Fluctuation effect on the incommensurate transition of quartz

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Fluctuation effect on the incommensurate transition of quartz

P. Borckmans, G. Dewel, D. Walgraef

To cite this version:

P. Borckmans, G. Dewel, D. Walgraef. Fluctuation effect on the incommensurate transition of quartz.

Journal de Physique I, EDP Sciences, 1991, 1 (7), pp.1055-1062. �10.1051/jp1:1991189�. �jpa-00246385�

Classification

Physics

Abstracts 64.70R 64.70

Fluctuation effect

_on

the incommensurate transition of quartz

P. Borckmans

(*),

G. Dewel

(*)

and D.

Walgraef (*)

Service de

Chimie-Physique,

Universitd Libre de Bruxelles, Bd du

Triomphe,

CP 231, B-1050, Brussels,

Belgium

(Received Jo December J990,

accepted

in

final form

2J March

J99J)

Rksumk. Dans la

description

_au

champ

_moyen de la transition _vers [es structures incommensu- rabies du quartz, les

phases

3-q,

J-q

et fl

apparaissent

_au mdme

point critique.

On montre que [es

fluctuations

critiques

[Event cette

ddgdndrescence

et transforment la transition du second ordre en

transitions du premier ordre. De

plus,

il existe _une

rdgion

de l'espace des

paramdtres

off la

phase

I-q

apparait

entre les

phases

fl et

3-q,

mdme _en l'absence de tension

appliqude,

_comme le

montrent des

expdriences

rdcentes.

Abstract. In the _mean field

description

of the incommensurate transition of quartz, ^the triangular 3-q, the

striped

I-q ^and the fl

phases

meet at the _same critical

point.

It is shown that critical fluctuations lift this

degeneracy

and transform the second order transition into first order

ones. Furthermore, there is _a

regime

of the parameter space where the I-q phase should appear

between the fl and 3-q

phases,

_even in the absence of any

applied

stress, as seen in recent

experiments.

The

a-p

transition in quartz ^{has been known for almost} a century

[Ii.

It is _a structural

phase

transition which

corresponds

to the rotation of

Si04

tetrahedra around the threefold axis. The order parameter _1~ can be taken to have _a

magnitude equal

to the

displacement

of _a

given

Si ion _as shown in

figure1 [2].

^It is _zero in the

p phase

and increases when the

temperature

is

lowered into the

a

phase

where it _can take two

equal

but

opposite

values

corresponding

to the so-called

Dauphinb

twin domains. More

recently,

_an incommensurate

phase

between the low- temperature a

phase

and the

high

temperature

p phase

has been observed

[3-6].

This is _one of the many

examples

where transitions _occur between normal

crystalline phases

and incommen-

surate modulated structures

[7],

but its interest lies in the fact that modulated structures of different

symmetries

_may be

simultaneously

stable.

Effectively,

the theoretical

analysis

based

on the Landau free _energy derived for this system

[8, 9]

concluded to the existence of

striped

modulated structures defined

by

_one

pair

of wavevectors

(I-q phase)

_or of modulated

structures of

triangular

_symmetry defined

by

^three

pairs

of wavevectors

separated by

2

ar/3 angles (3-q phase).

(*) Research Associates, National Fund for Scientific Research

(Belgium).

1056 JOURNAL DE

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I bt 7

3 y _~

x

~ 3

1

Fig. I. Basal plane

projection

of the silicon ions in the

Wigner-Seitz

cell of the quartz structure. The full circles indicate the

position

of the silicon atoms in the fl

phase

while the _arrows show the directions of the silicon displacements in the transition to the _a

phase.

In most

experimental conditions,

well-defined

3-q

structures of about 10 _nm

wavelength

have been observed which

persist

in _a small temperature range

(IA K)

around the critical

temperature (846K). According

to the _mean field Landau

description,

this

p

-

3-q phase

transition should be second order and _{occur at a} well-defined critical

point 7j [8, 9]. However,

in the presence of uniaxial stresses, a

striped I-q

modulated

phase

_appears between the

p

and the

3-q phases [10]. Furthermore,

recent

experiments [11, 12]

^concluded to the existence of the

I-q phase

between the

p

and the

3-q phases,

_even in the absence of _an

applied

stress, ^and

this _seems in contradiction with the theoretical

prediction

that the

3-q, I-q

and

p phases

should meet at the critical

point 1j.

The existence of uncontrolable local stresses or defects _can be invoked to

explain

this

phenomenon. However,

_an alternative

explanation

_may be found in the effect of critical fluctuations _on this transition.

Indeed,

ⁱⁿ a renormalization _group

analysis,

_no stable fixed

point

has been found in the _e

= 4 d

expansion [13]. Usually

this indicates that the transition is first order. On the other

hand,

it has been shown that critical

fluctuations _can

strongly

affect the nature of the transition between _a uniform state and _a

periodic

structure. It is the main purpose of this note to assess the effect of the order

parameter

fluctuations _near the critical

temperature 7j

^where the

p, 3-q

and

I-q phases

meet.

According

to the

phenomenological description

of the

a-p

transition of quartz

[14],

the free _{energy can} be

expanded

in _a power series of the Fourier components _1~~ ^{of the} order

parameter

^and of the strain field. After elimination of the elastic

degrees

of

freedom,

the

quadratic _part

of the free _{energy may} be

written,

ⁱⁿ scaled

variables,

_as

[8]

:

~2

"

I (~

_~ +

d(~~ _~~)~

₊

_{hQ~C°S~(~ ~} )) l'lq

_(~

(l)

q

where _e

=

(( T)/(, ( being

the

p-INC

transition

temperature

_{go is} the

length

of the critical modulation wavevectors and

j

^the

angle

between _q and the two-fold _x axis of the

a

phase (cf. Fig. I).

The

corresponding marginal stability

surface which limits the

stability

domain of the

p phase

in the

(e,

_{q space} is shown in

figure

2. One _sees that 6 unstable modes

are

expected

at T

=

Tj,

which

correspond

to modulation wavevectors of

length

_go and of orientation

j

=

(2n +1) ar/6.

Hence

single-

and

triple-q

structures may in

principle

be nucleated for T

< T; ^{but their}

stability

has _to be determined

by

the

higher

order terms of the Landau functional.

6

~.

qy

o

.s

-o

5 o

qx

Fig.

2.

Representation,

in the

(e,

_q) _space, of the surface which limits the stability domain of the fl

phase

of quartz,

according

to the Landau free _energy (Eq.

(I)) (d

= I, _qo

= I, A

=

0.5).

Effectively,

the

higher

order contributions to the free energy may be written _as

~~2

_"

I G( (q)

lJq lJq~ lJq~ ³

(q

_{+ q1+}

~2)

q,qi,q2

+

I H( (q)

_{lJq lJq~}

_lJq~lJ

q~ 3

(q

+ q1+ ~2 +

~3)

q.q].q2.q3

~ 2

1

~

~q i ''lq'

~ ~

i _'~qi'lq-q]

_~~~

q.qi

The detailed

dependence

of the coefficients

G, H,

K and

v on the elastic constants of the system can be found in references

[8, 9].

In

particular,

it may be

shown,

in the _mean field

approximation,

that the

angular q-dependence

of G allows the formation of stable

triple-q

structures via _a second order transition. The transition

temperature

is the _same for the

single-

and

triple-q

structures and _one should thus

expect

an extreme

sensitivity

of this

degenerate

critical

point

to thermal fluctuations.

Hence,

let _us

analyze

the influence of the most relevant fluctuations _on these two types ^of structures.

4.I

SINGLE-q

OR STRIPED PHASE. In the mean-field

approximation,

the existence and

stability

of _a

single-q

structure of wavevector q of

length

_qo and oriented

along

_one of the

directions defined

by j

=

(2

_{n +} I

ar/6

may

easily

be deduced from the Landau free energy

(1-2) [8, 9]. However,

critical fluctuations _are known to be able to affect the result of the mean-field

analysis,

and this is also the _case here.

Indeed,

if _ones takes into _account the contribution of these

fluctuations,

the

equation

of state may be

written,

at the Hartree

approximation [15, 16]

:

~ _~

~'~@~

~

~'lqo~ ~'l@)

^~ ~

~'lqo) l~/(

~2

~'lk 'l-k) (~)

where

(... )

_represents the

equilibrium

_mean, and where the

integral

is taken _over the

«critical

shell»,

I-e- _over wavevectors

pointing

in the domain defined

by

d(q~ _q()~

₊

_Aq

^~

cos~(3 j

< e.

1058 JOURNAL DE

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This

equation

is

generic

for _a second order transition from _a uniform state to one-

dimensional modulations. The coefficient _u may

easily

be obtained from the

corresponding

Landau free energy

(for quartz

or berlinite the evaluation of _u from the free energy

(1, 2)

_may

be found in

[8, 9]).

The correlation function

(1~~1~_~)

may be evaluated at the _same

approximation through

the self-consistent scheme

k~r

~~~

_~

~

r +

d(k~ q()~

₊

Ak~ cos~ (3 j )

r=-e+2u((1~~~)(~+u ~(2'r)

^~~

_{~(1~~1~_~) (4)}

where

k~

is the Boltzmann constant and _r

=

TIT.

As _a

result,

the

equation

of state may be rewritten _as

~l

i'lqol

~

l'~qol l'~qol

^~ ~

°

(~)

with

e = rj +

CmK(m) (6)

uk~

_r

where C

= and

K(m)

is the

complete elliptic integral

of the first kind with

~2

^ar

fi~

m~

=

~~°

~.

Since

K(m diverges

at rj =

0,

the

corresponding

modulated structure arises ri +

Ago

with _a finite

amplitude (I.e.

via a

first-order transition)

_:

'~% ^"

~~~ I

~

at ej =

e(rj~),

where

e(rj~) corresponds

to the minimum of the _curve

e(rj) given

in

equation (6),

and is thus defined

by

the condition _:

Cm/ _E(m~)

= ~ ~

(7)

2

Aqo(

I m~

)

where

E(m

is the

complete elliptic integral

of the second kind. In the weak

anisotropy

limit

(uk~

_r » d

~'~(Aq()~'~),

one has

~

3

luk~

^r

2'3

~

/$

~~~~

while in the

strong anisotropy

limit

(uk~

r «

d~/~(Aq()~'~),

_one finds

3ukB

_r

Aq/

~i =

(8b)

~ ~

/p~~

UkB T

One however needs to be _aware of the fact that the Hartree _or

one-loop approximation

consists in

neglecting higher

order contributions in the

expansion [17]. Following Brazovskii,

a contribution of order _n behaves _as

CmK(m ) (rj

₊

Aqj)~

~ and should thus be

negligeable

in the

region

of

interest,

I-e- where

uk~

_r

>

rj11/K(m).

_This

consistency

_argument,

supplemented by

the failure of the renormalization _group

analysis

to find _a stable fixed

point [13]

thus

suggests

^{that the transition should be first}

order,

and _we have here another

example

of the so-called Brazovskii effect

[16]

where fluctuations

modify qualitatively

the character of the transition.

4.2

TRIPLE-q

OR TRIANGULAR PHASE. A similar discussion may be

performed

for the

3-q

structures. In this _case, the cubic term of the free energy

couples

the

amplitudes

of the three modulations and behaves _{as l~j 1~~1~~ cos}

(3 j ).

The minimization process then shows that

cos

(3 j )

is

proportional

to

1~

(leading

to the observed rotation of the

triangular

structure and

a

slight

variation of its

wavelength),

_so ^that the cubic term

effectively

behaves _as

1~~, ^as shown in

[8, 9],

and the

potential corresponding

to these structures may be written _as

3

F

=

Z (-e+d(Ql-Ql)~ i~Jq,i~+( i~Jq,i~+2u ^i~Jq,i~ Z ~lq~l~)j

,=i j,,

j2 j2 j2

g

~~'

~~~

~~~

(9)

I l'lq,

,

If

needed, explicit

values for the coefficients _u and g may be obtained from the Landau free energy.

The

amplitude

of the three

components

^{of the order} parameter ^{which minimize this free} energy are

given by

0

(E

< 0

~~'~

~~~ ~~ ~~~

5 _u

~

2 _w ^{~ "}

~

'

~'~

~°' ~

~/

^~~~~

and the mean-field transition is thus second order _as for

I-q

structures

provided

w _<

~)

^For w _»

~)

^{the transition} should be first

order,

and _one needs _to consider

higher

order contributions _to the Landau free energy.

The

stability

domain of this _{structure may}

easily

be deduced from the free energy, and _one finds that it is

only

stable for _{w »}

(. Furthermore,

_on

including

the dominant contribution from critical fluctuations in the

equation

of state,

namely

the

diagonal

correlation

functions,

one obtains

[15]

_:

If the correlation functions _are evaluated at the _same

approximation

level _as in the

I-q

_case,

one finds

[15]

r~-

(u+2w) (1~~) (~=

⁰

⁽¹²⁾

and

e =

(I

+ y r~ ⁺

CmK(m) (13)

1060 JOURNAL DE

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where _Y

"~/+2w

Therefore the

3-q

modulated structure also arises with _a finite

amplitude

ll~q,I

=

J~ (~i

~ ^,

(14)

and _a finite rotation

angle j~, given by

cos

(3 j~)

_cc

J~~~

,

(15)

u + 2 _w

at e~=

e(r~~),

where

e(r~~) corresponds

to the minimum of the _curve defined

by equation (13),

^{and is thus defined}

by

the condition

Cm/ E(m~)

^'

~ ~ ~

2

Aq((

I

ml)

In the strong

anisotropy limit,

_one has _:

~

3

uk~

_r

~~

Aq((I

_{+ y)~'~}

~~~~~

/$~ ~~B

_~

while in the weak

anisotropy limit,

_one has

~~

2

uk~

r 2j3

~3 "

~(l

+ Y

~

_d

^(17b)

Hence,

^the thresholds for

I-q

and

3-q

structures are different

(for example,

in the strong 4

uk~

_r

anisotropy limit,

_{e~ ej}

=

In

(I

_{+ y}

)),

and

/$

e~ » e j for _w

< u

(18)

Thus,

_as shown _on the

phase diagram

sketched in

figure 3,

for ^~

< w < u, ^a

I-q

modulated 3

structure should _occur between the

p

and

3-q phases,

_even in the absence of uniaxial _stresses in agreement ^with recent

experimental

observations

[11, 12].

To

conclude,

_{we see} that critical fluctuations

provide

an intrinsic symmetry

breaking

effect which lifts the

degeneracy

of the _mean field transition

point

between the

p, I-q

and

3-q phases. They

also

modify qualitatively

the incommensurate

phase

transitions of quartz, ^which

should be

effectively

first order. In

fact,

this

phenomenon

should _occur

generically when, according

to the _mean field

approximation,

incommensurate

phases

of different

symmetries

meet at the _same critical

point,

_as in the _case of _some

layered

CDW

compounds

and _rare gas

monolayers

adsorbed _on

graphite [18].

Acknowledgments.

Fruitful discussions with Profs. S.

Amelinckx,

J.

Lajzerowicz,

J. Van

Landuyt,

G. Van Tendeloo and Dr. C. Leroux _are

gratefully acknowledged.

This work _was

supported

in

part by

the

Belgian

Government

through

the «Pbles d'Attraction Interuniversitaire»

program.

W .,~~

5u/ 2

'

3q a

~

~

u/3 ~q

o _~

W &,

~

~ j,

3q

~(

(

a _(b)

u/3

',~_~(

lq

E

Fig.

3. Phase

diagram

of the incommensurate

phase

transitions of quartz ^{deduced from the Landau} free energy

(Eqs. (1,

2)),

(aj

in the _mean field

approximation

and

(b) by taking

into account critical fluctuations (w, ^which is defined in Eqs. (9) ^and

lo),

represents ^the intensity of the nonlinear

couplings

between the wavevectors

underlying

the

triple-q

structure).

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