On the Landau theory of structural phase transitions in layered perovskites (CH3NH3)2MCl4(M = Mn, Cd, Fe) : comparison with experiments

(1)

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On the Landau theory of structural phase transitions in layered perovskites (CH3NH3)2MCl4(M = Mn, Cd,

Fe) : comparison with experiments

Michel Couzi

To cite this version:

Michel Couzi. On the Landau theory of structural phase transitions in layered perovskites

(CH3NH3)2MCl4(M = Mn, Cd, Fe) : comparison with experiments. Journal de Physique I, EDP

Sciences, 1991, 1 (5), pp.743-758. �10.1051/jp1:1991166�. �jpa-00246367�

(2)

Classificafion

Physics

Abstracts

62.20D 64.70

On the Landau theory of structural phase transitions in layered perovskites (CH~NH~~MCI~(M

=

Mn, Cd, Fe)

_:

comparison

with experiments

Michel Couzi

Laboratoire de

Spectrosccpie

Moldculaire et

Cristalline(*),

Universitk de BordeauxI, 33405 Talence Cedex, France

(Received11 December1990,

accepted

lfebruary

1991)

Rksumk. La

sdquence

de transitions de

phase 14/mmm

_~ Abma

~

P4z/ncm

(THT _~ ORT _~ TLT),

qui

est une

caract6ristique

_commune des

compos6s

(CH~NH~)~MCI~

(M ₌ Mn, Cd,

Fe),

est examin6e _sur la base de la th60rie de Landau, _en

comparaison

avec les r6sultats

exp6rimentaux disponibles.

Une attention

particulidre

est

port6e

sur le comportement _« anormal _» de la _constante

61astique

_c~.

D'aprds

le moddle

microscopique

de type champ moyen,

d6velopp6

ant6rieurement par Blinc, Zeks _et Kind, et k partir ^de consid6rations

dassiques

de th60rie des groupes, it

apparait

_que

l'dnergie

libre de Landau doit dtre

d6velopp6e

k l'ordre huit _pour obtenir _une

description

coh6rente de l'ensemble des donndes

exp6rimentales.

Abswact. The

phase

_sequence

14/mmm

_~ Abma _~

P42/ncm (THT

_~ ORT _~

TLT~,

which

is _a _common feature in

(CH~NH~)~MCI~ compounds (M

=

Mn, Cd, Fe

),

is discussed _on the basis of Landau

theory,

in

comparison

with available

experimental

results. Much attention is

paid

to the _« anomalous _» behaviour of the cm elastic constant. After the

microscopic

model of mean-field type,

developed previously by

Blinc, Zeks and Kind, and from classical _group theoretical considerations, it tums out that _a Landau

free-energy expansion

_up to the

eighth-order

is necessary ^to obtain _a coherent

description

of all

experimental

data.

1. Inkoducfion.

During

the last fifteen years, much attention has been

paid

to the structural

phase

transitions

occurring

in

perovskite-type layer compounds

with formula

(CH~NH~)~MCl4 (MAM

Cl for

short,

with M

=

Mn, Cd, Fe) [1, 2]

and cited

Ref.).

The structure of these materials consists of

double-layers

of

methylammonium

_groups

(MA)

sandwiched between

perovskite-type layers

made of

comer-sharing MCI~

octahedra. All structural modifications found in these systems ^derive ^from a

tetragonal high temperature

parent

phase (THT),

with space-group

(*)

U-R-A- 124, C-N-R-S-

(3)

14/mmm

and Z

= I formula unit in the

primitive

unit-cell of the

body-centred multiprimitive

cell. In this

phase,

the MA groups are

statistically

distributed between four

energetically equivalent orientations,

_so _as to reconcile the four-fold site symmetry ^with the three-fold

molecular symmetry

[3-7]. Then,

the

phase

transitions _are

govemed essentially by ordering

processes of the MA _groups,

coupled

with rotations of the

MCI~

octahedra within the

perovskite layers;

the

following phase

sequence has been

established,

in the order of

decreasing

temperature :

14/mmm

- Abma

-

P4~/nom

-

P21/b (Z

=

I

(Z

=

2) (Z

=

4) (Z

₌

2)

THT ORT TLT MLT

The THT~ORT transition is of

second-order,

while all other structural

changes

correspond

to first-order

transitions; also,

in MAMnCI and

MACdCI,

all structural modifications exhibit orientational disorder of the MA groups, of

dynamic

_nature,

excepted

the monoclinic low

temperature phase (MLT)

^which ^is ordered

[1-7].

In the _case of

MAFeCI,

the MLT

phase

has _never been observed

though experiments

have been carried out down _to

liquid

helium

temperature,

so that the

phase

_sequence is reduced to THT

~ CRT

~ TLT

[8] thus,

it _can be

thought

that the ordered state of MAFeCI

corresponds

to the TLT

phase.

Due to the ferroelastic character of the

phase

transitions in MAMCI

compounds,

_numerous ultrasonic _or Brillouin

scattering experiments

have been

performed

in order to

analyze

the behaviour of the elastic constants

through

the

phase

_sequence

[9-13].

All

experimental

data agree with _an

unexpected

behaviour of the elastic _constant c~~ in the TLT

phase, exhibiting

almost

complete softening

when the TLT _~ CRT transition temperature is

approached

from below

(Fig. I).

Different

approaches using

Landau

theory

have been made to

explain

this anomalous _» behaviour of c~~

[IO, 13, 14].

In

particular,

the

importance

of _an

hypothetical

orthorhombic low temperature

phase (OLT)

with space-group Pccn and Z

=

4 has been stressed

[10]

note that this

phase

experimentally,

but its existence

can be

predicted

_on

simple group-theoretical grounds [6, 7, 10]. Moreover,

different formulations for the Landau

free-energy expansion

_as

adopted by

_some authors have been the

subject

of

controversy [10, 14-16].

The aim of the

present

paper is _to reconsider this

problem

_we shall restrict ourselves to the

THT~ORT~TLT sequence, which is _a _common feature for all three

compounds

(M

=

Mn, Cd, Fe). First,

some basic aspects ^of Landau

theory

when

applied

to these systems

lo TLT ,CRT THT

, ,

,

5 ^'

~

, _,

i _,

, ,

0 ,

Temperature

Fig.

I.- Schematized

representation

of the temperature

dependence

of the _c~ elastic constant in

MAMCI

compounds

with M

= Mn, Cd, Fe

(composed

from Ref.

[9-13]).

(4)

will be

recalled,

_as well _as the essential results derived from _a

microscopic

model

by Blinc,

Zeks and Kind

[6, 7] (hereafter

referred to _as B-Z-K-

model)

which

actually provides

the best

description

of the transition mechanisms.

Then,

different formulations

already proposed

for the

free-energy expansion [10, 13, 14]

will be discussed in view of

experimental results,

and it

will be stressed the need for _a

complex development

of the

thermodynamic potential

including

terms up to

eighth~order.

2. Basic considerations and the B.Z.K, model.

2. I GROUP _THEORY. From the

only point

of view of group

theory,

the

phase

_sequence in MAMCI

compounds

can be

explained by

successive

freezings

of _a two-dimensional order parameter at the _zone

boundary point

X

(D~~ symmetry)

of the THT Brillouin _zone

[17~20].

The star of the _wave vector at

point

X contains two arms

(kx

=

00 and

kg

=

0)

and the

2 2 2

two-dimensional order

parameter

(1~_j, _1~

~)

^transforms

according

to the small

representations Xi _/B~~

for _l~j at X and

it _/B~~

for _1~~ at

I _(here

we use the notation of

Bradley

and Cracknell

[21]). Regardless

of the

stability conditions,

four different

phases

_can be

expected owing

to the

symmetry properties

of the order parameter components

(Tab.1)

Table I. The symmetry

properties of

the

primary

_(1~

j, 1~~) ^and

secondary

_(1~~,

f

order parameters in the

d%ferent phases of

MAMCI

compounds (the

notations

of Bradley

and Cracknell

[21]

are

used).

THT ORT TLT OLT

'll

~( ~/ ~/I'll

₊

_'l2) ~/

~~

it _Y( _r( _(~j _~j) _r/

'l3

~~ ~/ ~( ~/

f Z( Y( r/ r(

I)

the THT

phase (14/mmm)

is described

by

the trivial solution _l~j

= 1~~ = 0 _;

it)

the ORT

phase corresponds

to solutions such _as _l~j #

0,

equivalently by

l~i ⁺ 1~2 ~

0,

_l~i _'12 # 0

(antiphase domains)

iv) finally,

the OLT

phase (Pccn) corresponds

to the

general

solutions where _l~j #

0,

1~~ # 0 with

1~ j # _{1~ ~.} Note that Pccn is the _common maximal

sub~group

of the Abma

(Bmab)

and

P4~/ncm

_space _groups.

Still _on the

only

basis of

group-theoretical considerations, secondary

order parameters can

also be

introduced, designated

_as

1(rllB)

and

f(Z( /Bj~) [13, 17-20] then,

the THT

phase

is further characterized

by

_1~~

=

f

=

0,

the ORT

phase by

1~~ #

0, f

=

0,

the TLT

phase by

_1~~

=

0, f

_{# 0} and the OLT

phase by

_1~~ _#

0, f

_{# 0}

(Tab. 1).

2.2 LANDAU _THEORY.

Following

the method

developed by

Gufan

[22],

_see also Ref.

[23],

we introduce the full rational basis of invariants for the

image

of the

primary

order parameter

(C4v

in _our

case)

_we do not introduce here the

secondary

order parameters.

Then,

the

Landau

free-energy developed

_up to the

eighth~order

writes

[23]

:

F=AjIj+A~I)+A~I/+A~I)+BjI~+B~Ij+Cj~IjI~+Cjj~I)I~ ₍₁₎

(5)

~~~~~

~l

" 'l +

'l~

" P^~

~2

"16'l/~'l~)~-4 ~l'li" P~COS4

q~

with _l~j

addition,

the first-order ORT

~ TLT transition line is

distorted,

_so that this transition is _now

possible

at constant

Bj,

and sequences such _as THT

~ ORT

~ TLT _or

THT _w TLT _~ ORT _can be

predicted, depending

_on the

sign

of

Cj~ (Fig. 3)

note

again

that the OLT

phase

_never _appears.

Finally,

_a number of situations encountered with the full

/ j'

/

CRT TLT

/

Fig.

2. Phase

diagram

_as determined from the

potential (I),

with A~ = A~

= B~ =

Cj2

=

Cjj~

= 0

(A~ ~0). ^Dashed and continuous fines _are

respectively

lines of second and first-order transitions.

Hatched _areas

correspond

to

regions

of

instability.

H H

A2 A2~ _fiA~ A2~

B~ _B~

T/~TLT

CRT TLT

(al 16)

Fig.

3.-Phase

diagrams

_as determined from the

potential (I),

vith

A4~B2=Cj12=0. a)

A~ _> 0,

Cj~

_> 0.

b)

A~ _> 0, _C12 _< 0. Dashed and continuous lines _are

respectively

lines of second and first-order transitions.

(6)

development

_up to the

eighth~order,

with

A~

_>

0,

_are

represented

in

figure

4

[23].

The OLT

phase

_now _appears _as _a stable state

indeed,

the

eighth~order expansion provides

three different

(eighth~order) invariants, If, I(

and

I)I~,

able to make the discrimination between

the three

expected

low symmetry

phases, ORT,

TLT and OLT.

~~ ~----

CRT,

B~

,,TLT

B,

"iT/

TLT

)~li"

j ^t

i ^'

'

(a) 16)

Fig.

4.-Phase

diagrams

_as deterrnined from the potential

(I) (after

Ref.

[23]).

a)

A~~0,

A

= 4A2

82 C)2

_~ 0, C12 _~ 0.

b)

A2 _~ 0, A _> 0,

Cj~

_< 0. Dashed and ccntinous lines _are

respectively

fines of second and first-order transitions.

2.3 THE B-Z-K- MODEL. This model consists in _a mean-field treatment of the

ordering

process of the MA _groups in MAMnCI and MACdCI in _a

rigid

network of octahedra

[6, 7j.

According

to the structural data

[3-5],

these _groups _are

supposed

to be

statistically

distributed between four

energetically equivalent

orientations directed

along [l10], [i10], [l10]

and

[T10]

ⁱⁿ the THT unit~cell _(« orthorhombic _»

configuration [3, 4]).

Let _us call

#;(I=

I to

4)

the

mean

occupation probabilities

of the MA group in each of its

possible

orientations

then,

in the THT

phase

_one has

ii

=

i~

=

i~

=

i~

=

~. ₍₂₎

This model leads to the definition of symmetry

breaking

multidimensional

pseudo-spin

coordinates

[6, 7, 24]

:

&1=~(81-83)

2

~ ~ _~~~

~~~ ~~~

&~=~(bj-#~+8~-i~)

with

£#,=1.

In the C4v

Point

_group of the MA

site,

these coordinates

belong

to the

E( et, &~)

^and

82(&3) representations,

and

they

_can generate in the

14/mmm

space-group the

Xi (&j),

(7)

it _(&~)

_and

r( (&~) representations

_; _so

they

are associated with the

primary (l~j,

_1~~) and

secondary (1~~)

^order

parameters, respectively (Tab. I).

It is worth

noting

that _none of the

pseudo-spin

coordinate

(3)

is able to

generate

^the

Z( representation corresponding

to

f (Tab. I).

The _mean field B-Z-K- treatment,

including four-particles

interactions is able to

reproduce

the

phase

_sequence _as observed in MAMnCI and

MACdCI, excepted

that the authors could not get ^rid ^of ^the existence of the OLT

phase appearing

_as _an intermediate state stable between ORT and TLT

(Fig. 4),

even

though

its domain of

stability

could be limited to a very

narrow temperature range,

by using

a convenient set of coefficients

[7].

It is also

important

to notice

that,

in the frame of this

model,

the ordered state

(ground state) expected

&~ = 0 and &~

# 0) or

_to

OLT ^phase

(et

_#

0, &~

# 0, &~ #

0)

; _so,

ansition

should ccur leading ^ither to _the ORT or to

OLT

_phase as anltimate step.

Instead of _this,

the

ordered

MLT phase takes _place at low

temperature ; the transition

to

MLT is driven by an _additional

order

_parameter

with

Xi, ii symmetry _7, ^20] ^and xhibits

some character due to a change

of

^the

MA

^figuration _from

«

rthorhom-

bic » to « _onoclinic _»

[4].

_This^ransition will

not

be considered.

ncoherent eutron experiments _ith AMnCI [25] have been

interpreted

on

the

basis of

the B-Z-K- _model. It turned

parameter)

_are indeed of

major importance

in the ansition mechanism, but that

8~

(secondary

order arameter)

is almost

^gligible since, within

experimental

_errors, one

lways

has

#~

=

#~ =

~ in

the

ORT

_phase. _In

4

&~

The ^attering

performed with

shown that

coupling exists between

the pseudo-spin oordinates and

the

rotary

MCI~ octahedra.

Because

of the omer-shared

octahedra

arrangement, these

rotatory

modes

are

necessarily zone~boundary ^odes of the THT phase ; in particular,

around

axes contained

in thelayer planes

have

e Xi

and

i~ symmetry,

bilinearly _coupled with et and

&~,

respectively. In

contrast, there is

no

_octahedra _rotation

transforming

as

the ne-centre r( presentation,

so

that direct coupling ^tween

octahedra

rotation

_and 8~ is orbidden

by

_ymmetry._Structural data

[3-5],

show that octahedra _behave sentially as rigid odies (the ^r( octahedra distortion is small) and

this

certainly explains

the

_fact

that

8~

is

almost negligible. On _the

other

hand, there

mode in

the THT phase nsforming as Z(,

as

it was already

the

case for the pseudo-spin

coordinates. ^hus, in the

frame of

_the

B-Z-K- model, the secondary

order

^arameter _f

(Tab.

I)

has no physical eaning.

For

the sake _of

completeness,

let

phase

sequence in AFeCI

is

the _TLT

phase.

A _modified _version of

the B-Z-K- model,

ⁱⁿ

which

_the MA orientations lie along the

[100],

T00], [DID] and

[010] ^irections («

mono-

clinic » onfigurations [3, 4])

is

able

to

account

for

an ordered TLT ound state.

pseudo-spin

of

_the

same ^orm as (3) can be

onstructed, but now

&~

belongs to

the

Bi(C~~) _instead _of B~; _it ollows that &~

can

be _associated

with

the

econdary

order parameter

f(Z( ) _which is the _relevant _one _in _TLT (Tab.1).

(8)

3. Discussion.

We intent in this section to reconsider the formulations of the Landau

free-energy

_as

previously

introduced

by

different authors

[10, 13, 14]

ⁱⁿ order to

explain

the

peculiar

behaviour of the c~~ elastic ^constant in the TLT

phase

it will be shown that such _a behaviour

(Fig. I)

_can be

fully

understood with the

help

of _a

eighth-order expansion.

First,

let _us recall that

coupling

terms of the form

e~(1~)- _1~()

_are allowed

by

_symmetry

(improper ferroelastic),

where e~

designates

the e~~

component

^of ^the ^strain tensor with

r( _symmetry

in the THT

phase. Thus, using

_a fourth-order

expansion

of the

free-energy,

a

step

^like ^variation ^of ^the

corresponding

_c~~ elastic _constant is

expected

to _occur at the transition

temperature,

with _no additional

temperature dependence [26],

which is in contrast with the

experimental

results

(Fig. I).

Of _course, _one _may think about the influence of order

parameter

fluctuations

but,

at least in the TLT

phase,

there is _no

significant

difference between the data obtained for cu

by

_means of ultrasonic measurements at 5 MHz

[12]

and the Brillouin

scattering

ones obtained in the GHz range

[13]. Obviously,

additional terms should be introduced in the free _energy

expansion provided

that their

physical

_sense is in accordance

with the basic statements

presented

in section 2.

3.I GoTo _et al.'s FORMULATION.- Goto et al. have observed for the first~time the

« anomalous _» behaviour of cu in the TLT

phase

of MAMnCI and

MACdCI, by

_means of

ultrasonic measurements

[9, 10]. They

have considered the

following expansion

for the

Landau-free-energy [10]

_:

+fl~(-gl~~(1~)-l~j)+ ~coe~-hel~~-ke(1~)-l~j). ₍₄₎

2

In this

expression,

_l~j, _1~~ and

1~~ ^are the _same _as defined in section

2,

_e stands for

e~ and co for

c(~ (the

«bare» c~~ elastic

constant).

As

usual,

the authors have put

a ₌

ao(T- Tj).

The bilinear

coupling

term el~~ is able _to _account for _a

softening

of c~~

provided

that the

1~~ order

parameter

fluctuation _mode _also _softens. _Since _cu _exhibits _a strong

softening

in the TLT

phase

with

increasing temperatures (Fig. I),

_an unusual

temperature dependence

of the d coefficient has been

introduced,

such _as d

=

do(T~ T) [10].

There _are many

problems

with such _a

potential

_:

I)

Because of the unusual temperature

dependence

of the d

coefficient,

the

parent

^THT

phase

cannot be stable above T~, as stated

already by

Ishibashi and Suzuki

[14].

ii)

The

secondary

order parameter _1~~

plays

_a

prominent

role in

(4),

since it is introduced _as

a full order parameter ^with ^its own

(unusual)

temperature

dependence. Now,

_on the basis of

experimental

results

[25],

it has been stressed in section 2 that

1~~(&~)

is almost

negligible.

iii)

The authors assertion

[10] according

to which the introduction

of1~~

is the

only

_way to

explain

the soft-mode with

r( _symmetry

_as observed _on the Raman

spectra

^of the TLT

phase

[20]

is not correct. As shown in table

I,

such _a mode _can

merely

be

assigned

to the

(l~j -1~~)

_component of the

primary

order parameter fluctuation

mode,

_even

though

bilinear

coupling

with _an octahedra rotatory ^mode may occur

(see

Sect.

2).

3.2 YOSHIHARA _et al.'s FORMULATION. Brillouin

scattering experiments

_on MAFeCI

II

3) have

complemented previous

ultrasonic results

[12];

in

particular,

the behaviour of c66 in the TLT

phase

has been

confirmed,

and for the first

time,

data have been obtained in

(9)

the ORT

phase,

_as schematized in

figure

I.

Then,

Yoshihara et al. have

proposed

the

following free~energy expansion [13]

:

Again,

1~ j, 1~~ and

f

as introduced in

(5)

_are the _same _as defined in section 2 _as

usual,

the

authors have

put

_a

(T)

= a

o(T To).

^{This is} a fourth order

expansion

of the free _energy

and,

as established in section

2,

the ORT ~TLT transition is not

possible

if the fourth~order coefficients _are temperature

independent. So,

_a

temperature dependence

of _one of these coefficients is

indirectly

introduced

by taking

a =

ao(T Ti ) [13]. Indeed,

after minimization of G with

respect

to e~ and

f,

the renormalized fourth-order coefficients of the effective _»

potential

write

[13]

:

P'"P~~~~ ⁽⁶⁾

66

~

B~ Cl

~ ~n~

(8)

with

~

" ~

j

o 4 A^~

~~~ ~ Y

Then,

the

stability

conditions for the ORT

phase give

in

particular

_:

p'>0, y'>0 (9)

and those for the TLT

phase

_:

p'>0, -2p'<y'<0. (10)

Thus,

T~ represents ^the ^actual ORT _~ TLT transition temperature at which

y' changes

of

sign

and cu

(TLT)

_goes

exactly

to zero

[13].

The additional temperature

dependence

of c~~ ^on both sides of the THT

w ORT transition

temperature (Fig. I)

_are

assigned

to critical fluctuations

[13].

There _are

again problems

with the

potential (5)

I) According

to

(7), y' changes

of

sign

at T~

(ORT

_w TLT

transition)

but also it

diverges

at

Tj.

This results in

negative susceptibilities

for the order parameter fluctuation modes _as T~

Tj. Indeed,

because of

(lo),

the

potential (5)

is not bound

(the crystal

is

instable)

in _a range of

temperature Tj

_< T

<

T;

where

~2

T,

₌

Tj

+ _< T~

(I1)

2 ao

(p

⁺ y

2

and in the absence of

f~

_terms, the

potential (5)

is also instable below

Tj

because a<0.

ii)

As stated in section

2,

the order

parameter f

_can be _a relevant _one in

MAFeCI,

but has

no

physical

_sence in the _case of MAMnCI _or MACdCI.

Clearly, f

is not a

determining

(10)

parameter,

since the behaviour of the cu elastic constant is

quite

the _same in all three

compounds.

3.3 ISHIBASHI AND SUZUKI'S FORMULATION. In _a critical examination of Goto _et ai.'s

formulation

(4),

Ishibashi and Suzuki

[14]

have stressed the

importance

of sixth-order terms, in order to

explain

the

temperature dependence

_{of cu} ⁱⁿ both the ORT and TLT

phases

the

following thermodynamic potential

^is considered

[14]

_:

+

~U(~/~ ~i)

₊

(~/~ _~i)~ (~/+ ~i) _(l~)

In this

expression,

_qi and q~ stand for _l~j and

1~~

(as

^defined in Sect.

2), respectively;

co stands for

cl

_and

u for e~. For

convenience,

the authors have introduced in

(12) only

_one sixth-order invariant _out of the two allowed _ones. It should be

pointed

out that _an

algebraic

error is contained in this _paper _: the

equation

of _state for the TLT

phase

^should ^read

a +

(P

+

y) q(~T

=

0 instead of

a +

(P

₊

y) q(~T

=

0

[14],

_so that the simulated _curves 2

for the behaviour of

q(~T

and

cu(TLT),

_as well _as the calculated

phase diagram,

_are biased.

Nevertheless,

_as stated in section

2,

_a sixth-order

expansion

is indeed able to

reproduce

the

observed THT~ ORTWTLT sequence, and to introduce _a

temperature dependence

of

cu(ORT )

and

cu(TLT) [14]

_; we shall _come back _on the

implications

of such _a

potential

with the

help

of _a _more

general

formulation

presented just

'l

''ii) ₍₁₄₎

~~

~~~ ~~' ~~~ ~ ~~~ ~ ~~~~~ ~

~~~~~

_~ _~~~

+ Cj6~~ + C)2~l ~2 +

C)3(~l

~3 + ~2

to the symmetry ^of ^the

THT

phase

is

given by (15),

where the

e;'s (I

= I to

6)

_are the

components

of the strain tensor and the

cl's

_are the associated elastic constants ;

(16)

contains the

coupling

terms of lowest

order between strain and order parameter

components.

All coefficients _are

supposed

to be

temperature independent, excepted Aj

which is written under the form

Aj

=

aj(T- To).

As shown in the

preceding sections,

the

secondary

order parameters _1~~ or

fare

not the relevant

(11)

ones in the bahaviour of c~~, ^so that

they

have been omitted in

(13). Also, strictly speaking,

the

primary

order

parameter (l~j,1~~)

consists in linear combinations of

pseudo-spin

_an

octahedra rotation coordinates

(see

Sect.

2.3)

for the sake of transparency, we do not _come into such details

(coupled pseudo-spin-phonon mechanism)

and _we consider the order

parameter as a whole.

Minimizing hi (13)

with

respect

to the

e;'s

leads to an « effective _»

potential

of the form _:

+(A~+Cj~cos4q~)p~+ (A4+B~cos~4q~+Cjj~cos4q~)p~ ₍₁₇₎

where _:

a~=hAj+~kA~

~

hc(~-kc)~

(C/1 + C12) _C~3 ^~

C~

~

k(c)j

₊

_c(~)

2

hc)~

(Cil

+

C)2)

_C~3 ~

Ci

hi

=

~ 2

(~

It should be noticed that

(13)

is reduced to _a form similar to

(12)

when

a~=0

To),

it _comes _:

PTHT "

°

(18)

(l~~l )

~ (1~~2) ~ ~ ~41

(l,

E1°d~

(l~)

(xi~)-i

=

(xii)-

ⁱ

= o

(20)

For the ORT

phase (T< To,

_q~

=

nw/2),

_one obtains

~4l+2(A2+~l~~2~bl)P~RT+3(243+~12)P~RT+4(244+~2+~l12)P~RT~° (~~)

(l~~l

_~ ⁸ (242 + ~

l)

_P_~RT ₊ ~(243 +

~12)

(~4)

(

~e

)-

^I

(

~e

)-1

~

(~5)

12 21

~

hi

c~~(ORT)

₌ _c~~

(26)

242 +

~l

₊ ~(243 +

~12) P~RT

+

~(244

₊

~2

+

~l12) P~RT

(12)

For the TLT

phase (T< _To,

_q~

=

(2

_n ₊ I

) w/4),

~

~~~4

₊

~2 ~l12) P~LTI l~/

mode

+ ~

~12

_P

(LT

~

(~ ~2

~

II2) P~LT

in all

phases

are not still

available).

Nevertheless,

_one _can at least discuss _on the

general

trends

predicted according

to these

equations.

Thus,

in order to make the

junction

with section

3.3,

_we shall first consider _a sixth-order

expansion (A~

=

B~

₌

Cjj~

=

0)

with _a~

= 0 and

A~

₌

Cj~

_{# 0}

[14].

We have

represented

in

figure

5 the order

parameter

p ~, the inverse

susceptibilities (X[-)~

^and the c~~ elastic constant as a function of

At

=

aj(T- To),

for

arbitrary

sets of

parameters

^that ^make

possible

the

THT _w ORT _~ TLT sequence ^to take

place,

with a second order THT

~ ORT transition.

The

corresponding phase diagram

is

represented

in

figure

6.

The ORT

~ TLT transition is related to _a

softening

of the

Y( (ORT)

mode and of the

cu(TLT)

elastic constant, ^but this

softening

is not

complete

at the transition temperature

(first-order transition). Thus,

at least

qualitatively,

the sixth-order

expansion

is

already

able to

reproduce

^the ^behaviour ^of

cu(ORT)

and

c~~(TLT) (Fig. I),

with _a convenient choice of the coefficients

[14].

There _are however fundamental limitations with such _a

potential

_:

I)

In

figure 6,

_we have

represented

the

stability

limits of the ORT and TLT

phases,

I-e- the lines where the

Y( (ORT)

mode

frequency (line I)

and where

cu(TLT) (line 2) extrapolate

to _zero. The first~order ORT _~ TLT transition line lies in between it

corresponds

to

points

where the minimum value of

hi (ORT)

is

equal

to that of

A4 (TLT) (see

relation

(17)).

It

can be demonstrated that lines I and 2 _can _never _come upon,

excepted

at the

triple point

THT-ORT-TLT

(Fig. 6). Thus,

it becomes evident that the exact cancellation of

cu(TLT)

_on

the ORT _~ TLT transition line prevents ^the ^ORT

phase

to appear in the

phase

_sequence

(as

far _as

Bi

is

constant).

ii)

As stated in section

2,

the ordered

ground

state

expected

for MAMnCI and MACdCI _as T

- 0 is the ORT _or OLT

phase obviously,

the sixth~order

potential

is not able to account for this

implication

of the B.Z.K. model.

iii)

The concomitant

softening

of the

Y( (ORT)

mode and of the

c~~(TLT)

elastic constant

(r( represents premonitory

effects for the _occurrence of the OLT

phase (Tab. I),

which _can

never be stable.

Thus,

let _us _now examine the influence of

eighth-order

terms. As _seen from relations

(24)

(13)

, , ,

iau> TLT

'(

CRT THT

jau> TLT (CRT( _THT

,

_i

, '

o x-I

j~~j j~_~j ^'

'

' '

~~ _~+

,

I

C66/C~6

~

' '

-2 -I 0 A~ ^-2 ^-I ⁰ A~

aj

b)

TLT ' THT

jaul i

~ -~

~X2,~i

C66/C16

-2

Cl

Fig. 5.-Temperature dependence

of the order parameter, of the

susceptibilities

and of the cu elastic constant, calculated from relations

(18)

to

(30).

The

following

values of the coefficients have

been chosen _: A~

=

0, _A~

= A~ = Cj2 ₌ bj ₌ 1, A4

= 82 ₌ Cj12

=

0 and (a) Bj

= 0, (b) Bi ₌ 0.25, (cl

Bi

₌ 0.5.

(14)

~~ THT

~~

--~-

0 0.25 0.75 _B~

-0.5

CRT TLT

Fig.

6. -The

phase diagram corresponding

to the

therrnodynamic potential (13)-(17)

with the

following

values for the coefficients _a~

= 0, A~ ₌ A~ =

Cj~

=

bj

= I, A~ = B~ ₌

Cjj~

= 0. Dashed and continuous fines _are

respectively

lines of second and first~order transitions. The dotted lines

correspond

to the limit of

stability

of the ORT

phase (I)

and of the TLT

phase (2).

and

(30),

these _terms _can be

destabilizing

_ones for both the ORT and TLT

phases (and

_so

become

stabilizing

_ones for the OLT

phase) provided

that _:

28~+ Cjj~>0 (31)

2

B~ Cij~

> 0.

(32)

On the other

hand, positive

values of the discriminant A

=

4A~B~ C)~ (A~

_~

0) [23]

prevents

the ORT _~ TLT transition to occur, since OLT will be

always

stable between ORT and TLT

(Fig. 4). Thus, interesting

features _can be observed in the

phase diagram

in the

intermediate _case where A _< 0

(A~

_>

0)

and where relations

(31)

and

(32)

_are fulfilled

(note

that _one

always

has A _< 0 in the _case of _a sixth~order

expansion (B~

=

0

ii.

As _a matter of

fact,

^the lines I and 2 _now _can _come _across,

leading

to the existence of _a multicritical

(triple) point

at the

junction

of two second~order transition fines

(ORT

_w OLT and TLT

~

OLT)

and of _a first~order _one

(ORT

_w

TLT) (Fig. 7). Increasing

values of A make this

triple point

to _move towards the other THT~ORT-TLT

triple point,

and this latter becomes _a

four~phase point

for

positive

values of A

(Fig. 4).

Now,

the behaviour of the cu elastic _constant in both the ORT and TLT

phases

_can be

fully

understood with the

help

of _a

thermodynamic path crossing

the first-order ORT ~TLT transition line very close to the ORT~TLT-OLT

triple point,

I-e- at the limit of

stability

of the OLT

phase (Fig. 7).

In

figure 8,

_we have

represented

the behaviour of

quantities

of interest

(see

relations

(18)

to

(30))

in such _a situation. Marked differences _can be noticed when

compared

to

figure

5 in

particular, cu(TLT)

_goes

exactly

to zero at the

triple point,

whereas the ORT

phase

_may be stable in _a wide

temperature

range, as

expected.

At this

stage,

we

have not tried to go further in the

adjustment

of calculated _curves _{for c~~} to the observed _ones

(Fig. Ii,

for _reasons

already

mentioned. Let _us note however that introduction of the a~ coefficient

resulting

from the

coupling

with the volume strain

(h

_#

0,

k _#

0)

will

just

renormalize

A~,

^but that _a

higher-order coupling

term of the form

e((1~)+ _7~i) (which

has been

neglected

in

(16))

_may be of _some

help

for such

adjustments,

since it leads to an

additional

upward

_or downward

temperature dependence

of

cu(ORT)

and

cu(TLT),

depending

_on the

sign

of the

corresponding coupling

coefficient.

Also,

it should be

pointed

(15)

,

to TLT CRT THT

, ' ,

THT

, ,

,

0 0.75

,

,,

;

CRT

,',J

TLT ~+ f+

2. £

/

(ijJ -

,

, ,

'

-2 -I 0

= 1, 82 ₌ ^C

jj2 ^~ 0.125.

Dashed and continuous lines _are

respectively

lines of second and first-order transitions. The dotted- dashed line is the inferred

thermodynamic path

for the MAMCI systems.

Fig.

8.

-Temperature dependence

of the order parameter, ^of ^the

susceptibilities

and of the c~~ elastic constant, ^calculated from relations

(18)

to

(30).

The

following

values of the coefficients have

been chosen _a2

" 0, A2 ₌ A~ = C12 ₌

bi

= I,

Bj

= 0.25, A4 ₌ 1,

82

= C

i12 ⁼ 0.125.

out that Landau

theory

will _never account for _a

temperature dependence

of

cu(THT) (see

relation

(21))

_as observed

experimentally (Fig. I) thus, dynamical

critical fluctuations have to be introduced

[10, 13, 26]

for _a

complete description

of

experimental

data around the

THT _~ ORT transition.

Finally,

it _can be

predicted

from relations

(30)

and

(32)

that

c~~(TLT)

will cancel

again

at much lower temperatures

(where eighth-order

terms become

important),

_so that the

complete phase diagram

will include _a further transition _to the _reentrant OLT

phase. Depending

_on the value of

Bj

_<

hi (Fig. 7),

the

complete phase

_sequences

predicted by

the

eighth-order

2

expansion

_are either THT _~ ORT _~ TLT _~ OLT or THT _~ ORT _~ OLT _~ TLT _~

OLT,

which is

formally

in agreement ^with- ^the ^B-Z-K- ^model

[7].

4.

Concluding

remarks.

From the above

discussion,

it

clearly

_appears that

experimental

data obtained _on MAMCI systems cannot be accounted for in _a

satisfactory

_way

by

_means of _a fourth~order

expansion

of the Landau

free-energy, including secondary

^order parameters

[10, 13]

an

expansion

_up to the

eighth-order

in the

primary

order parameter ^is necessary, even

though

_a sixth-order

expansion provides already interesting

information _on the behaviour of the _cu elastic constant

[14].

(16)

In

fact,

the Raman

scattering

_spectra of MAMnCI and MACdCI obtained several years ago

[20]

have been

tentatively interpreted

in terms of OLT

phase

_« clusters » present ⁱⁿ both ORT and TLT

phases (short

_range

ordering phenomena).

This tentative

interpretation

finds _some

support

ⁱⁿ ^the

present analysis,

since the ORT _~ TLT transition

probably

takes

place

in the close

vicinity

of _a multicritical ORT~TLT~OLT

triple point.

A

major difficulty

encountered in the

interpretation

of

experimental

data is the _non~

observation of the OLT

phase

in MAMCI

systems,

whereas most of

previous analyses (and

also the present

one)

have stressed the

importance

of

premonitory

effects connected to fictitious ORT

~ OLT _or TLT _~ OLT transitions

[7, 10, 20].

Of _course, many attempts ^have been made to stabilize the OLT

phase.

In

particular, high

_pressure measurements with

MAFeCI have been unsuccessful

[12]

in the _case of MAMnCI and

MACdCI,

_a _new

high

pressure

phase

has been

evidenced,

but the mechanism _as established for the

high

_pressure

phase

transition has

nothing

to do with the _occurrence of the OLT

phase [27, 28].

Now, judging

from the value of

cu(TLT)

at the ORT _~ TLT transition

temperature [9, 13],

one can evaluate

qualitatively

the _neamess of the ORT-TLT~OLT

triple point

_; it tums

out that the smaller is the ionic radius of

M2+,

the closer from the

triple point

would be the ORT _~ TLT transition (r~~2+ _< rM~2+ <

rc~2+). So,

^after such

empirical considerations,

it

might

be

thought

that MAMCI

compounds

with smaller M2+ cations will

give

_some chance to

observe the OLT

phase.

As _a matter of

fact,

MAMCI

compounds

with M

=

Cu2+, Co2+,

Zn2+ have been

investigated.

In

MACUCI,

_a

strong

Jahn-Teller distortion of the

Cucl~

octahedra takes

place,

and the sequence of transitions is

quite

different from that considered here

[29].

On the other

hand,

MAZnCI

(and probably

also Co2+

compounds) belong

to the

A~BX~ family

of

crystals

with

p-K~SO~

structure, I-e- with

tetrahedrally

coordinated

M2+

cations

[30]

the _same holds _true for MACdBr

[3 Ii. Moreover, changing

the MA group

by

_n-

alkylammonium

cations

C~H~~~ jNH( (n

=

2, 3...)

also

changes

the

phase

_sequence

[1, 2].

Thus, apparently,

the THT~ORT WTLT sequence is the

unique

^fact ^of MAMCI

compounds

with M

=

Mn, Fe,

Cd

and, unfortunately,

_we _are still left with the non-existence of the OLT

phase.

Note added in

proof.-

Just _as this paper was

accepted

for

publication,

I learned from _one of the referees that _a similar discussion has been

already developed,

in the _context of the copper oxides

superconductors La~_~Ba~CUO~, by

Axe J.

D.,

Moudden A.

H.,

Hohlwein

D.,

Cox D.

E., Mohanty

K.

M., Moodenbaugh

A. R. and Youwen

Xu, Phys.

Rev. Lett. 62

(1989)

2751.

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