The Response to an External AC Drive of the Interface Between Solid and Liquid Helium Near the Roughening Transition

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The Response to an External AC Drive of the Interface Between Solid and Liquid Helium Near the Roughening

Transition

S. Giorgini, R. Bowley

To cite this version:

S. Giorgini, R. Bowley. The Response to an External AC Drive of the Interface Between Solid and

Liquid Helium Near the Roughening Transition. Journal de Physique I, EDP Sciences, 1995, 5 (7),

pp.815-845. �10.1051/jp1:1995170�. �jpa-00247106�

Classification Physics Abstracts

61.50Cj 64.00 67.80-s

The Response to

_an

External AC Drive of the Interface Between

Solid and Liquid Helium Near the Roughening llYansition

S.

Giorgini

and R-M-

Bowley (*)

Department of Physics, University of Nottingham, Nottingham NG7 2RD, UK

(Received

19 January1995, received in final form and accepted 17 March

1995)

Résumé. La théorie de la transition rugueuse développée _par Noziéres utilise _un programme de renormalisation tronquée _pour traiter de l'effet des fluctuations de l'interface liquide-cristal.

Pour _une direction facettée, la renormalisation s'arrête à la longueur de corrélation

en l'absence d'une force extérieure appliquée à l'interface. Cependant, le processus de renormalisation doit être arrêté plus tôt s'il _{y a une} longueur plus courte dan8 le problème. Nozières _a examiné de tels effets de taille finie soit _pour des surfaces vicinales, lorsqu'une force extérieure continue est

appliquée _{pour provoquer une croissance}

(ou

_une fusion) de l'interface à vitesse finie. Ici _on considère l'effet d'une force extérieure alternative qui introduit _une nouvelle échelle de longueur;

c'est la distance de diffusion d'une perturbation pendant le temps

w~~,

où _w est la pulsation de la force appliquée. Nous _avons étendu les relations de récurrence _pour traiter le _cas de courant

alternatif. Pourvu que la longueur de correlation soit plus grande _que les _autres échelles de

longueur, _{nous pouvons} résoudre les relations de récurrence et obtenir la réponse de l'interface,

une approche qui est exacte _au deuxième ordre

en potentiel d'ancrage. Quand la longueur de corrélation est plus petite _que les autres longueurs, _nous résolvons les relations de récurrence pour des échelles de longueur plus courtes que la longueur de corrélation; _pour des échelles de longueur plus grandes _nous résolvons l'equation de _mouvement renormalisée _{sans tenir} compte du bruit aléatoire. Cette approximation grossière _nous permet ^{de décrire} semi-quantitativement

le passage du comportement mobile _au comportement ancré de l'interface.

Abstract. The theory of the roughening transition developed by Nozières _uses the notion

of _a stopped renormalisation scheme when dealing with the effect of fluctuations of the crystal- hquid interface. For _a faceting direction, the renormalisation is stopped at the correlation length

in the absence of any externat driving force _on the interface. However the renormahsation _process needs to be stopped earher if there _{is a} shorter length scale in the problem. Nozières considered such finite size effects _arising from either vicinal surfaces _or from the effect of _an external DC driving force which

causes the interface to grow

(or

melt) at _a finite velocity. Here _we consider

the effect of _an applied AC driving force which introduces _{a new} length scale; this length ^is related to the distance _a disturbance _m the interface diffuses _{m a} time w~~, where _{w is} the

angular frequency of the applied drive. We have extended the recursion relations to deal with

the AC _case. Provided the correlation length is forger thon the other length scales _{we con}

solve the _recursion relations and obtain the response of the interface, _an approach which is

exact to second order

m the pmnmg potential. When the correlation length is smaller than

(*) Author to whom correspondence should be addressed

Q Les Editions de Physique 1995

one of trie other lengths, _we solve trie recursion relations for length scales shorter than the correlation length; for longer length scales

we solve the equation of motion renormalised _up to trie correlation

length

by ignonng trie random _{noise as} being small. This crude approximation

allows _us to describe

semi-quantitatively

the _crossover from trie mobile to trie pinned behaviour of the interface.

l. Introduction

The dassic work of Nozières and Gallet

iii

_on trie

dynamical broadening

of trie

roughening

transition _was

developed

in

parallel

^with trie beautiful

expenmental

work of

Gallet,

Balibar and

Rolley

_{[2] on} ^hehum

crystals.

In trie present paper we extend trie

theory

to describe trie

response of trie interface between solid and

liquid

helium to

applied

AC drives.

First let _us review trie

theory.

Consider trie interface between _a

crystal

and its melt for _a direction in which trie interface _can be facetted. The

Langevin equation

for the motion of the interface

is, according

to Nozières and Gallet

(lj

J~~~='fV~z+4l+R+~

Here J~ is _a friction coefficient which is related to trie

mobility,

K, of trie interface

by

trie relation

J~ ⁼ pc

/K

with _pc trie

crystal density;

_'f is trie surface

stiffness;

R is _a random force with _a white noise spectrum;

Fla

is trie

driving force,

related _to trie difference in chemical

potential

_across trie interface

Fla

₌

pc(~IL ~lc)

and 4l is trie force that _comes from trie

pinning potential (throughout

_we follow trie notation and

techniques

used

by

Nozières

iii ).

Trie

pinning potential

bas to be

periodic

with

period

_a, trie

spacing

of trie

crystal planes,

for if trie

crystal

_grows

uniformly by

_one lattice

spacing

trie

potential

is unaltered. Since trie

potential

is

periodic along

trie z-direction _{we can}

expand

it _{as a} Fourier series.

Only

trie lowest Fourier component is relevant in trie

region

^{of trie}

roughening transition;

trie other components ^vanish on

long length

scales and _are irrelevant.

Therefore,

Nozières

iii

took trie

pmning potential

to be

V(z)

=

-Vcos(27rz la)

so that trie

pinning

^force is

4l(z)

₌

-(27rla)Vsin(27rzla)

Nozières and Gallet

iii

^derived renormalisation

equations

^for _a ^{flat interface}

parallel

to trie

crystal planes. They

then extended trie calculation to vicinal surfaces whose direction lies close to trie

faceting

direction and to interfaces

subjected

to _a constant

driving

force. Trie introduction of vicinal surfaces, _or of _a finite

driving

force

produces

new

length

scales whose effect is to terminate trie renormalisation _group

equations

and broaden trie transition. Trie

driving

force introduces _a

length

TF "

(a~i/F)~~~ (l)

This

length

is denoted _as

Aj~ by

Nozières. We _con also write this

length

as TF =

l'ft/J~)~/~

where t

= a~J~

IF

is trie time for trie mobile

crystal

to grow one atomic

layer,

a.

For temperatures ^{above trie}

roughening

transition trie interface is

rougir;

for temperatures

below,

it is faceted. Trie

roughemng

transition

belongs

to trie Kosterlitz-Thouless

universality

class of

phase

transitions. Nozières and Gallet obtained _{a set} of recursion relations which

describe how trie surface

stiffness,

trie friction coefficient and trie

pinning potential

_are altered under trie renormalisation scheme.

Correlations of surface

displacements

_are described

by

trie correlation function

GIP)

=<

zir)zir

₊

P)

>

For temperatures ^below trie

roughening

transition trie correlation function

decays exponentially

as

G(p)

=

Ae~P/~

_where

(

is trie correlation

length.

For

length

scales shorter than

(

trie

pinning potential

is small and trie interface fluctuates

freely.

For

length

scales

larger

than

(

fluctuations

are blocked because trie cost in

pinning

_energy is too

large

to allow trie interface to _move from

one

crystal plane

to trie next.

Therefore, f

marks trie _onset of

faceting

_on

large Iength

scales.

Nozières and Gallet

iii give

trie

following picture

of trie

length

scale _TF- Under _an

applied

force _a free surface bas _{a curvature} R

=

ia/F.

An element of surface of

length

L bas _a vertical

displacement,

_a

bulge,

of order L~

/R

which is

superimposed

_on trie thermal fluctuations. There

are two _cases to be considered:

1) if trie

bulge

is _very much less than _a, it is irrelevant and trie surface element fluctuates

as if F _{were zero.}

ii)

if trie

bulge

is mach

larger

thon _a, then it averages trie

periodic potent1alto

_{zero, even} in the absence of thermal fluctuations. Trie surface behaves _as if it _were free. Trie

length

scale where trie

displacement equals

trie lattice _spacing is

just

_TF- We _are led to trie idea of

stopping

trie renormalisation program ^at trie

Iength

_TF

beyond

which trie

pinning potential

is irrelevant.

For temperatures

just

below trie

roughening

transition trie correlation

length, (,

is much greater than _{TF so} that it is _TF which terminates the recursion relations. The interface behaves

as if it _were

rougir,

_even

though

it should _appear to be faceted for these temperatures. Trie transition is broadened.

However,

when trie temperature ^decreases

further,

trie system reaches trie

region

where

(

is smaller thon _TF and trie

crystal

behaves _as if it _were faceted. Trie interface does not _move

freely.

^In DC

experiments

it grows

slowly through homogenous

nucleation of

terraces _on trie facets.

An

interesting question

_concems trie response of trie interface to _an

oscillating driving

force of trie form

(Fla)cos (wt).

Trie introduction of _a finite

frequency

Ieads to _{a new}

Iength

scale

rà =

ii/~~J)1/2

₁₂₎

This

length

scale

corresponds

to trie distance _a localised disturbance of trie interface _moves

along

trie interface in _a time of order w~~

The

Iength

_rd must be

compared

to both _TF and trie correlation

Iength

to _see which _one dominates trie behaviour of trie interface. In other words there is _a

competition

between these

Iength

scales. When w"~ is much

Iarger

than t

(trie

time to grow one atomic

Iayer)

then _rd is much

larger

than _TF

Trie situation is

simplest

in trie linear response

region

where _{TF is} very much

larger

than trie other two

length

scales. There _{are two cases to} be considered:

i)

If _rd is less than

f

the _recursion relations _are

effectively stopped

at rd and trie interface

behaves _as if it _were

rougir.

For

example

when _a

high frequency

_{pressure wave} is incident

normally

_on trie interface from trie

Iiquid

side there will be _very little transnlission of pressure into trie

crystal

if trie interface bas _a

large mobility;

this _{is a} charactenstic property ^of a

rougir

interface.

ii)

If _rd is

larger

than

(

trie recursion relations _are

effectively

stopped at

f

and trie interface behaves _as if it _were faceted. When _{a pressure wave} is incident

normally

on trie interface from the

liquid

side there is _a

large

transmission of pressure into the

crystal

_smce the solid and its melt _are

acoustically

similar. The transmission of low

frequency

sound _waves at the interface

depends

_on the ratio

rd/(. Altematively

_an

oscillating

_pressure could be

exerted _on the interface

by

_{a resonance}

technique

_[3]. The _{response at resonance}

depends

on the relative sizes of the three

length scales,

_{TF, rd} and

(.

Dur aim is _to

develop

_a

theory

of the AC _response which _can describe both the linear and the non-linear _response of the interface _near the

phase

transition. The non-linear _{case occurs} when the

Iength

_TF is

comparable

_or smaller than other

length

scales. The

theory

of the non-linear

response is

simplest

when the correlation

length

is

longer

than other

length

scales for then the

theory just depends

_on the ratio of _{rd to}

TF In this _case the interface is

effectively rough

and _moves

approximately

in

phase

^with the

driving

force. The response is

nearly

linear. We

develop

the renormalisation

equations

to deal

explicitly

with this _case. The situation is _more

complex

when

(

is

comparable

to the other

length

scales.

Effectively

the equation of motion for

long length

scales is dominated

by

the

pinning potential

which cannot be treated _as if it

were small. We propose a way of

dealing

with this

complication by abruptly changing

from the renormalisation _program to a method based _on the

equation

of motion of the interface

neglecting

fluctuations.

The

plan

of the paper is the

following.

In Section 2 _we set up the renormalisation scheme

followmg

the

techniques pioneered by

Nozières. In Section 3 _we derive the recursion relations _in the presence of _an AC drive and introduce the dielectric function. Section 4

gives

our numencal

results for both the hnear and the non-linear

regimes.

In Section 5 _we propose an

experiment

to test the

predictions. Finally,

in Section 6 _we discuss the

strengths

and weaknesses of the

theory

and directions for future theoretical work.

2. Renormalisation in trie Presence of _an AC Drive

The calculation _m this section is _an extension of that made

by

Nozières and Gallet

iii

to the

case of _an AC

driving

force and _uses the

technique

that

they

used. Consider _a flat interface which lies

parallel

to the

crystal planes.

An AC

driving

force

(Fla)cos(wt)

_causes the interface

to move. If there _{is no}

pinmng potential

^{the interface} moves at an average

speed

dzo

F

cos(wt)

$

aJ~

so at time t the

displacement

is

Fsin(wt)

zo =

aJ~w

The total

displacement

when there _is a

pinning potential

is _z

= zo + _zi where _zi

obeys

the

Langevin

equation

J~

~~~

=

iv~zi

^~~~ _sm ^{~~~~~ ~ ~°~} _{+ R}

₍₃₎

dt _{a a}

The

driving

force _appears to bave vamshed but it is hidden _m the _term which descnbes trie

pinning

force. Because _we have removed the

displacement

due to the

driving force,

all that

remains is the

displacement

due to fluctuations driven

by

the random force. Our atm is to renormalise equation

(3),

the

equation

of motion for the coordinate _zi driven

by

the random fluctuations.

The random force

R(x, y)

_can be Fourier transformed

R(x, y)

=

~j _Rke'~.~

k

into its components Rh- The random force has _a white noise spectrum <

Rk(t)Rk(t')

_>=

Gkô(t t').

The spectrum must have the form Gk _" 2J~kBTÙ(A

k)

in order to

satisfy

the fluctuation

dissipation theorem, something

which is

proved

in the

Appendix.

For convenience we consider _a

sharp

cutoff in Rh at k

= A

although

the

precise shape

of the cutoff is irrelevant

ÎIÎ.

The

length

scale A~~ is of the order of the lattice _spacing. In the renormalisation program

we

change

the cutoff from A to À and average over the Fourier components of the random force

with wavevectors in the range A to A. We do this in incremental steps

by gradually reducing

the cutoff wavevector, so that apart ^{from the}

spacing

of the

planes,

_a, all the parameters in

equation (3)

_are

changed

to renormalised

values;

these renormalised values _are denoted

by

_a

bar _over the top ^{of the}

symbol.

Dur aim is to get ^the

equation

~Î~ ~'f~~zi

^~~~ _~ ^{27r(21 + j}

~ in

o

~

~ ~

When _we have recovered the _same

equation,

_{we can} repeat ^the process

indefinitely

and _carry out a

large

renormalisation of the

equation

of motion, at each stage

removing

the effect of

fluctuations in _{a narrow} band of wavevectors. If _{we can carry out} the renormalisation program for all wavevectors then _{we can} account

completely

for the effect of fluctuations.

When

making

the renormalisation calculation _{we assume} that the

pinmng potential

is small in comparison to the other terms _so that _{we can use a}

perturbation

_expansion in the

pmning potential.

To obtain the

equations

of motion to describe the

roughening

transition it is

only

necessary ^to calculate _terms which _are second order in V. For temperatures below the transition the renormahsation

procedure

works well until _we reach _a

length

scale

(

₌

1/A~,

where the pinning

potential

becomes

large

and the renormahsation

procedure

breaks down. In this region

we can treat the

pinning potential

_as

large

and the fluctuations _as

being

small _m which _case

we can solve the non-linear equation ^of motion for ai

by neglecting

fluctuations.

We separate ^the random force into two parts

R=É+ôR

with É _{the part}

containing

wavevectors in the range 0 to À _{and ôR}

contaimng

wavevectors in the range À _{to A. We write}

é =

In(A Il)

We _are

only

interested in _an infinitesimal renormalisation at each stage, so the _range

À

_to A is small

compared

to A. Thus _we consider _a

tiny change

dé.

For any

history

of the random

force, R(t),

the

quantity zi(t)

is _a well-defined function of

R(t')

at earlier times t'. _{The average}

quantity

ai ^is defined _as

ai =<

ziÎÉ+ _ôR]

_>ôR

with the _average taken _over the fluctuations _m ôR. We _can write _zi

= ai +

ôzi

with

equations

of motion for both

ôzi

and ai

given by

and

dôzi

~ 27rV

27r(a

₊ ôzi 27rV

27r(a

+

ôzi

~- =

iv ôzi

+ ôR sin + sin

dt _{a a a a}

where a ₌ ai + _zo. V~e solve these

equations

for ai and

ôzi

_{as power serres} in trie

pinning potential, V, by wnting

ôzi

_{= ôz)°1} +

ôzl~l

+

where ôzl°~ is of order

(V)°,

_{ôzl~~ is} of order

(V)~

and _{so on.} Trie contribution to ôzi of zeroth order is

ôzl°~

(r, t)

=

/ _d~r' / _{dt'xc(r r',}

t

t')ôR(r', t')

where trie _response function

xo(p,

T) is

given by e-~P~/4~r

~°~~'~~

47riT

^~~~~

The random force ôR is distributed _{as a} Gaussian variable centred _{on zero.} It follows that ôzl°~ ^{is also} distributed _{as a} Gaussian with _{an average} value of _zero.

The correction to

ôzi

which is first order in the

pinning potential

is obtained

by

iteration of the equation of motion. Thus

ôz)~~ =

~~~

/ _d~r' / _{dt'Ko(r r',}

t

t') (sin

~~') sin

(~~~')

a a a

Here _we have used the dash _to indicate that _{we are} concemed with the _space time

point (r', t').

Thus

z'

=

2(r', t')

+ ôz)°~

(r', t')

Corrections of

higher

order in the

pinning potential

_can be obtained

by iteration; fortunately

we

only

need to determine contributions to ôzi to first order in the

pinmng potential.

In what

follows,

_we put ôzi " ôzl°~ + ôzl~~ and

ignore higher-order

ternis.

The

equation

^of motion of ai to second order _in the

pinning potential

is

~

ÎÎ _'~~~~~

~ ~

~~

~~~

Î~Î~ ~°~ ^~Î~~~

~ ~~~~

~~~~~

~~~

27rV

27r(a

+

ôzl°~)

_2~rôZÎ~~

à = cos

a a a

the average

being

taken _over the random force ôR.

Since the

quantity

ôzl°~ ^is distributed _{as a}

Gaussian,

_we have the exact relation

~

~~

~~~~Î~~

~-2w~<(ôz)°~)~>la~

~-nôg(0,0)

~

where the function

ôg(p,

T) is defined

by (see

Nozières's article,

Eq. (4.89)) ôg(p, r)

=

j _{jôzl°tir, t)ôzl°~ir}

p, t

r)j

ÎÎÎ~~~~~~

^~~~~~~

^~~~~~~~~

^~~~~~~

and where the

quantity

_{n is}

7rkBT

n =

la~

The term in the

equation

of motion for ii which is first order in V becomes

~

(ai

_{+ zo}

=

~~~

in

~~~~~~

_~°

27r -nôg(o,o) ~.

~~

~e

ⁱⁿ ~

a ~

with V

=

Ve~"~~(°?°l.

_The effect of the renormalisation at this stage is to leave the _pin-

ning potential

term with the _same form _as the

corresponding

term in

equation (3)

but with

renormalised values of _zi and V.

The term which is second order _m the

pinmng potential, à(~),

can be found

by substituting

for ôz)~~ ^which

gives

à(2)

=

~~~)~ / _d~r' / _{dt'xc(r r',}

t

t')

a

27r(z

_{+ ôz)°~}

7rz'j 27r(z

+ ôz)°~

7rz'j

cos sm cos sm

~ ~ ~ ~

Remember that the

primed

term z' refers _to the _space time

point (r', t')

whereas the

unprimed

terms describe the space time point

(r, t).

First consider the last term in braces _on the

right

hand side of this equation.

By taking

Gaussian _averages of terms

involving

_{ôz)°~ we} get

l127r(z

^{cos +}

^ôz)°1)

^sm

^7rz'j

^=cos

^7rzj

^sm

^7rz'j

^{e_~~~~~~ ~j?}

a a a a

Now consider the other term _m braces _on the

right

hand side. Before

taking

_averages we rewrite this _as

27r(z

+

ôz)°1) 7rz'j

1

27r(z'

+ z + ôz)°~ ⁱ

27r(z'

z

ôz)°1)

cos sm = sm + sm

~ ~ ~ _~ ~ _~

When _we take averages over ôR this becomes

C°S ~~

~ ~

~

Zi°~ ~~~,

~~~

~

^{= ~i~}

27r(2'

+

a)

2 _~

~_~~~~~~~~~ ^~

+ ~~

27r(2' a)

~ ~~~~~~~~

~ ^~ _~

e2nôg(p,~)~_~~~~ ^~~~

The _sum of the two terms _m braces _is therefore

given by

~~"Î~°'°~ SIR ~~~~l+

_~~

ie~"~~~~~~ i)

₊ SIR

~~~~l

_~~

ie~"~~~~~~

i)1

But the

quantity ôg(p,

_T) is small _since it is

proportional

to

dé;

it follows that _{we con ignore} terms in

ôg(p, T)2.

The final

expression

for

à(~)

_is

~~~

47r3V~

/

~

/ ^{27r(z' z) 27r(z'}

⁺

^z)

à

= ~

d _p

dTxo(p, T)2nôg(p, T)

sin _sin

a a a

where _we have written _p

= r r' and _T

= t t'. At this stage no approximation has been made.

The term which

depends

_on

sin(27r(z'+ z) la) corresponds

to _a

higher spatial

harmonic of the sort that has been

ignored

when

describing

the

pinmng potential.

It is irrelevant. It is the term in

sin(2~(z'- z) la)

which _we want, for this _is relevant close _to the transition. We write z'- _{z as}

ai

(r', t')

ai

(r, t)

^~

(sin(~at) sin(~a(t r))

= ai

(r', t')

_ai

(r, t)

_a

a~~J with _a

=

(2F/~~Ja) sin(~JT/2) cos(~J(t T/2)).

Nozières and Gallet

(ij

considered

only

the

w = 0 limit of this

equation

in which _{case a}

= Fr

la~.

The _new element of _our

approach

is to extend their

theory

to finite

frequencies.

We

pick

ont of

sin(27r(z' z) la)

the

slowly varying

mode which _we want and

thermally

average over the _rest of the modes which _are assumed _to be _m

equilibrium.

In this _{way we} get

27r(z' z) 127r (ii (r', t')

_ii

(r, t)

27ra 27ra _~~j~~ _~~

sm = cos -sm e ~

a a a a

where _we have written

The

slowly varying

mode which _{we want can} be

expanded iii

_{as a power} series in _T and _p.

After

integration

_over

d2p,

the lowest-order terms _are

ai

(r', t')

ii

(r, t)

=

-T~~~

+ ~

V~zi

dt

so that _we get ^{in lowest order}

Sm

~~~~l ~l

SQS

ll~ _1-T Î~

+

1v~zil

^COS

l~i~l

SIR

l~i~l1e-~"~~~~~

Of _course there are

higher-order

terms such _as

)T2d2zi /dt2.

These terms _are all

neglected

for

they

_are irrelevant in the

long wavelength

limit. For

example,

the term

)T2d2zi/dt2

alters the effective _mass of the

interface, something

which bas

already

been

neglected

since trie interface is

overdamped

in the

long wavelength

^limit.

Dur aim _{is to recover} the

original equation

of motion for _zi but with renormalised parameters.

By collecting

terms we get ^the

equation

~~~~

=

'fv~zi

^~~~ _sm

~~~~~

^~

°~)

+É

dt _{a a}

+

~~~~~~~ /d~p /dTxo(p, T)Jo(Ap)e~~~~~&e~~"~(P'~)

a

x

ll~ 1-T Î~ +1/4P~v~zil

^COS

l~i~l

^SIR

l~i~ Il

The first three terms _on the

right

hand side of this

equation

have the correct form. In the last term on the

right

^{hand side} there is _an expression

proportional

to dal

/dt

which renormalises

the friction coefficient there is also _a terni

proportional

to

V~zi

which renormalises trie surface stiffness coefficient. Both of these terms _are affected

by

trie

quantity cos(27ra la);

the presence of this term

changes

trie recursion relation

for'f

and _~ from those obtained

by

Nozières.

Moreover,

_we get an extra term _on trie

right

hand side which

corresponds,

_{as we} will _{see, to a} renormalisation of trie

driving force,

_or

altematively

to _a renormalised value of

dzo /dt.

We _use mathematical identities for

cos(fl cos(Ù))

^and

sin(flcos(Ù)) given by

Abramowitz and

Stegun (Sections

9.1.44 and 9.1.45 of Ref.

(4]).

^{It is convenient} to introduce trie

quantity

_y defined _as

47rF sin

(wT /2 47rr( sin(wT /2)

~ ~J~a~

r(

It follows that

ce

sin(27ra la)

= 2

~j(-1)~J2k+1IV)

cos

((2k

+

1)~J(t T/2))

k=o

and

ce

cos(27ro la)

₌ Jo_IV + 2

~j(-1)~

J2k

Iv)

_cos

(2kw(t

_T

/2)

k=i

We

only

need trie first _term in each of these

expansions.

Trie _{next term}

produces

_a correction to the

equation

^of motion which _are second order in trie

pinning potential,

_a correction that

oscillates at a

higher frequency.

This correction _can feed back and

give

rise to a contribution _at the fundamental

frequency,

but the

leading

contribution is fourth order _in the

pinning potential.

We _are

only working

to second order _so

only

the first term in each of these

expansions

is needed.

Thus _we take

sm(27ra la)StS2Ji

_{IV cos}

(w(t

_T

/2)

and

cos(27rala)StsJo(V)

The

neglected

terms oscillate _at

multiples

of _w. Remember _{we are} interested in the _response of the system at

frequency

_w and _{not at}

higher frequencies.

Dur immediate

objective

is to recover

the

original equation

of niotion for _zi

By collecting

terms we get the renormalised

equation

~Î _{'~~~~~ ~~}

~~~

~~~~~/

^{~~~ ~ ~}

~~~~~~~ /d~p /dTxo(p>

^T)

Jo(Ap)e~~~~~/~e~~"~(P~~)

^~~~

a

x2JiiY) cosjcojt r/2))

Here _we have renormalised _i

to1='f+di

where

dl

is given

by

dl

_"

~~~Î~

^~~~

/ _d~pp~ / dTxo(p> T)e~~"~(P'~) Jo(Ap)e~~~~~/~ Jo Iv)

a

~

and have renormalised _~ to q ₌

~+d~

where

d~

is given

by 16~4p~~dé

d~

~

= ~

d~p dTTxo(p> T)e~~"~(P'~)Jo(Ap)e~~~ ~/~Jo(1l)

a

Î Î

Both

expressions

contain the terni

Jo(Y)

^which

elfectively

cuts off trie renormalisation when _y becomes

larger

than umty.

Equation (5)

still is not of trie form _we

require,

^for there is an extra terni which oscillates at

frequency

_w. This extra term is

effectively

_an additiona1force which renornialises _zo to do.

To realise

this,

it is

simplest

to transform

equation (5) by writing

it in ternis of trie variable

z ₌ ai + zo. Thus

dz

~~

^2~V

j27rzj

flfcos(wt)

q- ₌ y z _sm + R +

dt _{a a} J~a

~~~~~~~

/d~p/dTxo(p>T)Jo(Ap)e~~~~~/~e~~"~(P~~)

a

x

2Ji(Y) cos(w(t T/2))

The last terni _can be

thought

of _{as a} contribution _to trie renormalised force; but it is

simpler

to define _a renormalised

velocity

of trie interface _as

dz/dt

₌ io+dào where trie incremental

change

in

velocity

is

dào _"

~~~Î~~~~~ /

_d~

p

/ dTXo(p, T)Jo(Ap)e~~~~~/~e~~"~(P~~)

Ji_IV)

cos(w(t T/2)

a il

When _we rewrite trie

equation

^of motion in terms of ii we recover equation

(4)

with do trie renormalised

displacement

of trie interface due to trie externat drive.

Trie renormalised

velocity

of trie interface _now contains _a part ^{which oscillates} as

sin(wt),

that is there is _a part which is _out of

phase

with trie extemal drive. It is convenient to express trie force _as

Fcos(wt

_{+ ç§).} Trie part of trie interface

velocity

which oscillates _as e~~~~ is

~°~"~ ~°~"~ ~aÎ~

We _con then write

d

Jnluolta))1

=

~~~(ÎÎ~~~ / _d~p / _dTxolP,

T)JOIÀP)e~~~~~/~e~~"~~P~~~Jilv)e~~~/~

3. Recursion Relations and trie Dielectric Function

3.1. REcuRsioN RELATIONS. Ii is convenient to

change

from variables _p and _r to dimen-

sionless variables

,~

~ 'lP~ ^~

~~

We also define trie dimensionless

quantities

r = rd

/rF

_fd

=

Àrd

The

quantity

r is

just

trie ratio of trie

length

scale _rd to TF In terms of these _new

quantities

trie parameter _y becomes

y = 47rr~

sin(xp~ /2fd~) ₍₆₎

The

pinning

_{energy per} unit _area is converted into an energy

given by

U

=

V/À~.

Trie recursion relations _express incremental

changes

in

U,

_'f>

~1 and

uo(w)

_as differential equations. In ternis of trie _new variables these _are

$

= 12

n)U Ii)

dl

~

2~~

~~ A(n,

td

r)

_~~~

(

=

~~[~

B(n,

td

r)

_~~~

d _{i a}

and

d

lin juo1°'))1

=

~~~~~

Gin,

_fd

r)

_~~°~

~é

72a4

~~~ ~~~~~~~~~ ~~ ~y~ ~~ c _are

given by A(n,

_td>

r)

= n

/ ~ _dpp~ / ~

_dz

e~~R~e~~"~(P'~) Jo(fl)e~P~~Jo(Y) Ill)

X

B(n,

fd>

r)

= n

/~ _dpp~ /~ dxe~~R~e~~"~(P~~)

_Jo

(P)e~P~~

Jo

Iv) (12)

o o

C(n,fd> r)

=

"~) /~ _dpp /~ ~~e~~R~e~~"~(P?~) _Jo (fl)e~P~~Ji(1l)e"P~/~~d~ (13)

7rr ~ ~ x

This is _a

complete

set of recursion relations which is valid for trie non-linear

regime provided

the

pinmng potential

is small _so that the second order calculation is sensible. For teniperatures

above the

roughening

transition the

quantity

_n is

always larger

than 2 for all é and the second- order calculation is valid for ail

wavelengths.

The _pinmng

potential

becomes

progressively

smaller _as é increases.

By making

_a succession of

tiny changes

of scale _we renornialise _over ail

length scales,

_a

procedure

_{we can carry} out without _any

problems

for temperatures above the

roughening

transition. The result is _a renormalisation of

U,

_i and _J~. The

siniplest

treatment is to

keep

the unrenormalised values of these pararneters on trie

right

^{hand side of}

equations (7), (8)

and

(9).

In _{a more}

sophisticated

calculation _{one uses} trie renormalised values _on trie

right

hard side of these

equations,

_a

procedure

which _goes

beyond

^trie second-order calculation. When this

approach

is

adopted,

trie _noise spectrum must be renormalised _at each stage in order to

ensure that trie fluctuation

dissipation

theorem is

obeyed.

This point ^{is discussed} _more

fully

in trie

Appendix.

Trie

velocity

of trie interface

given by

~~

uo(~J,é

₌

0)

~~

i~a4 ~~"'~~'~~

where

uo(w,

é

=

co)

is trie

velocity

after renornialisation to

infinitely long wavelengths.

We define _a "dielectric

constant",

_e,

by

trie ratio

~j~~

_~ ^{~01~°>é "}

o) uo(w,é

=

co)

This constant tells _us how the unrenornialised

velocity

is altered

by

fluctuations in trie random force _on all

length

scales. Thus

i~

l~l~°))

"

/~

dé~)))ÎÎÎ~ Cl", fd>T) 1")

All _we bave to do is to solve trie recursion relations _{so as} to get trie renornialised

pinning potent1al, friction,

surface stiffness and

driving

force _{as a} function of é and

integrate

trie

expression above to get trie dielectric constant.

There _are

only

two

Iength

scales for temperatures above trie

roughening

transition: _rd and TF If trie ratio _{r =}

rd/rF

_{is very} small then trie _{response is}

Iinear;

_as trie ratio increases and

becomes

comparable

with trie response becomes non-linear.

3.2. RESPONSE _FOR TEMPERATURES BELOW _THE ROUGHENING TRANSITION. For tem- peratures ^{below trie}

roughening

transition trie recursion relations break down for _a value of é = Ç

(a precise

definition of Ç is

given

later

on),

where trie

pinning potential

becomes _com-

parable

with

kBT.

As é increases _{one moves} from _a

regime

where trie

pinning potential

is small and trie fluctuations _are dominant to _a

reginie

where trie

pinning potential

is

large

^and

fluctuations _are small. This _{crossover occurs} at trie correlation

length, (

=

Ap~exp(Ç).

There

are three

length

scales _m this

regime:

_{rd> TF} and

(.

We _{can use} trie renormalisation niethod to

analyse

the response for

wavelengths

shorter than the correlation

Iength;

for

longer wavelengths

it is necessary to use a dilferent

technique.

We define trie dielectric constant for renormalisation up ^to Ç

by

trie ratio

éi~lta)

=

))))')f ₍₎

,

where _we renormalise trie

velocity

_over trie range 0 to Ç. Thus Ç

~~4uj~)2 inléi~lta))

₌

/

_dé

,y~~~4 CIn,fd,r)

o

At Ç trie equation of motion bas the

following

form

fl(Ç) ~~j)~~

⁼

Y(Ç)V~zi IL) ÎV(Ç)

_sin

_~~~~~~~~~

^{~ ~°~~~~~}

+

É(Ç)

where ail trie terms _are taken at the

point

^é ₌

Ç.

This

equation

_can

easily

be rewritten m terms of trie

quantity

z

= ii + do which

gives dé(Ç

~

27r

27rz(Ç) ^É cos(wt

+ ç§)

q(Ç)

₌

1(Ç)V 2(Ç) VIL

sin ₊

R(Ç)

₊

dt _{a a a}

~~~~~

É cosiLdt

+

')

=

iié~) ~j(C~

a

Here

Éla

_is the renormalised

driving

^force and

ç§ is _a

phase angle.

If _we

neglect

the randoni force _{we can} solve the equation of motion

analytically provided

the

pinning potential

is strong

enough

to allow

only

small

displacements

of the interface around the

equilibrium position.

Such _an

approach

is not correct when the renormalised

driving

force is too

strong, causing large displacements;

^{when that is trie} case we bave to solve trie

equation

of motion

nuniencally.

3.3. SMALL DISPLACEMENTS. Trie equation of motion for small

displacements

_can be

written _as

~ÎÎ ^'~~~~ Î

^~

~~ ~ ~ ~

~

~~~ÎÎ~

^{~ ~~}

which is _a

simple

hnear

equation

which _can

easily

be solved _[7] The Fourier coniponent of trie

velocity

due to trie renormalised

driving

force which oscillates _as e~~~~ is

~oj~a,

t

=

«)

= ~-

~~ ^'~

=

~

~-

uol~a,

^t =

t~)

(i

⁺

^1~~ ( ^~~~ _{(i +1~~} (

taira taira

The dielectric _constant is

In(e(w))

=

~~ dt~~(~~C(n,fd,r)

+ In

(1+

^~_

^~~~)

o 'f a wJ~ a

The

procedure

_now is to evaluate trie recursion relations _up to Ç and then to _use trie _renor- nialised values of

V(Ç)

and

fl(Ç)

to calculate the correction to trie dielectric constant for

large

é. This is _a rather crude

approxiniation,

^for _{we are}

ignoring

the thermal fluctuations of

long wavelength generated by

the renormalised random noise which oct to reduce

slightly

the

pin-

ning

potential.

Such _an elfect is

certainly

present, ^{but it is}

ignored

in _our

approach

_or rather it is subsumed in the choice of how to define

Ç.

If the choice of

Ç

is sensible _we expect ^trie

predictions

of trie

theory

to be

relatively

insensitive to small

changes

in

Ç.

The calculation

only

makes _sense when trie

displacement

^{of trie} interface is small in compar- ison with

a/27r.

If this condition _{is not} satisfied _we need to _{use a} numericalmethod to solve the

equation

of motion. The condition that the

displacement

is

sulliciently

small

depends

_on

trie three dilferent

length

scales. There _are two separate cases. When

(

is

bigger

than _rd, trie condition for small

displacements

is that _{rd is} less than _TF- When

(

is smaller thon _rd trie condition for small

displacements

is that

(

is less than _TF More

succinctly,

the

length

_TF must be

larger

than the smaller of the other two

lengths.

When this condition is not satisfied the

displacement

is

large

and _we need to _{use a} numerical method to obtain the dielectric constant.

3.4. LARGE DISPLACEMENTS. In _our

simplified

treatment we

neglect

trie random force

because its elfect is small in comparison to trie

pinning potentiaJ.

Therefore _{we propose} to solve trie equation of motion

~

ÎÎ ^'~~~~ Î

~ _~~~

~Î~

^~

~

~~~ÎÎ~

^~ ~~

numerically

when trie

displacements

_are

large.

^{Trie surface stilfness} term _can be

neglected

because _{we are}

considering only

_a

spatially

uniform

driving

force.

In order to

perform

trie numencal work it is convenient to recast this

equation

into dimen- sionless form

by using

the

length

scales _{rd, TF} and

(.

Let _us call

w =

27rzla

x = uJt

and _use the reduced variables [1,2]

21a~

2

~ 7rkBT _n

47rU

~ kBT

Then the

equation

of motion becomes

Î ÎÎÎÎ

^~~

^~~

^~_{~~~~~°~ ~ ~~~~~~~~}

^~°~~~~ _ÎÎÎÎ~ ^ÎÎÎÎ ~~~~~~

where the realand

imaginary

part ^{of the inverse dielectric function bave been} introduced. In trie above

equation

ail trie parameters are evaluated at

Ç.