Mapping of (1+1) D-Crystal Growth onto a 14-Vertex Model

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Mapping of (1+1) D-Crystal Growth onto a 14-Vertex Model

F. Hontinfinde, A. Levi

To cite this version:

F. Hontinfinde, A. Levi. Mapping of (1+1) D-Crystal Growth onto a 14-Vertex Model. Journal de

Physique I, EDP Sciences, 1996, 6 (7), pp.873-890. �10.1051/jp1:1996104�. �jpa-00247220�

J. Phys. I France 6

(1996)

873-890 JULY1996, PAGE 873

Mapping ^of (I ₊ I)D-Crystal Growth onto

_a

14-Vertex Model

F. Hontinfinde

(*)

and A-C- Levi

(**)

Dipartimeuto di Fisica and Istituto Nazionale _per la Fisica delta Materia, via Dodecaneso 33, 16146 Geuova, Italy

(Received ii January1996, received in final form 4 March 1996, accepted 19 March 1996)

PACS.02.70.Lq Monte Carlo and statistical methods

PACS.81.10.-h Methods of crystal growth; physics of crystal growth

Abstract. A restricted solid-on-solid

(SOS)

single- and double-step model is introduced and studied with Glauber dynamics. Kinetics and roughness of the growing crystal _are described

in terms of _a Markov _process whose states _are given by the crystal _upper edge profile that _we

map ^onto a 14-vertex model. We solve exactly the kinetic equation for small-size versions of the model. Extensive simulations _are performed to derive the large scale properties. ^The present study _{appears as a} further extension of Gates and Westcott's investigation of the single-step

model.

1. Introduction

Crystal growth, beyond

its

technological applications,

has _a

long

and rich

history

of studies in basic science

ill

and has

generated,

at a fundamental

level, interesting problems

of _non-

equilibrium

statistical

physics.

Its

study by

theoretical and

experimental

methods has

provided

a

deep insight

into solid state

physics

and

chemistry. Despite

the recent

significant

_progress

obtained

by using

statistical-mechanical models and methods

[2-4]

to

clarify

cluster and

crystal growth phenomena, high-dimensional growth

models _are still

complicated beyond

_any

hope

of

an exact

solution,

_even when well-known,

oversimplifying approximations (such

_as

SOS)

_are

applied.

In lower

dimensions,

the situation is less hard and _some important results _are available in the literature

concerning growth

kinetics. Garrod _[5] solved several two-dimensional

growth

models where

growth

_occurs

essentially

at kink sites. Gates and Westcott [6,7] ^studied

exactly

_a restricted SOS

single-step

model defined _{on a}

hexagonal

lattice and _a non-restricted SOS model defined _{on a} square lattice,

relying

_{on a}

dynamic reversibility

concept valid in these models.

Their results _on

growth

kinetics and

regimes supplement

those found in previous

investigations

on

polymer crystallization,

island _or cellular structures and 2-D

crystal growth

[8, 9j.

Recently,

Hontinfinde and Touzani introduced _a kinetic four-vertex model to

study

the

single-step

model in d

= I + I where atom

deposition, evaporation

and diffusion _on the surface take

place [lo].

The main result

they

obtained is _a reentrant

growth phase diagram similar,

but

probably

not equivalent to what is often observed

experimentally during

the

epitaxial growth

of metal

(*) On leave from LPT, Facultd des Sciences, BP 1014, Rabat Morocco and

IMSP(UNB),

BP 613

P/Novo

Benin

(**) Author for correspondence

(e-mail: levi@vaxgea,ge.infn.it)

© Les Editions de Physique 1996

874 JOURNAL DE PHYSIQUE I N°7

surfaces

ill].

In this _{paper, we} reconsider _some

important

features of

crystal growth

in two dimensions. We attempt _a further extension of Gates and Westcott's work [6,7]

by studying,

in d = I ₊ I, _a

single-

and

double-step

model defined _{on a} 45°-rotated square lattice where

single

and double steps are allowed _on the

moving

"surface" _(~

).

The

study

is made easier

by mapping

the interface

profile

onto a 14-vertex model. Net

growth

or

steady

decrescence of the

crystal

is described

by

_a

Kolmogorov

forward equation which _we solved

exactly

for small

samples.

The present ^{model is} not

dynamically

reversible

(see

Refs. [6,

7]); however,

_a

steady

state does exist and the

growth velocity

_can be

computed exactly

for _very small systems. The latter

helps

_us to build _a precise simulation

procedure

to

investigate larger

systems. Our results show that _some

asymptotic

properties of the model _can

already

be obtained _on

relatively

small system sizes, e.g., N

= 60. We find indications in the model for _a

rough-to-rough

transition [12] ^{where the}

growth

rate _or

velocity

_may be

size-independent.

Our model

depends

_{on a} parameter n related to the

spatial coupling

^between two next-nearest

neighbours.

We find

that,

for

increasing

n, the

growing

surface becomes smoother and _we suspect that for _{n-cc a}

negligibly

small

growth velocity

will result. The latter _appears to be consistent with the decrease of the time- exponent

fl

_[13] of the surface _{as a} function of _n. We

define,

at fixed

growth

parameters,

a

time-dependent

order parameter

ifi(t)

to report ^{results about the} step

density

fluctuations

on the

growing

surface. Such parameter ^{has been often used in} the

literature,

_e-g- to

study

the

dynamics

of

step-doubling

transitions [14]. ^{We find that its}

amplitude I(t)

is consistent with the

analytic

form

proposed by Hoogers

and

King [lsj

in _a

simple

model introduced _to interpret experiments on reaction kinetics. This order parameter has values

indicating perfect layer growth

at low temperature and

large driving

force _on

relatively large

systems, whereas _at

high

temperature, the

decay

of its

amplitude

_{expresses a} continuous mode where the

growing

surface exhibits _a

hill-and-valley

structure. We

give

_a

qualitative growth phase diagram

^for

n = 3,

relying

_on the Wilson-Frenkel [16] ^{behaviour of the}

growth

rate in the continuous mode.

We also

explore

the fundamental question ^{of effective diffusion in}

crystal growth theory

where

growth

_occurs

by

atom

deposition

and

evaporation.

When _an atom evaporates ^from a

site,

and

shortly afterwards,

in the next simulation step, ^another atom condenses in the

vicinity

of this site, this

coupled evaporation-deposition

_{process may} be considered _{as an} effective diffusion.

We show

by

_a statistical

analysis

that it is

possible

to find _a temperature ^domain where such effective diffusion follows _an Arrhenius law when the

evaporation

rate is

properly

chosen.

In Section 2, we describe _our stochastic model. Section 3

gives

its numerical solution

(exact

on small

samples

and

by

simulation _on

larger systems).

Section 4 is devoted to results and discussion.

2. The

Microscopic

Model

The model is defined _{on a} 45°-rotated square lattice formed

by

_a stack of discs

(Fig. I).

We

assume that the

growing crystal

is well-structured and the

solid-on-solid'(SOS)

condition is

fulfilled.

Single

and double steps are allowed _on the

surface;

therefore, the present model appears as an extension of the

single

step ^model studied

exactly by

Gates and Westcott

[6,7].

The essential feature of the model arises from the condition that the

height

^difference between any ^two surface nearest

neighbour

atoms can be either I _or 3 in unit of

a/2,

where _a is the

lattice repeat distance. The numerical

analysis

of the

growth

kinetics of models

including higher

steps ^{would be}

expensive

in

computation

time. We find convenient to

replace

the stack of discs

by

its upper

edge profile

and _we define the interface between the

crystal

and the fluid

phase

_as the line

connecting

the centers of surface _atoms of successive columns. Thus the

(~) ^For short, ⁱⁿ the following the model will be simply named the double-step ^model.

N°7 14 VERTICES IN

(1+1)D

GROWTH 875

o~o

~

jojo~ _~o

o~o o~o~o o~o~o~o~o~o~

o~o~o~o~o~ o~o~o~o~o~o~

o~o~o~o~o~o o~o~o~o~o~o~o~

o~o~o~o~o~o~o~o~o~o~o~o~o~

Fig. I. Mapping of the surface of _a two-dimensional crystal onto _a 14-vertex model. Simple and double steps are present _on the surface. The line connecting black discs represents ^the growing crystal edge profile which _can be thought of

as a connection of

some 14-vertex lines.

(>2

^{9 >o}

11

f~/ ^~

7 5 4

Fig. 2. The fourteen vertices used to map the crystal edge profile in Figure I. Vertices with small numbers _are those shown in the figure. Vertices with big numbers _are found by reversing all

arrows. At each vertex, the ice rule holds [17]. The line representation of these vertices is obtained

by replacing _arrows pointing to the south-west _or north-west by lines while others _are deleted. In this representation, vertices 3, 4, 5 and 6 coincide with their usual line representation in the 6-vertex

model

[iii.

lateral "area" of _a double step ^does not

belong

to the interface but to the

crystal

This line is identified to a line with

cyclic boundary

conditions in a

special

14-vertex model

(Fig. 2).

Other non-used vertices of the _same type are considered _as

non-physical

in the model. This line

mapping

establishes _{a one-to-one}

correspondence

between _a

given

state of the system and

a 14-vertex

configuration.

If Ah denotes the absolute value of the

height

difference between two surface next-nearest

(NN) neighbours,

the energy ev of _a

given

vertex _v is defined _as:

e~ =

ji/2)e~jih)n

₁₁₎

816 JOURNAL DE PHYSIQUE I N°7

where _e has the dimension of _{an energy, ~} is _a coefficient which introduces _a difference between vertices

by depending

_on the vertex

"length" (in

the line

representation)

_as follows:

~ =

12)~/~l/2a.

12)

Vertices 3,

4,

5 and 6 which

belong

to the 6-vertex model

(two-dimensional

version of the ice

model) II?]

have _~

= l. Even if _n

= I, the relation

(2)

does not favour double step formation at low temperature. ^In the

remaining

of the article, _~

depends

_on unless otherwise

specified. Expression ii

describes the

spatial coupling

between NN

neighbours

whereas _nearest

neighbours

interact with infinite _{energy. n}

=

1/2

_may

correspond

to the

standard/Gaussian

SOS model. The _{energy e} is defined _as positive and assumed not to

depend

_on the temperature.

The relation

ii)

insures:

ES " e6 " e7 = e8 = 0

(3)

Hence,

the

completely

flat

surface, corrugated

_on atomic scale with NN

neighbours lying

in the _same

layer

has _zero energy. The

ground

state of the system ^{is twice}

degenerated

with four

ground configurations,

since relation

ii) gives

to the "reconstructed" surface

(787878...)

and the non-reconstructed surface

(565656...)

the _same energy.

However,

the surface

(787878...)

has

larger

interface width and may evolve at low temperature to the structure

(565656...).

The 14-vertex energy of _a

configuration

_a is

given by:

E

=

~j

_N~e~

₍₄₎

J

where N~ ^{is the number of} vertex of type

j

in the

configuration.

Possible

configurations

of _a system ^{of linear size} N are

parted

into time-conserved classes characterized

by

S:

S=Ni+N3+2Nio+N12+2N13-(N2+N4+2Ng+Nii+2N14) (5)

related to the _mean

slope

of the surface. A class will be named "active" if the

corresponding configurations

_can

change according

to the

growth

rules

(see below), "passive"

otherwise.

The _case S

= o

implies

that the two ends of the interface have

height

difference

equal

to

a/2

_or

3a/2 (see Fig. I).

For N _even, S E

[-2N,

-2N ₊ 2,.., 2N 2,

2N],

whereas for N odd, S E

[-2N

+1,-2N + 3,..,2N

-1].

The

configuration

_a and its mirror

image (with

respect to an horizontal

axis)

_-a,

belong

to

opposite

classes. Pure

growth

on a

corresponds

to pure decrescence _{on -a,} but both classes have the _same

steady

state distribution

probability.

Accordingly,

the present

study

will be restricted to systems ^{with linear size N > 3 and to}

positive

and active

classes,

I.e, S E

lo,..

,

2N 2] for ^N even and S _E

[1,..

,

2N 3] for ^{N odd.}

N =1 and N

= 2 correspond to trivial _cases.

Classes in the model _are associated _to surfaces with well-defined Miller indices.

They

_are

naturally

divided into subclasses whose

configurations

differ

only by translations;

_e-g-, for N

= 2, class

lo),

_we have _ns

= 2

subclasses,

each

containing

two

configurations: (78, 87); (56, 65)

^{whereas for} N

= 4, _ns

= 11 and for N

= 8, _{ns =} 567. For N

= 7, class

(I),

_ns

= 187.

We find also

possible

to describe the interface

using

the

following procedure.

We label _as in references

[6,7],

unit steps of the interface

by

_az,

Ii

= 1, 2,

.,

N)

from the left to the

right

and set az

equal

to

1/3

if I is _a

single /double

_up step and _az

equal

to

II

3 if I is _a

single /double

down step.

Therefore,

the surface

configuration

^of

Figure

I which is, ⁱⁿ the 14-vertex formulation,

a =

(8, 7,14,

6,13, lo,..

,

II)

is

represented

as:

? "

(+3,

-3, -1, +1,

+3, +1,

-3, +1,..

N°7 14 VERTICES IN 11 ⁺

1)D

GROWTH 877

C~~E

_{5 6 5}

,», ^-~

Cl

_E

~~

b)

^{~ ~ ~}

c

I)

_{E 6 5 6}

,; - c

I j

E

~)

~~

d)

^~~~

Fig. 3. Example of two

depositionlevaporation

events _{on a} flat surface

(a,c)

and their correspon- dence in the 14-vertex formulation

(b,d).

Vertex numbers correspond to those listed in Figure 2.

The allowed surface

configurations

_are such that two consecutive values of _a~

equal

to 3

or -3 _are

forbidden; otherwise,

the model would become _a

triple

step ^model and much _more

N

complicated.

The

cyclic boundary

conditions

impose

_aN+i

= ai. When

~j

az = o, we recover

z

our usual class

(o).

Dynamics

is introduced in the model

through deposition

and evaporation of atoms _on the surface considered _as Markov processes. Vertices

8/6, 1/3, 2/4, 13/lo, 14/9, 6/5, 3/12, 4/11

and

5/7

show the presence of

growthlevaporation

sites. Condensation

(C)

and

evaporation (E)

at these sites

modify

the interface line _on three sites which define

growth

and evaporation kink

subconfigurations.

The different events

whereby

atoms

join

_or leave the

crystal edge

_are

expressed

in terms of transitions between these three-site kinks _as follows:

X A Y

ci jE

V B M

where

A/B

denotes the

previously

enumerated vertices. At each

position X,

Y, V, M, several vertices may lie. In the present

model,

_{we get} loo

growth

kinks

(X

A

Y)

and loo evaporation kinks

(V

B

M).

As

examples

of the

previous

transitions, _we

give

in

Figure

3 two events

leading

to an adatom creation

levaporation

_{on an} assumed

initially

flat surface

(5656...6).

If AE represents the

change

in the surface _energy

during

these

transformations,

there exist, for _{~ =} l and _{n =} I, _z transitions with AE

= +2e, z transitions with AE

= -2e and 2z

transitions with AE

= o, where in this _case, z = 25. In the

single

step

model,

the situation is rather

simple

with _z

= I, since

only

vertices 3, 4, 5 and 6 _are needed to describe the

moving

surface.

An

important ingredient

of _any

microscopic growth

model is the definition of

dynamical

rules. We model the

exchange

between the

crystal

and the fluid

phase by

_a non-conserved

Glauber-type

kinetics [18]. ^{The transition rates} are considered in the

following

^forms where _a bias is

applied

in favour of the

deposition:

~a

~

~~PIA~/kT)

j~j

I +

exp(AE/kT)

878 JOURNAL DE PHYSIQUE I N°7

for

deposition

and

j~a ~

j?)

I +

exp(AE/kT)

for

evaporation

of the kink of type

(a),

where _{a can} take loo different

values;

T is the temper- ature, k the Boltzmann factor and

A~

the

driving

force. These rates

satisfy

for

A~

= o the

detailed balance condition:

caja-a')

E~l-a- a')

~~~

with respect to the 14-vertex energy ^at the temperature considered. When

A~

= o, the model describes

equilibrium

of

crystal

^and

fluid,

whereas for

A~

> 0, net

growth

_occurs.

If _we denote

by q(a; a')

a transition out of _a and

q(a)

=

~jq(a; ^a'),

it appears that the

~, relation

q(a)

=

q(-a)

relevant for the

dynamic reversibility

of the model in the _sense defined

by

Whittle [19] ^{does not} hold for all _a. Our results show that there _may be _no reasonable

general

condition for the model to be _so. Such

dynamically

non-reversible model is not

easily

tractable

analytically. Therefore, finding

_an

expression

for the

growth

rate

depending

on

growth

parameters at the

thermodynamic

limit is not trivial. The above does not however

imply

the non-existence of _a

steady

state for the system. We check that _any initial distribution

probability tends,

when t-cc, to _a

steady

state distribution. The system is

normally

driven to statistical

stability

and this results from the stochastic nature of the transition rules. The

model shows that for _~

= l and _n

= I, the property:

~j _C~(AE#o)

=

~j _C~(AE

=

o) (9)

holds.

This relation is also valid in the

single

step model and coincides with _one of Gates and West- cott's conditions for the existence of _a

steady

state distribution

probability

of the system [6,7].

When _{e =} o, all surface

configurations

_are

ground configurations

and _{we recover a} random

deposition

model where

only

the restriction _on Ah could limit the

crystal growth.

This _case

obviously

^is

equivalent,

when

e#o,

to the infinite temperature limit of the model.

3. Numerical Solutions of the Model

3.I. EXACT STEADY GROWTH RATE CALCULATIONS.

They

_are

only possible

_{on very}

small

samples

since the number of

configurations

and subclasses increases

rapidly

with system size.

In the

model,

the

probability

of

adding

_or

_removing

_more than _one disc in _a small interval

time dt is set to _zero. Let _{us assume e-g- pure}

deposition.

If P~

it)

is the

probability

of _{a at}

time t,

q(a; a')

_the

probability

of _a transition out of _a,

g(t)

the unconditioned

expected

total number of discs at time t,

given

_a, then [6]:

glt

+

dt)

=

glt)+

<

qla)

>t dt.

(lo)

The

expected growth

rate

G(t)

_per surface site is:

Glt)

_"

N~~(glt)

"

N~~

_<

ql?)

_>t

Ill)

N°7 14 VERTICES IN (1+1)D GROWTH 879

In the I4-vertex

model, G(t)

has the

expression:

G(t)

=

N~~ ~j _{C~p[Pm(t) (12)}

where

p[

is the

multiplicity

of the kink of type

(a)

in the m~~ _subclass Sm whose

weight

Pm

it)

is defined

by:

Pmlt)

=

~j _{P«lt). l13)}

Pm

it)

is obtained

by solving

the

Kolmogorov

forward

equation:

d

(i~~

=

~

L~~,

ji~, T, e)p~, it) j14)

where Lmm, ₌

£~ C~(M(~, p[6mm>)

denotes the elements of the transition matrix L of surface

configuration

between subclasses

(M(~, being

^{the number} of _{ways a}

configuration

of

subclass m'

can

change

to _a

configuration

of subclass _m via processes

taking

place at _a kink of type

(a)).

The

steady

state solutions _are used to compute ^the

steady growth

rate on which

we will

mostly

focus in the

following.

3.2. MONTE CARLO SIMULATION The simulation avoids the difficult

problem

of subclass

classification. It is based _on the

ergodicity

of the model within _a class and makes the

study

of

relatively large

systems

possible

and _{even easy.} The

roughness

of the

growing

surface in the presence of

deposition

and

evaporation

_{processes may} be studied

using

the

following algorithm.

All active

subconfigurations

_are

listed, together

with their

coordinates;

then the total evolution

probability

of the surface is

computed:

Qla)

=

£IC~P$

+

E~q$) lis)

a

Let _{r =}

£~ C~p$ /Q(a)

denote the total condensation

probability.

^At each simulation step,

r must be

compared

to _a random number _ro

uniformly

distributed _over the interval

[o,I].

If _{r > ro,} condensation is

tried;

otherwise, evaporation is chosen. The

attempted

_move is

accepted

with

probability:

pmc " uJ

/Q(a) (16)

where _uJ is

equal

to C~

/E~

for

condensationlevaporation

_on the kink of type

(a).

Concerning

the

growth

kinetics, sites where in

principle

both

deposition

and

evaporation

events _are

possible

_are treated _as pure

growth

sites. Such avoided evaporation sites _{are more}

strongly

linked to their

neighbours

than those considered. We

study essentially

the kinetics under

deposition

and evaporation ⁱⁿ this restricted but

physical

case. For this

study,

the above

algorithm

needs _some modifications before

yielding

precise ^{results. One active}

subconfiguration

must be selected _among those listed and its evolution

accepted according

to the Monte Carlo

weights

in

expression (16).

The _mean

growth

rate per surface site of _a

given

active class after K simulations steps is defined _as:

K

G =

N~~t~~ ~j(2r ₁₎ (17)

~=i

880 JOURNAL DE PHYSIQUE~I N°7

7000

6000

sooo

4000

S=12

3000 S=2

2000

N " 8

IOOO ~ ~/

/

~~= j o

S=O

~

0.5 1.5 2 2.5 3 3.5 4 4.5 5

Temperature (kT/E)

Fig. 4. Dependence of the exact growth rate of _a system of linear size N

= 8 _on the _mean slope

6 of vicinal surfaces (see text) and temperature. The supersaturation is kept constant:

Ap/kT

= 10

and _{we use}

~y ⁼ I and _n =1.

where t is the total real simulation time defined

by:

t ₌

f I/Qla) l18)

The number K needed _to reach the

steady

state increases with

increasing

system size and

decreasing

values of the

supersaturation.

4. Results and Discussions

4.I. CASE _r

= I. The results

reported

in this section _are obtained in

far-from-equilibrium

conditions where atom

desorption

from the surface is

neglected.

We _are

mostly

interested in _some

time-dependent

quantities which could characterize the

growing

surface

morphology (roughness,

step

density fluctuations)

and which _are

experimentally

measurable

together

with the average

velocity

of the interface between the two

phases, commonly

called

growth

rate.

Steady

decrescence and

temperature-programmed multilayer desorption

could be studied

using

the _same

procedure.

Surfaces with odd size are

necessarily sloped

surfaces and _we do not consider them

explicitly

in _our

study.

In

Figure

4, we

give

exact results

(by solving

the master

equation)

_on the

dependence

of the

growth

rate for

initially

flat and

sloped

surfaces with linear size N

= 8 at constant

deposition

rate. The

growth

rate decreases with the macroscopic inclination b

(tan

b

=

SIN)

of vicinal surfaces at

high

temperature, _a consequence of the decrease of the number of active sites with

increasing

b. At low temperature,

preexisting

steps on

sloped

surfaces induce

higher growth

rate than _on a flat _one. In this case, the

growth proceeds layer-by-layer

_on vicinal surfaces

by

_step

propagation

as in

spiral growth

_[20]. This mode is

destroyed

at

high

temperature

by

N°7 14 VERTICES IN

(1+1)D

GROWTH 881

40

16

35 14 ~"8, C(CSS

(0)

12

30 lo

a

25 6

Rate ₄

20 2

~

~

2 2.22.42.62.8 3 3.23.43.63.8

4~

~

*

lo , ~

*

~ N ₌ 600

e '

~ ~$i-a,,~~

.

~ ~

~ ~ *

~

l 1.5 2 2.5 3 3.5 4 4.5 5

Disequilibrium Aq/kT

Fig. 5. The simulated growth rate m. supersaturation at different temperatures. ^The inset _curve

compares simulation results

(open circles)

and exact calculations

(lines)

for _a system of linear size

N = 8, class (0) at four temperatures: T

=

2.5e/k,

T

=

1.5e/k,

T

=

elk

and T

=

0.5e/k

from the top to the bottom. Calculations _are performed for

~y = I and _n

= 1.

nucleation events _on terraces. Surfaces with _no terraces

(S

= 14;

12)

_may then show

essentially

lateral

growth

at constant

velocity.

The above results _are consistent with those found

by Krug

and

Spohn

_[12] from the behaviour of the

growth velocity

_{as a} function of the local

gradient

T7h of the surface. For b small, the

following

relation

approximately

holds for _a system ^{of linear} size N:

G(b, N)

=

G(b

= 0,

N)

+ ~tb~

(19)

It has been

suggested

_[4] that the temperature ^where p vanishes and

changes sign

_may

correspond

to a

rough-to-rough (RR)

transition temperature where _a sudden

drop

of the

early-

time

scaling

exponent fl

(defined below)

of the surface is

expected.

In

microscopic growth

models, the effects of _a

change

of

physical

_{parameters on} the

growth velocity provides

the best

understanding

of the

growth

mechanisms of real materials.

Figure

5

(inset curve), displays

the dependence of the

growth

rate _on

supersaturation

at different temperatures. Simulation results

perfectly

_agree with exact calculations _up to several

digits.

Such

precise

results _are

expected

since the simulation

procedure

relies _on the basic ideas of the kinetic Monte Carlo

method;

therefore it may

obviously provide

_a

high-accuracy

solution of the master

equation (14).

The main

figure gives

simulation results for N

= 600, class

(o).

At low temperature and small

driving force,

the

growth

rate G

nearly

behaves _as:

G

+~

bA~/kT (20)

where the coefficient b, ^which

plays

the role of

mobility (corresponding

to the kinetic coefficient for

A~/kT

_-

o),

increases with

increasing

temperature and saturates around T _+~

1.5e/k.

When

T-o,

other results indicate that b tends

continuously

to o without _any

non-linear nucleation

behaviour,

in agreement with the _zero

roughening

temperature of the one-dimensional substrate. With

increasing

temperature, the

growth

rate

rapidly

saturates and

882 JOURNAL DE PHYSIQUE I N°7

BOOO

7000

h'9h

T

11

6000

~'~~ ~ ~

0,12 ~

4000

o.oa 3000

~~~~ Q ^{~'~~ 5 lo 15 20 25 30}

Q ~

AH/kT

~~~~

low T ~

o

Fig. 6. Dependence of the simulated growth rate _on system size at low temperature

(T

=

0.5e/k)

and high temperature

(T

=

4e/k)

for

Ap/kT

= 10. The high-temperature growth rates _are given in arbitrary units. Inset: behaviour of the coefficient K

m. the supersaturation at T

=

4e/k;

C

=

exp(Ap/kT).

Calculations _were performed for

~y = I and _n

= 1.

depends exponentially

_on the

supersaturation

at

large A~, following

the Wilson-Frenkel [16]

law: G

+~

exp(A~/kT)

I. In order to get

plausible

estimates of the infinite-size

limits,

_we

explore

finite-size effects _{in our} calculations.

Figure

6

displays

the

dependence

of the

growth

rate _on system size at low and

high

temperature under _a constant and

large supersaturation

of the fluid

phase.

The results _ensure that these effects _are rather weak for N > 60 at any super-

saturation. One _can describe the fast convergence of the

steady growth

rate to its

asymptotic

value

by

the

scaling

form [21]:

G(N)

=

G(cc)

+

KIN" (21)

where K and _uJ depend _on the

growth

parameters. ^{For N} > 60, the _{curves are}

approximately

linear and _uJ

+~ I _as in the

single

step ^model [21]. ^{For small sizes}

and/or

low temperature,

important

deviations from the

scaling

relation

(21)

_are obtained due _to

higher

order terms in I

IN.

The linear

region

increases with the

supersaturation.

A careful

analysis

of the results

reveals that there is _a critical temperature To at which K vanishes and

changes sign;

at

To,

the

growth

rate is almost

size-independent.

Such result indicates _a possible RR transition

[4,6,12,21].

A best estimation of To should be obtained at

large deposition

rate. In the inset of the

figure,

_we show the isothermal behaviour of the coefficient K with the supersaturation;

it emerges that Ii is _an

increasing

^function of the

deposition

rate and _no RR transition is

expected.

We

therefore,

_argue that in the

(A~/T,T) plan,

the RR transition line _may be

nearly straight

^and normal to the T-axis.

Another

interesting

result _concerns the difference in

growth speed

between the

single

step and the double step models that _we

display

in

Figure

7. At low

deposition

rate and

high

temperature, the

growth

rates _are

practically

the _{same, a} consequence of _a

nearly

identical behaviour of all surface active

subconfigurations. Hence,

the

single

and double step models have the _same

asymptotic growth

rate per active site: the

corresponding

formula is

given by

the relation

(3.18)

of reference [7]. ^{At low} temperature, most convex active

subconfigurations,

N°7 14 VERTICES IN

(1+1)D

GROWTH 883

i~oo N =64

n=1

1200

iooo

4.5

800 4

* 3.5

* 3

600

~

2.5 2

400 1.5

o.5

200 °

O.5 1.5 2 2.5 3

tem~erat/re (k)le)

^{~ ~}

Fig. 7. Comparison between simulation growth rates per active site for the single

(lines)

and double

(stars

_or

circles)

step models for _a system ^{of linear size N} = 64, class (0) at large supersaturation

Ap/kT

= 20

(main figure)

and low supersaturation

A~/kT

= 3.5

(inset curves).

absent in the

single

step

model,

_are less active than the _{concave ones} since their evolution is not

energetically favoured; accordingly,

most of them _are

rejected during

the simulations. Such

situation

obviously

^leads to

slightly

lower

growth

rate per active site than in the

single

step model. The double step nature appears ^at

large deposition

_rate and

high

temperature. There, the double _or

triple probability

of _some sites to increase

successively

their

heights,

influences the

growth speed

which becomes

larger.

We have

checked, using

the relation

(21)

and the

asymptotic growth

rate of Gates [7], ^{that the latter feature remains valid in the}

thermodynamic

limit. It exists _a well defined temperature

(main figure)

at which the two models are

strictly

identical

kinetically.

This temperature

slowly

increases with the

deposition

rate.

Below,

the

growing

surface in the

single

step model appears

rougher

than in the double step ^{model since} a

lower

growth

rate

corresponds

to less

potential

kink sites _on the surface. The situation remained

unchanged

for

increasing

n as shown in

Figure

8. Our results reveal that for _{any n,} there is _a temperature domain where almost _no

growth

_occurs at small

driving

^force. We found that the width of this

region

increases with _n whereas it decreases for

increasing A~.

The latter feature has been also observed in d

= 2 +1 [4] where _a real metastable state exists. A

sloped

surface in this

region

_may ^however _grow fast and

continuously

whereas _an

initially rough

surface will

show

extremely

slow

growth

once become

completely

flat. In d

= 2 +1 _or

higher dimensions,

fractal _or dendritic

growth

may occur if the nucleation barrier is

artificially

_overcome. The trivial random

deposition

model

corresponds

to _e

= o. There is _no correlation between surface sites and

particles

_are launched at

randomly

selected positions above the

deposit.

The

figure

shows that at low temperature, this model

yields

_very

rough

surfaces.

The

growth

rate at constant

driving

force

A~

and

varying

temperature shows the Wilson- Frenkel [16] ^{behaviour above} a well defined temperature. The latter is often considered in

a crude

approximation

as a transition point from the

layer growth

to the continuous _one.

Although

this criterion is

questionable,

_{we use} it to

display

_a

qualitative growth phase diagram

884 JOURNAL DE PHYSIQUE I N°7

o.6

~~ random

deposition

A/£/c=0.5

O.4 sing

~~~ ~

n =

O.3

O.2

n=

oi

~

0 0.5 1.5 2 2.5 3

temperature (kTle)

Fig. 8. Comparison between simulation results _on the growth rate defined _per active site for different models. The random deposition model is obtained by taking _{e =} 0.

a

7

6

(~ ^continuous

~

~4

~

N=64 2

n=3

layer-by- layer

o

~ ~ ~ ~ ~ ~~

temperature (kTle)

Fig. 9. Growth phase diagram of _a system ^of linear size N

= 64, class (0) obtained by taking _as

transition point to the continuous mode, the temperature which gives the maximum growth rate at fixed driving force.

in

Figure

^9.

Sufficiently

below the transition

line, layer growth

_occurs

by

birth,

spread

and coalescence of lD islands _or

by

the one-cluster mode. Well

above,

the

crystal

_grows

continuously

without _any nucleation barrier. As discussed in reference [4], ^the one-cluster mode is

expected

to

disappear

^with

increasing

temperature,

driving

^force or system size. The one-cluster mode

prevails

when the time between two consecutive nucleation events

(adatoms

formation _on the

terrace)

is

larger

^than one

monolayer completion

time t

by

extension from the first nucleus [22].

N°7 14 VERTICES IN

(1+1)D

GROWTH 885

A

simple

calculation

using

_~

= l and the relations

(1), (6), (17)

and

(18),

shows that at low temperature

(double

steps _are not

relevant)

and

relatively large driving force,

the

growth

rate per site behaves _as

N~~t~~,

I-e-:

~ '~

N(N ~l~~i(~e/kT))'

~~~~

One _can

easily

^check that G is

approximately proportional

to the nucleation rate, I-e-, to the rate of fixation of _new adatoms _{on a} flat surface:

exp(A~/kT)/(I

+

exp(2"e/kT)).

Such

proportionality however,

to _our

knowledge,

has _never been _proven

experimentally,

except in

electrocrystallization

_[23]. This mode

usually

exhibits

undamped

oscillations of the interface width W with

growth

time.

To get more

insight

into the

growth

modes and

roughness

of the

moving

surface, _we define

an order parameter

ifi(t)

to monitor step

density

fluctuations _on the

growing

surface:

ifi(t)

₌

~J _~~P(llrh~ (t))

~

(2~)

where h~

(t)

is _a

single-valued

function which denotes the surface

height

at substrate coordinate

j.

A similar order parameter has been considered in the literature to map the

dynamics

of

step-doubling

transitions [14]. ^Let us denote

by ((t)

the

quantity

I

(ifi(~(t),

and

by I(t)

the

amplitude

of (ifi(~(t). ^From an assumed initial

heights configuration (I,o,

I,

o,..,o)

of the

surface,

_one finds

(~l(~(t)

= o. To avoid such

situation,

we consider that all the flat surface

sites have the _same initial

height.

This

assumption

does not, of _course, affect the

physics

of

our results. The behaviour of

((t)

is

displayed

in

Figure

lo for different temperatures. ^When

I(t)

= I, lateral

growth prevails

and the surface evolves while

remaining

flat

(pure

one-cluster mode _on

average)

and this

happens

in the

figure

^for T

=

o.3e/k.

A

layer

is finished _to be formed before _{a new one} is started _on it. The

probability

that _a second nucleation _occurs in the

growing layer

_or above the

growing

^{island is of the} same order _as

A(t)

= I

I(t).

When

I(t)

is

just

constant after _{one or two}

periods

of oscillation of

((t),

the

growth

is

locally perfect

and takes

place

in _a very few

layers.

A _constant increase of

A(t)

_may indicate _a

precontinuous

mode where the

growth

is neither

completely

continuous _nor

layer-by-layer.

At very

high

temperature, _no oscillation is observed for

((t)

and the

growth

mode is continuous, even in the

early-stage

of the

growth.

The

amplitude

of

A(t)

is well fitted

by

the functional form [15]:

x(t)

=

x(~x~) 1 j

~~ ~~

(24)

where

X(cc)

= 0.90 and to is fitted to 8 MCS at T

=

e/k.

In _our calculations, to ^is approxi-

mately

the time for

((t)

to reach the value

o.5,

which _seems not to have _a

particular physical meaning. Although

the oscillation

periods approximately

coincide in Monte Carlo time at all temperatures iii _the

layer

_or

quasi-layer mode, they strongly

differ in real time which of _course

depends

_on

physical

parameters. We also

analyze

the influence of _{n on}

((t)

at fixed temper-

ature. We find for

n = 1, 2,3 _a fast

damping

of the oscillations. However, it turns out that

increasing

_n makes the surface smoother; ^{which results} are consistent with those

reported

in

Figures

8 and 9. A further

investigation

of the surface

morphology

is

performed by studying

the behaviour of the interface width

Wit)

defined

by:

W~(t)

=

ii IN) £(h~ _{it) ii IN)} £ _{h~(t))~ (25)}