Current Induced Faceting in Theory and Simulation

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Current Induced Faceting in Theory and Simulation

Harvey Dobbs, Joachim Krug

To cite this version:

Harvey Dobbs, Joachim Krug. Current Induced Faceting in Theory and Simulation. Journal de

Physique I, EDP Sciences, 1996, 6 (3), pp.413-430. �10.1051/jp1:1996166�. �jpa-00247194�

Current Induced Faceting in Theory and Simulation Harvey Dobbs (*)

^{and Joachim}

Krug (**)

Institut für

Festkôrperforschung, Forschungszentrum,

D-52425

Jülich, Germany

(Received

8

September

1995, received in final form 9 November,

accepted

10

November1995)

PACS.68.35.Ja Surface and interface

dynamics

and vibrations PACS.68.35.Rh Phase transitions and critical

phenomena

PACS.05.70.Ln Non

equilibrium thermodynamics,

irreversible _processes

Abstract. We consider the effect that _a directed

migration

of adatoms, such _as that which

arises due to _an

externally applied

electric

field,

bas _over the

morphology

of _a

crystal

surface.

Applying

linear

stability

arguments to _a continuum model _we find that the

stability

of _a surface is determmed

by

the

depeiidence

of the adatom

mobility

_on the surface orientation. In _one

dimension,

surfaces _may be either stable _or unstable. In two dimensions

however~

_we find that

a surface of general orientation _is always ^unstable. These results _are confirmed

by

computer simulations of solid-on-sohd

models,

which also show late time coarsemng behaviour in _cases of surface

instability,

such that the typical domam size is _seen to increase _{as a power} law of the

simulation time. Our numerical data demonstrate _a

growth

exponent

1/4

in

one dimension and

1/2

in two

dimensions,

which _can be

supported by

the continuum

theory.

1. Introduction

In 1938 Johnson

[ii, investigating

^{the surface}

morphology

of

tungsten

^{filaments in bumed-}

out incandescent

lamps,

made _a remarkable observation: the surfaces

appeared

smooth if the

lamps

had been

operated

on

altemating

current, ^but

developed

_a charactenstic

pattern

^of

ripples

under d-c- conditions. He

pointed

out that the effect could be

quahtatively explained by

the directed

migration

of

tungsten

atoms under the influence of the electric

field,

if it _were

assumed that 'the rate

of drift depends

_on the

type of underiying surface'.

At about the _same

time, Bagnold

_[2]

developed

his celebrated

theory

^of

ripple

formation

on wmd blown sand.

Despite

the

vastly

different scales

involved, closely

related mechanisms underlie the two

phenomena

observed

by

Johnson and

Bagnold.

In both _cases there is a flow of material

along

_a surface. If the flow _rate

depends

_on the local surface

slope

_in such _a way that the flo~v _is

larger

in the

uphill

direction than _in the downhill

direction,

then local

protrusions

are

amplified

and _a

npple morphology develops (Fig. 1).

For solid surfaces _a

slope dependence

of the flow would be

generally expected

because of

crystalhne anisotropy.

^{In the}

case of sand

ripples

the situation is somewhat _more subtle: there the _{main mass} flow

along

the surface is carried

by

_creeping

grains,

which receive their motion energy from collisions with

'saltating'

_grains

suspended

in the _air and carried

along by

the

wind;

the momentum transfer

(*)

Present address: The Blackett

Laboratory,

Imperial

College,

London SW7 2BZ, United

Kingdom.

(**)

Author for

correspondence (e-mail: jkrug©iff079.iff.kfa-juehch.de).

Present address: Fachbereich

Physik,

Universitàt GH Essen, D-45117 Essen,

Germany

Q

Les

Éditions

_de

Physique

1996

(a) _~b)

~ ~

/~

i

~ ~

i

fl

,

Fig.

1. Basic mechanism of surface

instability

due to _a

tangential

_mass flow.

(a)

If the

mass current is an

increasing

function of

slope,

then small

height

fluctuations _are

amplified

and the surface becomes

unstable

(dashed arrow);

in the opposite _case 16) ^the current stabilizes the flat

configuration.

from the

saltating grains

is

larger

_on the windward side of _a

protrusion

than _on the lee

side,

and

consequently

the

uphill

flow is enhanced [2]. The first

quantitative expression

^of the effect

(for

solid

surfaces)

_was

given by Frohberg

and Adam

[3],

^{who showed how the}

competition

between trie

destabilizing

surface flow and

smoothening

due to surface diffusion

gives

rise to _a selected

wavelength

at which the

instability

first appears.

Recent interest in current-induced surface instabilities has been

triggered by carefully

con-

trolled

experiments

_on ^surfaces vicinal to

Si( ii1),

where _a direct bulk

heating

current was found to give rise to a

variety

of step

bunching phenomena [4-8].

The characteristic

dependence

of the step

bunching

_on current direction

(relative

to the

miscut) implicated

a mechanism related to

surface eiectromigration _[9],

_as had been

originally proposed by

Johnson

[ii.

Inspired by

these

experiments, Stoyanov _[10] developed

_a one-dimensional

step-train

model of current-induced

step bunching,

_in which trie

electromigration

^flux induces _an

asymmetry

in trie

step dynamics

but trie motion of adatoms is not

explicitly

treated. This work _was

subsequently

extended to indude transverse step

meandering _[11]

and

advacancy

motion

[12],

and bas been shown to

provide

a

good description

of trie

experimentally

observed

phenomenology

_[8].

A certain drawback of

step-dynamical

models is their limitation to vicinal surfaces _{as we} bave

emphasized

in trie

preceding discussion,

current-induced instabilities _occur under much

more

general conditions, essentially

whenever there is _a directed flow of material

along

_a ^surface.

We bave

recently presented

_a continuum

theory

of current-induced

faceting,

which is better suited to describe trie uniuersai features of trie

phenomenon [13].

^Besides

providing

a

general

criterion for the

stability

of _a

given

surface orientation [3], ^the

theory predicts

the orientations of the facets that appear after the initial orientation has become

unstable,

and it allo,vs _one to address the

subsequent

_coarsening

dynamics

of the faceted surface.

The continuum

theory

is based _{on two}

orientation-dependent, macroscopic quantities

the surface stiffness and the adatom

mobility

which have to be obtained from _a

microscopic,

statistical mechanical model

[14].

^{~Ve have therefore}

supplemented

_our continuum

analysis

with àlonte Carlo simulations of _a solid-on-solid

(SOS model,

in which the standard transition rates for surface diffusion [15] are given a directional bios to account for the

electromigration

effect. Once the

correspondence

between

macroscopic

and

microsopic quantities

has been established

[14],

^the continuum

theory

_can be used to

predict

the

morphological

evolution of the SOS surface.

In this paper we

give

a detailed _account of the work announced in

[13],

^{and extend it}

in several directions. In Section

3,

_we show how the orientation

dependence

of the surface stiffness _can be included in the calculation of the selected

facets,

and illustrate the

procedure

for the standard SOS and the discrete Gaussian models [16] ⁱⁿ one

dimension;

_in Section 4 _we

analytically

derive the

t~R _coarsening

_{law that}

~&ras found

numerically

in _our one-dimensional

simulation

[13]; and,

most

importantly,

Section 5 contains

analytic

and numerical results for two-dimensional

surfaces, demonstrating

in

particular

that the facet structure _{coarsens as}

t~/2

in this _case, in accordance with recent

experiments

on

Si(1ii) _[17].

Section 2 introduces the

discrete and continuum models used in this

work,

and Section 6 offers _some

conduding

remarks.

2. SOS and Continuum Models

2.1. A ONE-DIMENSIONAL SO S MODEL. For

making microscopic

simulations _we

adopt

the usual SOS

description

of _a

crystal

surface in terms of

integer height

variables

h(1),

₌

1,. L,

with Hamiltonian

L

~ = K

~j _(h(1+1) h(1)(". (1)

~=i

We consider both the standard _n

= SOS model and the _n

= 2 Gaussian model. An average surface

slope

ù is

imposed by adopting

'helical'

boundary

conditions

h(L

+

1)

=

h(1)

₊

fiL, although only

with _an

integer

value for the vertical offset ùL.

During

simulation of surface

diffusion,

_an adatom _{on a}

randomly

chosen column

attempts

to

step

to a

randomly

chosen

neighbouring

column

j

₌

1+1,

so that

h(1)

_-

h(1)

-1 and

h(j)

-

h(j)

+1. Moves to the

right (j =1+1)

_are

attempted

with

probability

_p

,

and _moves to the left with

probability

1- p. An

attempted

_move is then

accepted

with _a

probability r~j,

known _as the transition rate.

For

equilibrium simulations,

_moves to the

right

_are

attempted

with

equal probability

as

moves to the

left,

_{p =}

1/2,

and the transition rates must

satisfy

detailed balance with

respect

to

~,

F~ /Fj~

₌

exp(-A7i~ (2)

where

A~~j

is the

change

in the value of ~ associated with

moving

_a

partide

from to

j (note

that the

temperature

^{has been adsorbed into the definition of the}

coupling

K in

(1) ). Apart

from these

requirements,

the

specific

choice of

dynamics

is unrestricted and in

principle

does not effect the outcome of the simulation.

The influence of

electromigration

can be induded

by considering

_moves of adatoms to the

right

with

probability

_p different from

1/2.

Thus for _{p >}

1/2

there is _a bias for _{moves to} the

right-

The choice of transition rates

r~j

is _Rot altered.

Microscopically,

an adatom _on the surface of _a

current-carrying crystal

_is

subject

to the combined effect of the electrostatic field E

driving

the

current,

^{and the 'wind force' due} to the current itself. The total force F _on the

atom can be wntten as

F ₌ eZE

(3)

which defines the effective valence Z. The bios _p is then

simply

related to F

by p/(1 p)

=

exp(Fa/kT)

where _a is the distance between

adjacent

surface sites. For

Cu(111)

at room

temperature

it has been estimated

[18]

^{that Z} m

-21,

_so that for _an electric field of1

mV/cm

the relative bias introduced into the

hopping

^of _an adatom

(p 1/2)

is

typically

of order

10~~ 10~~

_For

Si(111)

the valence is

expected

to be

considerably

smaller Z _m 0.01 ù-1

[8,19]

but with the

correspondingly higher

^fields used in trie

experiments (E

= 7000

mV/cm

in [8]

),

the bios is of the _same order of

magnitude.

With such _a small

bias,

surface instabilities

require

tens of seconds _or

longer

to

develop,

and simulation over such _a

length

of time _is

expensive. However,

we find the

phenomena

caused

by

the bios to be

independent

of its

magnitude

_in all

respects

other than the time-scale

they

take to

develop,

and _{so we} consider much

larger

biases

(p

= 0.7 _or

0.8)

in _our simulations.

Finally,

_we comment about _our choice of transition rates. For

dynamic

simulations of surface

diffusion,

rates chosen

according

to trie Arrhenius

prescription

_are

generally adopted, r~j

=

exp(-2Kn~)

where _{n~ =}

0, 1,

or 2 is trie lateral coordination number of trie adatom to be moved from column

[20].

^These rates _are based _on the

assumption

that ₁₎ surface diffusion _is

an activated _process, and

ii)

the activation barrier _is

equal

to the

binding

_energy at the initial

400

t=10° _t-io7

t=io6

zoo t=10 ~

B #

t=105

~ ~

t=lo~

° t-io4

-lùÙ t=10~

-200 -200

0 100 200 300 400 500

a) b)

Fig. 2.

Faceting

in the biased one-dimensional _n

= 1 and _n

= 2 SOS models. A

single

reahsation of trie surface _is

sho~vn,

at times measured in

attempted

_{moves per} site which _are indicated above the

profile. Subsequent profiles

_ai-e

displaced by

100 units in the vertical direction for

clarity.

For the

n = 1 model

la),

data for ù

= 0, K

= 2.0 and _p

= o-7 is sho~vn. For the _n

= 2 model

16),

_we have taken ù ₌ -o.25, K

= I.o and _{p =} 0.8.

site. The latter

clearly

constitutes _an

oversimplification,

since it

implies

that the

hopping

rate

r~j

is

independent

_on the site

j

= 1+ to which the adatom is

moved;

_{as a} consequence, the bios has _no effect ~vhatsoever _on the surface _structure

[14].

^In our simulations _we therefore _use

Metropolis

rates, ⁱⁿ which the barrier energy is identified with the energy

dijference

between final and initial site,

r~j

₌ min

[1, exp(-A~~)] (4)

While these rates _are not

necessarily

_more realistic thon Arrhenius

dynamics, they

reflect that

generically hopping

rates should

depend

_on both the initial and the final environment of the adatom [14] such _a

dependence

_appears to be crucial for the effects of interest in this _paper.

Figures

2a and 2b show the results of simulation of biased _n

= 1 and _n

= 2 models. Both simulations

began

from surfaces with _a small uniform

tilt, h(1)

₌

int(ùi).

For the _n

= 1

model,

the surface

develops

into regions of uniform

negative slope (facets), separated by

_narrow

regions

of

large positive slope (macrosteps),

which _are akin to the

step

^bunches observed in

experiments.

As the simulation progresses, the

regions

of

negative slope

coalesce and grow

larger although

their _average

slope

remains constant. In _contrast the

macrosteps

^do not grow

appreciably

in width but do _{grow in}

height.

thus _preserving the _average

slope

of the surface. For the _n

= 2

model,

the situation differs in that facets of _two

different,

finite

slopes

_appear- The

slope

of these facets does not

change

as the simulation continues, and there _{are no}

macrosteps.

This

mstability

of _a flat surface _is observed to _occur for all values of _p different from

1/2, although

the time taken for the facets to form increases _as the bias _is reduced.

Further,

_as far _{as one can}

tell,

the

slope

of the facets is

independent

of the bios _{as p}

-

1/2+ Trivially.

the

system

is invariant under simultaneous reversal of the bias _{p -} 1- p and the _average tilt ù _- -ù, so that _we will

only

consider _{cases p} >

1/2.

As well _as

investigating

the effect of the

degree

of

bias,

_{we can} also

investigate

the effect of

trie _average surface

slope

ù. Trie

picture

that emerges for trie _{n =} 1

model,

is that when facets

form,

trie

slope

of trie facets is

independent

of trie average surface tilt.

However,

trie surface will form facets

only

if the average

slope

is intermediate between that of trie facets and that of the

macrosteps (+oc).

Surfaces with _an average

slope

less than that of the facets _are stable and

remain unfaceted.

Thus,

the facet orientation appears ^to coincide with the

boundary

between surfaces stable and unstable _{to current} induced

faceting.

For the _n

= 2

model,

it is found that _a uniform surface of _any tilt other than

precisely

that of _one of trie facet orientations is

unstable. Trie surface will evolve into facets of two orientations: _one with _a

slope _greater

than and trie other with _a

slope

less than the _average surface

slope.

2.2. CONTINUUM MODEL. Since bath facets and

macrosteps

_appear on a scale

larger

than that of the

underlying lattice,

it is

possible

to describe their formation and

growth using

_a

continuum

theory

for the surface. This

replaces

the discrete column labels _i

by

_a continuous coordinate _~, and describes the surface in terms of _a continuous

height

variable

h(~,t).

Because

we

only

allow surface diffusion _processes to occur, the surface evolves

according

to _an

equation

of

continuity

ôth

+

ôzJ

" 0

(5)

where

subscripts

denote the differentiation

variable,

and

J(~,t)

_is the surface adatom _current.

This is 'driven'

by

local

gradients

of the chemical

potential

_{~1, as} well _as the

electromigration

force

F,

_so that

J =

a(ô~h) [-ô~~l+ Fi (6)

The constant of

proportionality

_a is called trie 'adatom

mobility'

and _we bave written in

its

dependence

_on the local surface

slope ô~h.

To relate the SOS mortel to the continuum

description,

_we need to evaluate both the chemical

potential gradient

and the

mobility

_in

terms of trie

microscopic parameters.

^Trie

procedure

for

this,

which _we outline

here,

_can be

found in _a

previous publication _[14].

To calculate the chemical

potential

in a

non-equilibrium situation,

we imagine an SOS surface with _a Hamiltonian _as in

equation il)

in which the _averages of the local

height

variables

(h(i))

are

slowly varying

with

time,

_so that

iocaiiy

the mortel is _in

equilibrium.

We then consider _an SOS mortel with _a

slightly

different Hamiltonian for which the average

equilibrium profile

is made to match that of the

close-to-equilibrium

surface

by adding

_an

inhomogeneous

chemical

potential

term to ~

~p

=

~j _{~1(1)h(1). (7)}

~

This _can be rewritten _in terms of the local

slope

variable

u(1)

₌

h(1+1) h(1)

and _a local

'slope potential' m(1)

which satisfies _~1(1)

=

-(m(1) m(1- 1))

~iv

₌

£mli)U(1). (8)

Since the local

slope

variables _{are not}

coupled by

^either 7i _or

7ip,

it is easy ^to evaluate their thermal _averages

(u(1)).

For the _n

= mortel this _can be clone

analytically

~~~~~~

coshll~lllm(il'

^{~ ~'} ~~~

For the _n

= 2

mortel, explicit

numerical evaluation _is

quite

_easy. ^{To calculate the chemical}

potential,

we now invert trie

relationship

between

m(1)

and

(vii)),

and take trie continuum

hmit for trie coordinate 1,

~ ~ ^~

~~~

~~~~ ÎÎÎ dÎ

d~2 ~~~~

where _we

drop

trie brackets

il

for trie averages

u(~)

and

h(~).

If _a surface is close _to

equilibrium,

this relation holds

although

with _a

partial

second derivative. Trie factor

multiplying

trie second derivative is

just

trie surface stiffness

1,

and it is _a function of trie local

slope u(~).

Trie calculation of trie adatom

mobility presents

more of _a

challenge. According

to linear response

theory, given

_a

specific microscopic

model and _a

specific

set of

dynamical rules,

trie

mobility

_can in

principle

be calculated from _a

knowledge

^of the _average

jumping

rate and

trie current-current correlation function in

equihbrium _[21].

In

practice,

it is

usually

_very

complicated

to calculate trie

dynamic

correlation function for trie surface currents. However trie average of trie

equilibrium jumping

rate can be calculated for one-dimensional models and this forms _{an upper} limit for trie

mobility

of ~(r~j)

_e

a+ (11)

It is easy ^to calculate

a+

_{for trie}

Metropohs dynamics

_{as a} function of either trie local

slope potential

_{m or} trie local

slope

_u. For trie _n

= 1 model

~+

_(y~~)

=

~°~~ _~

~~P(~~~

^~

j~~)

2 Siuh K

This

expression

^bas a minimum at m = 0

(u

= 0 and tends _to

1/2

for _m

- +K

(u

_-

+oc).

Simulations of trie

equihbrium roughening

of _a surface indicate that

a+

_{is in} fact _a fair

approximation

to trie

mobility _[14].

For trie _n

= 2 model

only

_a numerical evaluation is

possible, although

it is easy ^to see that both

a+

and

a are

periodic

functions of trie

slope potential

_m and trie surface

slope

_{u since} trie Boltzmann

weights

of trie

step

^variables

u(1)

for trie Hamiltonian 7i ₊

7ip

_are invariant up to _a factor under transformation of trie

slope potential

and trie

slope m(1)

_-

m(1)

+

2K~

u(1)

_-

vii)

+ 1. Hence

a(m

+

2K)

=

a(m)

and

~(m

+

2K)

=

u(m)

₊ 1.

Putting together

trie

equation

of

continuity (5)

and trie formula for the surface current

(6),

we thus arrive at the

required

continuum

equation

for the surface

ôth

=

-ô~ (a(ô~h)(ô~(1(ô~h)ôjh)

₊

F)) ₍₁₃₎

In fact _we will

study

the time evolution of the local

slope potential m(~),

because this _removes the stiffness from the

equation

and

gives

a linear second derivative _m the _expression for the

current. To do this we differentiate

equation (13)

with respect to ~,

giving

an

equation

for the

dynamics

of the local

slope u(~),

which _can then be converted

using

trie relation to trie

slope potential

ôtm

=

1)) ai ja(m) (à]m

₊

F) j. (14)

3. One-Dimensional

Analysis

With trie continuum

theory

of trie last

Section,

_{we now}

try

to understand trie simulation results.

Our first _{aim is} to look at trie circumstances under which _a surface _is unstable when _a field is

applied,

and _we do this

by

_{using a} hnear

stability analysis.

Dur second _{aim is} to

provide

a

description

of trie facets and

macrosteps

^that appear, in

particular

_we wish to

explain why macrosteps

appear in trie _n

= but not trie _n

= 2 model- To do this _we

analyse steady

states of trie continuum

equations.

3.1. LINEAR STABILITY. For _a flat surface of _average

slope

ù with _a small

periodic

distur- bance

superimposed

h(~,t)

₌ ù~ +

hq exp(~oqt) cos(q~) (15)

a

simple

lineansation of

equation (13)

shows that trie

growth

rate ~oq is

~dq ⁼

F~q~ a(ù)ilù)q~. _l16)

Although

the

sign

of the q~ term is

always negative,

that of the

leading quadratic

term _is determined

by

the

dependence

of the

mobility

_{a on} the average

slope

of the surface [3]. ^Thus if the

mobility

is _an

increasing

function of the

slope,

_~oq is

positive

for small values of _q and the surface suffers from _a

long wavelength instability,

whereas if _a decreases with

increasing slope

the surface is

linearly

stable. This is

precisely

the effect illustrated in

Figure

1- For

instance,

in the _case of the _n

= 1 SOS model for which the

mobility a(ù)

has _a

single

minimum at ù ₌ 0 and increases to _a finite limit for ù _-

+oc,

surfaces with

positive slope

are

linearly unstable,

and those with

negative slope

_are

linearly

stable.

This

simple macroscopic

^{criterion is}

reproduced by

the one-dimensional Burton-Cabrera- Frank

(BCF)

model for

stepped

surfaces

[10]. Quite generally

the

velocity

^of a

step edge

is

proportional

to the _sum of the currents

j+ impinging

_on the

step edge

^from the terrace below

(+)

and above

(-).

Thus trie

equation

^of motion for trie

position

^of trie i'tri

stop

_xi ^{has trie} form

Î~

=

-J+ix~ x~-i) J-(x~+i _x~) iii)

where _we take the _x axis as

pointing

_up the step ^train

ii-e-

the surface

slope

_is

positive).

The linear

stability

of _a uniform

step

train of terrace width 1is then determined

by

the

dependence

of the currents _on 1: for

stability

( (J+(1) -1-(1))

_> °.

l18)

The _term in brackets is minus the function

#(1)

introduced

by Stoyanov

in his discussion

[10].

In _cases without

desorption

of adatoms from terraces, the current is uniform _over the

length

of each

terrace,

so that

j+

₌

-j-

and is

equal

to the

macroscopic

surface current J. Thus the condition

(18)

holds when the surface _current is _a

decreasing

function of the surface

slope Ill-

For small

driving fields,

the surface current will be

proportional

to

F,

and _a surface

mobility

can be defined

accordingly

which then satisfies the _same

stability

criterion _as in the continuum

theory.

We also remark that _an

analysis

of

periodic perturbations

to _a step train shows that it is either stable _or unstable for all

wavelengths.

The lack of

stability

for small

wavelengths

_is due to trie absence of

capillarity

^effects in trie BCF model.

Although

this omission is not

important

for

determining

whether _a surface is unstable to

long wavelength perturbations,

it is _serious

when _one wishes to consider trie structure of _any step bunches that

form,

since trie surface

capillarity

is _a consequence of the

repulsive

forces

operating

between the surface steps-

So far _our discussion bas been limited to l1~lear

stability,

which is _{not a}

guarantee

^of

stability.

Thermal fluctuations _cause local

regions

of _a stable surface _to become tilted _over toward _orien- tations that _are

unstable,

_so that further

instability

_{may occur.} Hence it is

possible

^for surfaces which _are

linearly

stable to be in fact

unstable, although

the _converse is never true. As _an

example,

the simulations described in Section 2 _seem to

suggest

that the actual

boundary

be-

tween stable and unstable surfaces for the n = 1 model coincides with the selected orientation

of the facets

11

-30°

),

which lies well inside the

region

^of linear

stability (ù

<

0)-

A similar

situation _may have been observed in

experiments

_on surfaces vicinal _to

Si(001),

where step trains with

large step spacings (small slopes)

_were found to be unstable bath in the

'step up'

and the

'step

down' _current direction

[22]

^{within linear}

stability analysis,

_reversing trie current

direction should

àlways

transform _a stable vicinal surface into _an unstable _one, and _{mce versa.}

To go

beyond

the linear

stability,

_{we now} consider

'steady

state' solutions to

equation (13),

which allows _us to calculate the Lacet

slopes,

3.2. STEADY STATES. Because the

groin>th

rate of the facets that form after the initial

instability

in either the _{n =} 1 _or 2 models decreases _as the facets _grow, their

asymptotic shape

can be calculated

by setting

the time derivative _in

equation (13) equal

to _zero- Because of the one-dimensional nature oî the

problem,

it follows from the

equation

of

continuity

that the s~arface current is _a constant J* for all values of _x.

Using equation (6)

and the formula for the chenlical

potential,

the

quasi-static

surface

profile

is then determined

by

_a second-order

equation

for trie local

slope i~(~)

a(u) () _~i(u)))

+

F~

^{= J*.}

⁽¹⁹⁾

1 X ~

A

simple approximation

would be _to

ignore

the

slope dependence

of the surface stiffness

§ _[13].

However,

such _an

approximation

_is to be avoided in

studying

the formation of

macrosteps

since in the _case of the _n

= 1 SOS model the surface stiffness~vanishes for

large

surface

slopes-

Instead _{we con} rewrite

equation (19)

in _terms of the local

slope potential m(x)

(j

=

J*laIn~)

F.

12°)

In

solving

this

equation

_we look for

oscillatory solutions,

_so that there must be _an inflexion point at _some

repeated

value m*. This _is then related to the surface current

J*

=

Fa(m*). j21)

We _can

explore

the character of solutions to

(20) by mabing

_an

analogy

with the _motion of

a

partide

_{in a}

potential

well

V(m),

where _~

plays

the role of the tiine variable. In this _case

v(m)

= F

j~(1 ^a(m*) la(m'))dm'. (22)

Note that for

oscillatory solutions,

m* _must be _a minimum of this

potential

_so that

a'(m*)

> 0.

Typical shapes

of the

potential

for trie _n

= 1 and 2 SOS models _are shown _in

Figure

3. In both

cases the

mobility

is an even function of trie surface

slope,

so that

V(m)

_{is an} odd function of

m with _a ma~imum at -m*. For the _n

= mortel this _{maximum is unique,} whereas for the

n = 2

model,

the

potential

has

periodically repeated

_lraxima, due to the

periodicity

of the

mobility. Integrating equaiion (20)

_{gives a} first order

equation

for the _energy E

~Î

^~

~

~~~~

~' _~~~~

There _{are now} two free

parameters,

^the

slope potential

at the

turn1~lg point

^{m* and} the energy

V(-m*)

_> E _>

V(m*). However,

there _are two conditions _on the solution: the average

slope

oî the surface _is fixed

by

the

boundary conditions,

and the

period

of the

oscillatory

motion

must be chosen to

correspond

with the

wavelength

L of the

stationary

surface

profile.

o.oi

o.5

% _~

£ Î

> Ç

~2 -1 0 2 ~'~~~2 _{-1 0 2}

m m

a) b)

Fig. ^3. Potentials

V(m)

for trie classical mechanics

analogues

to the _n

= 1 ~ud _n

= 2 SOS models.

For the

n = 1

model,

the

potential

shown in

Figure

3a is calculated usmg the

analytic

_upper bound

a+(m),

_equation

(12),

with K

= 2.0. The value m*

= 0.89 _was chosen to

satisfy

the condition

(24), although

the

general shape

of the

potential

_is the _same for all m* _{> 0.} For the _n

= 2

model, Figure 3b,

the

potential

is calculated from _a numerical evaluation of a+ _For K

=

I.o,

m*

= I.à in order to

satisfy (26).

Considering

the _case of the _n

= model

first,

_{we see} that oscillations of

arbitrarily long period develop

when the _energy E

approaches

the maximum value of the

potential V(-m*)

The

partide spends

most of the time in the

vicinity

of this

point,

and

only

_a finite time at the

opposite

extreme of the motion. Thus trie surface

develops

facets of

slope u(-m*), separated

by

_narrow

regions

of

positive slope.

As trie

length

of trie facets

increases,

_so trie maximum

positive slope

that the surface attains must also increase to preserve the average

slope

of the surface. Thus _in this

limit,

_as the

negative

extreme of the

partiale

motion

approaches -m*,

the

positive

extreme

approaches

the surface

coupling K,

so that the surface

slope diverges

within the

macrosteps (compare

to

(9)).

Thus the

asymptotic

^value of m* is determined

by

the condition

V(-m*)

=

V(K),

and the facet orientation _can be calculated from _a

knowledge

of the orientational

dependence

of the

mobility

j~ _{(i a(m*) la(m))}

_dm

= o.

j24)

Note that there is _no

dependence

of the facet orientation _on the

magnitude

of the

driving

force

F,

_as confirmed

by

the p

independence

of the simulations. We _can estimate the facet

slope,

using the

analytic

_upper bound

(12)

for the

mobility

a+ We find for K

= 2.0 that m* _m

0.89, corresponding

to _a facet orientation of -23° which is rather smaller than the value of about -30° _seen in the simulations. The _reason for such

dîsagreement

is that the difference between the

mobility

and the _upper bound is

largest

^{for surfaces} at small

slopes,

for which the

mobility

is smallest and which

give

the most

important

contribution to the

integral

in

(24).

The _manner in which the extremes of the

partide

motion

approach

the

asymptotic

values

-m* and

K, depends

_on the _average surface

slope

ù,

although

such

oscillatory

solutions _can

always

be found

provided

ù >

~t(-m*).

For

~t(-m*)

< ù < 0 these solutions coexist with the

linearly

stable uniform solution h

= fix

(see

Sec.

3.1).

We have _{no means} of

deciding

which of

the two solutions will be

truly

stable in the presence of

fluctuations;

the simulations described

in Section 2.1 indicate that

faceting

_occurs

throughout

the

region

of

bistability.

Analysis

of the

equations

of motion show that for solutions

m(x)

with finite

period L,

there

are small finite size corrections to both the facet

slope

and the value of m* of order

L~~,

which _are related _to the behavior of the stiffness

1

=

dm/d~t

for

large slopes.

There is

also, by equation (21),

_an associated finite size correction _to the surface _current

J*,

which is most

conveniently expressed

in terms of the

height

A of the

macrosteps

J*lCC) J*l~)

"

~) _)~)~~~~~

+

°(~~~). ₁₂₅₎

This correction will be _very

important

when _we discuss the rate of

growth

of the facets after their initial

development

in Section 4.

For the _n

= 2 model the

potential V(m)

is

periodic

_as shown in

Figure 3b,

with _a

period equal

to twice the SOS

coupling K,

and with _maxima at m = 12K m* where1 is _any

integer.

The faceted surfaces _{seen in} the simulation

correspond

to

partide

motion between

adjacent maxima,

and the facet

slopes

that _{appear are} then 1+

~t(-m*).

Thus a

major

difference from the _n

= model is that _a surface of _any

slope,

other than that

corresponding

to _one of the facet

orientations,

is unstable towards

faceting.

The value of m* _con be calculated from the

condition that the _maxima of

V(m)

_are

equal,

_so that

/~~ ^{1 ~~j~}

^dm = 0.

(26)

For low values of the surface

coupling, 1faim)

has _a

small,

almost cosinusoidal

variation,

and

in this limit m*

=

3K/2. Using

this

approximation,

we find for K

= 2.0 facet

slopes

in the

vicinity

of _zero tilt of

approximately

-42° and

6°,

in

good correspondence

with the values

we can estimate from

Figure

2b

(-41°

and

5°).

Finite

period

corrections to the

slopes

and

currents _can also be

calculated, and,

_in contrast to the _n

= 1

model, they

_are

exponentially

small _as the

period

L increases.

4. Trie

Coarsening

Process

During

the _course of _a

simulation,

the average facet

size1increases steadily

both for the _n

= 1

and _n

= 2

models,

_{as can} be _seen

by inspection

of

Figures

2a and 2b- This process can be

quantitatively

monitored

by following

the behaviour of the real _space

height-height

correlation function- In _one dimension this oscillates _{as a} function of

distance,

and _a convenient _measure of the facet _{size is} the

position

of the first _zero

(.

In

Figures

4a and 4b _we show how this

length

scale increases as a function of simulation time t. In each _{case we} also show another _measure of the

faceting

_process. For the _n

= 1 model _we

plot

in

Figure

4a the _average

separation

of the macrosteps

d,

^which is

equal

to the _average facet

length

after _an initial

period

in which the macrosteps are

forming.

For the _n

= 2 model _we show in

Figure

4b the surface width _w, which is also

proportional

to the _average facet

length.

For both the _n

= 1 and _n

= 2

models,

it appears that the facets _grow

according

to _a power law

m~ t~ with the _same exponent s m 0.25.

In this section _we

investigate

this behaviour. For the _case of the _n

= 1 model _we find that the finite _size corrections to the surface current mentioned in the

previous

section _are

responsible

for the

coarsening,

_so that the evolution of the

system

^{after the initial} appearance of the facets

is

largely

deterministic. For the _n

= 2

model,

_we find that random fluctuations of the surface current _are

responsible

for the

coarsening.

^It is

coincidental,

that these different mechanisms

give

the _{same coarsening}

exponent.

100 d _

io

le+03 le+04 le+05 le+06 le+07 le+03 le+04 le+05 le+06 le+07

t t

a) b)

Fig.

4.

Coarsening

of the one-dimensional SOS models. In each _case the data is the average of 4 realisations of _a system of size 2048. For the _n

= 1 model

la)

the simulation _was

performed

with

ù = o, _{p =} o.7 and K

= 2.o. We show bath the first _zero of the

height-height

correlation function

(

and the average separation of macrostep

d,

which shows

an initial decrease while the macrosteps ⁱⁿ the system ^{form before}

increasing

_{as average} facet

length

increases. For the _n

= 2 model 16) _we teck ù = -o.25, p = 0.8 and 11'

= 1.o. Data for

(

and the surface width _{w are} shown. In bath

figures

the

dashed/dotted

lines have

slope 1/4.

In

studying coarsening

^behaviour there _{are a}

variety

of

approaches

that _may be taken.

A

systematic approach

_[23] is to consider _a

stationary, yet

^{unstable situation with} a finite characteristic

length

L and consider the

growth

rate of _a small

perturbation.

The

growth

law

can then be ascertained from the _manner in which this rate scales with

increasing

L. We

apply

this method to current induced

faceting, by considering

_a small

perturbation ni(x,t)

to the

stationary slope potential m(~)

of the

previous section,

and

expanding equation (14)

to

give

ou ₌

1)) lai ja(m)ôl

₊

Fa'(m) Ii) (27)

Although

this

equation

is

linear,

formal

analysis

^would be _very

heavy. However,

the

growth

rate for ni _can be estimated. All of the terms _on the

right

hand side _are

constant, except

within the

region

of _a macrostep ^{in the} n ₌ 1 model _or at the

boundary

between

adjacent

facets in the _n

= 2 model. We discuss each _{case in} turn.

For the _n

= 1

mortel,

the factors

a(m)

and

a'(m)

_remain finite for all _{~ as} the coarsening progresses, and the

only

term _on the

right

hand side of

equation (27)

which

changes appreciably

is the surface stiffness

dm/d~t.

This has _a broad minimum in the

vicinity

of _a

macrostep,

^which

con be shown to decrease with

step height

A _as

A~~. During

facet

growth

the

slope

and

slope potential change only

_in the immediate

vicinity

of the

macrosteps,

so that the linear mode ni

through

which coarsening

proceeds

is localised in the

regions

where

dm/d~t

is small. Its

growth

rate is therefore of the order of the value of

dm/d~t

at the

macrostep, proportional

to

A~~

_{Since the}

height

^of

macrosteps

^{and the facet}

length

_grow in

proportion

to _one

another,

we deduce that the average facet size

1should

_{increase with time t}

according

to im~

t~R

This result _can also be

reproduced by

another

simple argument

which relies _on the finite size corrections to the surface _current mentioned in the previous section. Consider two

adja-

cent macrosteps of different

height Ai

>

A2,

_{in a}

system

with

periodic boundary

conditions.

Because the current _over the taller

macrostep

is

slightly greater

than that _over the shorter _one

t=ixio(

__

00

ÎÎÎÎÎ17

~Î.~~ ..______~~~~~

-

"""...,

^~~~"

jf

⁰ ....,

-i oo

~~""'

"....,,

o 100 200 300

Fig.

5. In _a system of two macrosteps, the

Iarger

macrostep _grows at the _expense of the smaller.

The simulation shown

began

with _a macrostep ^of

height

100 at site

1= 100 and

one of

height

120 at

= 200, ^and _we teck p =

0.7,

K

= 2.0.

the facet 'downwind' of it will rise and the facet

'upwind'

will

fait,

_so that the diiference _in

height

of the two

steps

becomes

exaggerated,

until

eventually

the smaller macrostep vanishes

completely.

In

Figure

5 _we show the result of

simulating

such _a

system

^{which confirms} that this

is indeed what

happens.

For _a constant ratio

Ai /A2

and _a fixed _average

slope,

it follows from

(25)

that the diiference _{in current} between the two

macrosteps

is of the order of

A~~

r-

h~

The

disappearance

^of the smaller

step

_requires the

transport

of _a volume of order

Ai

r-

F

Thus the associated time scale is

f~,

which

reproduces

the _coarsening law

fr- t~R

For the _n

= 2

mortel, analysis

of the noiseless

equation (14)

shows that the

growth

rate of

perturbations

vanishes

exponentially

_as the _average facet

length increases,

and

analysis

of the facet currents shows

exponentially

small finite size corrections.

Thus,

the

coarsening predicted

is not a power

law,

but rather _a

logarithmic growth ^f

_r- In t

[23].

^{This does not} agree with the

simulations,

and _{we are} forced to consider the eifect that statistical fluctuations in the surface

current may have _on the

coarsening

behaviour. To do this _we add _a random term

JR(x, t)

to

the deterministic current

J(x,t)

in

equation (6).

Such _a term _anse

mainly

from 'shot noise' in the surface _current. Since fluctuations _at

separate positions

and times _are

uncorrelated,

_we

take

iJRix,t)i

₌ °

iJ~ix, t)J~i~', t'ii

=

J*oit t'iôix x'i.

₁₂₈₁

where J* is the uniform surface current of the

steady

state. The

equation

of

continuity

then becomes _a

Langevin equation

for the surface.

Although

_an

analysis

of this

along

traditional hnes

[24]

^{would be}

possible,

the consequences for the

coarsening

^behaviour can be understood

in

purely simple

terms

[25].

Consider the eifect of the current fluctuations _{on a}

single

faceted section of the surface.

During

_a time t there will have been _a random net influx of adatoms into trie facet with _zero

mean but with _{an rms} value

proportional

to

t~/~. Thus,

the surface of the facet is

displaced randomly

_{up or} clown _{as in}

Figure

6 and the 'mass' fi4 under the facet

represented by

the shaded _{area in} the

figure)

will be

similarly

reduced _or increased. If such _a fluctuation reduces

the _mass to zero, then the facet vanishes. In this way, the _mass under each facet of the surface