Elastic Interaction Between Modulated Steps on a Vicinal Surface

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Elastic Interaction Between Modulated Steps on a Vicinal Surface

B. Houchmandzadeh, C. Misbah

To cite this version:

B. Houchmandzadeh, C. Misbah. Elastic Interaction Between Modulated Steps on a Vicinal Surface.

Journal de Physique I, EDP Sciences, 1995, 5 (6), pp.685-698. �10.1051/jp1:1995160�. �jpa-00247094�

Classification

Physics

Abstracts

61.50Cj 05.40+1

68.55-a

Elastic Interaction Between Modulated Steps

_{on a}

Vicinal Surface

B. Houchmandzadeh and C. Misbah

Laboratoire de

Spectrométrie Physique,

Université

Joseph

Fourier

(Grenoble I)

lc

CNRS,

B-P.

87, Saint-Martin

d'Hères,

38402

Cedex,

France

(Received

28 November 1994, revised 14

February1995, accepted

6 March

1995)

Abstract. We

study

in this _paper trie elastic interaction between modulated steps, in both

homoepitaxy

and

heteroepitaxy.

The influence of this interaction _on the step fluctuation _spec- trum is discussed. There _are two

important

consequences that emanate from _our

analysis,

₁₎

in the

homoepitaxy

_case the elastic interaction favors _a modulated step

profile. However,

the

gain

in the elastic interaction does not

generally

_overcome the fine tension effect and

ii)

_more

importantly,

for the

heteroepitaXial

_case the elastic contribution to the energy modulation is of the same order _as that of the fine tension part.

Elasticity

results _{in an} effective fine _tension

(but singular),

which _may be either

positive

_or

negative depending

_on the

phase

shift between two

adjacent

steps. This

analysis

shows that the elastic contribution for _non-zero wavevectors

should be

incorporated

in any statistical

theory

of

fluctuating

steps, if _one wants ta account

properly

for

experimental

observations. We

point

eut similarities between this

problem,

and others

arising

in

quite

different systems. We suggest

that,

_m

general,

tensorial

(dipolar

and

multipolar)

interactions between

fines, surfaces,

etc. of

electrostatic, magnetic, hydrodynamic,

etc. nature, ^{should lead} ta the _same

generic

behaviour.

1. Introduction

Interaction between

steps

on a vicinal surface of

crystals

_{is a}

long-standing subject.

It is

a crucial

question

in many

important physical problems,

such _as trie

equilibrium shape

of

crystals,

trie formation of domains _on

crystal surfaces,

trie fluctuation

spectrum

of steps, ^trie terrace width distribution at

equilibrium,

etc.

Steps

_on vicinal surfaces _are interface defects

leading

to elastic _stress fields in trie bulk. Trie

long

_range elastic field of

steps

^interferes so that _an effective

step-step

interaction follows. Trie

problem

of interaction between

straight steps

in trie frarnework of continuum

elasticity

bas been trie

subject

of _vanous

investigations il, _2].

Trie _most

important

result which follows from these studies is that trie interaction energy of two

straight

steps,

separated by

_a

distance1,

is

+w

1/é~

in trie

homoepitaxy

case

(1.e.,

^{when trie last}

layer

of trie

crystal

bas trie _saine

equilibrium

structure _as trie

bulk),

and

+w lui in trie

heteroepitaxy

_case

(1.e.,

when trie last

layers

bave _a different structure than trie

bulk,

due to surface reconstruction _or

deposition

of other

species).

Several authors used trie result for

straight steps,

in _a mean-field

approach,

to

study

both trie

©

Les Editions de

Physique

1995

fluctuation of

steps

at

equilibrium

and terrace width distributions

[3],

or

step dynarnics

_m

step

flow

[4, Si. Incorporation

of

elasticity

in trie total _energy of

step

modulations

(1.e.,

^for situations

with _nonzero

wavevectors)

remains to be

formulated,

and its

far-reaching

consequences ^to be

elucidated.

Dealing

with these

questions

is trie main purpose of this _paper. This

requires extending

trie work of Marchenko-Parshin

iii

to modulated

step profiles.

When

dealing

with these

questions

_we bave also

recognized

that _our

investigation

_may be relevant to _a wide list of

physical systems,

as

explained

below.

Let _us outline trie main results to emerge from _our

analysis. Firstly, _contrary

to what could bave been

expected

à

priori,

m trie

homoepitaxial

_case

(where,

_{as we} shall _see, trie

step

^is a

location of force

doublets)

_a modulation of trie

step profile

results in _a decrease of trie elastic energy. An

inspection

of trie order of

magnitude

of trie elastic interaction

shows, however,

that litre tension should inhibit such _a

(potential) instability. Secondly,

for

heteroepitaxy (where

trie

step

is _a location of elastic forces in contrast to force

doublets)

trie elastic contribution to modes with _nonzero wavevectors results in _an effective line

tension,

which _can be either

positive

_or

negative, depending

on trie

phase

shift between two

adjacent

steps.

Moreover,

_an

inspection

^{of trie} order of

magnitude

shows that trie elastic contribution bas trie _same order of

magnitude

_as trie litre tension. As _a consequence, its contribution must not be

disregarded

when

dealing,

for

example,

with

step

fluctuations. We shall also _see that trie elastic

repulsion

between

straight steps (for

_q

=

0,

where _q is trie trie

wavevector)

_{may overcome, m some}

situations,

trie dilfusive attraction which

usually

leads to

step-bunching dunng

sublimation. In other

words, elasticity

_may inhibit such _an

instability.

Another

important

^{feature of this work is} trie

recognition

that trie

type

^of

(potential)

insta- bilities caused

by

elastic interactions may

arise,

m a somewhat similar way, in other

systems.

Trie

hydrated phospholipidic systems,

^which exhibit _a lamellar

phase,

_are

typical examples.

Indeed,

trie

dipolar

interaction between _two

layers

_is reminiscent of force-force interaction be- tween

steps.

^It comes out that _a modulation of trie

bilayers

red~ces trie _energy, and _may thus account for trie _appearance of

rippled phases

in these

systems.

We shall also

suggest

some ex-

perimental protocols

in other

systems (e.g., magnetic systems)

where trie interaction between

dipole

lines is

expected

to lead to _a spontaneous modulation of trie hnes.

Here _{is a} brief survey of trie

organization

of trie _paper. In Section

2,

we formulate trie

problem

of

step-step

interaction for _an

arbitrary step

modulation. We consider trie

homoepitaxial

and

heteroepitaxial

_cases

separately.

In Section 3 _we discuss trie order of

magnitude

of trie elastic interaction and its

implication

_on trie fluctuation

spectrum.

Section 4 is devoted _{to a} discussion of trie results. Section 5 _sums up our results and

presents

some outlooks. Details of trie basic

formulation

together

with _some technical

computations

_are

relegated

to

appendices.

2. Elastic Interaction of Modulated

Steps

To

study

trie

step interaction,

_we bave to establish _a close distinction between hetero- and

homo-epitaxy.

In trie first _case, trie atoms

belonging

to trie last

layer

are not of trie _same species as those in trie

underlying

substrate.

Since these

questions

_are not

usually

substantiated

enough,

_we bave felt it worthwhile to devote _a detailed discussion to this

problem.

Starting

from

elementary elasticity

_we shall _use

concepts

of force _or force doublet distribution

along steps

to derive

general expressions

for trie interaction energy between two steps

having arbitrary profiles.

We shall then

specialize

_our calculation to trie _case of small

perturbations

of trie step

profile.

This will be sullicient if _{one is}

only

interested in trie linear

stability analysis.

Let _us discuss trie

heteroepitaxial

and

homoepitaxial

_cases

separately.

_~~

t~

~Y

x

_f

Fig.

l. Flux of force

dipoles

_across the step give rise ta force localized

along

it.

2.1. THE HETEROEPITAXY CASE. First _we consider _an adatom _on trie substrate. This

adatom

locally

creates _a force _on trie substrate. Let f denote this force. Trie elastic

displace-

ment created

by

trie force _on trie

plane

is

given

^{in standard textbooks} of

elasticity [6],

^and con

be written _as

U~°~~~(r) =

A(r)f, (1)

where r is _a two-dimensional vector. A is _a second-tank tensor

where1 is trie unit tensor,

rr~

_trie second-rank tensor

(r~rj),

E trie

Young

modulus and _« trie Poisson coefficient.

Equilibrium implies

that the total force felt

by

the substrate _atoms is _zero.

However,

in

general,

the

dipole

is not _zero.

Thus,

the _presence of _an adatom is modelled

by

_a force

dipole.

More

precisely

in order to

compute

trie

displacement

induced

by

the

adatom,

_we bave to take

a force distribution fz at distance _az

(small compared

to

r)

around trie adatom located at _a

point

M

(Fig. 1),

with trie total force

equal

to _zero. fz is trie force exerted

by

trie adatom at

point

^M on trie substrate _atom 1in its

neighborhood.

Trie

displacement

at a

point

P

(Fig. ₁₎

is

easily

obtained from

equation il)

to be

u~~P°~~(r)

₌

(VA.az)fz

₌

VA.(azff)

= VA.D

(3)

where VA

=

(ôAzj /ôx~)

is _a third-rank _tensor and

(D~~)

₌

(az,~ fi)

is _a second-rank tensor of

dipolar

moments. It is understood that

repeated

indices _are to be summed _over. Since D is

symmetric iii

and does not

depend

_{on r, we can} transform

equation (3)

VA.D =

~~~JD~j)

=

(~~~J~J") ^{=div(A.D) (4)}

ôXv ôXv

To calculate trie total

displacement

at

P,

_we bave to

integrate

_over ail

dipoles

in trie half

monolayer

^hmited

by

trie step.

Applying

^trie

divergence theorem,

_we find

u~~P°~~(r)

=

/ /dTdiv(A.D)

⁼

/ _{dsA.D.n, (5)}

s

where _n is trie

in-plane

unit vector

perpendicular

to trie

step. So,

trie

displacement

caused

by

trie half

monolayer

is

equivalent

to that of _a force distribution f

= Dn localized

along

trie

step.

In trie

isotropic

_{case we} consider

here, Djj

=

Dôzj

^{and trie vector f} = Dn is

perpendicular

to trie

step.

^In

equation (5)

dT stands for trie surface element _on trie half

monolayer,

while ds

designates

trie

arclength

element

along

trie

step.

Thus it _appears that trie total elfect amounts to

considering

_a force distribution

along

trie

step.

This _is not surpnzing.

Indeed,

because trie Lamé

equations

are conservative

expressing

trie fact that _a volume interact

elastically

with its

surrounding only through

its

surface,

and thus trie total force must be conservative trie

integrated displacement

_over trie half

monolayer gives

a contribution

only through

trie contour, _i-e-, the step, m terms of _a force

dipole

distribution

flux _across the

step.

^This amounts to forces localized _on the

step.

In conclusion of this

discussion,

we reiterate that trie elastic

problem

reduces to

considering steps

as locations of force distribution.

A localized force

fi

_{on a}

given step

creates an elastic field _ui, whose expression is

given by equation (1).

The interaction _energy between two

points belongmg

each to two

neighboring steps separated by

_r reads

(where

trie forces located at the two

points

whose interaction _is

considered _are

fi,

and

f2)

E~°~~~

=

fiui (r)

=

f2u2(-r) (6)

where _uz is the

displacement

induced at the

application point

_{of fz}

by fj. Using expression (1),

we obtain

E~°~~~

=

~ ° °

fi

f2 +

(rfi (rf2 (7)

XE _{r r}

Note

that, fixing

_r, trie minimum energy is reached when ail the three vectors

fi,f2,r

_are

parallel

with

fi

"

-f2.

For

straight steps

on a vicinal

surface,

all forces point ^{to the} same

direction and this

configuration

bas trie _maximum energy.

Modulating steps

modifies this result

and,

_{as we} shall _see, lowers trie _energy under

precise

circumstances.

Let _{us now}

compute

^{trie total}

step-step

interaction energy. This is

obviously

_given

by mtegrating

E~°~~~ _over the two

step profiles

~~

~~~~ ~~

~~j~~~ / / _~81d82

~

~

(~l~2)

_~

~(~~l)(~~2)

,

(~)

~

where _{p is} the

density

^{of localized forces} at

steps (p

_+w

1la

where _a is trie atomic

spacing)

and

f

trie

strength

of localized

forces,

_r trie distance between two

points

at trie two

adjacent

_steps

during integration

and _nz trie local normal to step 1.

Expression (8)

is trie

general

form of trie interaction

irrespective

^{of the} step

profile.

Since _our atm _is to

investigate

trie

stability

of

straight steps against

infinitesimal

perturbations,we

shall confine ourselves to

computing

trie

step-step

_{energy up} to second order _m trie deformation. Since Fourier modes do not

couple

to this

order,

it suffices to consider _one Fourier

component. Moreover,

we shall allow for _an

arbitrary phase

shift between

steps.

More

precisely,

two

neighbonng step

modulations are

taken to bave trie

following

form

(see Fig. 2)

vi " e

cos(qxi)

_v2

" 1+

ecos(qx2

+

#) (9)

The

quantities

_{r, n,}

etc.,

^which _appear m

equation (8),

_are

easily expressed

in terms of _vi and _y2. Trie calculation _is

straightforward,

trie

algebraic

details _are

presented

in

Appendix

A.

We

find,

_per unit

length

of trie

step Ef~~~~P

=

~~

°~~~~~~~ (Ci In(ilL)

+

(eli)~ il iqKi (iq) cos11 (10)

XE

where

Ci

_"

2a/(1- a)

and

Ki

is trie modified Bessel function of first order. L is trie step

length.

Figure

3

represents

trie contribution

proportional

to e~

(that is,

_we take trie energy origm that

corresponds

to

straight steps)

in

Ef~~~~P

uersw

iq

for _vanous values of

#.

A few remarks

2~

q Q

Fig.

2.

Modelhng

of the elastic interaction of two modulated steps.

2.5

2.0 4"K

1 5

k4 -Q

1 0

____,

__---~_--

_-- _'

_-' -'

"

Ô 5 "

,' é"~'

,

0.0 ^'

0 1 2 3 4

iq

Fig.

3. Elastic interaction energy of two modulated steps vernis the _wave number _q, for different values of the

phase shift,

in the

heteroepitaxy

_case.

are in order. From

Figure

3 _{one sees} that _a modulation

automatically

leads _{to an} increase of trie elastic _energy, since ail _curves bave

positive

values. For

in-phase

fluctuations

(#

=

0)

trie _{energy is an}

increasing

function of _q, while it is _a

decreasing

^{function for out-of}

phase

fluctuations

(#

=

x).

More

generally,

for

#

>

x/2

trie energy

decreases,

and it increases otherwise.

It is instructive to

expand

trie _energy for small values of _q. To second order in q, trie _energy reads

~~~

~~~~ ~~

Î~~~~~ ~ Î~ ^~

_~

_~Î

_~ ~~ ~ ~

~~~~~~

_~~

~°~

Î

~~ ~~~~

where AE

designates

^trie energy

by taking

trie

straight step

as a reference. Trie

important

issue of trie

present

calculation _is that trie _energy behaves _as q~

lnq (to

be

compared

with trie line tension

term,

which behaves _as

q~).

^Since

qi

< trie factor

multiplying

_q~ bas _{a sign} fixed

by

that of

cos#.

For

#

<

x/2, elasticity

acts _{as a}

positive

line

tension,

whereas for

#

>

x/2,

trie elastic elfect _may be

thought

of _{as an} effective

negatiue

^line tension! If trie energy

magnitude

_were

large enough

vide

infra),

_one would expect to be in _a situation of spontaneow

eiasticity-ind~tced moTphological instability

when

non-equilibrium phenomena

_are involved. Trie basic _reason is that trie

(non-variational)

dilfusive mechanism

during

sublimation enforces trie

#

= «-mode. As _we shall _see

later, according

_to trie

exploration

of various

regions

^of

parameter

space made _so

far,

it _seems that diffusion of adatoms

(which

is

stabilizing during sublimation)

inhibits such _a

potential instability.

A remark should be made at this

point. Equation (5)

shows that _a

step

can be modelled

by

_a distribution of localized forces. This is _a

general

result: D represents trie

force-dipole

induced

by

adatoms _on trie last

layer

and trie localized forces f

along

trie

steps

are

simply given by

D.n.

Assuming

that D is

isotropic,

_we conclude that these forces _are

perpendicular

to trie

step.

^{It is}

possible

to extend this calculation to trie

anisotropic

_case. If trie exact form of D is

known,

we can

easily

deduce trie direction and

strength

of forces at trie steps.

An

important example

is

provided by Si(001)

where dimerisation _{occurs on} trie last

layer.

There _are two kinds of

domain,

_say A and

B, separated by

steps, ^{and trie direction} of dimeri- sation rotate

by

90° bewteen consecutive domains.

Taking

trie x-direction

parallel

to _one of trie dimensation

directions,

_{we can} model D

by

_[7]

D =

dll ^°

(12)

0

dl

^'

This amounts to localized forces f

=

~(djj dl

at

steps

whose direction is

opposite

from _one step ^to trie _{next one.} This situation contrasts with that considered above where D _was taken to be

isotropic.

Of _course, in

general

ail

crystals

_are

anisotropic,

and

strictly speaking,

D is

anisotropic accordingly.

The _case of

Si(001)

_is

spec1al

in _as much _as the

anisotropy

results in

an alternation of the force direction between two

adjacent

_steps due to the 1 X 2 and 2 X 1 reconstruction. The total

step-step

_energy is

given by

the _same

expression

as

equation (10) multiplied by

-1. This entails that

in-phase

_step fluctuations will be enhanced

by

elastic effects. This is trie _case studied

by

Tersoff and Pehlke _[8] who found indeed that

in-phase _step

fluctuations _were

unstable,

in

agreement

with _our

analysis.

It must be

emphasized, however,

that these authors confined their

study

to trie _case of

#

= 0. Dur calculation treats

arbitrary

fluctuations with

regard

to trie

phase.

As is

clearly

_seen from

Figure

4 trie most unstable

configuration

is obtained for

#

= x and q =

0,

^and not

#

₌ 0 and _q

= q*

(where

_q* is _a finite value obtained from _a

compromise

between

elasticity

and line

tension;

_see

Fig. 4(b) ).

Otherwise

stated, elasticity

leads to

(deterministically straight) step-pairing. Therefore,

their conclusion

regarding elasticity-induced morphological instability

in _a realistic situation

(1.e.,

^{where ail}

phases

_are

permissible)

is called into

question.

We must however be somewhat cautious: their

calculation

(including

trie bulk

response)

is

fully anisotropic,

and _our present ^remark would be conclusive

only

after trie authors bave included _an

arbitrary phase

in their treatment.

Before

proceeding farther,

let _{us now} bave _a

digression.

Since trie interaction energy between two forces behaves _{as m}

1/r,

its

integration

_over trie

straight steps diverges.

This is trie

analog

of trie interaction between two

charged

lines _m electrostatics. This term _is not

eztensiue,

_i.e., the interaction energy of N

steps

grows as NlnN. This

longstanding puzzle

is still _a matter of debate.

2.2. THE HOMOEPITAXY CASE. In this subsection _we deal with

homoepitaxy.

The most

important point

is that the adatoms of the last

layer

_are of trie _same nature as those of trie substrate. Besides trie

fact,

which holds for all

situations,

that trie total force exerted

by

trie last

layer

atoms on their

neighbors

is zero, here trie total

dipole

vanishes too. It should be

emphasized, however,

that _an isolated

adatom,

be it of trie _same nature _as trie

underlying

iq°

-05 ', _,, , .

, , , ,'

' ,

"»,__1_,,*"

' .

~.5

.15

.15

0 1 2 3 4 0 1 2 3 4

1q _lq

Fig.

4.

Step

_energy in trie _case of

Si(001) (Solid

fine:

#

= x; dashed fine:

#

=

0) (a)

elastic

interaction; (b)

elastic interaction

plus

fine tension contribution.

substrate _or

not,

creates a non-zero force

dipole (see

last

subsection).

Trie difference arises

only

when _one considers _a half

monolayer, Indeed,

in trie

present

case trie

monolayer

is not constrained

(because

there is _no

misfit).

It behaves

exactly

_as trie bulk. We _can

present

_ouf

reasoning by imagining

trie

crystal

structure to be _a result of _energy minimization with

respect

to trie lattice spacing. Since trie half

monolayer

atoms are m their natural bulk

position,

this amounts to

requiring

trie force

dipole

to vanish. One _can, if need

be,

convince oneself

by resorting

to a broken bond model.

Thus,

_an adatom

belonging

to trie half

monolayer

bas

a zero force

dipole.

Trie first

non-vanishing

contribution _comes then from

force q~tadrupoles only.

A remark is in order. When _we refer _to trie natural

position

of trie atom

belonging

to trie last

layer,

_we bave in mind trie

position

in trie

plane

of trie surface

crystal.

Of _course, trie vertical

position

is affected

by

surface

relaxation,

which decreases

exponentially

with trie

penetration distance,

_as

compared

to trie

algebraic decay

of

displacement

field due to

in-plane

forces. The contribution

stemming

from surface _{relaxation can thus be}

ignored

for ail

practical

purposes [9].

Having

^{shown that} an adatom _on trie surface _can be modelled

by

_a force

quadrupole,

_we

are in _a

position

to determine trie

step-step

^elastic _energy. Trie

procedure

follows

exactly

that

presented

above for

heteroepitaxy.

Trie total elastic _energy associated with trie

monolayer

_can

be written _as

being

due to

force

do~tblets localized

along

trie

step,

since trie flux of

quadrupoles implies dipoles along

trie

integration

contour. This distibution of force doublets

perpendicular

to trie

step

is what _was

initially proposed by

Marchenko and Parshin

iii

and rederived in _a

more

rigourous

_manner

by

Andreev and Kosevitch [2].

An

important

^{remark should be made. All} we discussed here _was about surface defects vith

zero total moment.

However,

as

pointed

out

by

Marchenko and Parshin

iii, steps,

as surface

defects,

must bave

non-vanishing

total moments to

compensate capillary

forces. Consider _a

large

radius

region

around _a

given step. Capillary

forces

acting

_on trie

edge

of this

region

create _a moment ~fa per unit

length,

where

~f is trie surface tension of trie substrate and _a trie

step height.

Intemal

stress,

^modelled as force

dipole

with _non-zero total moment and _peTpen-

dic~tlar ta the

s~trface,

should

compensate

trie moment induced

by capillary

forces.

However,

in trie

followmg,

we

neglect

this effect.

Actually,

_{we are} interested in trie interaction energy of modulated

steps,

as a correction to trie zeroth order

approximation

of

straight steps.

^Trie

derivation of this _non-zero total moment

dipole

_was based _on

capillary

forces

acting

on a

large

region around trie

steps,

^and does not

depend

_on the

morphology

of trie

step. Therefore,

_we

neglect

it in trie

following.

Trie interaction energy between two force doublets

(each

on a different

step) separated by

_a

distance _{r is}

given by

~~~~~~~~ ^~~ ~

~~~'~~

~~ ~Î ^~~~~l)~

⁺

^(~Ll2)~j

⁺

~)~(rLll)~(rLl2)~~

~~~°~

^~~

(rni)(rn2)(nin2)

+

~°

~

(nin2)~ ₍₁₃₎

r r

For

complementary

information about details

leading

to trie above _{expression see}

Appendix

B

(compare

with

Eq. (7)).

Note that

here, f'

is the force

doublet, 1.e.,

bas trie dimension of _a force

multiplied by

_a

length.

For _a fixed _r, trie interaction energy is minimum when

fi'

_and

f2'

are

parallel

and make _an

angle

with _r. This

angle

is

given by:

cos~

=

~(

~

~

(14)

which is _m

x/3

for

typical

values of _{a, a tz}

1/3.

This effect _may favor

modulating steps (see below).

To obtain trie total step-step interaction energy,

expression (13)

is

mtegrated

_over trie two

step profiles, exactly

_as in trie

heteroepitaxy

_case. When _we

spec1alize

_{our expression} to small deviations

(with

_one Fourier mode _as

before)

about trie

straight configuration,

we obtain

E~~~~~~

=

~~~

)j~Î~~'~~

+ ^{~~ ~}

)j(Î'~~~~ ^{1(1 a)(6 (iq)~)+}

l((1- 7a)(iq)~ a(iq)~) Ki(iq)+

(i a)(-3(iq)2

₊

(iq)4)K~(<q)j cosij. _(is)

As

before,

_p is trie number of doublets _per unit

length

of trie step. ^{The first} term _m the above

expression

is trie interaction energy of two

straight steps,

_as

given by

Marchenko and Parshin

iii.

This is trie classical

lli~ _repulsion

between

steps.

We _are interested here _m trie e~ term in this

expression

which

corresponds

to trie contribution due to trie modulation.

Figure

5

displays

trie behaviour of

AE~~~~~P (where

trie contribution from

straight

steps ^lias been

subtracted)

_{as a} function of

qi.

A

striking

feature of trie outcome of this calculation is that

elasticity

destabilizes

straight steps,

above _a certain value of

qi

m 1- 3;

(AE~~~~~P

becomes

negative).

This is _{a new}

qualitative result,

discovered here for trie first

time,

to trie best

knowledge

of trie authors. This is _a

surprising

^{feature masmuch} _as two

straight _steps having

trie _same

sign repel

each other. That is to say, contrary to what could bave _a

priori

been

expected

on trie basis of trie well-known

repulsion

of two

straight steps,

their modulation

results in _a decrease of energy.

This result _can

easily

be

recognized

from

equation (15) by noting

that modified Bessel functions decrease _as

exp(-iq)

for _{q »}

1Ii,

_so that

AE~~~~~P

+~

-q~

for

large

values of _q, 1-e-,

AE~~~~~P

_{is a}

decreasmg

function of _q

Elasticity

at

eq~tilibr~~tm

should lead to a

spontaneous

step modulation if it _were

acting

alone.

However,

_any

step

modulation is

accompanied

with

a line tension energy. Thus, _a modulation _can take

place only

if trie elastic energy

amplitude

overcomes trie hne tension effect. In

fact,

_{as we} shall _{see m} trie next

subsection,

trie hne tension

is

sufficiently large (typically

two _or three orders of

magnitude larger

than trie elastic

energy)

so that it

penalizes energetically

_any modulation.

We would like here _to

anticipate by making

trie

following important

^{remark. Trie} _expression of the interaction energy between forces

(as

_m the last

subsection,

_see

Eq. (7))

_or force doublets

8 o.o

-0 2 4

~-0.4

_~4

~ ~ ⁰

0 6

-4 -0 8

-1 o -8

0 0 0 5 10 15 2 0 0 1 2 3 4

1f/ 1f/

Fig.

5. Elastic interaction _energy of two steps _m the

homoepitaxy

_{case versus} the _wave

number,

for dilferent values of

phase shift, a) #

₌ 0;

b)

dashed fine: #

=

r/2,

solid fine:

#

_{= r.}

(Eq. 13)

_{are very} reminiscent of

dipole-dipole

and

quadrupole-quadrupole

interactions in electrostatics.

Therefore,

the

implication

of _our results may be

generic

in the _sense that

they

con

apply

to

electric, magnetic, hydrodynamic,

etc.

systems

_as it will be documented in trie conclusion of trie

present

_paper. In

particular,

_an

interesting system

is trie _pure

phosphohpidic

membranes which _may

organize

themselves for _a certain range of

temperature

and water

content _m lamellar

phases

of the

amphiphilic bilayers

where trie molecules have _a

polar

head

[11].

^Trie

dipolar

interaction energy between _two consecutive

layers

leads to _a similar

instability [12],

which _can manifest itself since such _a

system

^{is devoid of} a surface tension. We shall _corne back _to this

point

^later.

Let _{us come} back to expression

(15)

^{For small} _q, ^{trie e~} term reads

Taking

_a

typical

value of _{a m}

1/3,

_we

easily

find that for

cos#

<

2/3,

1.e.

approximately

#

>

x/4, AE~~~~~P

_{is a}

decreasing

function of _q for small _q. Here

again

one sees that

elasticity plays

the role of _an effective line tension.

2.3. ORDER _OF MAGNITUDE OF ELASTIC INTERACTIONS. A

potential instability

caused

by elasticity

_may ^{manifest itself}

only

if it is

capable

of

overcoming

the line tension elfect. It is therefore of

great importance

to

provide

orders of

magnitude

of the elastic contribution. Let _us

concentrate _on the

small-q regime,

^{where the} elastic contribution

(due

to the

step modulation)

is +w

q~.

The q~

prefactor

_{is our} definition of the

amplitude (which

_we

eventually

_compare to the line

tension).

Let _us consider

heteroepiatxy

and

homoepitaxy separately.

. In

heteroepitaxy,

it is clear from

equation (Il)

that this

amplitude,

denoted

by Ai

is given

by

Ai

+w

~(~~ (17)

and

f designates

a force. We _can

give

a crude

approximation

of this

amplitude

based

on a dimensional

analysis. f

is _a

force,

and

depends

_on trie mismatch between atomic

spacing

in the surface and in trie

bulk, ôala.

Trie

only

_way to construct a force _is

(with

the additional

assumption

^of linear

elasticity

which

implies

that

f

+w

ôala) f

~-

(ôala)Ea~. ₍₁₈₎

Taking

p1

1la,

ôa

la

_m

^3.10~~

and for E _a

typical

value for

silicon,

E1 10~~

Pa,

_{a m} 3

À,

_we obtain

Ai

m

10~~~ J/m

The

question

of order of

magnitude

is not documented

enough

in the literature. Alerhand et ai. [7] and Webb ^et ai. [13] ^{used trie}

spontaneous

^formation of stress domams _on

Si(001)

to evaluate the force

strength

at steps. ^Their

analysis yields pf

= 0.07

eV/À~. Using

this

value,

_~ve obtain

Ai

_m

^10~~~ J/m,

which is coherent with _our dimensional

analysis.

This

amplitude

is to be

compared

to trie line _tension. A dimensional

analysis (a typical

cohesion energy, a fraction of _an eV per an atomic

length)

leads to _~f

+w

10~~° -10~~~ J/m.

This is consistent with the value measured

by

Alfonso et ai.

[14]

^{from trie}

eqmhbrium

fluctuation spectrum on

Si[111].

^In trie _case of

Si[001], however,

where reconstruction occurs, this value is much smaller. Trie value measured

by

Bartelt et ai.

[15],

^{lv~hich is} based _on trie

study

of trie step fluctuation spectrum, is m trie range of1-10

mev/Àfor

trie two

types

^of

steps, SA

and

SB,

_i-e-,

^10~~~ -10~~~ J/m,

which is very close to trie

amplitude

of elastic interaction.

. In

homoepitaxy,

trie

amplitude

of elastic interaction _is obtained from

equation 16) (we

recall that _{we are} interested in the q~

term)

Ad

_+~

~jj/~ ⁽¹⁹⁾

~vhere

f' designates

a force doublet. Note here that the

interstep

^{distance 1enters} the

amplitude,

~vhich is _a consequence of trie

algebraic

interaction

(as opposed

to the

logarithmic one).

From _a dimensional

analysis,

_we obtain

f'

+w

Ea~. (20)

Taking

_a

typical

_step

spacing

^1m 100 atornic spacing we obtain

Ad

_~3

10~~~ J/m.

Alfonso et ai.

[14],

^{studied trie} terrace width distribution _on clean

Si(111)

surface.

They give i~Ad

_m

10~~°

J _m. For _a

step

_spacing

of100a,

_we find

Ad

_"

10~~~ J/rn,

~vhich is

not too far from the value derived from dimensional considerations.

3. Discussion

The main result obtained here _is

that,

at least in trie

hetero-epitaxy

_case, values of trie hne tension and elastic interaction of

steps

are close. This should bave _an important influence _on trie

analysis

of trie fluctuation

spectrum

of

steps

at

equilibrium.

For

large wavelengths [10],

trie fluctuation

spectrum

^of a train of

steps

reads

w +~

~~~~

~~°~

~~

+

(A ie~~~

_cos

( +'i)q~. ₍₂₁₎

1

where _we have

disregarded

the

In(iq)

which

origmates

from trie elastic

part.

The two

important

results _can be stated _as follows:

At low values

of1, however,

and for ont-of

phase fluctuations,

the _energy cost is

larger

than for

in-phase

fluctuations. As _a

result,

_one is

tempted

to

expect

that for small 1the

in-phase

fluctuation modes

prevail.

In _a such _case

elasticity

results in _an

(effective) _positive

line tension.

2- The elastic term is

important only

for _{q <}

é~~.

_{In this}

regime, elasticity

and line tension

are additive. For

#

_>

x/2, elasticity

acts _{as an}

effective negatiue

fine tension. For such _a

type

of modes

elasticity

_{may cause a}

spontaneo~ts modulation,

if it is

large enough.

Whether this

type

^{of mode} may be relevant _{or not} is at

present

_{an open}

question

which

requires

_a

detailed statistical

study

of _a

fluctuating

train. For _q »

é~~,

trie elastic _term is

unimportant,

as

represented by

trie

exponential

^decrease of trie Bessel function for

large arguments.

Trie

interstep

distance é

plays

trie role of _a cut-off

length

below which

elasticity

_is irrelevant with

regard

to

step

modulations.

Usually,

in the

experimental analyses

^of the

step

fluctuation

spectrum,

the elastic renormal- ization of the line

tension,

which _can attain

large values,

is not taken into _account. While the

theoretical

study

of _a train fluctuation is _a

problem

which is

relatively

hard to tackle _even at

equilibrium,

it is _an

important

^{task for future}

investigations

to

properly incorporate elasticity

m trie model

equations.

4. Conclusion and Outlook

Ta

conclude,

we extended trie

problem

of elastic interactions to trie _case of modulated

steps

with

arbitrary profiles.

For

homoepitaxy,

_we found that

elasticity

favors modulated

steps!

This is _a

striking

feature inasmuch _as two

steps having

trie _same

sign repel

each other. This

phenomenon

_occurs

irrespective

of trie

phase

shift. While line tension should

penalize step modulations,

this result is

interesting by

itself _on trie

conceptual

level.

Realizing

that the

general

^{form of trie} force-force and

dipole-dipole

interaction

energies

bear

a

strong

resemblance with electrostatic

dipole-dipole

and

quadrupole-quadrupole interactions,

we were

naturally

led to ask whether trie ideas

put

forward here could

apply

to a wide list of systems. ^Of

particular

interest _seem to be trie

hydrated amphiphilic

systems ^which form for _a certain range of water contents and

temperatures

lamellar

phases,

where the

bilayer

entities form _a

parallel

_array.

Upon

variation of the central

parameter (e.g.,

the

temperature)

trie lamellae become

rippled _[11].

TO

date,

trie

origm

^{of trie}

rippling

transition bas remained

obscure,

albeit _some

key ingredients

are

begmning

to emerge. Since trie

amphiphilic

molecules bave _a

polar head,

it _was natural to _see whether trie

dipole-dipole

interaction between

adjacent bilayers

_may lead to _an energy reduction due to _a modulation. This is indeed what _cames Dut frein _Dur calculation

[12].

Because trie membrane bas

(virtually)

_{a zero} surface tension

(but

_a

rigidity, giving

rise to _a q~ contribution instead of

q~),

_we

expect

^trie

dipolar

_energy to become

competitive

with the

rigidity

_energy. ^While trie order of

magnitude

of trie

ripple wavelength

which _we calculated is smaller

by

one order of

magnitude

than that

observed,

this idea _may open an

interesting

line of future

inquiry.

Other situations where _our ideas _may stimulate _{new progress concern a}

variety

of

systems, going

from electrostatics to

hydrodynarnics.

Beside the above cited _{case, one can,} for

example,

conceive of ways of

testing

_our ideas _on

macroscopic magnetic

systems

by setting

_{up expen-}

ments with lines

supporting magnetic

_campasses. On the other

hand,

in

hydrodynamics,

it is

known that two _von Karmann _vortex lines

[16]

are

susceptible

to modulations. Since the

hy- drodynarnic equations

are

,

for _a two-dimensional

flow,

reminiscent of electrostatic

equations,

it is

possible

to formulate trie line-fine interaction in trie

present

terms. Other various

examples

may

be, probably,

^found in trie _case of vortex lines in

superconductors,

etc.

Thus,

it appears that

anisotropic

interactions

(of vectonal,

and tensonal think of

multipoles origins)

_may

embrace _a

disparate variety

of

physical systems.

Trie other line of

investigation reported

here _was trie

discovery

of trie relevance of trie elastic contribution to trie total energy modulation in trie

heteroepitaxy

_case. Future works _on step fluctuations should

incorporate

such _an effect with trie aim to

interpret expenmental

results

properly.

For

phase

shifts

larger

than

x/2,

trie effect of

elasticity

is

opposite

to that of trie line

tension;

^it

plays

trie role of _a

negative

^fine tension. It is not

yet

clear whether in

equilibrium

the modes with

#

>

x/2

_are relevant _{or not.} This

question

^is

currently

under

investigation.

What

might

trie consequences in trie out-of

equilibrium regime

be? It is known that

during step

^flow

growth

_{[17] the}

straight step

_may ^become

morphologically unstable,

the mode with

#

= 0 [18]

being

the most

dangerous

_one.

During sublimation,

the

straight step profile

_is

stable,

while the vicinal surface sufiers _a

step-bunching instability (mode

_q

= 0 and

#

₌

x).

Because

during

sublimation the difiusive

phenomenon

enforces the mode

#

= x, we may expect

elasticity (recall

that for

#

_{= x} it

plays

the role of _a

negative

line

tension)

to lead to _a

morphological mstability.

The

incorporation

of elastic efiects in the Burton-Cabrera-Frank

(BCF)

_[19] model results in _a modified condition at the steps

[12].

^So

far,

_we have found that

during sublimation,

diffusion

(which

is

stabilizing) precludes

_an elastic induced

morphological instabihty.

This _is

attributed to the fact

that, generally,

the diffusion "force" _is much

larger

than the elastic _one.

In

conclusion,

trie work

presented

here should _{serve as a} basic framework for future _inves-

tigations dealing

with the

problem

of

fluctuating _steps.

On the other

hand,

it has led to the

identification of the relevance of the

(tensorial)

interaction _m other systems ^{that look}

extremely

diverse. What _we have learnt _so far is that these

types

^of interactions

(e.g., dipolar interaction).

familiar _as

they

_{may appear,} lead to

unexpected

results.

Appendix

^A

Interaction

Energy

of Two Modulated

Steps:

The

Heteroepitaxy

Case

We derive here the interaction energy of two lines of forces

(Fig. 2). Step

modulations are

defined

by

vi " e

cos(qxi _),

_v2

" 1+

e

cos(qx2

₊

_çi). (A.1)

Their interaction energy reads

(Eq.(7))

E~~~P ₌

~

°~~~~~

dsi ds2

^°

(nin2)

+

(rni)(rn2)

,

(A.2)

XE

~

r r

where fz ₌

fnz,

and _{p is} the force

density

at

steps.

^For two

points Pi (xi, vi), P2(x2,v2)

_on

steps,

_we

develop

terms _m

integral

at the second order _{m e.} To shorten the

notation,

_we use:

(

_"

(Xi X2)/t,

1"

(Ii, _#

"

tq,

A

=

cas(qxi) cas(qx2

_{+ çS)}

=

cas(qx2

+

#() cas(qx2

+ çS), II

=

sin(qxi) sin(qx2

_{+ çS)}

=

sin(qx2

₊

#() sin(qx2

_{+ çS),}

Z

=

sin(qx2

₊

#()

₊

sin(qx2

_{+ çS),}

gi(i)

" 1(1 +

2)(1

₊

i~)~i~~~~/~ i(i

₊

i~)~i~~~~/~ (A'3)

Integration

of the first term in

equation (A.2)

reads

Il _dsids2 ^~~~~~~

r

=

/ / _{d(dx2 ((1+ (~)~~/~}

+ ê~

(H(~(1+ ^(~)~~/~

⁺

(1/2)A~gi(()j

= -L

In(t IL)

₊ L

il fKi (#)

_cas

#] ^ê~. (AA)

where L is trie

step length.

Integration

of the second _term in

equation (A.2) gives

Surprinsingly,

ê~ terms in

equation (A.5)

vanish

exactly.

For this

term,

^the _energy ^decrease due to modulation

compensates exactly the

_energy increase due to

augmentation

^{of forces} per

projected length.

Adding

these two contributions

(A.4-A.5),

_we

get

the

expression given

in

equation (10).

Appendix

B

Interaction

Energy

of Two Modulated

Steps:

Tl~e

Homoepitaxy

Case

Here _we

compute expression là), using

the _energy of _two force doublets

given by equation (13).

We _use notations of

Appendix

A. Trie interaction energy of two modulated

steps

^reads

E~~~P

=

/ / _dsi

d82E~°~~~~~

(fi, il, r). (B. Ii

Integration

of different terms of

equation (13) gives

(<2

IL) j j _{dsi ds~}

r

= 2 +

(6

+ #~ +

(f~K2(#) #~K3 (#)j

cas

#)

ê~

= 2 +

(6

+ #~

(#~Ki (#)

+

3#~K2 (f)j

cas

çi) ^ê~. (B.2)

(t~ IL) / / _{d81ds2 ((rni}

)~ +

(rn2)~j

r

= 8

/3

+

(24

_{+ 2@~ +}

(-4@~Ki (#) 4#~K2 (4)

₊

_4@~K2(4)

+

3

10#~K3 (#) 2#~K4(#)j

cas

çi)

^ê~

=

8/3

₊

(24

+

2#~

+

(-6#~Ki (#) 12#~K2(4)+

3

2#~K2 là))

cas

çi) ^i~. (B.3)

(i~ IL) / / _{dsi ds2 (rni}

)~

(rn2)~

r

=

16/15

₊

j148 4#~

+

18#~K~14) 121~K~14)

+

91~K4(@-

(~K5(4)j

_cas

çi)

^ê~