(3)

is the

Laplacian squared,

I-e- v~ = 0.

Finally,

_our results for the _n

=

I model _are consistent with the choice v~ ~ 0. If v~ ~

0,

then all

higher

order terms in the

Langevin equation (3)

_are irrelevant in the

renormalization group sense

(as

_can be shown

by

_power

counting

in the

corresponding

Martin-

Siggia-Rose [13] functional)

and _we

immediately

find _z

= y

=2,

consistent with _our

simulations.

We _note that these

exponents

are the _{same as} those found for the Edwards-Wilkinson model

j14]

in which the

Laplacian

term with

positive

_{v~ is} due to

gravity.

This is also the _case for _a

model studied

by Family [15]

in which relaxation is

explicitly

due _to

gravity

rather than diffusion because

particles

_move

only

downhill. In _our model there is

nothing

like

gravity

and

we conclude that the

Laplacian

term with _a

positive

coefficient v~ ^must be

produced

at

large length

scales

by

a renormalization effect.

Remarkably,

a

simple change

of the

parameter

n in

equation (I)

_seems to

produce

flow to

quite

different fixed

points.

We also

point

out that in the

case of

equilibrium

fluctuations

(diffusion

but _no

deposition)

this

Laplacian

term is not

generated

under renormalization and that the

dynamic

_as well _as static

scaling properties

of the model _are

independent

of _n. The existence of the

Laplacian

term can also be traced to a

slope- dependent

surface current which is caused

by

the

breaking

of detailed balance

by

the

deposition

events

[17]. However,

neither the

sign

of v~ nor its

magnitude

is

easily

obtained from the

microscopic

details of _a

given

model _a derivation of v~ would

require

a derivation of the

Langevin equation

from the

appropriate

Master

equation.

The result _y

=

2 for the _n

= I model _means that the width of the surface

diverges only logarithmically

in two dimensions. This

implies

that this model is _a

good starting-point

for _a

description

of molecular beam

epitaxy,

a process that _at low temperatures can

produce

atomically

flat films. For _a 128x 128

substrate,

the width in the

steady

state at

1376 JOURNAL DE

PHYSIQUE

I N° 6

T

=

o is

only

a few

layers,

even for the

high deposition

rate used in _our simulations. The

steady

state is attained after many thousands of

layers

have been

deposited

and _at

early

times

(ten

to one hundred

layers)

the interface is

essentially

flat _a result also obtained

by Jiang

and Ebner

j16]

who studied the _n

=

I model

(with evaporation

and

smoothing by

a

locally varying

deposition

rate rather than

diffusion)

for very short times. In this _sense the _n

= I model is very different from

previously proposed

models

j2-6]

that result in very

rough

surfaces.

Acknowledgments.

We thank Joachim

Krug

for

stimulating

conversations. This research _was

supported by

the

NSERC of Canada.

References

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[4] SIEGERT M. and PLISCHKE M.,

Phys.

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[5] YAN H., Phys. Rev. Lett. 68 (1992) 3048.

[6] KESSLER D. A., LEVINE H. and SANDER L. M.,

Phys.

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published.

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Phys.

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[9] In (2) and

throughout

the _rest of this article _we neglect the

dependence

of correlation functions _on the direction of k. This is

justified

in the

scaling regime.

[10] GOLUBOVIC L. and KARUNASIRI R. K. P.,

Phys.

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[13] MARTIN P. C., SIGGIA E. D. and RosE H. A., Phys. Rev. AS (1973) 423.

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[15] FAMILY F., J.

Phys.

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published.