ttf)~i~ complet...superconductivity. t~ varies like m~~ where m stands for the elemental mass. the...

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 ddplacementsisotopiques 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 isotopiquesupdrieur 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 pairbreaking 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.

872 JOURNAL DE PHYSIQUE I N° 3

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 valuesa

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 longerenters 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 superconductorslike 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 superconductorswith 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 presenceof magnetic impurities. This is supported, for example, by the rather low Dingle temperatureand 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 regularisotope shift [7], consistent with the elementary BCS theory, although one order of magnitude

AT~larger,

-=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 conductingproperties 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 andK

phases of (ET)~X salts.

With deuterium substitution, p~-(ET)~I~ where p~ labels the superconducting phaseobtained 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 isotopesubstitution [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-phaseswith 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 experimentalconditions. 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 usingthe 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 mayreflect 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 agreementwith 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 antisymmetricalmode 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 spectrar~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

b)g

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~

OO

,~°

0.70

:~°~

'~

a' ~'~ ~~~

~ 0O%

~$ O

~ 0-5°~ O~

~ e a

~~

O

~

e

~~~~

°O O O O o

13~

:

~

~ ~ ~ ° ° O~~C

O O

o i~

.

'~

i~

.

~' ° O O OC

.

". ,

12"

~

o oo°

~°'~~.o7.5 8.o 9.5

TEMPERATURE(K)

al

flH(BEDT-TTF)~l~

o

j~~~~~C ,..OO°~''jli"~

~ _o'°°~ ~

~

~~"'~E o° oo.

~ e e"

o te~ ie

° °

- o .

~e e

.". ,-

> O.02 O

12~ ,.'"~ ,'b' °

'

~i "

/bJ£~ O O

~

o, o

°.°°7,oa-o a.5 g-o g.5TEMPERATURE(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 probablyattributable 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 resistivityin 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 stretchingvibrations 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 adjacentring 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 13Cones 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 BCSw

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 Migdaltheorem) 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 ofa.

Strictlyspeaking, 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 theoryitself, 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 singlehalf-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. 0f~wc

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 anyproximity 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-quantitativeanalysis 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 coefficienta

[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 awill 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~-wDI

=

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


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 coefficienta =

1/2 d In T~/d Inw ~ =

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 integrationlimits 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 valuea =

0.43 is also found. One therefore

concludes that an important increase ofa cannot originate from a density of states effect.

1.io

a=.43

1.00

u~'

o-go

hoo

-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 possiblemicroscopic 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 problemis 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 ofa

shown in

figure 7 one observes thata can become extremely large if rQ becnmes sufficiently close to

.5

a

i,o

o.5

0.0Q-Q 0.2 0.4 0.6 Q-S I-Q

~1H,cw/Ti«

Fig. 7. Isotope coefficienta 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 anda ~ 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 typeof 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 ofa

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 notinghere 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 paramagneticimpurities. 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 forK

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 byabout 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, usingthe same 13C-labeled ET molecule. We could not detect any reliable isotopic shift of


T~, but the resistive transition of thisK

phase is not narrow enough (I K or so) and only too few

samples have been tested up to now to be the matter of a separate publication.It would certainly be of great interest to perform an analogous study in the Bechgaard salt

superconductor, (TMTSF)~PF~ since pairing may not have similar origins in one or two

dimensional organic superconductors. It has been claimed that the proximity between a spindensity wave ground state and superconductivity in (TMTSF)2X could suggest the existence of

a non-phonon mediated pairing. Finally, it is tempting to extend these ideas to the pairing in an

other family of molecular superconductors ; the (alkali metals)3C60 conducting salts which

also show Raman active modes in the same energy range. Isotopic shift studies similar to those

reported in this article are now in progress.

Acknowledgments.

One of us (C.B.) would like to thank the Conseil National de Recherches en Sciences et en

G£nie du Canada (C.R.S.N.G.), l'organisme F-C-A-R- du gouvemement du Qu£bec, and

l'Universit£ de Paris-Sud d'orsay for their financial support. Several discussions with

A. M. Tremblay and L. G. Caron are also greatly acknowledged.

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ttf)~i~ complet...superconductivity. t~ varies like m~~ where m stands for the elemental mass. the...

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