Successive phase transitions in the organic conductor (TMTSF)2PF2O2

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Successive phase transitions in the organic conductor (TMTSF)2PF2O2

S. Ravy, Jean Pouget, R. Moret, F. Wudl

To cite this version:

S. Ravy, Jean Pouget, R. Moret, F. Wudl. Successive phase transitions in the organic con- ductor (TMTSF)2PF2O2. Journal de Physique I, EDP Sciences, 1991, 1 (5), pp.703-720.

�10.1051/jp1:1991164�. �jpa-00246365�

J.

Phys.

I1

(1991)

703-720 _MAT 1991, PAGE 703

classification

Physics

Abstracts

61.50K 74.70K 64.70

Successive phase transitions in the organic conductor

(TMTSF)~PF~O~

S.

Ravy ('),

J. P.

Pouget ('),

R. Moret

(')

and F. Wudl

(2)

(~) Laboratoire de

Physique

des Solides

(*),

Bit. 510, Universitd Paris-Sud, 91405

Orsay

Cedex, France

f)

Institute for

Po1ynlers

and

Organic

Solids and

Departrnent

of

Physics, University

of

Califomia,

Santa Barbara, CA 93106, U-S-A-

(Received

20 December 1990, accepted 5 February 1991)

Abstract. We present an

X-ray study

of the

organic

conductor

(TMTSF~~PF~O~.

At ambient pressure this

compound undergoes

a metal-insulator transition at about 137 K but, at variance with sHflar materials, _a metallic state is not restored in the whole temperature range under pressure lower than 14.skbar. We show that

(TMTSF~~PF~02 undergoes

two successive structural

phase

transitions at T~,j4 =136.3 K and T~,j~ =

135.3 K. Between 2~jj~ ^and

T~j,

_we

observe _a

phase

characterized

by

the _presence in the diffraction pattem ^of superstructure reflections of reduced _{wave vector} qj~ =

(1/2,

±

JR, 0).

At T~,j~ ^{a first order}

phase

transition suppresses this

phase

and _a superstructure ^with qjj~ =

(1/2,1/2,1/2)

is stabilized _as for _most

(TMTSF~2X

salts built with tetrahedral anions. We have studied the behaviors of the associated

thermodynamical quantities (order

parameters,

susceptibilities)

that _a Landau model with two

biquadratically

coupled order parameters

qualitatively

explains. We discuss the nature and the

competition

of these two

phases.

1. Introducdon.

Since the

discovery

of

organic superconductivity [I]

in radical cation salts based _on the

tetramethyltetraselenafulvalene (TMTSF) molecule,

_numerous studies have been devoted to the so-called

Bechgaard

salts

[2] (TMTSF~~X,

^{where X is} a monovalent anion. In addition to the

superconducting properties,

these materials exhibit _an unusual

variety

of

competitions

between

metallic, antiferromagnetic

_or dielectric

insulating phases.

Salts

containing

_cen-

trosymmetrical

octahedral anions Eke

PF~

have _an

antiferromagnetic ground

state at ambient pressure

[3, 4]

and become

superconducting

^with the restoration of the metallic state under modest pressures

[I].

With the

exception

of

(TMTSF~~CI04,

^which

presents

a

superconduct- ing

state at I bar

[5],

salts

containing non-centrosymmetrical

tetrahedral anions like

Re04,

have _a dielectric

insulating ground

state at ambient pressure

[6].

^This

insulating

state

disappears

under pressure and the

superconductivity

is observed

[7l.

(*)

CNRS URA 2.

Crystals

of

(TMTSF~~X

consist of chains of molecules

forming

_a

quasi-one-dimensional

_»

(quasi- ID)

conductor in the _a direction. The anions X and the TMTSF molecules form sheets in the

(a, b) plane

and these sheets alternate in the _c direction. If

superconductivity

and

magnetism,

which _are present ^{in salts} with

centrosymmetrical

and _some

non-centrosymmetri-

cal

anions,

arise from instabilities _on the TMTSF

stacks,

the dielectric

insulating

state, observed

only

with

non-centrosymmetrical

tetrahedral anions is due to the interaction between the anions and the molecules.

Actually,

the metal-insulator

phase

transition

(arising

at T~~) ^is connected with _a structural

phase

transition

[8].

The superstructure stabilized below T~~ and characterized

by

the reduced _wave vector

qj/~

=

l/2

a* ₊

1/2

b* ₊

1/2

c*

(hereafter

noted

(1/2, 1/2, 1/2)),

consists both in _an orientational

ordering

of the anions

(disordered

at ambient

temperature)

and in _a

displacive

modulation of the

organic

chains

[9, 21]

at

precisely

twice the Fermi _wave vector 2

kr(= 1/2 a*)

of the

quasi-

lD electron gas.

However, although

2

kr charge density

_waves

(CDW)

_are formed _on the

organic

chains below T~~, the close link between the anion

ordering (AO)

and these CDW rules out the existence of _a conventional Peierls

instability,

driven

through

the

electron-phonon coupling by

the

divergence

of the

electron-hole _response function

(although

_a

vanishing 2kr

CDW

instability

is observed in

centrosymmetrical

anion salts like

(TMTSF~~PF~ [10]).

The

qj/~ instability

is most

likely

driven

by

the 2k~ ^CDW response function

[I I] through strong

anion-electron

[12]

^and anion-

cavity

distortion

coupling.

However the qj/~

ordering

does not minimize the Coulomb interactions between

adjacent

tetrahedra

[6, 8],

which would rather favor _an uniform

ordering along

the chains. The balance between the direct Coulomb interactions and the electron-

mediated 2

kr

ones is reversed in

(TMTSF~~Re04

under pressure,

probably

because of _an increase of the former interactions and of _a further decrease of the

2k~

CDW _response function. The

qj/~ ordering

is then

replaced by

the q~ =

(0,1/2,1/2)

_one

[13], leading

to _a metallic and

superconducting

electronic

ground

state

[7].

This

competition

between altemate and uniform

ordering along

the chains has _a drastic influence _on the electronic

properties

of the

Bechgaard

salts with tetrahedral anions. Here _we want to illustrate with

(TMTSF~~PF~O~

a new kind of structural

competition involving

dif§erent interchain

periodicities

at ambient pressure.

2.

Physical properties

of

(TMTSF)~PF~O~.

After the

observation,

from electrical measurements, ^of an onset of _a

superconducting

transition at about 2 K under 6 kbar in

(TMTSF)~FSO~ [14],

the influence of

dipolar

anions

on the

superconducting

state has been

questioned [15].

In that purpose, the

Bechgaard

salt with the

dipolar

anion

PF~O~

_was

synthetized.

(TMTSF~~PF~O~ undergoes

_a metal-insulator

phase

transition at about 137 K at ambient pressure

[16], temperature

which decreases under _pressure. But _{even at} 14.5

kbar,

the

highest

pressure

reached,

the metallic state is not restored in the whole

temperature

range,

contrary

to what is observed at these pressures for other

Bechgaard

salts.

Moreover,

_{a very}

intriguing temperature dependence

of the sound

velocity

_was

reported

at ambient pressure

[17].

Unlike

(TMTSF~~FSO~,

which shows _a sudden increase of the sound

velocity

at the

qij~

^AO

phase transition, (TMTSF~~PF~O~

^exhibits an unusual

dip

of sound

velocity

between 137 K and 139 K. In

addition,

a structural

study performed

at 125K failed to detect the

expected superlattice

reflections at the

qj/~

wave vector

[18].

For these _{reasons we} decided to

perform

_a

new structural

investigation

of

(TMTSF~~PF~O~

at ambient pressure.

The above

quoted

structural

study [18]

shows

that,

at room

temperature, crystals

of

(TMTSF~~PF~O~ belong

to the space group

Pi,

with lattice parameters

quite

close to those of

(TMTSF~~FSO~

and

(TMTSF~~CI04.

The anion

PF~O~,

which has the

geometry

^of a

distorted

tetrahedron,

is

orientationally

disordered at this

temperature.

M 5 SUCCESSIVE PHASE TRANSITIONS IN

(TMTSF)2PF202

705

3.

Experilnental.

Several

crystals,

with _a needle

shape

and _a few _mm

long

_were used for the structural

study.

They

_were from the _same

preparation

_as those used in reference

[17].

The

crystals

_were all found to be

twinned,

_as is _common in

(TMTSF~~X salts,

with two dif§erent types ^{of domains} which share the _{same a} axis and

correspond through

_a 180° rotation about this axis. The relative fraction of the domains

ranged

from

20/80

to

40/60.

This _was deduced

by comparing

the intensities of reflections from the two

types

^{of domains. Most}

crystals

_were found _to have

a

relatively

broad mosiic. The best _{one was} used for the dif§ractometric

study.

This

study

_was

performed

with the

CUK~(1.54181)

radiation obtained after

(002)

reflection of the

X-ray

beam _{on a}

doubly

bent

graphite

monochromator. A

qualitative

_survey

between 10K and 300K _was done

first, using

the fixed-film

fixed-crystal photographic

method. The

quantitative study

of the

phase

transitions _was

performed

_{on a} normal bearn-

lifting-detector type

diffractometer installed _{on a}

lligaku

12-kW

generator.

The

crystal

was

mounted in _an helium-filled container where the

temperature

was controlled within 0.I K.

The

experimental

resolution measured

by

the half-width at half-maximunl

(HWHM)

of several

Bragg reflections,

_was : AQ~~ = 0.018

1~ ', _AQ~

=

0.0141~

' and AQ~~ ₌

0.0141~

4. Results.

4.I PHOTOGRAPHIC _SURVEY. The results of the

photographic

_{Survey are} Summarized in

figure

I.

They clearly

show the existence of _{two very} close

phase

transitions _at

T~jj~(~

¹³⁵

K)

~~~

1°cl/4(~

^'~~

~)

Above T~j/4

(high temperature phase)

broad diffuse

scattering

spots are observed _on the

reciprocal

lattice

planes

of index h

= n +

1/2 (n

is _an

integer).

Between T~j/~ and T~j/4

(intermediate phase) sharp

satellite reflections _{appear at} the reduced _wave vector

qj/4

=

(1/2, ±1/4,0) (the corresponding phase

will be named

I/4

»

thereafter).

Below T~j/~

(low temperature superstructure)

these spots

disappear

and _new

sharp superlattice

reflections _are observed at the reduced _wave vector

qj/~

=

(l/2, 1/2, 1/2) ~phase

named

1/2 »).

The close

vicinity

of

l~j

/~ and T~j_/4 lead _us to

perform

_a

quantitative study

of the

competition

between the two superstructures.

4.2 STUDY OF THE PHASE TRANSITIONS AT T~j_/~ AND T~j/4. ^ThiS

Study

_was

performed

_{on a}

crystal

different from that used in 4.I. The transition

temperatures

were determined with _a better accuracy and found to be :

l~j/4

^=136.3 ± 0.I K and T~j/~ ^{=135.3 ± 0.I K. Above} T~j_/4, q_j/~

quasi-isotropic

fluctuations _were observed until about 160 K and _qj

/4 ^ones until about 200K. Between T~ij4 and T~i/~

qij4-superlattice

reflections _were found with _{an average}

intensity

of 103

counts/s,

100 times less intense than the fundamental

Bragg

reflections.

Reflections with _wave vector 2

qj/4

^{were also observed with a maximum}

intensity

of about 10

counts/s.

The

position

of the

qj/4

superstructure spots was refined and

qj/4

was found to be

(0.5±0.01, 0.25±0.01,

0±

0.02).

Below

l~j/~, qj/~ superlattice

reflections _were observed with _{an average}

intensity

of about

104 counts/s,

10 times less intense than the fundamental

Bragg

reflections.

Figure

2

gives

the

temperature dependence

of the

intensity

of

superlattice

reflections associated with each

superstructure,

as measured _on

cooling.

4.2.I Order parameter ^{behavior. From} the data shown in

figure 2,

the T~j/4 transition appears ^to be continuous. The second order character of this transition is also

supported by

(TnTSF)~ PF~ o~

~~cli4

~~cl12

Fig.

I.

-X-ray

patterns ^from

(TMTSF)2PF202,

above T~j~~ between 7~jj4 ^and 2~jj~ and below 2~jj~. ^The

superlattice

reflections at the reduced _{wave vector} q,j~ ^and qjj~ ^are indicated

by

_arrows. The

chain axis _a is vertical.

the

divergence

of the

pretransitional

fluctuations _:

susceptibility

and correlation

lengths (see

Sect.

4.2.2). l~j/4=136.3±0.I

K is defined

by

the

temperature

at which the fluctuations

diverge.

Within

experimental

_errors there is _no

hysteresis.

The T~j/~

phase

^{transition is}

clearly

first order.

Figure

2 shows that the

intensity

of the

qj/~ superlattice

reflections

jumps

at T~jj~. ^In addition there is _an

hysteresis

of about 0.8 K at this

transition,

which masks

nearly completely

the _«

I/4

_»

phase

_upon

heating.

In addition

some of the lattice

parameters (see

Sect.

4.2.3) present

^anomalies at T~jj~.

M 5 SUCCESSIVE PHASE TRANSITIONS IN

(TMTSF)2PF202

707

q~a~2.2$7)

136

a)

qm2,i~as)

q~

=

5000

4.2.2 Pretransitional

fluctuations.

From the

temperature dependence

of the

peak intensity

of the diffuse

scattering I(q)

measured above the

background,

the

generalized susceptibility X(q)

_can be obtained from the classical limit of the fluctuation

dissipation

theorem

[19]

_:

1(q) ~kB TX(q) (1)

In

practice

_x

(q,j4)

^{has been obtained from I}

(G

_{+ q}

_ij4)

and _x

(qi/~)

^{from I}

^(G

+

q1/~),

^where

G is _a vector of the

reciprocal

lattice. _X

(q)

has

generally

a Lorentzian

q-dependence

around its maximunl at q~ :

x

(q)

= x

(qc)/(I

₊

(q

_q

c) P(q _{qc)) (2)}

The half width at half maximunl of the diffuse

scattering

in the _q direction

gives,

after _a resolution

correction,

the inverse correlation

length f~

in this direction

[19].

We have measured the correlation

lengths

of the

pretransitional

fluctuations of the _«

I/4

_» and _«

1/2

_»

phases along

the

a*, b*,

and c*

reciprocal

axis directions.

4.2.2.1 qjj4 fluctuations. The

temperature dependence

of

x(qjj4)~~

is

given figure

3. It shows _a Curie-Weiss behavior between

T~,/j

^{and 137 K}

and,

at this

temperature,

a sudden

decrease of the

slope.

The

slope

above 137 K

strongly depends

_on ^the

experimental

run. In

particular,

there is _some indication of kinetic ef§ects which have not been studied in detail.

The 137 K

anomaly

in the temperature

dependence

of the

qj,prettansitional

fluctuations is also observed _on the correlation

lengths

measured in the three directions

a*,

b* and c*. It is

clearly

illustrated

by figure

4

showing

the temperature

dependence

of the HWHM of the

qj/4-diffuse scattering along

b*. The correlation

lengths

_were found _to be

isotropic along

the

a*,

b* and c* directions.

qj/4

fluctuations under the form of broad diffuse spots were also observed below T~j/~ ^when the

qjj4 long

_range order is

replaced by

the

qj/~

^one.

4.2.2.2 qjj~ fluctuations.

qj/~

fluctuations _are

clearly

observed above T~j/~ in the

«1/4»

phase

and

they

extend above T~j/4.

Figure

5

gives

the temperature

dependence

of

x(qjj~)~~

^This

^quantity

^{shows also} a well defined Curie-Weiss like behavior above T~j_/4 followed

by

_a

large

decrease of the

slope

in the _«

1/4

_»

phase.

No detectable

anomaly

_can

be noticed at 137 K. The correlation

lengths

_were also measured above T~jj~. ^A

typical

temperature

dependence

is shown

by figure

6 from the measurement

performed along

a*.

This measurement _was done at the end of the

experiment

when the _« I

/4

_»

phase

had almost

4

q=(0.S,2.25,7)

§

#

$

133 T(K)

Fig.

3. Temperature

dependence

of

T/I proportional

to

(X

_ij4)~~, where I is the

peak intensity

in

(0.5,

2.25,

7)

for the q,j~ fluctuations above T~ij4 and below _T~jj~. Note the break _at 137 K in the

slope.

M 5 SUCCESSIVE PHASE TRANSITIONS IN

~TMTSF)~PF~O~

709

q=(fiio25,4)

R=Q0l4k

170 190 TIK)

al

q~l6$Q2$4)

R

~Kl

b)

Fig. 4. a) Temperature

dependence

of the half-width at half-maximum

(HWHM) along

b* for the qjj~ pretransitional fluctuations

(63,

0.25, 4). b)

Magnification

of the region near T~jj~ ^{which shows}

more

clearly

the break at 137 K.

q=(@15,o.5)

0

135 136 137 138 T(K)

Fig.

5.

-Temperature dependence

of T/I,

proportional

to

(Xij2)~ ',

_{where I is the}

_{peak intensity}

_at

(U,

1.5,

0.5),

for the qi/~ fluctuations above T~jj~. Note the

change

of

regime

of fluctuations _near

~lf4.

disappeared.

This could be due to the _numerous thermal

cycling

_or to irradiation defects

[34].

Therefore

figure

6 cannot be used to

analyze

the

competition

between the

qj/~

^and

qj/4

^order parameters.

iso

T(K)

Fig.

6. Temperature

dependence

of the HWHM.

along

a* for the qjj~

pretransitional

fluctuations at

(63,

1.5,

0.5) (upper curve)

and of the inverse correlation length after correction

using

a Gaussian resolution function

(lower

curve).

4.2.3 Lattice parameters. ^The lattice

parameters

^of

(TMTSF~~PF~O~

have been measured

previously

at _room

temperature [18].

At

T~i/~

a

large

variation of the

angular position

of the main

Bragg

reflections incited _us to determine which lattice

parameters

are

mainly

affected

by

this transition. The six triclinic

parameters

a,

b,

_{c, a,}

p,

_{y were} determined

using

_a least- square refinement of the

angular position

of _a set of

Bragg

reflections. Their temperature

dependence

between 90 K and 300 K is

given

in

figure

7. The

angles

_a and

p

show the

strongest

^{anomalies at}

l~ij~.

^{Weaker anomalies} are observed for _y and _c. A

change

of

slope

in

the temperature

dependence

of b can be also detected. Within the accuracy of _our

measurement

(5 fib),

a does not

present

any

anomaly

at T~j/~. ^The

temperature dependence

of the lattice

parameters

^{and the} nature of the anomalies

(with

the

exception

of

p)

_are similar to those

previously reported

for the

qj/~

^{first order transition of}

(TMTSF~~Re04 [20].

The

magnitude

of the anomalies is however

quite

different for most of the lattice parameters.

Within

experimental uncertainties,

_no lattice parameters

anomaly

is observed at the

qi/4 phase

transition.

5. Discussion.

5.I NATURE _OF THE PHASES.

In

analogy

with the other

Bechgaard

salts

(TMTSF~~X

with tetrahedral anions which present the

qj/~

^low

temperature superstructure

^{and for which structural} refinements have been

performed (case

of

Re04 [9, 21], BF4 [22]),

the

«1/2 phase

of

(TMTSF~~PF~O~ certainly

consists in _an alternate orientational

ordering

of the

PF~O~

tetrahedra

along

the three directions _a, b and c as

schematically

shown in

figure

8 in the

(a, b) plane.

Furthermore and like in the latter _cases, the

ordering

of the anions inside the TMTSF lattice cavities is most

likely accompanied by

_a

displacement

of the anions and

by

_a 2

kr

distortion _wave of the

organic

stacks. This _wave distorts also the cavities which however remain all

equivalent by

symmetry.

M 5 SUCCESSIVE PHASE TRANSITIONS IN

(TMTSF)2PF202

711

~~

lt~

*~l7.2

~

~~

loo 150 200 250 300 T(K)

~~ ~

loo 150 200 250 300T(K~

§77

,~

'

~~

loo 150 200 250 300TlK~

13.3 ~

loo 150 200 250 300 T(K)

.S~ ₇₁

~13.2 ~/~

loo 150 200 250 300 TIK) ₇₀

"~

r690 ~~

loo 150 2o0 250 300T(K~

~~~ loo 150 200 250 300TlK~

Fig. 7.-Dependence

_upon temperature ^{of the tridinic lattice} parameters and volume of

~TMTSF)2PF202

between 90 K and 300 K.

l~lj j,lj

Fig.

8. Models for the

(1/2,

1/2) and

(1/2, 1/4)

orientational orders of the

PF202

anions in the

(a, b) plane.

The

symbols

₊ and

symbolize

the values of the

Ising-like

order parameter.

We have tried to

perform

_a structural refinement of the

I/4

»

phase

of

(TMTSF~~PF~O~

from the collection _at 136 K of the

intensity

of 200

qi/4 superlattice

reflections. We have

only

succeeded to show

(although

with _{a poor}

reliability

factor of 25.5

fib),

^that the TMTSF stacks

are

distorted,

with _an

amplitude

of distortion and _a

polarization

sinfilar to those found in the structural refinement of

(TMTSF~~Re04 [9]. Unfortunately, introducing

the

ordering

of the

PF~O~

anions _as well _as their

displacements

did not

improve

the fit. This _can be

explained by

the _poor

quality

of _our data

(a

variation of

temperature

^of 0.I K _near 136 K

changes

the

superlattice intensity by

10 fib) ^{and the}

relatively

small

scattering

factor of

PF~O~ (9

times less electrons than in _a TMTSF

molecule).

However the

cooling

rate

dependence

^of the behavior of

x(qi/4)

_see

(Sect. 4.2.2), _suggests by analogy

with kinetic ef§ects

previously

observed in

(TMTSF~~CI04 [23, 24],

that the anions _are also involved in the nature of the _«

I/4»

superstructure.

Figure

8 shows _a

possible

anion

ordering pattern compatible

with the

doubling

of the lattice

periodicity along

_a, its

quadrupling along

b and the

quasi-absence

of 2 q,_/4

superlattice

reflections.

Within the above models for the nature of the _«

I/4

_» and

«1/2

_»

phases,

the associated transitions involve _an

ordering

of the anions and

displacements

of the anions and of the molecules. A

description

of such

phase

transitions would then in

principle

take into account the three associated order

parameters Ising-like

variable for the orientational

degrees

of

freedom and

amplitude

of modulation for the _two

types

^of

displacements.

In the

following,

_we

shall _assume

that,

for each ordered state, two of these order

parameters

^{have been eliminated} and _we shall describe each

phase by

a

Single

order parameter. ^We

present

_now a

simple

phenomenological

model of this _sequence of

phase

transitions.

5.2 PHENOMENOLOGICAL MODEL. in this model _we Shall

neglect

the fluctuations and

consider

only

the order

parameter

at the critical _wave vectors

q,/~

^and

qj/4.

the order

parameter

1~~~~~

has _one

component

^that we shall _name l~jj~

the order parameter

1~~~~~

has two

components (the

_wave vectors

qj/4

^and

q,/4

^{are not}

equivalent)

and _can be

put

^{under the form}

l~j/4exp(I@).

The free energy can be

expanded

as a function of these two order

parameters

^{in the}

vicinity

of T~j/~ ^and T~j/4. ^This

expansion

is

justified by

the small difference

(~

l

K)

between the two critical

temperatures. Up

to fourth

order,

_{we can} write _:

l~16'l

1/2' 'l _l/4 eXp

(

I

), T)

= at_{/2 'l} ^/2 +

(h1/~~)

'l

'/2 ~)

+

~l/4'l

_~/4 ⁺

(h(/4/~)

_'l

_'/4

+

(h()4/~)

'l

'/4C°S (~

^°

) ~) (3)

+ ~ 'l /2

'l ~/4 C)

The terms in

(3a)

_are

straightforward

^for _a one dimensional order parameter. ^In

(3b)

_we

have included the

b()4 umklapp

term

(1~(~~~+1~~~~~~ ⁼

21~)/4cos (4 @))

because

4qj/4

is _a

reciprocal

lattice _wave vector. The lowest order

coupling

term between the two order parameters ^is

biquadratic

and

given by (3c).

For _a triclinic Bravais lattice the

only

_symmetry

to consider in the free energy

expansion

is the translational invariance.

Choosing b()4~0,

a first Ininimization _with

respect

to

gives @=nor/4 (with

n

=1,3,5,7), corresponding

to the four

possible phases

between the lattice and the

qj/4

modulation. With these conventions and

defining bjj4=b(/4-b()4

the free _energy

becomes

l~l'l1/2,

_{'l l/4,}

l~)

" al_{/2 'l ~/2} +

(bl/~~)

_{'l ~/2} +

~I/4

_{'l ~/4} ⁺

(b1/4/~)

'l

'/4

⁺ ~ 'l /2

'l ~/4 As usual for such _a Landau

expansion,

_we shall _assume that

bjj~ bj/4

^and A _are

slowly varying

functions of the

temperature

^{and that}

hi

/~ ^~

0, hi

_{/4 ~} 0 and

~ ~ ⁼

(bj/~ hi /4)'/~,

which insures the existence of aIninimwn of the free energy. We define also two temperatures

Tjj~

^and

Tj/4

^{from the} temperature

dependence

of aj/~ ^and aj/4.

"1/2(l~ l~l/2)

" al/2 ^~~d "

I/4(l~~

_{I~I/4) ~}

_~I/4,

that _we shall relate later to T~,/~ and T~j/4. With these

conventions,

_our free energy is

exactly

of the _same form _as that

already

studied

by Imry [25].

We shall

briefly

recall

Imry's

results. In addition to the trivial l~j/~ =l~j/4 = 0

(O phase) Jfigh temperature

^{Ininimum of}

f

three distinct solutions _can be obtained _:

phase 1/2

_{: 1~}

jj~ # 0

, 1~ jj4 ⁼ 0

corresponding

to _a

1~

j/~-ordering

,

Ft 5 SUCCESSIVE PHASE TRANSITIONS IN

(TMTSF)~PF202

713

phase 1/4

_:

1~ jj~ ⁼ 0

, 1~ j/4 # 0

corresponding

to a « 1~

j/4-ordering

»

,

phase 1/2

₊

1/4

_{: 1~}

j/~ # 0

, 1~ j/4 # 0

corresponding

to _{a «} nfixed

1~ j/~-1~-j_/4

ordering

_».

The domains of

stability

of these

phases

_are determined

by

three

parameters

:

The ratio of the temperature t ₌

Tj/~l~/4,

^{which determines the first}

^ordering

^to occur

when the temperature ^is lowered.

The

strength

of the reduced

coupling

_p

= A

IA

~,

which controls the existence of _a Inixed 'l_l

/2~'l

_1/4

°~~~£lD~.

The ratio of the free _energy of the

separate 1/2

and

I/4 phases

at T

= 0 _:

f

~

l~l'l1/2,

_{'l I/4 ~}

°,

l~

" °

)/l~l'l1/2

_~ ^°

,

'l l/4, l~

"

°

,

"

(~ ~~/~~ hl/2)/(~

_~~/4/~

him)

_~

(" I/~"l/4)~ t~(bl/4/hl/2)

,

which characterizes the most stable low

temperature ordering.

Figure

9a shows the domains of existence of the various solutions in the _p T

plane.

We shall restrict _our discussion to the _case

f

_~ l and t _~ l. The succession of

phase

transitions

encountered in

(TMTSF)~PF~O~

is well accounted for

by

the _case of «strong

repulsive

coupling

_»

(I.e.

_p

~

l)

_as shown

by

the vertical solid line. In that _case, the values of the order parameters

(calculated by minimizing I~

and of the associated

susceptibilities (defined by

x[/

=

(a~Fla1~)/~)

^and

xjj

=

(a~Fla1~)/4))

^{in the different}

phases

_are

given

in table I.

Table 1.

~ ⁱ

phase

_'l

i/2 X ~/~ '~ '~ ~~

° ° ~

~l/2

^{° ~ ~} _l/4

1/4

0 2

~l/2

^{~ ~}

~l/4/bl/4 (~

~l/4/bl/4)~~~ ^~

~l/4

~/~ (~ ~l/~bl/2)~~~

^~ ~ l/2 ° 2

~l/4

^{2 ~}

~l/~blf2

GTE

1/~+i/,

@

2 0

Fig. 9.

a) (p,

T)

phase diagram.

The vertical solid line

corresponds

to the _case of

~TMTSF)~PF~O~.

b) Temperature

variation of the separate ^free

energies

in the case of

~TMTSF)~PF~O~.

In the O

phase,

the two order parameters 1~,/~ and

1~1/4 ^are

equal

to zero and the associated inverse

susceptibilities

tend to vanish

linearly

at

Tj/~

^and

Tj/4 respectively.

The first

phase

to be

stabilized at

l~jj4= _Tjj4

is the

I/4

one

(t~l condition).

In that

phase l~jj~=0

and

l~j/4

grows as

(Tj/4- Tl'/~

when T decreases. Below

T,/4,X[/

varies _as

(T- T(4),

where

T(4

_~

_Tj/4

is the lowest linfit of

stability

of the _«

I/4

_»

phase.

On further

cooling,

because the mixed

1~ j/~ ^1~j/4

phase

cannot exist

(the repulsive coupling

is _too strong, p ~

l)

and since the

phase 1/2

» is _more stable at lower

temperature ~f

~ l

),

_a first order

phase

transition takes

place

at

T~

^towards a

separate

_{1~ j}

/~

ordering. T~

^{is the}

temperature

of

equality

of the

separate

free

energies,

which it is assumed to _occur

only

_once in the whole temperature _range

(see Fig. 9b).

At

T~,

_l~j/~

jumps

to _{a non zero} value and increases _as

(Tj/~- T)~/~

when T

decreases,

_l~j/4 goes

abruptly

to _zero and fluctuates

again. xjj

tends to vanish at

T(~ (upper

limit of

stability

of the

«1/2» phase)

with _a

negative slope.

The

T(4, T(~

^and

T~ temperatures

can be derived

by

the

following

formulas _:

$~^~

t~(1~~4 l~l/4)~

~

f(1°(4

_I~I/2)~

$~~f(l~(2 _l~l/2)~

"

t~(l~(2 l~(4)~

t~(TE _l~l/4)~

"

f(I~E

_I~I/2)~

and

tl~/4

~ I~I/2

Figure

10 summarizes the variations

of1~)j~,

_1~)j4,

xj/, xj/ together

with the

experimental

temperature

dependences.

For _reasons which will appear in the

following,

_we have not tried to fit _our data with the results of this crude

model,

but _a

good

_agreement is obtained between

calculated and

experimental quantities especially

from the behavior of the

susceptibilities (from

the

experimental

values of

Tjj4=136.3K, Tj/~=135.9K, T(~=135.6K

and

T(4

₌ 135

K,

_{we can} obtain the values

T~

₌ ^135.4

K,

_t

=

0.997, f

= 3.365 and _p

=

1.268,

used in

Fig. 9).

However there _are several features which _cannot be accounted for

by

such _a

simple

model _:

I)

^{The break observed} at 137 K in the behavior of the

susceptibility

and of the correlations

lengths

of the

qjj4

fluctuations

(Figs.

3 and

4).

This effect could be _more

likely

due to _a crossover between two different

regimes

of fluctuation than to the

crossing

of _a disorder

line,

where

only

the correlation

lengths

_are affected

[26, 27].

It could be

tentatively

ascribed to a

crossover from _a normal short range

Ising

critical behavior to a

dipolar

critical behavior

[31],

the

dipolar coupling occurring

between the

PF202

anions. Nevertheless kinetic ef§ects remain to be understood.

ii)

The

anomaly

in the

temperature dependence of1~j/~

^{and the saturation}

of1~j/4

near

T~j/~

(Fig. 2) require

to go

beyond

the _mean field

description

and to consider the mutual ef§ects of the fluctuations of the order

parameters through

the

coupling

term

(3c).

At the

lowest order its ef§ect is to

change

In the

phase «1/4

_:

~l_/4 ^{l~ ~}l/4 ⁺ ~ ~

3q 'l ~/2

(fi

_~

))

In the

phase «1/2

_:

ajj~

ⁱⁿ a j/~ ⁺ A

li~q (1J j/4(8q )

_'l

i/4(8q ))

In these

expressions 8q

_are the _{wave vector} fluctuations around

qj/~

^and

qj/4 respectively.

The thermal _average

(... )

is

proportional

to the q

dependent

dif§use

intensity given by (I)

and

(2).

The correction is

given by

_an

integration

of _x

(q

+

8q

which has been

performed

for

a

displacive-type transition,

in the mean-field

approximation,

in reference

[28].

Its critical behavior scales with the _energy

density [29].

It does not

diverge

at T~,

temperature

at which

M 5 SUCCESSIVE PHASE TRANSITIONS IN

(TMTSF)2PF202

715

T*m

,,,Tim

¹³⁷ _TjK~

',,'

,

l

135

TE

^{136 137} TjKi

)

orrection has a sign

opposite

substracted

_to

the

_mean field behavior

of1~)/~ and 1~)/4

thexperimental observation ^of _figure 2. _This eduction of satellite intensity can ^be

alternatively

analyzed

in term _of

Debye-Waller

^factor. _Instead of1~

j/4

one

can

troduce

in

the

structure factorof

the «

_I/4»

superlattice^eflections~j/4exp(-Wj/~),_here

the

Debye- Waller

factor :

T/~

~

iul/~) ~

Iaq

^I li1/~(8q ))

takes into

ccount the _lattice

fluctuations

of the

«

1/2 ^{» phase.} The ^bscripts_must

for the tructure factor

of

the «1/2

^»

phase.

iii) The

5.3. RELATIONSHIP _wiTH THE ELECTRONIC AND ELASTIC PROPERTIES. As in the others

Bechgaard salts,

the

qjj~

^anion

ordering correspondsto

an

insulating

electronic

ground

state

[16]. However,

in the _case of

(TMTSF)2PF~O~,

it is _not known whether the metal-insulator transition _occurs at T~j/~ ^or

l~j/4

^{because these two}

temperatures

are too close to each other.

However these two

phase

transitions _can be

clearly ditinguished

in the sound

velocity

measurements of reference

[17]

but with _a shift of about 2 K between these data and _our results.

For _a triclinic

symmetry

^the

coupling

between _an orientational order

parameter

_1~ and _a lattice deformation _«e», is at the lowest

order,

of the form

Ff~

~ ⁼

Ci l~~e. ₍₄₎

Terms of this type must be added to the free energy ^F considered in

(3)

and to an elastic energy of the form _:

F~

=

1/2 ^Ke~ (5)

The Ininimization of

F(~~

~

+

F~

with

respect

to _e

gives

e ₌

(ci/K) ~2 (6)

The lattice deformation

(or

_more

precisely

its additional deformation with

respect

to the normal thermal lattice

contraction)

behaves like the _square of the order parameter

(I.e.

like the

superstructure peak intensity).

This is

qualitatively

the _case below T~j/~ for ^the triclinic

angles

_a and

p (Fig. 7).

This behavior has also been observed for the extra contraction of the

parameter a below the

(0,1/2,0)

anion

ordering

transition of

(TMTSF)~Cl04 [30].

The

linear-quadratic coupling given by (4)

has further consequences

[31]

_:

I)

^It decreases the coefficient b

of1~

^~

by

C

)/2

Kin the free energy when the variable _e has been elimitated. The value of the free energy obtained after all the

nfiniInisations,

a~/2 b,

thus decreases _more

rapidly

^with temperature ^{than in} absence of

coupling.

In the

case of _a

strong

value of

Cj,

this

coupling

_{can even}

change

the

sign

of the

1~

~-coefficient

_which

for _a

negative sign

leads _{to a} IS'order

phase

transition. This effect could be

important

for the

understanding

of the first order nature of the

qj/~-anion ordering phase

transition in

(TMTSF)~Re04

and

~TMTSF)2FSO~.

ii)

It leads to _a

sudden-drop

of the elastic constant _« K _»

by

C)16 ^at the critical temperature of _a second order

phase transition,

which could account for the sudden

softening

of the sound

velocity

observed below T~j/4 ⁱⁿ

(TMTSF)~PF~02 [17].

The next order elastic

coupling

term :

F(~~

~ ⁼

C~I~ ^~e~ (7)

must be considered for deformation in chains direction when the anion

ordering

transition is

accompanied by

a metal insulator transition because the

opening

of _an electrical _gap

(proportional

to _1~) in the ID electron gas, decreases the electronic

screening

contribution at the in chain elastic constant K'. This term is essential to understand the sound

velocity stif§ening

observed at the

qj/~-transition

^of

(TMTSF)~PF~O~

and of

(TMTSF~~FSO~ [17].

Finally

it is

interesting

to compare the elastic deformations observed at the

qjj~

^anion

ordering

transitions of

(TMTSF)~PF~O~ (Fig.7)

and of

(TMTSF)~Re04[20].

In

(TMTSF)~Re04

the

(a, b) plane

of

Re04

anions is

mostly

distorted

by

the

ordering:

b decreases and _y increases

sizeably

at 180 K. In contrast, for

(TMTSF)~PF~O~

the relative

arrangement ^of two

neighboring

anion

layers

^is

mostly perturbated

at T~jj~. ^{a increases and}

p

M 5 SUCCESSIVE PHASE TRANSITIONS IN

~TMTSF)2PF202

717

decreases

sizeably

and _c decreases too. In this

salt,

the elastic deformations within the

(a, b) plane

of

PF~O~

anions _are much weaker than between

neighboring planes.

5.4 ANALYSIS OF THE RELEVANT STRUCTURAL PARAMETERS. Even

though

it is

beyond

the scope of this section to propose a detailed mechanism for the anion

ordering

transitions in this series of

compounds,

_we would like _{to use} the

(TMTSF)~PF~O~ study

to

point

out _some

characteristic features which could

help clarify

such _a mechanism.

Longitudinal

_component

ofthe

_{wave vector.-}

It _was very soon realized

[8]

^{that the alternate}

ordering

of anions

along

_a does not minimize the direct Coulomb interactions between anions of tetrahedral

symmetry.

A

quantitative

calculation _was

performed [6]

in the _case of

(TMTSF)~Re04 although

the alternate shift of

the

Re04

inside the

organic

cavities _was onfitted. Therefore the alternate

ordering

corresponding

to the

2k~ periodicity

in stack direction is

clearly

stabilized

by

the

gain

of electronic energy due to the

opening

of _an energy gap in the band

spectrum

^{of the lD electron}

gas. The metal insulator transition is however not driven

by

the

divergence

of the

2k~

CDW response function of the conventional Peierls

transition,

_as

suggested by

the

observation of _a

vanishing

2

kr

CDW response function

x~(2 kr)

in

~TMTSF)~PF~

and

ASF~

[10].

The alternate

ordering

_more

likely

involves _a mediated interaction between anions

through

the CDW response function of the

organic

stack

showing only

_a weak

anomaly

at

2kr [10].

Since the anion

ordering

transition _occurs in the temperature range where

Xp(2 k~)

^is

small,

it

requires

_strong

eleptron-anion [12] and/or anion-cavity

deformation

[I I]

couplings.

Transverse component

along

b*

As shown

by

the observation of

incipient

2k~ ^{diffuse sheets in}

(TMTSF)~PF~

and

ASF~

[10],

the electron gas is one-dimensional in the

temperature

range where the anion

ordering

occurs. This rules out mechanisms where the transverse _wave vector

component 1/2

b* _or

I/4

b* is stabilized

by

Ferlni surface

nesting

effects.

Thus,

this component ^{is due} to direct transverse interactions between the anions such _as the Coulomb forces.

Within the

(a, b) layer

of

anions,

these interactions _are difficult to

calculate, especially

for directions different from _a because of the

coupling

with molecular

displacements.

Never-

theless,

their effect _on the

ordering

_can be

phenomenologically analyzed using

_an

Ising

model with

/ coupling

constants. The

ordering

in the

(a, b) layers depends

_on the relative

sign

and

magnitude

of these constants. As a~b and _y

~70(

there _are three

nearest-neighbour

interactions

along a(Jj ),

b

(J~)

and _a b

(J~)

_as defined in

figure

I la.

Moreover,

the _presence of _a

I/4

b*

component

^{for the}

qj/4

wave vector leads to take into account second

neighbours

interactions. As _we shall _see, the

simplest

_{way to account} for a

(1/2, 1/4)

order in the

(a, b) plane

is to consider the interactions

along a-2b,

I-e-

J4.

In the _case of donfinant

antiferroelectric interactions

along

_a

(/ ~0), figure

llb shows the influence of the

J4

and

J~ J~

constants _on the values of the

ordering

_wave vector

[32].

In this

figure (1/2, 1/2)

and

(1/2, 1/4)

_are the first two

components

^{of the critical} wave vectors

qjj~

^and qj/4 ^of

(TMTSF)2PF202.

The

(1/2, 1/4)

order is stabilized

by

the

J4

ferroelectric interaction

when there is

nearly compensation

between

J~

and

J~.

This

J4 coupling

could

originate

from the

long

range

dipole-dipole

interaction.

However,

the accuracy of the atomic coordinates

given

in

[18]

is not sufficient to evaluate the

sign

of this interaction and to test this

hypothesis.

The

good agreement

between _our results and the

phenomenological

model of section 5.2 allow _{us to} discuss _more

physically

the

parameters

t, _p and

f

As

emphasized above,

the transition

temperature

of such

systems

is

mainly

determined

by

~

l

j~

(al J,

_j

^j

_{~i ij}

' 2'2

~_j

ji j

³

I

~>0

16)

Fig.

ll.

a)

Definition of the

coupling

constants

/

in the

(a, b) plane

of anions.

b) Superstructures

stabilized _{as a} function of J~ ^and J~ J~ ^for a dominant antiferroelectric

coupling along a(Jj

_>

0).

the

strength

of

the

anion-mediated

coupling through

the 2

kr

response

function,

this latter _one

beeing

the _same for the _two instabilities. The

hypothesis

that the weak

dipole-dipole

interactions _are relevant in the stabilization of the _«

I/4

»

phase

could

explain

that this _one appears ^at a

slightly higher temperature

than the

«1/2» phase.

This is consistent with t _~ '

(I~I/2

~

l~l/4).

Due to the absence of _a relation between the

qi/~

^and

q,/4

wave vectors, the lowest

coupling

term between the two order parameters ^is

biquadratic.

Moreover it is

impossible

to order the anions with these _{two wave vectors}

simultaneously.

This makes _an

important

distinction with the _case of pure

displacive

modulations. We suggest ^{that this is}

responsible

for the strong

repulsive coupling

between the two order

parameters. Although

the exact calculations have not been

performed

in this

special

_case, it bears _some

analogy

with the

problem

of the

competition

between the

incompatible

uniform and alternated

ordering

of

pseudo-spins

in

p-(BEDTTTF)~I~ [33], ^whire

the exact entropy derivation

yields

_p

=

6.

Finally,

_we have _seen that the

qj/~

^{order parameter is}

strongly coupled

_to the elastic

degrees

of freedom. This

coupling

stabilizes _more

rapidly

the

«1/2» phase by decreasing

the

corresponding

free energy. This could

explain why

the

separate

^free

energies

_cross at _a

temperature

close to T~i/4, ^as shown

figure 9b,

and hence

why

the

phenomenological parameter f

is

greater

than I. These

interpretations

_are consistent with _our

X-ray

data and with

previous

results _on

(TMTSF)~Re04 [20]

and

(TMTSF)~Cl04 [30] showing coupling

between the orientational and elastic

degrees

of freedom.

6. Conclusion.

In

agreement

^with an earlier report of sound

velocity

measurements, _we have observed two structural transitions at T~j/~ 135.3 K and T~j/4 ^{136.3 K} in

(TMTSF)~PF~O~

that _we have

tentatively assigned

to different

competing

orders of the

dipolar

anion

PF~Oj.

The

superstructure

^{stabilized between} _T~ij~ ^and _{T~i/4 has} ^the

qj/4

=

(1/2, 1/4, 0)

_wave vector and that stabilized below T~jj~ ^the

q,/~

=

(l/2,1/2,1/2)

_one. ^These transitions _were also charac-

terized

by

the measurement of the

pretransitional

fluctuations and of the

superstructure intensity.

The

phase diagram

is well accounted for

by

_a Landau model with _a

repulsive

biquadratic coupling

between the two order parameters. ^Additional

improvements

of this model

taking

^into account the fluctuations of the order parameters ^{and the}

coupling

to elastic

degrees

of freedom _were also considered.

M 5 SUCCESSIVE PHASE TRANSITIONS IN

~TMTSF)2PF202

719

In the

light

of the

previous results,

the

physical origin

of these transitions _was also discussed in the anion

ordering hypothesis.

The _wave vector

component 1/2

a*

= 2

kr,

_common to both

transitions,

is the result of _a

coupling

between the

organic

stack and the anionic array. This induces 2

kr

CDW _on the

organic

stack and then _{a gap} in the electronic spectrum. ^It has been

proposed

that the

qj/4 ordering

is the result of

dipolar coupling

between the anions. This

coupling

could be

responsible

for the _crossover observed _at 137

K, just

above T~,/4 in the

temperature dependence

of the

q,/4

fluctuations. The low

temperature

stabilization _energy of the usual qj_/~

ordering

is

proposed

to be enhanced

by

the

coupling

of the AO order parameter with the elastic

degrees

of freedom.

These

interpretations emphasize

the subtle influence

played by

the nature of the anions and their

coupling

with the molecular array on the

physical properties

of this series of

compounds.

Acknowledgments.

We have benefited from useful discussions with T.

Garel,

R. C. Lacoe and M. Vallade.

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