Decay of Grooves Cut in a Surface with Singular Orientation when the Neighbouring Orientations are Unstable

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Decay of Grooves Cut in a Surface with Singular Orientation when the Neighbouring Orientations are

Unstable

Christophe Duport, Anna Chame, W. Mullins, Jacques Villain

To cite this version:

Christophe Duport, Anna Chame, W. Mullins, Jacques Villain. Decay of Grooves Cut in a Surface

with Singular Orientation when the Neighbouring Orientations are Unstable. Journal de Physique I,

EDP Sciences, 1996, 6 (8), pp.1095-1125. �10.1051/jp1:1996118�. �jpa-00247233�

Decay of Grooves Cut in

_a

Surface with Singular Orientation when the Neighbouring Orientations

_are

Unstable

Christophe Duport (~*),

^{Anna Chame}

(~,**)

,

W-W- Mullins

(~)

and

Jacques Villain (~)

(~) CEA,

Département

de Recherche Fondamentale _sur la Matière

Condensée,

SPSMS, 38054 Grenoble Cedex 9, France

(~)

Carnegie-Mellon University, Department

of Materials Science and

Engineering, Pittsburgh,

PA 15213, USA

(Received

19

February

1996,

accepted

10

April 1996)

PACS.68.35.Ja Surface and interface

dynamics

and vibrations

PACS.05.40.+j Fluctuation

phenomena,

random processes, and Brownian motion PACS.68.35.Bs Surface structure and

topography

Abstract. The

decay

of _a

grooved profile

when trie average orientation is

singular

_is treated in the

particular

_case ^when the orientations close to the

singular

_{one are} unstable. The system is assumed to

exchange

atoms with its vapour. The step fluctuations, ^which allow the

profile decay,

_are treated

by

_a

partially

exact transfer matrix method. The time T to

peel

the topmost

layer

is obtained

as a function of the width 1of _a terrace: f _c~

iexp(~i),

where the constant _~

depends

_on the temperature.

Il. Introduction

~rhe

smoothing

of _grooves

artific1ally

made in _a

crystal

^surface is _a classical

problem (Blakely il ],

130nzel _[2] and Preuss

[3])

^solved

by

Mullins [4] in the case of _a

non-singular

orientation. If the final

(or average)

orientation of the surface is

singular,

the non-linear

problem

which arises is

ontrovers1al.

Indeed,

it is not clear whether the facets which

temporarily

_anse in

experiments

<ire a result of _a miscut as claimed

by Lançon

and Villain

[Si,

or intrinsic

properties

_as claimed

1)y

Spohn

_[6] and

Hager

et ai. [7]. ^The

problem

_is somewhat

simpler

if the orientations close la the

singular

one are

unstable,

because in that _case the

topmost layer

is

peeled

without

perturbing

too much the lower _ones. The

present

^{work is} an

investigation

^of this situation.

At

equihbrium,

the

parts

^with

singular

orientations form facets with

sharp edges.

It is

i easonable to _assume that the

top

and the bottom

parts

of the

decaying profile

also have facets with

sharp edges (Fig. l). ^Indeed,

^the

^{sloping parts}

^{of the surface are formed of}

^steps

^which

interact

through

an attractive interaction

(in

contrast with the usual _case discussed

by

Bonzel

<md Preuss [2] and then

by

Rettori and Villain [8] and then

by others).

The

exchange

of matter

(atom

_per

atom)

_may be either

by

^diffusion on the

surface,

inside the solid _or

by exchange

with the vapour [4]. ^{The last process,} called SALK

(surface

attachment

limited

kinetics),

_is much

simpler

than the other _ones. We shall restrict _our attention io this

(*)

Author for

correspondence le-mail: [email protected])

(**)

Permanent address: Universidade Federal Fluminense, Instituto de Fisica, 24210-340, Niterôi RJ, Brazil

Q

^Les

Éditions

_de

Physique

1996

- 1+

Fig.

1. Initial state of the profile.

Fig.

2. A

typical

structure with

meandering

topmost step. The lower

layers

_are not disturbed.

case. This _{process is}

supposed

to

apply

not

only

to true

evaporation/condensation

of _a

crystal

in contact with _{a vapour}

phase,

but also _in the _case of surface diffusion _on the

crystal

surface

(no exchange

with

vapour)

when

exchange

of the mobile _species

je.

_g.

adatoms)

with _sources and sinks

je-g-

kinks and

steps)

is _so dilficult _{as to} be _rate

limiting.

Then the mobile _species may be considered to have _an uniform chemical

potential

~1 as if it _{were a} vapour

bathing

the surface. The

exchange

of matter between this mobile

layer

and the _{sources or} sinks is assumed

proportional

to the difference between

~land

the local chemical

potent1al

_{~1+ ô~l} determined

by

surface _energy

changes resulting

from surface

displacements.

It is not clear whether this

limiting

_case

actually

_exists, but it has been used

extensively

as a basis for model

calculations [9].

In _a realistic model of the

profile

considered

here,

the two topmost steps should be assumed to fluctuate between their lowest _energy

positions.

The other steps

might

be assumed to be

straight. However,

this

simplified

model is

fairly complicated,

and _we will

study

_a

_yet simpler

model with _a

single fluctuating _step (Fig. 2).

The system ^{will be} assumed to

exchange

atoms with its vapour, so that it is sulficient _to consider _a

single ridge.

In Sections 2 and

3,

the relaxation mechanism is

analyzed

and the

problem

_is reduced _to the

equilibrium

statistical mechanics of _a

single fluctuating

fine attracted

by

_a fixed fine. This

problem

_{is a} dassical _one and its

thermodynamical _properties

will be recalled in Section 4. Its

dynamical behaviour, however,

_{requires a}

special analysis

which is given in Section

5,

where the relaxation time _is estimated. In Section 6 the

principal

results _are

presented.

A

dictionary

ai

t=t

b)

~_~

2

Cl t=~£

1?ig. ^3. Successive states of the

peehng

of trie topmost

layer: ai

at time ii a

junction

^with the immobile step _occurs ^{and the} two steps

loops

created

begin

to recoil

(trie

topmost

layer begin

to

<~vaporate), b)

at time t2 ^another

junction

_occurs ^and

ci finally

at T two opposite

recoiling loops

_merge

~nd the

layer

is

peeled.

>Jf the notations used is at the end of the text. The most

complicated part

of the

algebra

_is

,zontained in the

appendices.

^In

^Appendix

^{A and B} some

equilibrium properties

_are evaluated

asing

the transfer matrix method and in

Appendix C,

the

dynamic problem

is treated _using

an alternative

approach:

a Fokker-Planck

equation

is derived and solved

approximately,

and the

typical

^{times involved} are estimated.

2. Characteristic Time and

Length

If the

topmost

step ^is in its

ground

state

(or nearly soi

at time t

= 0 and

begins

to

fluctuate,

one has to wait _a

typical

^time fi until it makes

junctions

with the immobile

step (Fig. 3a).

After _a

junction

^{has taken}

place,

both ends of the

topmost

terrace

evaporate

and recoil at the

velocity

_~, which is

only

_a function of the terrace width

1,

^{and of the fine tension}

WI la

of the

step.

^If one considers _one of the

recoiling

step

loops,

it will merge

(Fig. 3c)

after _a time f with another

loop recoiling

at the

opposite velocity,

which has been created _at time t2

(Fig. 3b).

Let

Li

be the distance between the

points

where the

junctions

at times fi and t2 have taken

place.

Let _pi be the

probability

_per unit

length

^and time to have _a

junction.

^The time

fi

is the time _one has to wait before

observing

_a

junction

in _a

length

of order

Li Therefore, piLiti

Ci 1. But

(assuming

_{t2 >}

fil

the time t2 fi is also the time _one has to watt before

observing

_a

junction

in _a

length

of order

Li

Therefore _pi

Li(t2 fi)

_CÎ 1. Thus:

fi CÎ t2 Ci t2 fi CÎ

j _(2.1)

Pi ₁ The

quantity (f fi)

is

comprised

between

Li/~ (if

_t2

=

f)

and

Li/(2~) (if

_t2

=

fi).

Therefore

f fi CÎ

~~ (2.2)

~

The left hand side f fi of this

equation

is of _course

larger

than t2 fi, but cannot be much

larger; otherwise,

this would

imply

that the

probability

of _a

junction

in _a

length Li during

a time much

larger

than t2 fi is

negligible,

and this would be _in contradiction with the definition of _pi

Therefore,

f fi _CÎ t2 ti

Comparison

with

(2.1)

and

(2.2) yields

Li

fi CÎ

(2.3)

Now, (2.2)

reads f cf fi +

Li /~

_or

inserting (2.3) jnd (2.1):

f _Qf

~~

Qf

(~ ~)

~

~Î~Î

Then,

it is clear that

1Ci

(2 5)

fi

Thus,

the time f necessary ^to

peel

the

topmost layer

will be known if _{we are} able _to calculate

~ and _pi The

velocity

_~ is

proportional

to the difference between the

evaporation

rate _a of the curved

step

^{and the}

evaporation

rate où of the

straight step.

This difference is itself

proportional

to the local _excess ô~l of the chemical

potential.

The

precise

^formulae _are

~ =

a(a où)

₌

aaoflô~l (2.6)

where _{a is} the interatomic distance and

fl

is

1/(kBT).

The _excess chemical

potential

is the

excess free _{energy per}

partiale.

Since _a

loop

of width 1and

length

~dt has _an energy

Wi~dt la

and contains

i~dtla~ particles,

it follows

ô~l= Wia Ii

and

(2.6)

reads

~ oe

a~aofl ^~ _{(2 7)}

3. The Junction

Probability

_pi

Ii)

The

junction probability

_can be evaluated

ignoring

the

non-fluctuating step.

In

reality

when the

fluctuating

fine touches the immobile _one, the _process becomes irreversible and the _two

separated

_parts of the topmost

layer begin

to recoil. In _our calculation _we will

simply

obtain

an estimate of the

typical

time after which the

fluctuating

fine touches the other for the first time.

Consider _a

part

^of

length

L of the

fluctuating step.

At time t

=

0,

it is

supposed

to be

straight

at

position xiv, 0)

= 0 for _any value of _y. At time

t,

the maximum value

h(t)

of

xiv, fi

will be called the

elongation.

We want to know the

typicaltime Tilt, L)

after which

1

- ~

a) ^L

(h -'- y

~)

---

---'

L

x

~y -~-~---~~à-~-

i~

- -W

c) L

1?ig. ^4.

Typical

excursion

(of elongation h)

of the top ^{of the}

profile

_{on a}

length

L

ai explicitly iepresenting

the lower steps and the _non

fluctuating

step ^and

b) ignoring

them. A _excursion with

elongation larger

thon1, the _size of the terrace, _is ^shown in

ci.

For this

configuration

_{we can} ^define

lhe distance Y

along

^{which all the} points ^have a position x >1.

h(t)

reaches the value1for the first time

(Fig. 4a).

This time should

satisfy

pijtiL~ijt,Li

_~ i

j3.ii

Insertion of this

equation

into

(2.5) gives

us

y _~

@13.21

The immobile

step

^has no

part

in the calculation of

Ti

Therefore,

it _can be

ignored.

We

now consider _a

single

step

(Fig. 4b).

Rather than

calculating Tilt, L),

it may be _easier to calculate the

typical

time _To

Ii) during

which the

elongation

h remains

larger

than 1 if it is

larger

than 1 at _a given time. It tutus out that both

quantities

_are related

by

_a

simple

relation which _{uses an}

equilibrium property, namely

the

probability p)(L,1)

that the

elongation

at _a

particular

time is

larger

than _a

particular

value1

mjch larger

than the. thermal _average

elongation. Indeed,

the

periods

in which the

elongation~is _larger

than 1have _a

typical

duration _To

Ii)

and _are

separated by

_a time interval of order

Tilt, L).

Therefore

PllL'i)

_~

~)j~j~ ^13'3)

1 ,

The

quantity p) IL,1)

is not easy ^to calculate

exactly.

In order to _use

lin

_an

approximate

way)

the exact data which _are

available,

_we introduce the

equilibrium probability pj(L,

_Y,

_h)

that,

_{on a} fine of given

length L,

a number

Yla (and

_no

more)

of

points

^have a

position

_x > h.

Obviously

p)(L,1)

=

~j pj(L,Y,1) _(3.4)

Y>o

On the other

hand,

consider _a number

ôÎ

of

replicas

of the system

(1.e.

a fine of

length L)

taken from the canonical ensemble with the canonical

probability.

The number of

points

of _a

given

ordinate _y which have _a

position

_z > h is

ôlp)(h),

where

PI _(h)

=

~j _{p(z) (3.5)}

~>h

is the

equilibrium probability

that for _a

given

_y,

x(y)

> h. The

probability p(h)

is known

exactly

_as recalled in the next section. The total number of

points

of the fine which have _a

position

x > h is obtained

by multiplying

the number of points of _a

given

ordinate _y which have _a

position

x > h

by

the number L

la,

and therefore _it is

equal

to

(Lla)ôlp((h).

_{On the}

other

hand,

the number of points ^of a given ordinate _y which have _a

position

_x > h _can also be written _as

N

£

^~'

_{pj (L, Y, h) (3.6)}

y~o ^a

Equating

this _to

IL la)ôlp) (hi,

_one obtains

£ Ypj(L, _Y, hi

=

Lp) _(3.7)

Y>o

This formula _may be

exploited

in _an

approximate

_way if Y is

replaced by

its average value

1:

P £ pj(L, _Y, hi

=

Lp) _(3.8)

Y>o Insertion into

(3.4)

_now

yields

p)(L, hi

=

~~~~~~

(3 9)

Y

Now, using (3.2), (3.3)

and

(3.9)

_we obtain the basic formula for the

peehng

time:

T ~

fiÎ 13.i°)

where _{~ is}

given by (2.7)

_as

pointed

out before and

p) _ii)

_is

_exactly

_{known. Its value will be}

recalled in Section 4.

Before

estimating

Y and _To

Ii)

_we define _more

precisely

_our model and recall _some well-known

equilibrium properties.

4.

Equilibrium Properties

of _a

Fluctuating Step

In the

previous sections,

the

problem

has been reduced to the statistical mechanics of _a fine which _can fluctuate around _a fixed fine which attracts it. As will be _{seen, an}

"unbinding"

transition can occur for this

system:

^the two fines _cari be bound

(localized solution)

if the tem-

perature

^is

sulficiently

low _or, for _a fixed temperature, ^{if the attractive}

potential

is

sulficiently strong.

^The

firit

_exact

theory

^of

binding

^of two watts _in d

= 2 _was done

by

^Abraham

[loi iising

the transfer matrix method. Chui and Weeks [11] ^treated the _same

problem

under _a

solid-on-solid condition. A different

derivation,

^with an extension to d

= 3 _was

proposed by

Vallade and

Lajzerowicz _[12].

The

problem

^{treated in}

^Appendix

^{A is the}

^study

^{of the} fluctuations of _a

step

^{which has} a

stiffness

+~

(defined by (5.1) below)

and is bound

by

_an attractive

potential -Wo

to a

straight

fine.

We will _assume

throughout

the paper that the distance between kinks

along

the

step,

^{which is}

of the order of _+~a~

/kBT,

is

large

with

respect

to a, that is to say +~a »

kBT.

This

assumption

,ùlows _one to _use

equation (A.3c),

that

is,

to consider

only

transitions of _one

step

_away or

loward the

binding

fine.

In

Appendix A, by using

the transfer matrix

method,

it is shown that the

density

of

proba- lJility p(x)

_to find _a

point

^{of the} step at a distance _x of the borderline _is, for _x

# 0,

Plx)

=

iaoi~exPl-21~x) 14.1)

where _~ is _a real number.

Since the

integral

of the

probability

is

unity,

the constant in

(4.1)

is

ao[~

oe 2~

(4.2)

The

probability

to find _a

point

of the

step

at _a distance

larger

than _x of the borderline _is

jjiven by integrating (4.1):

p)(x)

_ce

exp(-2~x) (4.3)

The width of the

topmost layer

is assumed to be

large

with

respect

to the size of the thermal fluctuations which is of the order

of1/~,

then i~ » 1.

Moreover,

the continuum limit

implies

that _~a < 1.

Thus,

the relative order of the

length

scales is 1»

1/~

» a

It

might

be

argued

that the

assumption

~a < 1 is very

spec1al,

_since it will be valid

only

on a very small _range of

temperatures.

^This

assumption imphes

that a fluctuation of

large elongation

^{would be}

relatively

_easy. If _a

large

value of

Wo (say Wi/2)

is

chosen,

which would be

physically reasonable,

it could make _~a

comparable

to

unity

and would restrict allowed

fluctuations to a few times _a.

However, experimentally,

^{for various} reasons

je-

_g.

feasibility),

_{we are} interested

by

_macro- or at least

mesoscopié profiles.

The

profile

width 1 then must be much

larger

than _a.

Therefore,

the

product

_~a should be

small,

otherwise the

product

~i will be

huge

and the relaxation time too

(as

will be _seen

later).

On the other

hand,

since _~

changes rapidly

with

T,

the

temperature

range where ~i is small is very narrow. In this _way the

assumption

1 »

1/~

» a is

really physical.

In this

connection,

in

Figure

5 _a

plot

of _{~ versus}

Wo

is

presented (obtained by solving Eq.

(A.25) ),

in which _{~ goes} to _zero as

Wo approaches

the

limiting

^value

given by equation (A.26b) (as

_an

equality).

Since

1/~

is the order of the average

elongation,

this

plot gives

the reader _a

feel for the

dilficulty

of _a

fluctuating step

to _cross the top ^terrace.

The transfer matrix method demonstrates the existence of _a transition at a certain tempera- ture

Tc

at which the constant _~ vanishes. Above this

temperature,

^the

step prefers

to increase

,

/

' /

/

/

,

0.8 /

/

' ,

/ /

/

~'~ Î _/

/

~Î t'

~ / ^~~Î

^~~

0.4 ^' / ~~ ^~'~

/

/

fiw =1.5

i

/

_fiwi ₌₂₀

°'~

' /

fiw

=2.5

,

/

fiw = 3.0

0

0 0.2 0A 0.6 0.8

)W

Fig.

5.

~ as function of

Wo,

the

binding potential.

The order of the average

elongation

of _an excursion is

1/~.

The solid

curve

(flwi

=

o-à)

_may ^be unreliable because

exp(-flwi)

_{would be too}

big.

its

entropy

rather than its

potential

_energy, and thus is not bound to the terrace

edge.

This transition _is called

unbinding

transition _or

delocalisation

transition. When T _>

Tc,

there _are

no forbidden orientations

in the

vicinity

of the average surface

orientation,

and this is the _case discussed in references _{[5] to}

[8].

^{In the} present

work,

the condition T _<

Tc

is assumed. Then

in the _continuum

hmit,

which

corresponds

to _~a <

1,

_we obtain the relations

(see

in

Appendix A, Eqs. (A.23)

and

(A.25),

which define _~, and

Eq. (A.46))

Wo

<

kBT (4.4a)

~~~~'~~~

~

ÎÎ~Î~ ^~~'~~~

The transition

temperature

_is

given by

the condition _~

= 0 and therefore

kBTc

"

@~ _(4.4c)

5. Estimate of

1

Dur next task is to calculate Y and _To. In this section _an estimate for

1

_{is obtained}

using

_a

simple

and intuitive

argument

and in

Appendix

B it is evaluated

through

_a detailed calculation.

The evaluation of_To will also be

performed

in two ways: in Section 6

through

_a

simple argument

and in

Appendix

C.

A

simple

_way to estimate Y is

through

the

evaluation,

for _two

limiting

_cases, of the _average

quadratic

difference

(ôx~ )y

^{between the horizontal}

positions

of _two

points

of the hne

separated

by

_a distance _y.

If _{y <}

1,

then

(ôx~)y

must be

approximately

the _{same as} in the absence of _an attractive

potential, namely

jôx2j~

₌

j _jà.ii

On the other

hand,

if _y

»1,

_{the attractive}

_potential

_must be

considered,

and

(ôx~)y

₌

/ ~ Î~ _lx x')~pja;nt(y;

_x,

x')dxdx' (5.2)

where pja;nt

(vi

_x,

x')

is the

joint probability

that

x(0)

= x and

xiv)

=

x'

simultaneously.

It _can

lJe written as the _sum of _an uncorrelated

probability

and _a

correction, namely

Pjo>nul, X>

X')

_"

AIX)piX')

+

ôpi§,

_X,

X') 15.31

]If_y »

1,

the correlation term is

negligible

and

m m

~~~~~~

Î

~~

~'~~PixiPix')dxdx~ j~,~~

which _can be written _as

(ôx~)y

= 2

/~ _x~p(x)dx

₂

/~ ^p(x)dx)

^~

(5.5)

o o

Using (4.1)

and

(4.2),

this

yields

~~~~~~

12

~~'~~

V

_{can then} be defined _as the _crossover value between the behaviours

(5.1)

and

(5.6),

_so that

Y

=

~( (5.7)

2~

A _more elaborated evaluation

of1

is done in detail in

Appendix

B and the final result is ,£most the _{same as} the

previous

_one:

~ Y

=

0.45~) ^(5.8)

The above result is consistent with the

assumption1

_»

$

_{in the hmit}

~a <

1,

that is _to iay, this result _is vahd when

1

_is much

greater

^{than the vertical distance between}

kinks,

and

this _occurs for

large

values of the

elongation.

Since1

_{is defined}

as

(see Eq. (3.9)

^for

instance)

the average

length along

which the

points

of the fine have

positions

x > h for _{an excursion} with

elongation larger

than h

(where

h is much

larger

^than the thermal _average

elongation),

the fact that

1

does not

depend

_on ^h is

surprising.

In

general1

would

depend

_on

h,

however the result

(5.8)

shows that

1

does not

depend

_on h if the

elongation

^is

sulficiently large.

6. The

Typical

Time _TO

The last unknown

quantity

is _TO

Ii).

A

simple

_way to estimate this

quantity

^is to consider the recoil of _an

exceptional

excursion of

amplitude

h.

Let _a be the

evaporation probability

from the

fluctuating

_{curve per} site and _per

second,

and let

rp

be the

adsorption probability

from the _vapour when this vapour has _a

density

P " Po

exp(fl~l) (6.1)

The

quantity ~lis

the vapour chemical

potential

counted from _an

appropriate

_origin. At

equilibrium, adsorption

and emission

mutually compensate

and

a =

rpo expjp~) j6.2)

In

reality,

_{an excursion} of

large amplitude

h is _{not at}

equilibrium

with the _vapour which has

a fixed

density

_{po, so} that the

adsorption

rate per ^site has _a fixed value

où "

rpo (6.3)

On the

contrary,

the emission _rate per site has _a value

given by (6.2)

_{as a} function of the chemical

potential ~1(h)

^that the _vapour would have if it _were in

equilibrium

with the excursion of

amplitude

h:

a = où

exp(fl~l) (6.4)

This value will be assumed to be uniform _on the whole _excursion.

In this way, the

velocity

of recoil of the excursion is

given by

(

"

°o(1

_eXP

lfl~))a (6.5)

Assuming

that

fl~l

<

1,

one can write ôh

1 ^°°~~~

~~'~~

The chemical

potent1al

~1 can be written _as

(+~a~)/R,

where R is the radius of _curvature of

the _excursion. The

quantity

which _appear in this

expression

^for _~1 is the hne stiffness

+~ rather than the fine tension

Wi la

_{as in}

(2.7)

because this

expression

for

~1 is valid for weak

slope

fluctuations while

(2.7)

is related to _a macroscopic

loop.

To estimate R _we recall _a result from

Appendix A,

where the

typical

extension of _a fluctuation of

elongation

h is obtained. One finds that

~~~~ àÎ~ _~~'~~

Assuming

the excursion _{is an arc} of ciEcle _or

parabola

of base

length

and

amplitude h,

_one finds R

=

(fl~+~~h) /~~ multiplied by

_{a constant} of order

unity.

Insertion _into

(6.6) yields

ôh

ao~~a~

j _16p~h ^(6.8)

The fact that h _> 1 at t

= 0

imphes

that the

typical elongation ^[

that exceeds 1lies between 1 and 1+

) Therefore,

the

typical

time _To necessary for

[

_{to become smaller than is}

1/(~ () ).

Replacing () _by

its expression

(6.8),

_one obtains that

T0(h)

t

~j~

~

j6_gj

ao~ a

7. Results

In this

section,

the

assumptions

and the

principal

results of the

preceding

sections _are

recalled,

in order _to obtain _a final

expression

for the time f to

peel

the

topmost layer.

We assumed that the

step

^{which fluctuates is bound} to the

hmiting

borderline of the

topmost layer,

that is _to say, the

step

is below the

unbinding

transition: the wavevector of the bound

state is _~.

The

topmost layer

is éissumed to be

large

with

respect

to the _size of the thermal fluctuations which is of the order

of1/~,

then i~ _» 1.

Moreover,

the continuum limit

implies

that _~a < 1.

In this _way the relative order of the

length

scales is1»

1/~

» a.

Finally,

_we have

supposed

that the

typical length

between kinks

along

_a

step,

fl+~a~, ^is

larger

than the interatomic distance _a and that _{we are near} the

unbinding

transition in order that

the

large

excursions _are not too

exceptional.

In this _{way +~a »}

kBT

_»

Wo.

The

quantity

_we want to calculate is the

peeling

time of the

topmost layer,

which is

given by (3.10).

It is sulficient _{to insert} into that formula the

expressions (2.7), (4.3), (5.7)

and

(6.9)

to obtain the final result

~ ~

210a2

fi~~~~'~~~

^~~'~~

where _~ is defined

by (4.4b).

All these results refer _to the _case when the

topmost

terrace is limited

by

_a

fluctuating

step and _a

rigid

_one. If it is limited

by

two

fluctuating

_{steps, a} formula

analogous

to

(3.10)

should

apply, namely

" ~

i~ ~~~~

where

p)o(h)

is the

probability that,

for _a

given

_y, the distance between both

topmost steps

is

greater

^{than h. This}

probability

_is

given by integrating

the

probability pro(h)dh that,

for _a

given

_y, the distance between both

topmost steps

^is between h and h + dh. Now

Poolhl

"

/ _{Plh qlPlqldq (7.3j}

or

using (4.1)

and

(4.2):

pro(h)

=

4~~hexp(-2~h) (7A)

Integration

now

yields

m cc

p(o(1)

=

pro(h)dh

=

4~~

h

exp(-2~h)dh

=

(2~i

₊

1) exp(-2~i) (7.5)

Equations (2.7), (5.7)

and

(6.9),

which

respectively yield

~,

i

and _{To, are} still

expected

to hold.

Inserting

these relations and

(7.5)

into

(7.2),

_one obtains the

peeling

^{time of} a

layer

in the _case of two

fluctuating steps

as

y~ r-

_2~t3aoa2 fiexpj~ti) ^{ah 17.6)}

Acknowledgments

This work _was initiated in Jülich

(Germany).

W-W- Mullins

acknowledges

the

hospitality

of the Institute für Grenzflàchen und

Vakuumphysik

and

support

from the _von Humboldt

Sitftung,

and J. Villain the

hospitality

of the

Hochleistungrechenzentrum.

A. Chame

acknowledges

_a

grant

^{from the Brazilian} National Council for the Scientific and

Technologica1Development (CNPq)

and the

hospitality

of the Centre d'Etudes Nucléaires de Grenoble. We thank Prof.

Lajzerowicz

for fruitful discussions.

Appreciation

is

expressed

to

Xing

Chu for

preparation

^of

Figure

5.

Notation

Dictionary

. a: interatomic distance.

. kB~ Boltzmann constant.

. i: terrace width.

. L:

length

of _an

arbitrary _part

of the

system.

.

Lit

distance between the

points

where the

junctions

at times fi and t2 have taken

place.

.

L(A):

number of

adsorption

sites.

. N: number of lattice sites in the _x direction.

. pli

junction probability

_per unit

length

and unit time.

.

pj(L,Y, h): equilibrium probability that,

_{on a} fine of

given length L,

_a number

Y/b

of

points

have _a

position

_x > h.

.

p(h): equilibrium probability

that for _a

given

_y,

x(y)

=

h, independent

of _y.

. fil time _one has _to watt before

observing

_a

junction

_{in a}

length

of order

Li

.

p((h): equilibrium probability

that for _a

given

_y,

xiv)

> h.

.

p((L,1): equilibrium probability

that the

elongation

at _a

particular

time is

larger

than

a

particular

value 1 much

larger

than the thermal _average value of _x.

.

p(y;

_x,

x') joint probability

that

x(0)

= x and

xiv)

= x'.

.

p(~~ ~(y,

_x,

x'):

conditional

probability

that

xiv)

= x' if

x(0)

= x.

.

pro(h): probability that,

for _a

given

_y, the distance between both

topmost steps

is h.

.

p)o ii) probability that,

for _{a given y,} the distance between both

topmost steps

^is

greater

than 1.

.

PL (h, t): probability

that the

elongation

at time t of _a fine

segment

of

length

L is h.

. T:

temperature.

. ~: recoil

velocity

of the

topmost

terrace.

.

Woi

attractive

potential

due to the

binding

line

(Wo

>

0).

.

Wi step

^free _{energy per} atom.

Wi la

is also the line tension at low

temperature (Wi

>

0).

.

i(h):

_average value of the

length Y, along

which the points have

positions

x > h for _an

excursion of

elongation larger

than

h,

where h is much

larger

than the thermal _average

elongation.

~

p,

¹

kBT'

. ~t: line stifIness.

. r:

adsorption probability

from the vapour per ^site and _per second when this _vapour has

a

density

_p

= 1.

. e:

exp(-flwi).

. < defined

by equation (A.25).

.

À(x):

extension of _a fluctuation of

elongation

_x; vertical distance after which the line has lost the memory and _come back to its thermal

position.

. /~; vapour chemical

potential.

. pot vapour

deisity _(fixed).

. a:

evaporation probability

from the

fluctuating

_{curve per} site and _per second.

. où

adsorption probability

from the vapour per site and per second.

. f: time necessary ^to

peel

the

topmost layer

when _one of the topmost

steps

is unable to fluctuate.

. f21 ^time necessary ^to

peel

the topmost

layer,

when both

topmost steps

can fluctuate.

.

Ti(1, L):

^{time interval between the}

^periods

^{in which the}

elongation

of _a

particular step segment

^of

length

L is

larger

than 1.

. To

Ii): typical

time

during

which the

elongation

remains

larger

than1 if it is

larger

than 1 at _a

given

time.

Appendix

A

Unbinding

llkansition in 1+1 Dimensions We consider

(Fig. 3a)

_a fixed

straight

wall

ix

=

0)

and _a hne which is at the

position xiv)

> 0.

The line has _a stifIness which maintains it

straight

at the

temperature

^T =

0,

^and it is attracted

by

the fixed wall due to _a

potential V(x)

< 0 for _x

= 0.

It is easier to suppose that _x and _{y are} discrete

variables,

_x

la

and _y

la being integers,

where _a is the interatomic distance. It is also _easier to use

periodic boundary

conditions. The _{energy is}

jN-1)a jN-1)a

U((X), (VI)

#

~j _Vl(X(§))

₊

~j V2(X(iÎ)> X(iÎ

+

~)) (~.~)

y=0 y=0

where

Aix)

=

-woôo,~ (A.2)

with

Wo

> 0 the attractive

potential

and V2

corresponds

to the stifIness. We _can write

V~(x,x)

= 0

(A.3a)

V2(x,

x +

a)

₌

V2(x

+ a,

x)

=

Wi

> 0

(A.3b)

and for the other

values,

V2(x, x')

= oo

(A.3c)

Equation (A.3c)

_means that

x(y) x(y -1)

_can

only

have the values

0,

_{a or -a,} otherwise the Boltzmann factor is

equal

to zero.

A.I. THE PARTITION FUNCTION. The

partition

function is

Z =

£ _{exp[-flvi (xo}

_Il

_{exp[- flV2 (xo,}

xi

Il exp[- flvi (xi

_Il

~o,~i, ,xN-i

x

exp[-

_flV2

(xi,

_x2Il

exp[- flV2 (xN-i,

_xoIl

(AA)

where xp =

x(pa).

It is useful _to introduce the notation

(x[8[x')

= exp

-~Vi xii

exp[-fl%(x,x')]

_exp

(-~Vi(x')j ^(A.5)

2 2

Now

(AA)

_can be written

Z "

L iXoi8iXiiiXii8iX2i. IXPIÔIXP+ii. IXN-iiÔiXoi iA.61

The

quantity (x[8 lx')

_can be considered _{as an} element of _a real Hermitian matrix 8. For the

moment _we do not

give

the vector1al

meaning

to the notation

xi.

We will do it

later,

but _now

this notation is

only

a part ^of the

quantity (xl

8

lx'),

which _{we can} call

8xx,. Equation (A.6)

is _written

Z = Tr

8~ IA.?)

The trace _is easy ^to calculate if the

eigenvalues 0k

_are known. Then Z =

£ 0f _(A.8)

k

The

interesting

_case is the limit N

= oo. In this _case,

by supposing

that there _{is a} gap between the

greatest eigenvalue

00 and the others

eigenvalues, equation (A.8)

is written

Z =

0/ _(A.9)

This

expression

is _not

universal,

because there _is not

always

_{a gap,} but is

quite general.

The calculation of the

eigenvalues, instead,

is

specific

of each

problem.

A.2. EIGENVALUES. An

eigenvalue

0 is defined from the

equation

jjxjejx'iijx'i

=

oijxi jA,ici

where _we have omitted the indice k.

From

equations IA-à)

and

(A.3c), equation (A.10)

reduces _to

lx181x)41x)

₊

lx181x

+

a)41x

+

a)

₊

lxie(z a)j(z ai

=

ojjz) jA.ii)

For _x >

2a,

from

equations (A.5), (A.2)

and

(A.3),

_{one can written}

_equation (A.Il)

_as

#ix)

₊

e#ix

₊

a)

₊

e#ix a)

₌

0#ix) iA.12)

where

f = exp

i-flwi) iA.13)

Equation (A.12) obviously

has the

following

solution for _x > _a

~k(X)

_{" ak}

eXp(-kX) _(A.14)

with the

eigenvalue

0k

_" 1 +

e[exp(ka)

+

exp(-ka)] ^(A.15)

Since the matrix 8 is

Hermitian,

its

eigenvalues

must be real and then there _are three

possibilities:

_{1) k is}

real, ii)

k is

purely imaginary, iii)

the

imaginary

part of k is

equal

to 17r.

The third

possibility yields

an

eigenvalue (A. là)

which is smaller than _one and _can not be the

largest eigenvalue.

We will _see that k _can be _or not a real value

(and only one).

To understand the

importance

^of this

fact,

let _us calculate the distribution of

points

^of the

fluctuating

line.

A.3. DISTRIBUTION OF THE FLUCTUATING LINE. The

probability p(x)

that

x(y)

has _a

value _x is

independent

^of _y ^{and is}

given by pjx)

=

j expj- pvi ix)j expj-pV2 ix,

_{xi )i}

expj-pvi (xi )i expj-pV21xN-1, x)j

~

xi,...,x~-i

where Z is

given by equation (AA). Transforming

this

equation

in the _{same way as we} had transformed

equation (AA),

_we obtain the

following expression,

similar to

equation (A.6):

Plx)

=

j _L lx181xi)lxi181x2). lxp181xp+1) lxN-i181x) lA.16)

To obtain _a

compact

^{relation similar} to

equation IA.?)

it is suitable to introduce the vectors

[xi

and the

operators

^{X defined from}

jx'jxi

=

ôxx, jA.17ai

and

(x'[X[x)

=

ôxx,x (A.17b)

This is _a little abstract. For _a

physicist

familiar with

quantum physics,

it is useful _to consider

the vectors

[xi

_as

representing

the states of _a

partiale

that _moves

along

the _{x axis}

satisfying

the

equation

of

Schrôdinger IA.10).

From

equation (A.16)

_we obtain

Plx)

=

jlx18~ix) ^lA.18)

To utilize this

equation,

we have to introduce the

eigenvectors [k)

^{of 8.}

Substituting

8 =

£ jk)0k(kl lA.191

k

in

equation (A.18),

_we obtain

Plx)

₌

j L0ilxlk)lklx) lA.20a)

If the condition of

validity

of

equation (A.9)

is

satisfied, equation (A.20a)

can be written

Plx)

=

j0tlx10)101x)

⁼

^{lx10)101x) lA.20b)}

where _we have used

equation (A.9)

to write the second

equation

and _we have introduced the

eigenvector loi

which

corresponds

to the

greatest eigenvalue 00.

This

eigenvector

^is given

by equation (A.14),

where k has _a

particular

^value _~, ^{that is to} say,

loi

=

fl _lx1401xl

=

aoDl0l

+ ao

L _{[xl exPl-~xl} lA.211

x x#o

where D is _{a constant.}

Inserting equation (A.21)

into

equation (A.20b), using equation (A.17b),

and

supposing

_~ to be

real,

_we

obtain,

for

x diflerent from

0, equation (4.1).

If _{~ is}

positive,

the

fluctuating

fine is localized. In this

article,

this condition is assumed to be fulfilled.

If ail the k _were

imaginary,

the

general

term of

equation (A.20a)

is

independent

of _x. The

fluctuating

fine is delocalized.

A.4. SEARCH _{FOR A} BOUND STATE. A bound state

corresponds

to the solution

(A.14)

with real k.

Equation (A.14)

must also be solution of

equation (A.Il)

for _x

= 0 and _x

= a,

that is to say,

A~#o10)

+

eA#oia)

=

0#o10) jA.22a)

#oia)

+

e#o12a)

+

eA#o10)

=

0#oia) iA.22b)

where

A =

exp(flwo/2) _(A.23)

By replacing

into

equation (A.22) #o(0), #o(a),

and

#o(2a)

calculated from

equation (A.21),

we find

A~D

_{+ CA}

exp(-~a)

= 0D

(A.24a)

AD

= 1

(A.24b)

Inserting

the value for 0

given

in

equation (A.15),

_we obtain the

following equation

of second order in

exp(~a):

A~

₊

eA~ exp(-~a)

= +

e[exp(-~a)

₊

exp(~a)]

or

f(e~~) ~2

" e~~~ e~~

(A~ -1)

= 0

(A.25)

e

The solutions of this

equation

in

exp(~a)

_are

always

real and with

opposite

_sign. The

positive

solution is the

only

_one which

corresponds

to _a real value of k. It

corresponds

to a localized solution if _~ is

positive,

that is to say if

exp(~a)

is

greater

than

1,

that is to say if

f(1)

_< 0.

Then

f(1)

=1- 1+

f~ (A~ -1)

_< 0

or

A~ _> 1 ₊

~ _(A.26a)

or

+ 2

exp(-flwi

(A.26b)

~~P~~~°~

_~

l ₊

exp(-flwi

We observe that the two lines _are bounded if the

temperature

_is

sulficiently

low _or for _a fixed

temperature,

^{if the} attractive

potential

is

sulficiently strong.

A.5. JOINT PROBABILITY. The

probability

that

x(y')

= x and

simultaneously x(y'+y)

= x'

is

p(y;

_x,

x')

=

£ ~j exp[-flvi lx)] exp[-flV2(x,

_xi

)1exp[-flvi (xi

~ )1

xi, ,xyla-i xyla+i, .,xN-i

~~~~~~Î~~~ÎÎ~~~~~~fI~~Î~~~~~~~ ~~~~"~~~~~~~~~~~~ ~~~~Y/~+~)Î

-,

or

piy;

_x,

_x~i

=

j _L ixieixiiixiieiz~i iz~/~-iieiz~i

xi,..,x~/~-i

£ l~'lÔl~yla+11" l~N-llôlX)

~yla+>;. ~N-1

or

Plv;x,x')

=

jlx18~/~lx')lx'l8~~Y/~lx)

For y much smaller than the size Na of the

system,

this

expression

can be written

plu; x,x')

=

fl ())

^~~~

lxlk)lklx'llx'101101xl ^lA.27al

or

fl Yla

pjy.

_x

x')

=

~j

^~

(x[k)(k[x')[ao[~ exp[-~(x

+

x')] (A.27b)

~

flo~

An useful check is obtained for _y

=

0, using equations (A.27a), (A.20b)

and the relation

£~ [k)(k[

₌ 1:

p(0;

_x,

x')

=

£(x[k)(k[x')(x'[0)(0[x)

k

=

(xix')(x'ΰ)(°Îx)

_"

ô~~'P(x) (A.28)

It is convenient to

distinguish

between the localized state

loi

and the unbound state

[q).

Equation (A.27)

is written

P(Yi

_X,

X')

_"

P(X)P(x')

+

ôp(y;

_x,

x') (A.29)

with

ôPlv; x,x'l

=

L ())

^~~~

lxlqllqlx'llx'101101xl ^lA.30)

q#o ^°

The _wave function of the unbounded state _is written

[q)

=

Dqoq[0)

+ oq

£ _sin(qx

+~q)

lx) (A.31)

~#0

where

Dq

^and _+~q are solutions of the

system A~Dq

+ CA

sin(qa

+~q) ⁼

0Dq (A.32a)

sin(qa

+~q) + _e

sin(2qa

+~q) +

eADq

₌ ⁰

sin(qa

+~q)

(A.32b)

or,

replacing

0

by

its

expression IA-là),

A~Dq

+ CA

sin(qa

_+~q)

=

il

₊ 2e

cos(qa)]Dq (A.33a)

ADq

_{= sin+~q}

(A.33b)

Eliminating Dq,

_we obtain

A~

₊

eA~[cos(qa) sin(qa)

cot

+~q] ⁼ 1+

2ecos(qa) (A.34)

or

~2

_{1 +}

f

COS(qa)(À~ ~)

~~

_~~~

~~t +~~ ⁼

~~2

sin(qa)

We _see that

+~q is

proportional

to q if _{q is} small.

The other parameters are determined:

Dq,

from

equation (A.33b);

_oq, from the normaliza- tion condition. In

short,

for _an infinite system in the _x

direction,

ail values _{of q are} allowed.

However,

it may be

preferable

to _{assume a}

large

but finite size L

=

Na,

_so that _q takes discrete values

which,

_as

usual,

_are the

integer multiples

of 27r

IL. Then,

the normalisation condition takes the form

Î~QÎ~

l~

~~~~~~~ _'~Q) ~

~~2'~~j ~~'~~~)

~

~

which,

for

large L,

reduces _to

laql~

Cf

) _lA.36b)

By using (A.30)

and

(A.31)

_we obtain

o Yla

ôPlY> x,

x')

=

lac

_l~

£ _laq

l~

(() _smjqz

~f~)

sinjqz'

_~f~)

expj-~jz

₊

z')j

~

o ^Yla

=

lac

_l~

£ _laq

l~ ^~

coslqjz z')i expl-~jz

₊

z')j

~

o

~ Yla

~l£X01~

£1£XQl~ (() _C°Sl~lX

+

X')

_2'fQl

~~Pl~~lX

_{+ X')1}

_lA.37él)

~

or,

substituting

the _expression

(A.15)

for 0

ôp(y;

_x,

x')

=

[oo

_Î~

~j _Îaq

[~

~ ~~

°~~(~~

^~~~

_cos[q(x x')] exp[-~(x

+

x')]

A

37b)

~

l + 2e cosh

~a)

~

_~ ~ +

z'

j~~

_j2

j jaq

_Î~

i~/Î~ÎÎÎÎ~)

^~~ ~~~~~~~ ~

~'~

_~'~~~

_~~~~

~

It is

pretty hard,

from this

formula,

_{to recover}

(A.28)

for _y

= 0. As _{a matter} of

fact,

formula

(A.37b)

is not very convenient for small y.

However,

_{if y is}

large enough,

the

quantity

Il ^{+2ecos(qa) ~f~}

+ 2e

cosh(~a)