Isotope effect in the organic superconductor βH-(BEDT-TTF)2I3 where BEDT-TTF is bis (ethylenedithiotetrathiafulvalene)

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Isotope effect in the organic superconductor βH-(BEDT-TTF)2I3 where BEDT-TTF is bis

(ethylenedithiotetrathiafulvalene)

P. Auban-Senzier, C. Bourbonnais, D. Jérome, C. Lenoir, P. Batail, E.

Canadell, J. Buisson, S. Lefrant

To cite this version:

P. Auban-Senzier, C. Bourbonnais, D. Jérome, C. Lenoir, P. Batail, et al.. Isotope effect in the

organic superconductor βH-(BEDT-TTF)2I3 where BEDT-TTF is bis (ethylenedithiotetrathiafulva-

lene). Journal de Physique I, EDP Sciences, 1993, 3 (3), pp.871-885. �10.1051/jp1:1993169�. �jpa-

00246764�

J.

Phys.

I France 3 (1993) 871-885 _MARCH 1993, _PAGE 871

Classification

Physics

Abstracts

71.20 74.20 74.70K 78.30J

Isotope effect in the organic superconductor fl~.(BEDT- TTF)~I~ where BEDT.TTF is bis

(ethylenedithiotetrathiafulvalene)

P. Auban-Senzier

(I),

C. Bourbonnais ('>

*),

D. J£rome

(I),

C. Lenoir

(I),

P. Batail

(I),

E. Canadell

(2),

J. P. Buisson

(3)

and S. Lefrant

(3)

(~) Laboratoire de Physique des Solides (*), Universitd Paris-Sud, 91405

Orsay

Cedex, France (2) Laboratoire de Chimie

Thdorique,

Universit£ Paris-Sud, 91405

Orsay,

France

(~) Laboratoire de Physique Cristalline, IMN, Universit6 de Nantes, 44072 Nantes, France

(Received 3

August

1992,

accepted

in

final

form ¹⁴ October 1992)

Rksulm4. Nous

pr6sqntons

une dtude simultande d'effet

isotopique

sur la transition supraconduc- trice _et les spectres ^Raman dans le supraconducteur

organique fl~-(BEDT-TTF)213

(T~ ₌ 8 K).

Pour cela, _{nous avons}

synthdtisd

le

compose

dans

lequel

les _atomes de carbone de la double liaison centrale de la moldcule BEDT-TTF sont substituds par l'isotope ^13C. Les

ddplacements isotopiques

mesurds par

spectroscopie

Raman _sont bien

expliquds

_par la

dynamique

moldculaire

standard.

Cependant,

la

tempdrature critique

est abaissde de 0.2 K dans le mat£riau enrichi _en '3C.

Nous dtudions les

origines

possibles de cet effet qui permet ^d'obtenir un coefficient

isotopique supdrieur

h la valeur BCS. Des calculs de la densitd d'dtats effectuds par la mdthode de HUckel dtendue pour les deux bandes HOMO du

composd

montrent que, dans le cadre d'une th£orie de

couplage

faible, _son

importante

variation h I'£chelle de w~ ne peut

expliquer

l'effet observd.

D'autre part, nous expliquons comment la diffusion

dlectronique

indlastique observde _en rdsistivitd

juste

au-dessus de T~ peut ^{conduire via} un mdcanisme de brisure de

paires,

h une

augmentation significative

du coefficient

isotopique.

Abstract. We have

performed

the simultaneous

investigation

of the

isotope

effect _on the

superconducting

transition and _on the Raman spectra ^{in the}

organic

superconductor

flH-(BEDT-

TTF)~I~ (T~ = 8 K). ^{For this} purpose, we substitute '3C for 12C _on the carbon sites of the central

double bond of BEDT-TTF molecule. The

isotope

shifts measured

by

Raman

experiments

_can be

fairly

well explained by standard molecular

dynamics.

However, the critical temperature ^is lowered by 0.2 K in the '3C enriched material. We analyse the possible sources of this remarkable

downward shift which leads _{to an} isotope coefficient

higher

than the BCS value. The extended- Hiickel calculations of the

density

of _states for the _two HOMO bands of

fl~-(BEDT-TTF)~I~

do show that, within the framework of _a weak

coupling

theory, ^{its sizeable variation} on the scale of

w~ ^{cannot account} for the observed isotope effect. On the other hand, _we discuss how inelastic

electronic

scattering

observed in

resistivity

measurements

just

above T~ can lead

through

_a

pair breaking

mechanism _{to a} sizeable increase of the

isotope

coefficient.

(*) Associd _au CNRS.

(*) Permanent address _: Centre de Recherche en

Physique

du Solide,

Ddpartement

de

Physique,

Universitd de Sherbrooke, Qudbec, Canada JIK-2Rl.

Introduction.

The

finding

of _an

isotope

shift for T~ ⁱⁿ conventional

superconductors

has been _a

major

argument ^{in favor of the role}

played by phonons

in the

theory

for

superconductivity.

T~ varies like M~~ where M stands for the elemental _mass. The value _a «1/2 is

actually

obtained when the electron-electron attraction in the

Cooper pair proceeds

via low energy acoustic

phonons ⁽⁼ Debye energy)

characterized

by

the energy scale wD, where the condition

hw~

«

E~

is fulfilled which is the _case in most s or p band metals. In the usual BCS formulation T~

depends

on the elemental _mass

solely through

the

prefactor

of the relation

T~ w~ exp

[- I/AN(E~)] ^(I)

where _w~ defines _a characteristic energy scale around

E~

in which the attractive

coupling

constant A is _non-zero. Here

N(E~)

is the

density

of states at the Fermi level. Values

a <1/2 _are well known

[I]

to result from the

repulsive

screened Coulomb

pseudo-potential

E~

i

p ^* = p l ₊ MN

(E~)

^In

(2)

°'D

which is _w~

dependent

and _enters in the above BCS

expression (Eq.(I)) using

the

transformation A _~

i

= A p ^* In wide band metals

hw~ «E~,

^then

^I

^does not deviate

from A very much and for not too small T~, ^the conventional range ^{of values} a s1/2 is

obtained.

However, remarkable deviations from the classical BCS formulation _{are met} when the Fermi level is close _{to a van} Hove

singularity (divergence)

of the

density

of _states

[2].

Such _a situation

is encountered in A15

superconductors

and also in _two dimensional half-filled band

superconductors. Thus,

the

explicit

_energy

dependence

of the

density

of states must be taken into account when

solving

the

integral equation

for the gap. The elemental _{mass no}

longer

enters the definition of T~ in _a

straightforward

_manner and the

isotope

effect _on

T~ becomes _{a more} delicate

problem.

This is

likely

_{to occur} for _narrow band

superconductors

like the

(BEDT-TTF)2X

series where

preliminary

self-consistent electronic band _structure

calculations

[3]

made for the X

= I~

compound

do show

important

variations of

N(e)

on a

energy scale smaller that the bandwidth.

As

recently pointed

out

by

Carbotte _et al.

[4],

another _source of strong modification of the

isotope

effect is the existence of _a

pair-breaking

mechanism _as it _{can occur} for

superconductors

with

paramagnetic

centers. This lowers the value of the critical temperature ^{which increases} the

amplitude

of _a. This effect tums out to be still present even when the contribution of the

electron-phonon

interaction _to

pairing

is weak.

Organic superconductors

like

(BEDT-TTF)~I~

are well known to be characterized

by

a

high degree

of

purity however, ruling

_our the presence of

magnetic impurities.

This is

supported,

for

example, by

the rather low

Dingle temperature

and the fine details of the Fermi surface revealed

by magnetotransport experiments [5].

As

noted

by

Lee and Read

[6] however,

inelastic electronic

scattering

which _{acts as a true} life time effect for electrons that _are involved in the

Cooper pair formation,

is also

pair-breaking.

Experimentally,

this mechanism becomes

clearly

manifest when _a strong temperature

dependence

of the

resistivity

is _seen

just

above

T~.

^Such an anomalous temperature

dependence

is

precisely

a common feature of

organic superconductors

and in

particular

for

(BEDT- TTF)~I~,

^and therefore deserves _to be

analysed

in connection with the

isotope

^effect. The

investigation

of the

isotope

shift of T~ can

provide

much

insight

into the role of attractive and

repulsive parts

^{of the} interaction and also on the

dimensionality

of the electron gas.

N° 3 ISOTOPE EFFECT IN

fl~-(BEDT-TTF)213

873

Several

isotope

effect

investigations

have

already

been undertaken in

organic superconduc-

tors.

They

have involved deuterium for

hydrogen

^{and 13C} for 12C substitutions.

However,

_no firm conclusions could be reached _so far.

The substitution of lD _for lH in the

methyl

_groups of

(TMTSF)~Cl04

has led _{to a}

regular isotope

shift

[7],

consistent with the

elementary

^BCS

theory, although

one order of

magnitude larger,

AT~

-=

0.13,

than what _can be foreseen from _a

straightforward application

of the T~

model.

As far as the series of

organic superconductors exhibiting

two-dimensional

conducting properties

are concemed,

namely

those built around the BEDT-TTF molecule, called ET from

now on,

isotope

^{shifts studies} of T~ have been carried _out with

p

and _K

phases

of

(ET)~X

salts.

With deuterium

substitution, p~-(ET)~I~

where

p~

labels the

superconducting phase

obtained

by cooling

the

sample

down to low temperatures (T~ ₌ ^{I. I K without} any pressure

cycling,

the

sign

of the

isotope

effect is

opposite

to the

prediction

of the BCS formulation

[8].

However,

when the

p~ phase

is stabilized _at low

temperatures

under pressure

(P

= 0.5 kbar

the

sign

of the

isotope

effect _agrees with the BCS

prediction [9].

Similarly,

_no firm conclusion could be reached

by

deuterium substitution in the

K-phase

series with anions such _as

Cu(NCS)2 l10], Cu[N(CN)~]Br [((i

and Cu

[N(CN)~]Cl [12].

A recent

study

of the

isotope

^{shift of}

K-(ET)~Cu(NCS)~

_upon substitution of13C for 12C in

the

ethylene

_groups of the ET molecule has shown that T~ is almost unaffected

by

the

isotope

substitution

[13].

The

interpretation

of

isotope

shift

experiments

in

organic superconductors

must be treated with great ^caution as many extrinsic effects may influence the determination of T~.

I)

The

isotope labelling

of the

methyl

_groups located _at the outskirt of the molecule in the

(TM)2X

series may result in _a

significant

volume effect with _a concomitant influence _on T~ since the pressure coefficient of T~ ^{is known} to be very

large

in

Bechgaard

salts.

ii)

_{T~ is very} sensitive to

alloying

and

(or)

disorder. This is _true in

particular

for the

K-phases

with X

=

Cu(NCS)2 l14]

_as well _as in the

p-phase

because of the pressure occurrence of an

incommensurate lattice distortion _at low temperatures.

iii)

The

isotopic

substitution _must be

performed

on those sites where the

charge density

is

largq ^enough.

As _we tend _to believe that all

problems

raised above had not been

properly

solved

simultaneously

in

previous

studies we have decided _to take them into consideration in the present

study

of the

isotope

effect in _an

organic

conductor.

The present ^work reports the

study

of the

isotopic

shift of T~ in the

organic superconductor (ET)21~ fulfilling

three criteria _: the absence of any volume

change resulting

from the

isotopic

substitution,

the

high purity

of the material and the

exchange

of atomic sites which _are known to be active for the electronic

properties

of the

conducting

salt. Furthermore the effect of the

isotopic

substitution has been

probed by

Raman spectroscopy.

Experimental background.

First,

the

study

was carried out on a member of the series

(ET)~X superconductors

_as this

family

of 2-D conductors

provides

the

highest

values for T~ among

organics.

Then,

given (I)

that the

largest

_p~ carbon atom orbital contribution to the HOMO of the ET molecule _are those of the central double bond and

(it)

the former well documented evidence of

a strong

coupling

of the

symmetric

vibrational mode of this central C

=

C bond with the energy of MO levels

[15],

_we chose _to substitute 13C for 12C _at these carbon sites

only, thereby

JOURNAL DE PHYSIQUE I T 3. N'3, MARCH IW3 30

affecting

the

dynamics

of _a chemical bond central to the electronic

properties

of the cation radicals in

p-ET( Ii.

Finally,

the

system p-(ET)~I~

_was chosen since

single crystals

of this material _can be

prepared

with _a

high degree

of

purity. Also,

the

particular cooling procedure (Orsay

_process

[16])

enables the stabilization of the

p~ phase

at low temperatures ^{free from} any incommensur- ate distortion. In this

respect

^the observation of

giant magnetoresistance

oscillations in this

p~ phase [5]

have been

recognized

_{as a} manifestation of the remarkable

purity

which _can be

attained in this

superconductor.

Parallel,

small scale

syntheses

of the standard and

13C-enriched

_{ET molecules}

were

conducted

following

the Larsen-Lenoir

procedure [17]

under

strictly

identical

experimental

conditions. 500 mg

of13CS~

from

Cambridge Isotopes

Inc. _were

engaged

to

yield

370 mg of 13C-ET after two recristallizations in chlorobenzene. The

degree

of

isotopic

enrichment of the neutral molecule is that of the

starting material, typically

^{99 fb.}

Likewise, single-crystals

of

p- (ET)~I~

and

p-13C(ET)~I~

were grown in identical electrochemical cells

by

oxidation at _a

platinum

^wire anode of

180mg

of the

corresponding

neutral donor in loo ml of

I,1,2-

trichloroethane

containing

_{I g of}

BU4NI~

at 5

~Amp

and 20 _± 0.5 °C for

21days.

As _an additional verification _we have checked that lattice parameters ^{and EPR linewidth} are

similar in both the

regular

and the 13C substituted

p-(ET)213

salts and

equal

to the values known in the literature

[18].

Transport experiments

were

performed

_on

single crystals

of size 1.5 _x 0.5 _x 0.05

mm~

using

the standard four _contacts AC

technique (1

₌ ⁵⁰

~A ).

The

p~ phase

was stabilized

by increasing

the helium gas pressure up to 1.5 kbar _at T

= 300 K _;

cooling

the pressure cell under constant pressure down _to

= 70

K, releasing

_pressure to I

atmosphere

and further

cooling

down to 4.2 K with _a

cooling

rate

kept

below 0.2 K/min in the range 15-4.2 K.

The temperature ^{of the} pressure cell _was measured with _a calibrated silicon diode _sensor and the temperature ^{difference between the} top ^{and the bottom of the} pressure vessel monitored

by

a differential

copper-constantan thermocouple

_never exceeded 0.05K below 20K. No

significant

differences between

cooling

and

heating

_runs were observed.

Raman spectroscopy

experiments

were carried _out with _a

microprobe

Raman set-up

using

the excitation CW argon laser radiation A

=

514.5 _nm and

equipped

with _a

microcryostat

for the low temperature conditions. The

degradation

of the

sample by

the laser beam _was

prevented by using

a power as low as

possible (=

5

mW).

Raman

experiments

have been carried _{out on} both ET and 13C-enriched ET molecules in

order to

probe

the

isotope

effects _on the intramolecular a~ vibrations. Most of the Raman

experiments

_were

performed

_on the neutral

compounds

^{since the Raman}

signal

is _more intense in these _cases than in

conducting

salts. On the other hand, due to

charge

transfer

effects, only

_a

small

frequency

difference for the Raman bands is observed in 13C enriched

p-(ET)~I~

and standard

p=(ET)21~

_as illustrated in

figure

I

(note

that the spectrum

(a)

in this

Fig.

I may reflect fortuitous

polarized

observation

conditions).

If _we focus _on the main features of the Raman spectra, ^{recorded under}

unpolarized light,

the

standard ET

sample

exhibits

peaks

at

1495,

1512 and 1555

cm-I,

in excellent agreement

with

previous

results

(Fig. 2a).

The strong

peak

observed at 1512 cm-I is

expected

to be _a combination

mode,

as

suggested

in reference

[19]

_or

altematively

due _to the

antisymmetrical

mode of the C

=

C

ring

stretch

[20].

In '3C enriched

ET,

the main Raman bands _are

peaked

at

1468 cm-' and

1521cm-' (Fig. 2b).

Two additional weak bands _are also observed _at 485 cm-I and 495 cm-'

Superconductivity

in the

p~ phase

_was detected

resistively

on two

samples

_run simul-

taneously

in the pressure cell

(one

'2C and the other '3C

substituted).

Data for two

'2C

_{and two}

'3C samples

_are

displayed

in

figure3a.

The value

N° 3 ISOTOPE EFFECT IN

fl~-(BEDT-TTF)~I~

₈₇₅

a) b)

1200 1800 1200 1800

Raulau shift (cm

.l~

Fig. I. Raman spectra

r~corded

at _room temperature ^with A

~~~ ⁼

514.5 _nm of al standard fl-(ET)~I~

b) '3C enriched fl-(ET)213.

lexc."

514.snm _i«s

C)

S

0v~i~*

~

~

E

~

#

WI g b)

a)

1350 1450 1550 1650

t0

(cm.i) Fig.

2. Raman spectra obtained at T

=

77 K with _A~~~

=

514.5 _nm al

unpolarized

spectrum ^of '2C- ET molecule, b)

unpolarized

_spectrum of '3C enriched ET molecule, cl

polarized

spectrum ^{with incident} and scattered

light

parallel to the main axis of the '3C enriched ET molecule.

i o

o 90

~~~~~~~

)(~'~

_.. . . . .

Ii,~i,i,«o.-~>'~

°

~ ~

* O ~

/

O

O

~ ~

O

O~

O O ,~°

0.70

:~°~

'~

a'

~'~ ~~~

~ ^{0 O%}

~$

^O

~

0-5° ^~

O~

~ ^{e a}

~

^{~ O ~}

e

~~~

~

°

O O O O o

13~

:

~

~

^{~ ~ ° ° O ~~C}

O O

o i~

.

'

~

i~

.

~' ° O O O C

.

" . ,

12

" ~

o oo°

~°'~~.o

7.5 8.o 9.5

TEMPERATURE(K)

al

flH(BEDT-TTF)~l~

j~~~

_~~C

,..OO°~''jli"~

o

~

_o'°°~

^~

~

~~"'~

E _o°

oo.

~ e e"

o te

~ _° ° ^ie

- ^{o .}

~

e e ."

. ,-

> O.02 O

12~

,.'"

~ ,'

b' ° '

~i "

/

bJ

£~ O O

~

o, o

°.°°7,o

a-o a.5 g-o g.5

TEMPERATURE(K)

b)

Fig.

3. al

Superconducting

transition measured

by resistivity

in _{two sets} of

samples

'2C and '3C enriched

flH-(ET)213

measured

simultaneously

in the pressure cell at P

= I bar. Resistances _are normalised to their values _at 9 K and

only cooling

_{runs are} shown for each sample. b) Resistivity _versus temperature ⁱⁿ two

fl~-(ET)21~

samples ^12C and 13C enriched. The calculation of the resistivity for the 12C

compound

takes into _account the

penetration

depth ^{of the} current along the _cross section. Cooling

and warming runs are shown for each sample. ^Insert : the resistivity ^is plotted against ^T~.

N° 3 ISOTOPE EFFECT IN

fl~-(BEDT-TTF)21~

₈₇₇

T~ = 8.0 ± 0.05 K for

'~C _p ~-(ET)~I~

is in very

good

agreement ^with that of the literature

[16].

Superconductivity

of

'~C p~-(ET)~I~

gave T~ =

7.8 _± 0.05

K,

I-e- _a shift of 0.2 K

(±

0.I

K)

below the value for standard

'2C samples,

which leads to

~~~

= -2.5fb

(±1.25fb).

T~

T~ is defined

by

the

temperature corresponding

to the mid-resistive transition.

The accuracy in the evaluation of T~ is limited

by

the different

spreading

and

shape

^of the resistive transitions from

sample

to

sample

and

by

the fact that the _onset of the transition for '2C

crystals

is broader than for 13C

samples.

The data

presented

in

figure

3a

correspond

to the best

samples,

I-e- with the

sharper

transitions. This is

generally

associated with the

absence,

_on

resistivity

_curves, of

jumps

caused

by

microcracks in the

crystal occurring during cooling

_or

pressure

cycling.

The resistive tail observed in _some

samples

at low temperatures ^is

probably

attributable to the existence of _some

macroscopic

defects sometimes

iiduced by

these

microcracks.

However,

_even in these defective

samples (around

five different

crystals

of each

batch),

the _onset

temperature

^{remained similar} to those of

high quality samples:

T~~~~~~~~=8.2±

^{o-I K for '2C}

samples

and

T~~~~~~~~=7.90±0.05K

^{for 13C} substituted

samples.

This

isotope

shift is still consistent with the result obtained from the

midpoint

critical temperatures.

The resistances in

figure

3a _are normalised _to their value _at 9 K because of the difficulties _to evaluate the actual resistivities. This is

essentially

due _to the

anisotropy

of the

resistivity

and

the _occurrence of microcracks. For two

samples,

one 12C and the other 13C substituted which

have

presented

resistance measurements without any

jump

we tried to compare the actual

resistivities. We have calculated the

penetration depth

A from the relation

[21]

A

=

L/2(«J«~)-1'2

where L is the distance between current

injection

contacts in order _to compare it with the thickness _e, of both

samples. Using

the

anisotropy

ratios

«~/«~

=

780 _at

room temperature ^{and around 200 in the}

p~ phase, (between

lo K and loo

K) [22],

_{we get} for the '2C

sample (L

= 1.7 _{mm, e}

= I lo

~m)

A

(300

K

)

= 30 _~m and

A(

lo K

= 60 _~m, and

for the 13C

sample (L

= 2 _{mm, e}

= 20

~m)

: A

(300

K

= 35 _~m and

A(

lo K

= 70 _~m. This

means that the first

sample

with _a thickness

larger

than the

penetration depth

of the _current

cannot present p

(T)

curve free from

anisotropy

effects.

By replacing

the thickness

by

A for the 12C

sample

we obtained the same values for both

samples

: _{« =} 40-50

(Q.cm)~

at room

temperature ^{and P} ₌ ^{I bar and observed the} same behaviours _at low

temperatures

^{in the}

p~ phase,

_as shown in

figure

3b. Above the transition

(between

lo K and 40

K),

the

resistivity

in the

p~ phase

follows _a law of the type p = p~ +

AT~

where p~ = 15

~Q,cm

is the residual

resistivity

and A

= 0.3

~Q.cm/K~ _(see

_{the insert of}

_{Fig. 3b).}

Discussion.

According

to

previous dynamical

calculations

performed by Meneghetti

et al.

[23]

_on '2C ET

compounds,

the 1495 cm-' and 1555 cm-' modes _are

assigned

to C=C

stretching

vibrations

involving

both intemal and extemal C

= C bonds.

Based _on similar

dynamical calculations,

_we have extended this

study

to the '3C enriched

compound.

Since _our main purpose is _to

assign

the different vibrational modes observed

experimentally,

_we have made the

following hypothesis

_:

We have considered _a

planar

^molecule

(symmetry D2h)

^and

neglected

the

hydrogen

atoms.

The geometry parameters have been taken from

p-(ET)~I~ projected

onto a

plane [24].

We have taken force _constants

directly

from refined calculations

performed by

^Bozio et al.

[25]

for the TTF

molecule,

whereas additional _{ones were} introduced for ET

(extemal rings)

with

physically

^{reasonable values. No} additional fit _was needed _to obtain _a

good

agreement ^{with the}

experimental

values. For

instance,

the force _constant relative _to the C-S stretch of the extemal

ring

in the ET molecule has been taken close _to that of the intemal

ring.

Also, the valence force field for the extemal

ring

does not influence the C

= C

stretching

vibrations in _a

significative

way. As _a consequence, the P-E-D-

(Potential Energy Distribution),

which is _a relevant

parameter to express

simply

the contribution _{to a} vibrational mode

coming

from the different force constants, is not

expected

to be

strongly

affected

by

a small

change

of these force field

parameters. ^{In table}

I,

_we have collected the different

experimental

and calculated values for both central and

ring

C

= C

stretching

vibrations

together

with the P-E-D-

values,

determined from _our calculations.

Table 1.

Observed Calculated P-E-D-

(fb)

frequencies frequencies

C

= C C

=

C C-S

adjacent

ring

central C

=

C

'2C 555 551 27 62 9

BEDT-TTF 495 494.5 74 26 4

'3C enriched 521 523 78.5 17.5 2.5

BEDT-TTF 468 462 23 71 lo

From these

calculations,

it appears

clearly

that the vibrational modes observed _at

1555 cm-' and 1495 cm-' in

'2C

ET and 1521cm-' _{and 1468} cm-' in

'3C

_{enriched ET}

are mixed and

coupled stretching

vibrations of both

ring

and central C

= C bonds. In

figure 4,

we have shown the atomic

displacements

for '3C enriched ET.

Also,

from the P-E-D-

determination,

the substitution of the 12C _{central atoms with} 13C _{ones induces}

an inverse

contribution to the _two observed modes

coming

from the two types ^{of C} = C bonds. This

corroborates

experimental

results obtained in 13C enriched ET in

polarized light (see Fig. 2c)

1462 cm~l

1523 cm'l

Fig.

4. Calculated

stretching

modes for the '3C enriched ET molecule. The _arrows indicate the atomic displacements ^{associated to these modes.}

N° 3 ISOTOPE EFFECT IN

flH-(BEDT-TTF)~I~

₈₇₉

in which

only

the 1468 cm-' mode is observed. In

addition,

_{we can} show that such _a

substitution does _not induce any

significant

shift _on the a~ ^{modes associated to C-S bonds.}

Also,

the force constants associated to the C-S bonds

adjacent

to the C

= C central bond contribute very

weakly

to the _two main modes observed

experimentally (Tab. I).

These

superconductivity

and Raman shifts data _are very

suggestive

of _a strong involvement

(at

= 0.2

eV)

of the

high

_energy C

=

C vibration modes in the

pairing

interaction.

The

experimental

data

presented

here have shown that the

isotope

shifts of the C

=

C mode vibrations _can be

fairly

well understood in _terms of standard molecular

dynamics.

However the

observed shift of the Raman modes

~°'=-

l.8fb leads

(within

the canonical BCS

w

formulation, Eq. (I))

to an

isotope

shift for T~ ^{which is about} two times smaller than the observed

experimental

value.

Since the

frequency

of the boson excitation is

typically

of the order of

E~,

the usual BCS

approximation (w~«E~)

breaks down and _vertex corrections

(inapplicability

of

Migdal theorem)

can

strongly modify

the _structure of the

theory.

Taking

into account the uncertainties on the measured T~ and

AT~,

one

gets

^the

following

range a = 0.35 1.05

(a

= 0.7 _± 0.35 for the observed

isotope

effect coefficient. Such _a range of values

justifies

to look at

possible

sources of

significant

increase of _a.

Strictly speaking,

the involvement of intramolecular

phonons

in

superconductivity

for

(ET)~I~

should

not make any difference in the

isotope

effect.

Among

the different _Sources that can

significantly

alter the

prediction

for the

isotope

coefficient _as well _as the structure of the

theory itself,

the rather

high

_energy scale

(w~

= 0.2 eV of the

exchanged

boson

compared

to the

width W=0.5 eV of the half-filled conduction

(antibonding)

band

[26] (see

also

Fig. 5)

certainly

deserves _to be discussed.

High

_energy

phonons

for the

pairing

mechanism will decrease the ratio

E~/w~ thereby affecting

the reduction of the Coulomb

pseudo-potential

p ^*

according

to the well known Morel-Anderson formula

(Eq. (2)).

From the above

single

half-filled band

picture

where the Fermi energy

E~

=

W/2,

the reduction of _p would be

essentially

absent for an intramolecular

phonon

_energy of 0.2 eV. As

pointed

out

by

Varma et al.

[27] however,

_p * would reach much smaller values close to those found in wide band

metals

(p

^* N

(E~

= 0. I

ill,

if _one takes into _account the contribution of several bands which

are known _to be

relatively

close to each other in energy for molecular materials like the

organics [28] (see below).

-7.o

~ -8. 0

f~

w c uJ

-9.o

o-o 5.o lo-o

oos

Fig.

5. Calculated

density

of _states DOS (electrons _per eV per unit cell) for the two HOMO bands of fl- (ET)213 at 4.5 K and 1.5 kbar. The dashed line refers _to the Fermi level.

The range taken

by

the ratio

E~/w~

also

brings

_us to the

problem

of _vertex corrections and the

applicability

of the

Migdal

theorem. In this

respect, by performing

Monte-Carlo

simulations _on the 2D Holstein model which consists of _a two-dimensional square lattice of

tight binding

electrons

coupled

to a

high

_energy Einstein

phonon,

Scalettar _et al.

[29]

have demonstrated

that,

whenever the

nesting properties

of the entire Fermi surface _are

weak,

the

large

_wave vector

density

_wave fluctuations and in tum vertex corrections _are irrelevant _so that

the solution of

Eliashberg equations

which _are based _on the

Migdal

theorem remains _an

excellent

approximation

for the

description

of

superconducting

correlations for this model. The closed Fermi surface extracted from the extended-Hiickel band calculations of

Whangbo

_et al.

[26]

for the

pL-(ET)21~

do

support

^{the absence} of

nesting properties

of the Fermi surface. We

have confirmed these results

by performing

the _same type ^of calculations for the

p-(ET)~I~

structures determined

[30]

at 4.5 K and 1.5 kbar and 6.I K and 4.6 kbar. Another strong support to the weakness of

nesting properties, however,

is

brought by essentially

all

experiments

made _on both

p~

and

p~ phases

of this

compound

which do _not show any

proximity

with _an

antiferromagnetic

_{or a}

charge density

wave

phase

in the

phase diagram

as

well _as any related precursor effects in the normal state

[28].

One _can therefore expect that the ladder

summation, though

less accurate than the full solution of the

Eliashberg equations,

is still _a

physically meaningful starting point

to obtain the w~

dependence

of the critical

temperature

^{in weak}

coupling

and in _tum for _a

semi-quantitative analysis

of the

isotope

effect in

(ET)~I~. Moreover,

in the absence of _vertex corrections and for

sizeable T~

(=

lo K

),

_p ^* should

only

_act to favor _a

slight

reduction of the

isotope

coefficient

a

[I]

so that without _a controlled determination of the ratio

E~/w~ entering

in

(2)

for _a series of

bands,

the effect of _p ^* _{on a} will be

neglected. Adopting

this

point

of

view,

_our

analysis

will then focus _on the evaluation of the critical

temperature according

_to the t-matrix

expression

t(Q, wm)

₌ Al

(i

^{AT- '}

^{it G°(k+ Q,}

^{wn +}

^{wm) G°(-}

^k,

^{n)j (3)}

~ ~~

for the electron-electron

propagation

in the

Cooper

channel.

G°(k+ _Q,

_{w~ +}

_w~)

is the bare electron propagator ^with the fermion Matsubara

frequencies

_w~

=

(2

_{n +} I

)

arT and

Q

and w~ =

2 marT _are the external momentum and

frequency

of the

pair, respectively (h

= kB ₌ ^I

).

In the Holstein

model,

A is the effective electron-electron interaction induced

by

_an

intramolecular

phonon exchange

and it is attractive and unretarded within _an energy shell of the order of _{w~ on} both sides of the Fermi level.

DENSITY o~ STATES EFFECT ON THE ISOTOPE COEFFICIENT. In the usual way, the temperature

at which the normal _state becomes unstable is the _one

leading

to the

simple pole

^of

(3)

when uniform

Q

=

0 and static _w~

=

0 conditions

prevail. Taking G°(k,

w

~ ⁼

[i

_{w~ e}

(k )]~ ',

and after the

frequency

summation, one gets ^{the familiar condition for} T~, ^{that is}

~ E~-wD

I

=

N(e)tanh [(e -E~)/2 T~]/(e -E~). (4)

~

E~+wD

Here N

(e)

is the

density

of states at the energy e. Since _w~

= 0.2 eV is not _a small energy

scale,

N

(e)

is

likely

to vary

appreciably

_over the interval 2 _w~

[3].

In order _{to test} this

point,

we have carried out

tight-binding

band structure calculations _on

(ET)213 using

the structures determined in reference

[30].

An effective one-electron Hamiltonian of the extended-Hiickel type

[31]

_was used. The

off-diagonal

matrix elements of the Hamiltonian _were calculated

according

to the modified

Wolfsberg-Helmholz

^formula [32]. The exponents and parameters used in _our calculations were the _{same as} in _a

previous

article

by Whangbo

et al.

[26].

The

N° 3 ISOTOPE EFFECT IN

fl~-(BEDT-TTF)~I~

88

calculated

density

of states, N

(e ), (in

electrons per eV per unit

cell),

associated with the _two HOMO bands of

p-(ET)213

for the _{structure at} 4.5 K and 1.5 kbar is shown in

figure

5. From the results of

figure 5,

it is clear that the energy

dependence

of N

(e)

_can

play

_a role in the

evaluation of T~. ^In

addition,

one also observes that there is _no gap between the

bonding

and the

antibonding

bands which supports ^the argument

given

above that _more than _one band should be taken into _account for the reduction of the Coulomb

pseudo-potential [27].

^{In the} presence of sizeable

changes

for N

(e),

this leads _{to an extra}

dependence

_on N

(E~

±

w~)

in T~ which can differ

appreciably

from N

(E~).

Such _a difference is well known to affect the value of the

isotope

coefficient

[2].

From the calculated N

(e ),

we

give

in

figure

6 _a numerical

evaluation of T~

given by (4)

_{as a} function of _w~ on a

logarithmic

scale. The results have been

obtained

by taking

for the reduced

coupling

_constant AN

(E~)=0.217,

which

yields

a

T~ of 8 K at w~ = 0.18 eV. The variation is found _to be

essentially

linear and this leads _{to an}

isotope

coefficient _a

=

1/2 d In

T~/d

^In _w

~ ⁼ 0.43, which is smaller than the BCS value 1/2.

This value _can be

easily

understood if _one realizes that _a

frequency

shift _{8 w~} in the

integration

limits of

(4) only

^affects the contribution to the

integral

in the

vicinity

of

E~

± w~. From

(4),

one can then derive the

approximate expression

« =

IN (E~

₊

w~)

+

N(E~ w~)j/N(E~) (5)

at small 8

w~/w~. Using

the results of

figure

6 the value _a

= 0.43 is also found. One therefore concludes that _an

important

increase of

a cannot

originate

from _a

density

of _states effect.

1.io

a=.43 1.00

~'u o-go ho o -4

o-so

Log

_wo

Fig.

6. -Variation of the calculated T~ (Eq. (4)) _{versus w~ on a}

logarithmic

scale. The value of

a = O.43 for the isotope shift is obtained.

PAIR _BREAKING CONTRIBUTION TO ISOTOPE EFFECT. A remarkable feature found for the

organic superconductor p~-(ET)~I~

as well _as for other members of the series is the

strong

temperature

dependence

of

resistivity

above T~

[28, 16] (see

also

Fig. 3b).

This indicates that elastic

impurity scattering

does not

play

_any

significant

role in the transport

properties

above T~ but ^{rather that} inelastic

scattering

is dominant and

responsible

for the temperature

dependent

resistivity.

As

previously

noted

by

Lee and Read

[6]

in the context of

high-T~ superconductors,

this

temperature dependence

introduces _an inelastic life time r~~ that is

sufficiently

short

(rQ

m T~

)

which acts as a

pair-breaking

mechanism for the formation of the

Copper pairs

and thus for the critical temperature ^{itself. In the}

following,

_we do _{not want to} discuss the

possible microscopic origin

of _r;~

(electron-electron interaction, spin fluctuations, etc.)

but _{we are} rather interested in how it _can induce _a

significant change

in the

isotope

coefficient if _one

assumes its existence _on

experimental grounds. Actually,

^it tums out that the present

problem

is

quite

similar _to another _one where the

pair-breaking

^{is induced}

by

electronic

scattering

_on

paramagnetic impurities

which have been shown

theoretically

to be at the

origin

of _a dramatic

change

in the

amplitude

of the

isotope

coefficient

[4]. Indeed,

the presence of _a finite r;~ will «fuzz _{out »} electronic energy

thereby cutting

off the

logarithmic singularity

in

equation (3).

This life time effect _can be

incorporated

in the

equation

for T~

by writing

~

=

Re

i~

^{~~ ~~ tanh}

^{$ (6)}

N

(EF)

A

~~ ⁸ + I r 2

Tc

where r

=

rQ

~.

Subtracting

_a similar

expression

in the limit r

~ 0 _on both sides of the above

equation

and

expressing

N

(E~)

A in _terms of the critical temperature T~ ^{for r} _~

0,

_{one gets}

in

(T~/Tc)

" P

li/2

₊

(2 2rTc T,n)~

~l P

li/2j (7)

where

$r(x)

is the

digamma

function. As

pointed

out

by

^Carbotte et al.

[4],

this kind of reduction of T~ ^{due to}

pair breaking

^{effects will} lead _{to an} increase of the

amplitude

of the

isotope

coefficient. From the definition of _{a, one} indeed

gets

" "

"Oil (2

_{WTC ~>n)~}

#'i'/2

₊

(2

_WTC

T,n)~ ~ii~ ₍₈₎

where ao = 1/2 is the BCS limit for r;~ ~ oJ. From the

resulting

variation of

a shown in

figure

7 _one observes that _{a can} become

extremely large

if

rQ

^becnmes

sufficiently

close to

.5

a

i,o

o.5

0.0

Q-Q 0.2 0.4 0.6 Q-S I-Q

~1H,cw/Ti«

Fig.

7.

Isotope

coefficient _{a versus} the

pair-breaking

ratio

T~~_~/T~~.

N° 3 ISOTOPE EFFECT IN

fl~-(BEDT-TTF)213

₈₈₃

the critical value

rol~~

=

arT~/2

_y

(y

= 1.781.. where T~ =

OK and

a ~ oJ. For

rQ,'~~/rQ

^' = 0.7 _{we see}

~that

_one

easily

reaches the range a = I. For T~ = 15

K,

_one

gets

^for

example,

the reasonable value

rj

^' m 9

K,

which

according

to the

analysis

made in reference

[3]

is consistent with the observed

slope dp/dT

of

resistivity

for

p~-(ET)~I~.

Concluding

remarks.

The observation of

important isotope

effects in _a non-conventional

superconductor

like the

organic

system

(ET)~I~

is of considerable

importance

for the clarification of the mechanisms that _can lead _to the

phenomena

of

organic superconductivity.

It demonstrates for the first time at least for these

quasi-2D

materials that

high

_energy intramolecular vibrational modes _can be

directly

involved in the

pairing

formation.

Furthermore,

the attractive interaction between electrons that results

being totally symmetric,

it would favor the stabilisation of _{an s-wave}

type

of

pairing

with the absence of zeros for the

superconducting

_gap on the Fermi surface. Besides the apparent relevance of these intramolecular modes in

superconductivity

of

p~-(ET)~I~,

the

amplitude

of the

isotope

effect which is

larger

than the BCS

prediction

is unusual. In order to

try

^and understand this anomalous

feature,

_we have

explored

two different _avenues.

First,

_we have evaluated from the extended-HUckel band calculation method the energy

dependence

of the electronic

density

of _states and its sizeable variation _on the scale of the intramolecular

phonon frequency

_w~, from which _we calculated in weak

coupling

the _w~

dependence

of the critical temperature.

Owing

to some asymmetry ^{in N}

(e

at

E~

_{± w}

~ with respect to the Fermi

level,

the influence of the related _{states on} the

isotope

coefficient tends _to compensate ^each

other and

only

a

slight

decrease of

a from the BCS result _was found. In the second stage ^of our

analysis,

_we considered the influence of

pair-breaking

_on the

amplitude

of

a due _to inelastic electron

scattering.

The presence of _a finite

rQ

^' in

p ~-(ET)~I~

and other

superconductors

of the

same series is

supported by

_a

temperature dependent resistivity

^above T~. ^{It is worth}

noting

here that from

tunneling experiments [31]

made _on the similar

compound p-(ET)~AUI~

(T~ = 3.8 K

),

the ratio

A/T~

was found to be four times the BCS value which is consistent with the range

rQ

m T~

[34].

The effect of a finite lifetime for electrons _on the

isotope

shift _tums

out to be

analogous

with that

recently investigated

for

superconductors

with

paramagnetic impurities.

We have shown that for reasonable values of

rQ~ compatible

with

resistivity

measurements, it _can

give

rise to an

isotope

shift of

magnitude comparable

to that observed.

As this work _was

completed,

a

study

of the

superconducting isotope

shift _was

performed

_on

the

K-phase superconductors

enriched with 13C _atoms in the central double of the ET molecule

[35].

Based _{on ac}

susceptibility

determination of the

transition,

_no shift of T~ ^{could be detected}

within _an accuracy of I fb in T~ in

K-(ET)~CufN(CN)~]

Br. The difference of behaviour

between _K and p

phases

is indeed very

intringuing

since the

quality

of the labeled ET

molecules

giving

rise _to similar shifts of the Raman modes _cannot be

argued.

^The

finding

of _no T~

isotopic

shift

(or,

_at least, of _one much smaller than the value which _can be derived from the

measured shifts of the C

=

C vibrations

frequencies)

would

imply

that these modes are not at all

coupled

to the electron energy levels. This is

possible

but _not in

agreement

^{with the}

argumented

discussion in references

[25, 28].

We may

point

out that

although

two-

dimensionality

is _{a common} feature for

K and

p phases

the

respective

band structures are

noticeably

different. In

particular,

it is not understood

why

there exists

only

a 20 fb _{or so} difference between T~ of the _two

phases

whereas the calculated values for

N(E~)

differ

by

about _a factor two

[36].

This

discrepancy

may

actually

hide _more subtle

problems making

_K

phase

not similar _to

p phase superconductors. Finally, during

this

work,

_{an attempt} has been

made _to determine the

isotope

effect in

K-(ET)~Cu(SCN)~ by

transport measurements,

using

the _same 13C-labeled ET molecule. We could not detect any reliable

isotopic

shift of