Fluctuation forces and the Devil's staircase of ferroelectric smectic C*'s.

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Fluctuation forces and the Devil’s staircase of ferroelectric smectic C*’s.

R. Bruinsma, Jacques Prost

To cite this version:

R. Bruinsma, Jacques Prost. Fluctuation forces and the Devil’s staircase of ferroelectric smectic

C*’s.. Journal de Physique II, EDP Sciences, 1994, 4 (7), pp.1209-1219. �10.1051/jp2:1994195�. �jpa-

00248038�

J. Phvs. II Franc-e 4 (1994) 1209-1219

JULY 1994, PAGE 1209

Classification Ph,vsics Abstrac/s

61.30 64.60

Fluctuation forces and the Devil's staircase of ferroelectric smectic C *'s.

R. Bruinsma and J. Prost

Groupe de

Physico-Chimie Thdorique,

Ecole Supdrieure de

Physique

et Chimie Industnelles, 10,

rue

Vauquelin,

75231Paris, Cedex 05, France (Received J0 January 1994, accepted 23 March 1994)

Abstract. We show that there exists _a novel long range interaction in ferroelectric Sm-C*

materials due _to thermally excited polarization _waves. We propose that this interaction is the long range

repulsion

whose existence had been postulated ^earlier to

explain

the

long period

ferrielectric phase~ observed _near the ferroelectric to antiferroelectric phase transition.

1. Introduction.

The formation of

large-scale

structures in

simple physical

systems otherwise dominated

by

short-range

forces has

intrigued physicists

for many years

[Il.

In solid-state

physics

such extended _{structures are}

repeatedly

encountered. for instance in

binary alloys,

near

hcp-fcc

structural

transitions,

in intercalated

graphite

and in many other systems. ^{The link}

connecting

these _cases is the presence of _some form of frustration in those parts ^{of the}

phase-diagram

where the _{structures are} encountered. At the

point

where the dominant

ordering

force of _a system

happens

to

change sign

_a

large

number of alternative structures may have

exactly

the

same free energy. This

degeneracy

_can be lifted either

by

^weak

long-range

forces

previously

obscured

by

the

ordering

force _or

by

thermal fluctuations. The

importance

of frustration

points

lies in part in the fact that

they

allow _us to

study

^delicate,

long-range

interaction effects which elsewhere in the

phase-diagram

have little effect. Statistical mechanics models have been

developed

which

examplify

these two different _ways of

lifting

the

degeneracy

: the one-

dimensional

Ising

model with

long-range repulsion

ⁱⁿ a field and the so-called ANNNI model with

competing

nearest and next-nearest

neighbor coupling [Il.

For the _case of the _one-

dimensional

Ising

model with

long

_range

repulsion,

Bak and Bruinsma

(BB)

found [21 an

infinite sequence of structures, with

arbitrarily large

^modulation

lengths,

whose

stability

intervals form a so-called

complete

_« ^{Devil's Staircase} _». A very similar sequence appears in

the ANNNI model.

In the

physics

of

liquids,

one would _not expect to encounter frustration

points

^{since it} seems easy ^to lift any

degeneracy by displacing

the molecular

positions

in _some way. Nevertheless. a

series of

optical

studies

[3-51

have revealed that smectic-C*

(Sm-C*) liqiA crystals

exhibit _a

sequence of ferrielectric

phases

around the transition

point

between ferro- and antiferroelec-

tricity,

with modulation

periods

of at least _seven

layers.

Takanishi et al.

[31,

and Hiraoka et al. [41

Proposed

that the observed

phase

sequence of MHPOOCBC and MHPOCBC

(and

their

mixtures)

under variation of temperature

[3]

_or electrical field

[4]

could be _an

example

of the BB model. However, the BB model

requires

the presence of _a

long-range repulsive

forces and because of the absence of any obvious

candidates,

Yamashita and

Miyazima [6] suggested

that the ANNNI model is

really

a more natural

explanation (although

that still

requires

_a rather strong next nearest

neighbor coupling).

To

explain

the relevance for ferroelectric Sm-C* materials of these

simple Ising

models, recall that such materials consist of rod-like chiral molecules

arranged

in

layers.

The

long

^axis of the molecules makes _an

angle

o with the

layer

normal. The average

in-plane projection

of this

long

axis

provides

_a

preferred

direction

along

the

layer,

called the c-director. The

molecules carry an electrical

dipole perpendicular

to both the

layer

normal and the c-director.

We will in the

following

_use a coordinate system ^{with the z-axis}

along

the

layer

normal, with the _x axis

along

the

c-director,

and with the _y axis

along

the

polarization

^direction.

Suppose

_we

rotate a

layer by

_ar around the c-director. This would

flip

the direction of the

polarization

to lie

along

the _y

axis,

while the

long

axis of the molecules makes _an

angle

of ar-o with the

layer

normal. A stack of such

layers

would constitute _a ferroelectric with

opposite polarization.

It is

now natural to

assign

an

Ising spin

variable

S,

₌ ± I to each

layer,

with S, ₌ ^{I for} a

layer

with

polarization along

+ y and _S~

= I for _a

layer

with

polarization along

_y. The two

possible

ferroelectric states would

correspond

to

S,

₌ I _or

S,

₌ I for all I. An antiferroelectric

« Chevron»

phase,

such _as the

Sm-Ct phase,

would

correspond

to an

altemating spin

variable _: either

S,

₌

(-

I )' or

S,

₌ (- I

I'.

A

ferroelectric,

like the

Sm-C( phase,

would

correspond

to a structure with _a certain basic sequence of up and down

spins repeated

ad infinitum.

The

problem

of

predicting

the

phase

behavior is _now reduced _to that of

constructing

the

appropriate

_« Hamiltonian

» for these

spin

variables. If there exists _a strong

long-range

_« anti-

ferromagnetic

_»

coupling

between the

spins,

then the BB model would be _a

good

candidate. In the absence of any

long-range coupling, only

thermal fluctuations could

provide long-range

correlation and the ANNNI model is the natural

description.

At first

sight,

it appears ^to be easy to find

long-range coupling

mechanisms. The _most obvious candidates _are the

dipole-dipole

interaction and the _van der Waals interaction. In addition, the elastic

deformability

of _a smectic medium could also lift the

degeneracy.

As discussed

below,

neither the direct

dipole-dipole

interaction _nor

the

van der Waals

coupling

force _can contribute _to the effective

coupling

of any

prospective

BB model while elastic distortion

provides

a weak attractive

coupling (so

it has the wrong

sign).

We will demonstrate that Sm-C* ferroelectrics

actually

do carry a

significant long-range repulsion.

The mechanism is

specific

^{for Sm-C*} ferroelectrics and results from the presence of

thermally

excited

polarization

fluctuations. Like for any Sm-C

material,

the energy of _a Sm-C*

is

degenerate

^for

global

rotations of the c-director around the

layer

normal. This allows for _a spectrum ^{of low} energy modes

involving long wavelength

variations of the rotation

angle

of

the c-director. What is

peculiar

to _a Sm-C* is that this mode

automatically produces

_a

Polarization,

which in _tum leads to Coulomb

coupled charge

fluctuations. These

polarization

fluctuations of Sm-C* materials have been measured

[7]

and _are known to be

important

for the dielectric behavior.

2.

Dipolar,

van der

Waals,

and elastic forces.

The _most obvious candidate for

long-range coupling

in _a ferroelectric is the

dipole-dipole

interaction

Vdp

N° 7 DEVIL'S STAIRCASE OF FERROELECTRIC SMECTIC C*'s 121

Vdp = E~

i l~' ^~

~

3 ~'

~~' '~ _~

~~' '~

(2, i)

<.J

R, Rj ^R, Rj

where _e is the short-distance dielectric constant and where the summation _{runs over}

pairs

of

dipoles

_on different molecules. In

evaluating

this _{sum, we must}

distinguish

three different

cases

I) pairs

of molecules in the _same

layer.

This contribution _cannot affect the

stacking

sequence and thus

plays

no role _;

it) pairs

of molecules in

adjacent layers.

Due to

positional

correlation between molecules in

adjacent layers, dipolar coupling

between

adjacent layers

_can be

significant.

We will include it

as a

phenomenological

parameter ⁱⁿ section 4

iii) pairs

of molecules in

layers

with

spacing

of _more than one

layer. Scattering

studies of

smectics reveal that

positional

correlations between molecules in

layers

which _{are not}

adjacent,

_are

completely negligible.

This _means that the

dipolar coupling

between

non-adjacent layers

and

j,

_V~~

(I j ),

can be

computed by treating

each

layer

_{as a} two-dimensional _«

liquid

_» of

dipoles.

The interaction between two such

layers

^is

V~~(I j )/A

=

S,

_S~

~ _d~p

~ ~ ~ ~~~

3

~

~ ~~~

~ ~~~

~ (P + d

(I j (p

₊ d-

(I j )

=

0.

(2.2)

Here, A is the area of _a

layer

and

S,P

is the

dipole-moment

_per unit _area of the I th

layer (along

the y

axis).

The cancellation is strict

only

in the

thermodynamic

limit of infinite A.

Subleading

_« surface _» corrections are present ^{and could}

play

a role for very narrow,

needle-shaped samples,

which _we will exclude. We conclude that there is _no direct

dipolar coupling

except ^between

adjacent layers.

This cancellation is less

surprising

if _we recall from Maxwell's laws that _an infinite slab of

material,

which is

uniformly polarized perpendicular

to the surface

normal,

has _no electrical field outside the surface _so two such slabs, with

parallel

orientation, do _not interfact.

We _{now turn to} the _van der Waals interaction which describes the

gain

in

dipolar

_energy due to correlated

polarization

fluctuations of

pairs

of molecules which _must be

computed

quantum-

mechanically.

Consider two of _our molecules, I and 2, with _an intermolecular

separation Rj~.

We will first _assume that there is _no thermal motion. The two molecules then _can have

only

one of the _two orientations

dependent

on whether

they

are part ^of an

S,

=

I _{or an}

S,

₌ I

layer.

Let each molecule have N electrons. The

dipolar coupling

between the two molecules is still

expressed by equation (2.

I), except ^{that the} sum must be taken _over all

pairs

of electrons _on the _two molecules, with p, the

dipole-moment

operator ^{of the i'th electron. Let}

0)

be the

many-electron groundstate

of _a molecule and let _m

)

stand for the excited _states. If

we use

perturbation theory

to compute ^{the effect of the}

dipolar coupling,

then the first-order term

gives

the direct

dipolar coupling

between the two molecules. We saw earlier that

summing

over all

pairs

of molecules leads _{to a} cancellation. Second-order

perturbation theory gives

for the non-retarded _van der Waals interaction energy between the two molecules

E(Rj~ )

=

/ _jj

^'

x

*E~R~2

mj, m~

~

~o _Emj

_Em~

x

(0~ (0~( (pi

_p~ 3 p~

ij~

_p~

r~~) (m~) (m~)

_(~

(2.3)

where

Eo

^and

E~

are,

respectively,

the

energies

of the

many-electron groundstate

and excited

N

states and where _p

=

jj

_p, is the

dipole

moment operator ^of the whole molecule. Define the

,

frequency-dependent

tensor

a, (w

=

f _{(w (E~ Eo)/h) (0(}

p,

(m) (m

_p~

(0) (2.4)

n,

Inserting equation (2.4)

into

equation (2.3) gives,

~~~'2~

fi2

2)j~

~~°

'

~~°2

w~ w~

~

x

~~ ^~~'

~~°

~~~~°~~~

~ _~~~

~~'

_~~°

~~~~°~~~

_~~~ ~

l(2.5)

+ ~

(i12'~l (W1l'~12) (i12' ~2(W21'~12)

The tensor, which is

proportional

tu the

frequency dependent polarizability

tensor of the

molecule,

must have _one

principal

axis

along

_{the y}

direction,

since this is the direction of the broken symmetry, while the two

remaining principal

axes must lie in the _>--z

plane. Recalling

that the

polarizability

tensor is

symmetric,

it follows that for _an

S,

₌ I

layer

& must have the

general

tensorial structure

o o i

&j(w

₎

=

tYd(w I

+

a~~(w

o o o

(s,

=

1) (2.6)

o o

with

a~(w

^and

a~~(w) frequency dependent

functions. The

polarizatibility

tensor for _a

molecule in _an

S,

₌ I

layer

must be related _to that of _an S, ₌ ^I

layer by

a

similarity

transformation

corresponding

to a rotation

by

_ar around the x-axis

(since

this is the

operation

which relates the two

polarization states).

This

operation flips

the

sign

of the

off-diagonal

contribution. We thus _can encompass both _cases

by writing

0 0

I,(w

=

ad(W I

+

a~~(w S,

0 0 0

(2.7)

0 0

If _we insert this

expression

into

E~~.

we find contributions

independent

of S~, ^terms

proportional

_to

S,,

and bilinear _terms of the form

S,

_S~.

Summing

_over all

pairs

of molecules in

layers

I and

j

and

again neglecting positional

correlations between molecules results in _an energy per unit _area

V(I j

):

V(I j

)

=

c/d[

₊

s,

_s~

fi

dw~ dw~ "°d~~°'~ "Od~~°2~

h F Wj + W~

~

12

^(x2 +

d(

) (~d,~⁾²

x d _p ~

~ ~ ~

6

~ ~ + 36

~ ~ ~ ⁼

C/d,~ ^(2.8)

(p

+ dj~

(p

+

d,

~

) (p

+ d,~

with C _a constant, _v the number of molecules per unit _area and with

d,~ ⁼

d(I j

). ^{The first}

term is the usual _van der Waals attraction between two

layers,

^{which does} not

depend

_on the

polarization

direction. The second term would be

contributing

to the interaction term of _a BB

N° 7 DEVIL'S STAIRCASE OF FERROELECTRIC SMECTIC C*'s 1213

model but it

happens

_to be _zero. Thermal motion would allow the instantaneous

principal

_axes

of _a

given

molecule _to differ from the average

principal

_axes. However, one finds that this

only

means that _we must

replace

the

polarizability

tensor

by

its thermal average, which has the _same

symmetry

properties

_as

equation (2.7)

_{so our} conclusion remains the _same. Besides the contribution of quantum fluctuations of the

dipole

moment to the _van der Waals

interaction,

there is also _a thermal contribution. It is found _to

again give

_no contribution _to the BB model

for the _{same reasons.}

This

vanishing

of the _van der Waals _{term cannot} be

justified quite

_so

easily

in _terms of Maxwell's laws, and _we have not found _a

convincing physical

argument to demonstrate

why

it

happens

to be _zero.

Finally,

_we must consider

long-range

elastic forces. The elastic free energy H of _a Sm-A type ^{material has the} well-known form _:

H

=

ld~r

₂

^K(A~

^{u)~ +} ₂ ^{B(d~u)~ + C}

^(d]u

^{)~ +}

^(2.9)

with _u the

layer displacement,

K the Frank

bending

_energy, B the bulk

modulus,

and C _a second-order elastic modulus

setting

the range of

layer

^dilation

compression.

The

length

scale ~

is of order the

layer

^thickness. Assume _we have _a ferroelectric and

flip

the

sign

of

~B

the

polarization

of _two

layers,

say the i'th and

j'th layers.

^The two

flipped layers

will feel _a pressure due to the mismatch with the

adjacent layers.

This pressure will lead to a dilation _{or a}

compression

of the _two

layers. Mathematically,

this _means that each of the two

layers

act like _a

function _{« source}

» in d~u on the elastic free energy. The elastic deformation

produced by

such _{a source}

decays exponentially

with ~

as

decay length

_so it does not

produce

_a

long

~B

range force.

Thermally

excited elastic _waves do however

provide

a

long-range

interaction. If the _two

flipped layers

would be treated _as

infinitely rigid,

then the interaction energy per unit _area is known _to be

[8]

V(I j )

=

'~~~

~ ~~~

~~

^~

(2.10)

16 _ar K

d(I j(

This is _a serious overestimate since in

reality

the

flipped layers

will have _a

flexibility comparable

to the other

layers.

It _can be shown that this will reduce the interaction energy

by

a

factor of order

~ ~

The

resulting

attractive interaction is then

quite

weak

compared

to

d

Ii j

the interaction considered in the next section and, furthermore since it is attractive it could not

produce

_a Devil's Staircase in the BB model. We conclude that Sm-A type ^elastic waves do _not

play

a role.

3. Fluctuation force.

Suppose

we write the

polarization

of the I th

layer

as :

P, (r

= P

o

(sin

_q,

(r

), cos q,

(r ))

₊ &P

(3.

I

The first term is the

macroscopic,

spontaneous

polarization, making

_an

angle

_q,

(r)

with the y axis. This

angle

_can fluctuate

smoothly

_across the

sample, producing

the

long wavelength

polarization

_waves referred to above. The _reason that such coherent

polarization

fluctuations

are

only important

in the

long-wavelength

limit is due _to the fact that in that limit the

restoring

JOURNAL DE PHYS<QUE T 4 N'7 JULY 1994 46

force of the mode goes to zero, due to the rotational invariance around the _z axis. As _a consequence, ^the

amplitude

for such orientational fluctuations

diverges

in the

long-wavelength

limit. The second _term describes incoherent quantum ^{and thermal} fluctuations of the

polarization.

This is

just

the _van der Waals interaction discussed above and _we will

drop

it in the

following.

The

position dependence

of the

angle q,(r) produces

a

charge-density

V P~ per unit _area in the i'th

layer.

The associated Coulomb energy cost is

H~

_m

~° ~ jj

S, _S~

d~p~ d~p~ d,

_q,

(#, _d,

_{q~ (k~}

_(3.2)

J )R~ R~

with

R,

₌

p,

+ I dl. We assumed in

equation (3.2)

that the rotation

angle

is

small,

I.e. that the

polarization

direction remains close _to the y axis.

We must add _to

equation (3.2)

the energy cost of _a

long wavelength

orientational fluctuation of

unpolarized

Sm-C materials. This is well-known and it takes the

(discretized)

form

Hsm

c ⁼

I d2p,

_(Kjj

(vw,

)2 +

K~ (d j

_w,

~

w,1)~) ^(3.3)

,

where the Frank _constants

Kj

and

K~

are or order

k~

TAG with TAG the Sm-A _to Sm-C transition temperature. ^In

principle

the modulus

K~ depends

on the

product

_S,

~

S,.

We can

ignore

this

dependence

however without

changing

the

physics

of the

problem.

For _an

arbitrary stacking

_seqence

(S, ),

the full Hamiltonian H

=

H~

+

Hs~c

for orientational fluctuations is then

H=)zld~P, lKi(vw,)~+Ki(diw,+<-w,i)~l

₊

+

~~ jj

_S,

S~

d~p, _d~p~

^~~

^~~~~'~

^{~~ ~~}

~'~

(3.4)

E

,,j

jk~ k~

with _an associated free energy

F

(S,

=

k~

T In

fl Dq,

e-

~~.

(3.5)

,

If _we could

exactly

evaluate this free energy, we would know the contribution of the

orientational fluctuations to the

stacking

_{sequence energy.} But,

eventhough

we are

only

discussing

Gaussian fluctuations,

equation (3.5)

cannot be evaluated for _a random

stacking

sequence. It is for instance well-known that

problems

of this class _can lead _to subtle

localization effects in the

eigenmodes

of the Hamiltonian.

3.I PERTURBATION THEORY.

By considering equation (3.4),

one sees that _{we can use}

perturbation theory

if the parameter ^A =

P( dleKjj

is small

compared

to one. If _{we use}

typical

values for Sm-C* materials, then A is of order J0-' _to J0-3 _so it would appear that

perturbation theory

suffices.

First, define the Fourier transform

wi(r)

=

~ _{i (w,(q)}

_e~~'~

+

c-c-) (3.6)

N° 7 DEVIL'S STAIRCASE OF FERROELECTRIC SMECTIC C*'s J2J5

with A the _area of _a

layer. Using equation (3.6)

into

equation (3.5),

first-order

perturbation

in

the Coulomb contribution _to the mode energy

gives

for the free energy correction

AF _:

AF

=

)) _isi

Sj d~P

j

_~

~

~

i iql

e~~.P

(w,(q) _w~*(q))

₊

c-c-1.

~,

J p + d

(i j)

_q

(3.7)

The thermal averages in

equation (3.7)

must be

computed using

the Sm-C Hamiltonian

equation (3.3). Using

the relation

I,q.p

^{-qd j,-jj}

d~p

^~

=

2 ar

~

(3.8)

N/P~+d2(I j)2

q

this

simplifies

to

AF _ar ^°

jjS~

_S~

jj~'e-Q~

-J

(q~(q) q~*(q))

_{+ c.c.}

(3.9)

~

, ~

The thermal

expectation

values of the orientational fluctuations in the Sm-C

phase entering

in

equation (3.9)

_are most

easily expressed

in terms of the full Fourier transform

with N the number of

layers.

The correlation function is then _:

~ ~~' _{~ ~~}

Kjj q~ + 2

K~ ~~[

l _cos

(k)

~~'~

_~~

with _a free energy correction

p2

_k T

AF

= ar

fl ~j ^S,

_S~

~j

q ^{e~ ~~ ~J cos}

(k j

)

~

~

NE ~

,~ ~ ~

Kjj q + 2

K~

d~

[I

_cos

(k)1

(3.12)

For

large

distances, _{we can}

expand cos(k)

with the result

p2

s s

kBT

o

~

^J

(3.13)

~~~ ~~~ _~

jKl

~

~

^{~ ~~} ' 'J ~ ^~ l ₊

,

Ki Ki

~i

To

appreciate

the

strength

^{of this}

coupling,

it is useful to compare it with the usual _van der Waals attraction between two dielectric

layers

of thickness

d,

which is of order kB

T/d~

_(I

j

)~

(recall

that this force

actually

does not contribute _to the BB

model).

The _van der Waals force is

bigger

than the fluctuation force

by

a factor A for

adjacent layers.

However, because the _van der Waals force

decays

more

rapidly,

for

layer spacings

ⁱⁿ excess of order A- ^'~~ the fluctuation

force would exceed the _van der Waals force. If _we compare the fluctuation force with the

elastic

coupling

then the fluctuation force

again

exceeds it in

strength

^for

larger layer spacings.

3.2 STRONG-COUPLING. If _we examine the convergence of the

perturbation series,

_we find that _even for small A, the series stops

converging

for

large enough layer spacings.

To _see what will

happen,

we consider here the strong

coupling regime

of

large

A. First,

perform

_a so-called

Mattis transformation _:

~i,(p,)

=

si w,(p,) (3.14)

In _terms of the transformed

variables,

the Hamiltonian is

H

=

jj d2p, _(Kjj (v~,

)2 +

K~ (j~,

~ j

~, l/d)2)

₊

~ l

~(

~ ^~2

_~

^~2

_~ ^~Y

^~'~~i

^{) ~y #~j}

^(~

2 e

,

J

R,

_R~

K~

d- ^~

jj

_(S~

S,

~ j I d~ _p, #r, #r,

~ j

(3.15)

,

The first _{two terms} of

equation (3.15)

constitute the free energy of _a

polarization

fluctuation of

a

purely homogeneous

ferroelectric Sm-C*. The information _on the

stacking

sequence order is

now all contained in the last term, which is _{non zero}

only

at _« defect _» sites of the ferroelectric

structure

(I.e.

where

S, S,

~ ⁼

l).

If the Coulomb

coupling

is strong, ^then we can use a

perturbation expansion

in powers of A- ^' To lowest order in

perturbation theory

one finds

AF/A m Cte K

~

d~ ^~

(#r ~) jj S, S,

~ j

(3.16)

where the

expectation

^value of the

angular

fluctuations _{now must} be

computed using

the first

two terms of

equation (3.15).

If _{we are} far from any

para-electric phase,

then

(#r~)

must be

small

compared

to _{one so} we obtain

only

_a modest,

nearest-neighbor

ferroelectric correction.

Thus, paradoxically,

if _we increase the

coupling

constant A, _we must lose the

long

_range

fluctuation force.

Higher

order _terms

produce multi-spin coupling

terms.

In

general,

the

perturbation expansion

stops

converging

when

(I j

^d exceeds _a distance f

given by

(Id

= A- ^'

=

eK/P(

d

(3.17)

which is of order 100

A

_{to I}

~Lm. Since in strong

coupling

there is _no

Jong-range interaction,

f sets a cut-off for the

power-law

interaction

equation (3.13).

The power law is also truncated if any free

charges

_are present. Free

charges,

due to

impurities,

would lead _to

screening

of the

Coulomb interaction and

consequently

of _our power law force

equation (3.13).

For

reasonably

clean

samples

^the associated

Debye length

is of order _a few

thousand1.

4. Phase

diagram.

We _now will consider the

phase-diagram

of the Sm-C* _{as a} function of electrical field and temperature ^{in view of the above results.} In the ferroelectric

phase,

Sm-C* materials

normally

have _a helical _structure but many

experiments

are

performed

in _an electrical field

E which is

large enough

to

quench

the helix. We will _assume this _to be the _case. We also will be

assuming

that the

layer polarization Po

is

sufficiently

weak _{so we are} allowed _{to use} the

perturbative

^result

(Eq. (3.13)).

An electrical field

obviously

will create a gap in the excitation spectrum ^{but this}

provides only

another cut-off for _our

power-law,

now at

length

of order

N° 7 DEVIL'S STAIRCASE OF FERROELECTRIC SMECTIC C*'s J217

/~

,

with E the

applied

electrical field. For

typical

electrical

fields,

of order

EPO

V/~Lm,

this cut-off is of order thousands of

A.

The energy cost per unit _area of _a

given stacking

_sequence

(S,)

will be modeled _as

F=-AjjS~S~~~+rjj~_~'i~~-HjjS,. ^(4.I)

, ,,j I-J

,

The first _term describes the

coupling

between

adjacent layers.

For

positive

A,

adjacent layers prefer

to have the _same

polarization (ferroelectricity),

while for

negative

A

they prefer

to have

opposite polarization (antiferroelectricity).

The A parameter ^{includes steric}

effects, favoring ferroelectricity,

and the short-distance

dipolar coupling

which favors

anti-ferroelectricity.

The various effects

entering

in A _are discussed in _more detail

by

Koda and Kimura

[91.

As noted in the introduction, modulated structures should be

expected

around the frustration

point

A

=

0. The A parameter ⁱⁿ

general

will

depend

on temperature ^and concentration, but should be

independent

of the electrical field. The second _term is _our

long-range

fluctuation force with

~

~~ ^~

~~d ^~~'~~

l ₊

/~'

^~

^fi

Ki

The r parameter

again

could

depend

_on temperature and

composition,

in

particular

as we

approach

_{any para} electric

phase. Finally,

the electrical field is included

through

the last _term with H

=

EPO.

This model is similar but _not identical _to the BB model. The BB model

requires

«

convexity

_». This _means that if

J(I j

is the

spin-spin

interaction then _{we must} demand that

J(K+

I)+J(k- J)-2J(k)m0. (4.3)

This

inequality

is

obeyed

if rm ~~

A. Thus. for

negative

or

weakly positive

I I

A, the condition is _met. However, _as we enter

deeper

into the ferroelectric

phase

^{it is}

likely

to be violated and _a rather different

phase-behavior

is encountered. We start with the first _{case :}

4. I A/r _< I I/18. We _can

directly apply

the results of the BB model in this

regime.

Assume

we vary the electrical field. Consider _a repeat group of _n

layers

^with m up

spins.

For every rational number _q

= m/n, there is _a finite

stability

^{interval AH}

(q

as a function of the electrical field

given by

i ~ i

AH ~Q~ ~ ~

~ji

_~'~

~Pn

+

1)~

^{~ ~~} _{~~'~ ~~} ~~

~

~

(4.4)

n~

for n # 2. The series converges

sufficiently rapidly

that the

precise

value of the cut-off of the power law is _not

important.

The

stacking

_sequence inside the repeat group is discussed in

reference

[101.

Some

typical examples

_are

llilllillllilllill _q=2/9 illillilllill _q=4/13

iiiliiiliiil q=3/4 illlllllllllll q=1/15

As _a function of the electrical field, the

typical stability

_range is thus of order

k~

^TA

(4.

5)

~~ ~~~

d~P

o

n~

which for smaller _n is in the range of

V/~Lm assuming

A of order 10- ^~ This is

encouraging

_as it is the order of

magnitude

^{of the electrical fields used in the}

experiments.

A

special

case is the

statility

_range of the antiferroelectric

phase.

^{This is the}

only

structure whose

stability

interval

actually depends

on the A

parameter

AH(1/2)

=

A ₊ r

jj

_{2 p}

~ + 2 _p

~

(4.6)

pwj

(2p +1) (2p-1)

p

If _we increase A, the

stability

_range ^{decreases but} at the threshold A/r

=

I1/18 where the BB model loses

validity,

it is still _non-zero.

4.2 A/r _m I I/18. We have found _{no exact} solution of the

phase

_sequences in this range and

had to resort and to

explicit comparison

of the

energies

of different structures. We will

only

consider _zero electrical field.

By

direct calculation _we find the

following energies

_{per «}

spin

_» :

EA~=A-0.81r (I()

E~~

= A ₊ 1.54 r

I (4.7)

E~A~=-0.20r (11(().

There is, _{as a} function of A, _a first-order

phase

transition from the antiferroelectric to the _so- called doublet antiferroelectric

(DAF)

_at A/r

=

I1/18.

By computing spin flip energies

and domain wall

energies

it _was found that the antiferroelectric indeed remains

locally

stable

right

up to the threshold, while the DAF is stable

beyond

the threshold. This behavior is

radically

different from the BB

phase

_sequence where _no first-order

jumps

occur. Moreover, the DAF is in fact absent from the BB

phase

_sequence. Further increases in A would appear to favor the ferroelectric

phase

around A/r of order 1.74. However, if _{we use}

equation (4.

I) to compute ^the energy ^cost of _a domain wall in the ferroelectric

phase

we find

m m+N-i j

AE=4A-4r

jj jj

j

m=i n=m

~

m4A-4rln

(N) (4.8)

with N the number of

layers.

^The ferroelectric

phase

^is thus

predicted

to be unstable for any value of A

(at

least in the absence of _a

field).

What

happens

if _we increase A is that

phases

appear which _are similar in structure to the doublet antiferroelectric except ^{that the number of} repeat

spins

_per block increases. The blocks

N° 7 DEVIL'S STAIRCASE OF FERROELECTRIC SMECTIC C*'s J219

are

separated by

domain walls of the ferroelectric

phase.

If _{we assume} that the block size is d, then the energy is of the form

R

=

~ ~

4 r ~~ ~~~~~

(4.9)

The first term acts like _a chemical

potential

of _a ferroelectric domain wall and the second _term acts _as a

logarithmic

attraction between domain walls.

Minimizing

with respect to d

gives

an

optimal

block size

d ^*

~

e~~~

(4.10)

so as A inreases _we indeed do

approach

the _true

ferroelectric,

without _ever

reaching

it. Recall

though

that for _zero electrical field helical _structures appear so this

prediction

must be treated with caution. It _can be shown that the repeat

period

of the helix _acts

again

_{as a} cut-off _{on our}

interaction, so when d* reaches this repeat

length,

the ferro-electric _or rather the heli-electric could appear.

In fact two other cut-off

lengths

_may ^stabilize the ferroelectric _state the

Debye screening length

A~, ^{and the}

length

which defines the range of the fluctuation interaction. For

d* ~ f or d* _~ A

~, that is _more

precisely

for A _m r In

(inf (f, ~)/d),

_{we expect} the ferroelectric _{state to} have the lowest energy. These block antiferroelectric _states will _not be easy ^to

distinguish

from

simple

antiferro _states.

X-rays

are not

expected

to

yield

_easy

identifications, because

similarly

to our arguments

concerning

van der Waals and Casimir forces the _contrast between antiferro and ferro interfaces should be

vanishingly

small in the domain of

phase

_space we consider. A last word of caution concerns the _use of

Ising

variables

in _{our treatment:} in

principle

_one could

imagine

other _structures

involving

o and

q variations

[lll.

Our

approach

^{holds in the} parts ^of

phase

space where _a

straight

Landau

expansion predicts

_a direct ferro-antiferroelectric transition.

In summary, we have found that there does exist _a

long

range

repulsion

which could account for the observed

periodic

structures in Sm-C* materials. It results

quite naturally

^{from the}

long

wavelength

fluctuations of the c-director in _a ferroelectric material, and is

reasonably large.

The absence of the _true ferroelectric for _zero electrid field, the appearance of the doublet

antiferroelectric,

and the breakdown of the Devil's Staircase for

positive

A are

specific

features of the

analysis proposed

ⁱⁿ this paper and should be useful _{as a}

diagnostic

for

distinguishing

the

present approach

from the ANNNI model.

References

[Ii Bak P., Phys. Today 39 (1986) 38.

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Phys.

Rev. Lett. 49 (1982) 249.

[3] Takanishi Y., Hiraoka K., Agrawal V. K., Takezoe H., Fukuda A., Matsushita M., Jpn J. Appt.

Phys. ³⁰ (1991) 2023.

[41 Hiraoka K., Takanishi Y., Skarp K., Takezoe H., Fukuda A., Jpn J. Appt. Phys. 30 (1991) L 1819.

[51 ^Isozaki T., Hiraoka K., Takanishi Y., Takezoe H., Fukuda A., Suzuki Y., Kawamura I., Liquid Crystals 12 (1992) 59.

[61 Yamashita M.,

Miyazima

S.,

preprint.

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[81

Adjari

A., Peliti L., Prost J., Phys. Ret,. Lett. 66 (1991) 1481.

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[10] Hubbard J.,

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