# Isotope effect in the organic superconductor bH-(BEDT-TTF)I3 where BEDT-TTF is (ethylenedithiotetrathiafulvalene)

Denis Jerome, Pascale Auban-Senzier, Claude Bourbonnais, P. Batail, Enric Canadell, S. Lefrant

► To cite this version:

Denis Jerome, Pascale Auban-Senzier, Claude Bourbonnais, P. Batail, Enric Canadell, et al., Isotope effect in the organic superconductor bH-(BEDT-TTF)I3 where BEDT-TTF is (ethylenedithiotetrathiafulvalene). Journal de Physique, 1993. hal-04989894

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Physics Abstracts

71.20 — 74.20 — 74.70K — 78.30J

# Isotope effect in the organic superconductor $\beta_H$-(BEDT-TTF)$_2I_3$ where BEDT-TTF is bis (ethylenedithiotetrathiafulvalene)

P. Auban-Senzier (1), C. Bourbonnais (1, *), D. Jérome (1), C. Lenoir (1), P. Batail (1), E. Canadell (2), J. P. Buisson (3) and S. Lefrant (3)

(1) Laboratoire de Physique des Solides (*), Université Paris-Sud, 91405 Orsay Cedex, France

(2) Laboratoire de Chimie Théorique, Université Paris-Sud, 91405 Orsay, France

(3) Laboratoire de Physique Cristalline, IMN, Université de Nantes, 44072 Nantes, France

*(Received 3 August 1992, accepted in final form 14 October 1992)*

**Résumé.** — Nous présentons une étude simultanée d'effet isotopique sur la transition supraconductrice et les spectres Raman dans le supraconducteur organique $\beta_H$-(BEDT-TTF)$_2I_3$ ($T_c$ = 8 K). Pour cela, nous avons synthétisé le composé dans lequel les atomes de carbone de la double liaison centrale de la molécule BEDT-TTF sont substitués par l'isotope $^{13}$C. Les déplacements isotopiques mesurés par spectroscopie Raman sont bien expliqués par la dynamique moléculaire standard. Cependant, la température critique est abaissée de 0.2 K dans le matériau enrichi en $^{13}$C. Nous étudions les origines possibles de cet effet qui permet d'obtenir un coefficient isotopique supérieur à la valeur BCS. Des calculs de la densité d'états effectués par la méthode de Hückel étendue pour les deux bandes HOMO du composé montrent que, dans le cadre d'une théorie de couplage faible, son importante variation à l'échelle de $\omega_D$ ne peut expliquer l'effet observé. D'autre part, nous expliquons comment la diffusion électronique inélastique observée en résistivité juste au-dessus de $T_c$ peut conduire via un mécanisme de brisure de paires, à 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 $\beta_H$-(BEDT-TTF)$_2I_3$ ($T_c$ = 8 K). For this purpose, we substitute $^{13}$C for $^{12}$C 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 $^{13}$C 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-Hückel calculations of the density of states for the two HOMO bands of $\beta_H$-(BEDT-TTF)$_2I_3$ do show that, within the framework of a weak coupling theory, its sizeable variation on the scale of $\omega_D$ cannot account for the observed isotope effect. On the other hand, we discuss how inelastic electronic scattering observed in resistivity measurements just above $T_c$ can lead through a pair breaking mechanism to a sizeable increase of the isotope coefficient.

<footnote>(*) Associé au CNRS.</footnote>

<footnote>(*) *Permanent address* : Centre de Recherche en Physique du Solide, Département de Physique, Université de Sherbrooke, Québec, Canada J1K-2R1.</footnote>
**Introduction.**

The finding of an isotope shift for $T_c$ in conventional superconductors has been a major argument in favor of the role played by phonons in the theory for superconductivity. $T_c$ varies like $M^{-\alpha}$ where M stands for the elemental mass. The value $\alpha \le 1/2$ 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 $\omega_D$, where the condition $\hbar\omega_D \ll E_F$ is fulfilled which is the case in most s or p band metals. In the usual BCS formulation $T_c$ depends on the elemental mass solely through the prefactor of the relation

$$T_c \sim \omega_D \exp[-1/\lambda N(E_F)] \quad (1)$$

where $\omega_D$ defines a characteristic energy scale around $E_F$ in which the attractive coupling constant $\lambda$ is non-zero. Here $N(E_F)$ is the density of states at the Fermi level. Values $\alpha < 1/2$ are well known [1] to result from the repulsive screened Coulomb pseudo-potential

$$\mu^* = \mu \left( 1 + \mu N(E_F) \ln \frac{E_F}{\omega_D} \right)^{-1} \quad (2)$$

which is $\omega_D$ dependent and enters in the above BCS expression (Eq. (1)) using the transformation $\lambda \to \tilde{\lambda} = \lambda - \mu^*$. In wide band metals $\hbar\omega_D \ll E_F$, then $\tilde{\lambda}$ does not deviate from $\lambda$ very much and for not too small $T_c$, the conventional range of values $\alpha \le 1/2$ 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 $T_c$ in a straightforward manner and the isotope effect on $T_c$ 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 = I₃ compound do show important variations of $N(\epsilon)$ on a energy scale smaller than 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 $\alpha$. This effect turns out to be still present even when the contribution of the electron-phonon interaction to pairing is weak. Organic superconductors like (BEDT-TTF)₂I₃ 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 $T_c$. Such an anomalous temperature dependence is precisely a common feature of organic superconductors and in particular for (BEDT-TTF)₂I₃, and therefore deserves to be analysed in connection with the isotope effect. The investigation of the isotope shift of $T_c$ can provide much insight into the role of attractive and repulsive parts of the interaction and also on the dimensionality of the electron gas.
Several isotope effect investigations have already been undertaken in organic superconductors. They have involved deuterium for hydrogen and ¹³C for ¹²C substitutions. However, no firm conclusions could be reached so far.

The substitution of ¹D for ¹H in the methyl groups of (TMTSF)<sub>2</sub>ClO<sub>4</sub> has led to a regular isotope shift [7], consistent with the elementary BCS theory, although one order of magnitude larger, $\frac{\Delta T_c}{T_c} \approx -0.13$, than what can be foreseen from a straightforward application of the model.

As far as the series of organic superconductors exhibiting two-dimensional conducting properties are concerned, namely those built around the BEDT-TTF molecule, called ET from now on, isotope shifts studies of T<sub>c</sub> have been carried out with β and κ phases of (ET)<sub>2</sub>X salts.

With deuterium substitution, β<sub>L</sub>-(ET)<sub>2</sub>I<sub>3</sub> where β<sub>L</sub> labels the superconducting phase obtained by cooling the sample down to low temperatures (T<sub>c</sub> ≈ 1.1 K) without any pressure cycling, the sign of the isotope effect is opposite to the prediction of the BCS formulation [8]. However, when the β<sub>H</sub> 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 κ-phase series with anions such as Cu(NCS)<sub>2</sub> [10], Cu[N(CN)<sub>2</sub>]Br [11] and Cu[N(CN)<sub>2</sub>]Cl [12].

A recent study of the isotope shift of κ-(ET)<sub>2</sub>Cu(NCS)<sub>2</sub> upon substitution of ¹³C for ¹²C in the ethylene groups of the ET molecule has shown that T<sub>c</sub> 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<sub>c</sub>.

i) The isotope labelling of the methyl groups located at the outskirt of the molecule in the (TM)<sub>2</sub>X series may result in a significant volume effect with a concomitant influence on T<sub>c</sub> since the pressure coefficient of T<sub>c</sub> is known to be very large in Bechgaard salts.

ii) T<sub>c</sub> is very sensitive to alloying and (or) disorder. This is true in particular for the κ-phases with X = Cu(NCS)<sub>2</sub> [14] as well as in the β-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 large 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<sub>c</sub> in the organic superconductor (ET)<sub>2</sub>I<sub>3</sub> 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)<sub>2</sub>X superconductors as this family of 2-D conductors provides the highest values for T<sub>c</sub> among organics.

Then, given (i) that the largest p<sub>n</sub> carbon atom orbital contribution to the HOMO of the ET molecule are those of the central double bond and (ii) 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
affecting the dynamics of a chemical bond central to the electronic properties of the cation radicals in $\beta$-ET$_2^+$ $I_3^-$.

Finally, the system $\beta$-(ET)$_2$I$_3$ 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 $\beta_H$ phase at low temperatures free from any incommensurate distortion. In this respect the observation of giant magnetoresistance oscillations in this $\beta_H$ 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 of ¹³CS₂ 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 %. Likewise, single-crystals of $\beta$-(ET)$_2$I$_3$ and $\beta$-¹³C(ET)$_2$I$_3$ were grown in identical electrochemical cells by oxidation at a platinum wire anode of 180 mg of the corresponding neutral donor in 100 ml of 1,1,2-trichloroethane containing 1 g of Bu₄NI₃ at 5 µAmp and 20 ± 0.5 °C for 21 days.

As an additional verification we have checked that lattice parameters and EPR linewidth are similar in both the regular and the ¹³C substituted $\beta$-(ET)$_2$I$_3$ salts and equal to the values known in the literature [18].

Transport experiments were performed on single crystals of size 1.5 × 0.5 × 0.05 mm³ using the standard four contacts AC technique ($I$ = 50 µA). The $\beta_H$ 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 1 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 0.05 K below 20 K. 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 $\lambda$ = 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 $a_g$ 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 $\beta$-(ET)$_2$I$_3$ and standard $\beta$-(ET)$_2$I$_3$ as illustrated in figure 1 (note that the spectrum (a) in this Fig. 1 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 1 495, 1 512 and 1 555 cm⁻¹, in excellent agreement with previous results (Fig. 2a). The strong peak observed at 1 512 cm⁻¹ is expected to be a combination mode, as suggested in reference [19] or alternatively due to the antisymmetrical mode of the C = C ring stretch [20]. In ¹³C enriched ET, the main Raman bands are peaked at 1 468 cm⁻¹ and 1 521 cm⁻¹ (Fig. 2b). Two additional weak bands are also observed at 1 485 cm⁻¹ and 1 495 cm⁻¹.

Superconductivity in the $\beta_H$ phase was detected resistively on two samples run simultaneously in the pressure cell (one ¹²C and the other ¹³C substituted).

Data for two ¹²C and two ¹³C samples are displayed in figure 3a. The value
<figure><img src="image_2.png" /><figcaption>Fig. 1. — Raman spectra recorded at room temperature with λ<sub>exc</sub> = 514.5 nm of a) standard β-(ET)<sub>2</sub>I<sub>3</sub> ; b) <sup>13</sup>C enriched β-(ET)<sub>2</sub>I<sub>3</sub>.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. — Raman spectra obtained at T = 77 K with λ<sub>exc</sub> = 514.5 nm : a) unpolarized spectrum of <sup>12</sup>C-ET molecule, b) unpolarized spectrum of <sup>13</sup>C enriched ET molecule, c) polarized spectrum with incident and scattered light parallel to the main axis of the <sup>13</sup>C enriched ET molecule.</figcaption></figure>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

Fig. 3. — a) Superconducting transition measured by resistivity in two sets of samples ¹²C and ¹³C enriched $\beta_H$-(ET)₂I₃ measured simultaneously in the pressure cell at P = 1 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 $\beta_H$-(ET)₂I₃ 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 T².
$T_c = 8.0 \pm 0.05$ K for $^{12}$C β<sub>H</sub>-(ET)<sub>2</sub>I<sub>3</sub> is in very good agreement with that of the literature [16]. Superconductivity of $^{13}$C β<sub>H</sub>-(ET)<sub>2</sub>I<sub>3</sub> gave $T_c = 7.8 \pm 0.05$ K, i.e. a shift of 0.2 K ($\pm$ 0.1 K) below the value for standard $^{12}$C samples, which leads to $\frac{\Delta T_c}{T_c} = -2.5\%$ ($\pm$ 1.25%).

$T_c$ is defined by the temperature corresponding to the mid-resistive transition.

The accuracy in the evaluation of $T_c$ 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
$^{12}$C crystals is broader than for $^{13}$C 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 induced 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 :
$T_{c(onset)} = 8.2 \pm 0.1$ K for $^{12}$C samples and $T_{c(onset)} = 7.90 \pm 0.05$ K for $^{13}$C 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 Λ from the relation [21] Λ = L/2 (σ<sub>a</sub>/σ<sub>c</sub>)<sup>-1/2</sup> 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 σ<sub>a</sub>/σ<sub>c</sub> = 780 at room temperature and around 200 in the β<sub>H</sub> phase, (between 10 K and 100 K) [22], we get for the ¹²C sample (L ≈ 1.7 mm, e ≈ 110 µm) : Λ(300 K) ≈ 30 µm and Λ(10 K) ≈ 60 µm, and for the ¹³C sample (L ≈ 2 mm, e ≈ 20 µm) : Λ(300 K) ≈ 35 µm and Λ(10 K) ≈ 70 µm. This means that the first sample with a thickness larger than the penetration depth of the current cannot present ρ(T) curve free from anisotropy effects. By replacing the thickness by Λ for the ¹²C sample we obtained the same values for both samples : σ ≈ 40-50 (Ω.cm)<sup>-1</sup> at room temperature and P = 1 bar and observed the same behaviours at low temperatures in the β<sub>H</sub> phase, as shown in figure 3b. Above the transition (between 10 K and 40 K), the resistivity in the β<sub>H</sub> phase follows a law of the type ρ = ρ<sub>r</sub> + AT<sup>2</sup> where ρ<sub>r</sub> ≈ 15 µΩ.cm is the residual resistivity and A ≈ 0.3 µΩ.cm/K<sup>2</sup> (see the insert of Fig. 3b).

**Discussion.**

According to previous dynamical calculations performed by Meneghetti et al. [23] on ¹²C ET compounds, the 1 495 cm⁻¹ and 1 555 cm⁻¹ modes are assigned to C = C stretching vibrations involving both internal and external C = C bonds.

Based on similar dynamical calculations, we have extended this study to the ¹³C 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 D<sub>2h</sub>) and neglected the hydrogen atoms.
The geometry parameters have been taken from β-(ET)<sub>2</sub>I<sub>3</sub> 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 (external 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 external
ring in the ET molecule has been taken close to that of the internal ring. Also, the valence force field for the external 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><caption>Table I.</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Observed<br>frequencies</th><th rowspan="2">Calculated<br>frequencies</th><th colspan="3">P.E.D. (%)</th></tr><tr><th>C = C<br>ring</th><th>C = C<br>central</th><th>C-S adjacent<br>to C = C central</th></tr></thead><tbody><tr><td><sup>12</sup>C</td><td>1 555</td><td>1 551</td><td>27</td><td>62</td><td>9</td></tr><tr><td>BEDT-TTF</td><td>1 495</td><td>1 494.5</td><td>74</td><td>26</td><td>4</td></tr><tr><td><sup>13</sup>C enriched</td><td>1 521</td><td>1 523</td><td>78.5</td><td>17.5</td><td>2.5</td></tr><tr><td>BEDT-TTF</td><td>1 468</td><td>1 462</td><td>23</td><td>71</td><td>10</td></tr></tbody></table>

From these calculations, it appears clearly that the vibrational modes observed at 1 555 cm⁻¹ and 1 495 cm⁻¹ in ¹²C ET and 1 521 cm⁻¹ and 1 468 cm⁻¹ in ¹³C 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 ¹³C 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)

<figure><img src="image_6.png" /><figcaption>Fig. 4. — Calculated stretching modes for the ¹³C enriched ET molecule. The arrows indicate the atomic displacements associated to these modes.</figcaption></figure>
in which only the 1 468 cm⁻¹ mode is observed. In addition, we can show that such a substitution does not induce any significant shift on the a<sub>g</sub> 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 $\frac{\Delta\omega}{\omega} \approx -1.8\%$ leads (within the canonical BCS formulation, Eq. (1)) to an isotope shift for $T_c$ which is about two times smaller than the observed experimental value.

Since the frequency of the boson excitation is typically of the order of $E_F$, the usual BCS approximation ($\omega_D \ll E_F$) 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 $T_c$ and $\Delta T_c$, one gets the following range $\alpha \approx 0.35$. 1.05 ($\alpha \approx 0.7 \pm 0.35$) for the observed isotope effect coefficient. Such a range of values justifies to look at possible sources of significant increase of $\alpha$. Strictly speaking, the involvement of intramolecular phonons in superconductivity for (ET)<sub>2</sub>I<sub>3</sub> 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 ($\omega_D \approx 0.2$ eV) of the exchanged boson compared to the width $W \approx 0.5$ 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 $E_F/\omega_D$ thereby affecting the reduction of the Coulomb pseudo-potential $\mu^*$ according to the well known Morel-Anderson formula (Eq. (2)). From the above single half-filled band picture where the Fermi energy $E_F \approx W/2$, the reduction of $\mu$ would be essentially absent for an intramolecular phonon energy of 0.2 eV. As pointed out by Varma et al. [27] however, $\mu^*$ would reach much smaller values close to those found in wide band metals ($\mu^* N(E_F) \approx 0.1$) [1], 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).

<figure><img src="image_7.png" /><figcaption>Fig. 5. — Calculated density of states DOS (electrons per eV per unit cell) for the two HOMO bands of β-(ET)<sub>2</sub>I<sub>3</sub> at 4.5 K and 1.5 kbar. The dashed line refers to the Fermi level.</figcaption></figure>
The range taken by the ratio $E_F/\omega_D$ 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 turn 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-Hückel band calculations of Whangbo *et al.* [26] for the $\beta_L-(ET)_2I_3$ 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 $\beta-(ET)_2I_3$ structures determined [30] at 4.5 K and 1.5 kbar and 6.1 K and 4.6 kbar. Another strong support to the weakness of nesting properties, however, is brought by essentially all experiments made on both $\beta_L$ and $\beta_H$ 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 $\omega_D$ dependence of the critical temperature in weak coupling and in turn for a semi-quantitative analysis of the isotope effect in $(ET)_2I_3$. Moreover, in the absence of vertex corrections and for sizeable $T_c$ ($\approx 10$ K), $\mu^*$ should only act to favor a slight reduction of the isotope coefficient $\alpha$ [1] so that without a controlled determination of the ratio $E_F/\omega_D$ entering in (2) for a series of bands, the effect of $\mu^*$ on $\alpha$ 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

$$t(Q, \omega_m) = -\lambda / \left[ 1 - \lambda T^{-1} \sum_k \sum_{\omega_m} G^0(\mathbf{k}^+ \mathbf{Q}, \omega_n + \omega_m) G^0(-\mathbf{k}, -\omega_n) \right] \quad (3)$$

for the electron-electron propagation in the Cooper channel. $G^0(\mathbf{k}^+ \mathbf{Q}, \omega_n + \omega_m)$ is the bare electron propagator with the fermion Matsubara frequencies $\omega_n = (2n+1)\pi T$ and $\mathbf{Q}$ and $\omega_m = 2m\pi T$ are the external momentum and frequency of the pair, respectively ($\hbar = k_B = 1$). In the Holstein model, $\lambda$ 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 $\omega_D$ on both sides of the Fermi level.

**DENSITY OF 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 $\mathbf{Q} = 0$ and static $\omega_m = 0$ conditions prevail. Taking $G^0(\mathbf{k}, \omega_n) = [i\omega_n - \varepsilon(\mathbf{k})]^{-1}$, and after the frequency summation, one gets the familiar condition for $T_c$, that is

$$1 = \frac{\lambda}{2} \int_{E_F + \omega_D}^{E_F - \omega_D} N(\varepsilon) \tanh \left[ \left( \varepsilon - \frac{E_F}{2} \right) T_c \right] \left( \varepsilon - \frac{E_F}{2} \right) \quad (4)$$

Here $N(\varepsilon)$ is the density of states at the energy $\varepsilon$. Since $\omega_D \approx 0.2$ eV is not a small energy scale, $N(\varepsilon)$ is likely to vary appreciably over the interval $2\omega_D$ [3]. In order to test this point, we have carried out tight-binding band structure calculations on $(ET)_2I_3$ using the structures determined in reference [30]. An effective one-electron Hamiltonian of the extended-Hückel 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(ε), (in electrons per eV per unit cell), associated with the two HOMO bands of β-(ET)<sub>2</sub>I<sub>3</sub> 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(ε) can play a role in the evaluation of T<sub>c</sub>. 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(ε), this leads to an extra dependence on N(E<sub>F</sub> ± ω<sub>D</sub>) in T<sub>c</sub> which can differ appreciably from N(E<sub>F</sub>). Such a difference is well known to affect the value of the isotope coefficient [2]. From the calculated N(ε), we give in figure 6 a numerical evaluation of T<sub>c</sub> given by (4) as a function of ω<sub>D</sub> on a logarithmic scale. The results have been obtained by taking for the reduced coupling constant λN(E<sub>F</sub>) ≈ 0.217, which yields a T<sub>c</sub> of 8 K at ω<sub>D</sub> ≈ 0.18 eV. The variation is found to be essentially linear and this leads to an isotope coefficient α = 1/2 |d ln T<sub>c</sub>/d ln ω<sub>D</sub>| ≈ 0.43, which is smaller than the BCS value 1/2. This value can be easily understood if one realizes that a frequency shift δω<sub>D</sub> in the integration limits of (4) only affects the contribution to the integral in the vicinity of E<sub>F</sub> ± ω<sub>D</sub>. From (4), one can then derive the approximate expression

$$ \alpha \approx \frac{1}{4} \frac{[N(E_F + \omega_D) + N(E_F - \omega_D)]}{N(E_F)} \quad (5) $$

at small δω<sub>D</sub>/ω<sub>D</sub>. Using the results of figure 6 the value α ≈ 0.43 is also found. One therefore concludes that an important increase of α cannot originate from a density of states effect.

<figure><img src="image_8.png" /><figcaption>Fig. 6. — Variation of the calculated T<sub>c</sub> (Eq. (4)) versus ω<sub>D</sub> on a logarithmic scale. The value of α ≈ 0.43 for the isotope shift is obtained.</figcaption></figure>

PAIR BREAKING CONTRIBUTION TO ISOTOPE EFFECT. — A remarkable feature found for the organic superconductor β<sub>H</sub>-(ET)<sub>2</sub>I<sub>3</sub> as well as for other members of the series is the strong temperature dependence of resistivity above T<sub>c</sub> [28, 16] (see also Fig. 3b). This indicates that elastic impurity scattering does not play any significant role in the transport properties above T<sub>c</sub> 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-$T_c$ superconductors, this temperature dependence introduces an inelastic life time $\tau_{in}$ that is sufficiently short ($\tau_{in}^{-1} \ge T_c$) 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 $\tau_{in}$ (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 turns 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 $\tau_{in}$ 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 $T_c$ by writing

$$ \frac{2}{N(E_F) \lambda} = \text{Re} \int_{-\omega_D}^{+\omega_D} \frac{d\epsilon}{\epsilon + i\Gamma} \tanh \left( \frac{\epsilon}{2T_c} \right) \quad (6) $$

where $\Gamma = \tau_{in}^{-1}$. Subtracting a similar expression in the limit $\Gamma \to 0$ on both sides of the above equation and expressing $N(E_F) \lambda$ in terms of the critical temperature $T_c^0$ for $\Gamma \to 0$, one gets

$$ \ln (T_c^0/T_c) = \psi [1/2 + (2 \pi T_c \tau_{in})^{-1}] - \psi [1/2] \quad (7) $$

where $\psi(x)$ is the digamma function. As pointed out by Carbotte et al. [4], this kind of reduction of $T_c$ due to pair breaking effects will lead to an increase of the amplitude of the isotope coefficient. From the definition of $\alpha$, one indeed gets

$$ \alpha = \alpha_0 [1 - (2 \pi T_c \tau_{in})^{-1} \psi' [1/2 + (2 \pi T_c \tau_{in})^{-1}]]^{-1} \quad (8) $$

where $\alpha_0 = 1/2$ is the BCS limit for $\tau_{in} \to \infty$. From the resulting variation of $\alpha$ shown in figure 7 one observes that $\alpha$ can become extremely large if $\tau_{in}^{-1}$ becomes sufficiently close to

<figure><img src="image_9.png" /><figcaption>Fig. 7. — Isotope coefficient $\alpha$ versus the pair-breaking ratio $\tau_{in, cr}/\tau_{in}$.</figcaption></figure>
the critical value $\tau_{\text{in, cr}}^{-1} = \pi T_c^0 / 2 \gamma$ ($\gamma = 1.781..$) where $T_c = 0$ K and $\alpha \to \infty$. For $\tau_{\text{in, cr}}^{-1} / \tau_{\text{in}}^{-1} \approx 0.7$ we see that one easily reaches the range $\alpha \approx 1$. For $T_c^0 = 15$ K, one gets for example, the reasonable value $\tau_{\text{in}}^{-1} \approx 9$ K, which according to the analysis made in reference [3] is consistent with the observed slope $d\rho/dT$ of resistivity for $\beta_H-(\text{ET})_2\text{I}_3$.

**Concluding remarks.**

The observation of important isotope effects in a non-conventional superconductor like the organic system (ET)$_2$I$_3$ 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 $\beta_H-(\text{ET})_2\text{I}_3$, 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-Hückel band calculation method the energy dependence of the electronic density of states and its sizeable variation on the scale of the intramolecular phonon frequency $\omega_D$, from which we calculated in weak coupling the $\omega_D$ dependence of the critical temperature. Owing to some asymmetry in $N(\varepsilon)$ at $E_F \pm \omega_D$ 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 $\alpha$ from the BCS result was found. In the second stage of our analysis, we considered the influence of pair-breaking on the amplitude of $\alpha$ due to inelastic electron scattering. The presence of a finite $\tau_{\text{in}}^{-1}$ in $\beta_H-(\text{ET})_2\text{I}_3$ and other superconductors of the same series is supported by a temperature dependent resistivity above $T_c$. It is worth noting here that from tunneling experiments [31] made on the similar compound $\beta-(\text{ET})_2\text{AuI}_2$ ($T_c \approx 3.8$ K), the ratio $\Delta/T_c$ was found to be four times the BCS value which is consistent with the range $\tau_{\text{in}}^{-1} \ge T_c$ [34]. The effect of a finite lifetime for electrons on the isotope shift turns out to be analogous with that recently investigated for superconductors with paramagnetic impurities. We have shown that for reasonable values of $\tau_{\text{in}}^{-1}$ 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 κ-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 $T_c$ could be detected within an accuracy of 1% in $T_c$ in κ-(ET)$_2$Cu[N(CN)$_2$] Br. The difference of behaviour between κ and β phases is indeed very intriguing since the quality of the labeled ET molecules giving rise to similar shifts of the Raman modes cannot be argued. The finding of no $T_c$ 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 κ and β phases the respective band structures are noticeably different. In particular, it is not understood why there exists only a 20% or so difference between $T_c$ of the two phases whereas the calculated values for $N(E_F)$ differ by about a factor two [36]. This discrepancy may actually hide more subtle problems making κ phase not similar to β phase superconductors. Finally, during this work, an attempt has been made to determine the isotope effect in κ-(ET)$_2$Cu(SCN)$_2$ by transport measurements, using the same ¹³C-labeled ET molecule. We could not detect any reliable isotopic shift of
$T_c$, but the resistive transition of this $κ$ phase is not narrow enough (1 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, $(TMTSF)_2PF_6$ since pairing may not have similar origins in one or two dimensional organic superconductors. It has been claimed that the proximity between a spin density 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)$_3C_{60}$ 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 Génie du Canada (C.R.S.N.G.), l'organisme F.C.A.R. du gouvernement du Québec, and l'Université de Paris-Sud d'Orsay for their financial support. Several discussions with A. M. Tremblay and L. G. Caron are also greatly acknowledged.

### References

[1] MCMILLAN W. L., *Phys. Rev.* **167** (1968) 331.

[2] LIE S. G. and CARBOTTE J. P., *Solid State Commun.* **34** (1980) 599;
SCHACHINGER E., GREESON M. G. and CARBOTTE J. P., *Phys. Rev. B* **42** (1990) 406;
TSUEI C. C., NEWNS D. M., CHI C. C. and PATTNAIK P. C., *Phys. Rev. Lett.* **65** (1990) 2724;
HORSCH P. and RIETSCHEL H., *Z. Phys. B* **27** (1977) 153.

[3] KUBLER J., WEGER M. and SOMMERS C. B., *Solid State Commun.* **62** (1987) 801.

[4] CARBOTTE J. P., GREESON M. and PEREZ-GONZALES A., *Phys. Rev. Lett.* **66** (1991) 1789.

[5] KANG W., MONTAMBAUX G., COOPER J. R., JÉROME D., BATAIL P. and LENOIR C., *Phys. Rev. Lett.* **62** (1989) 2559.

[6] LEE P. A. and READ N., *Phys. Rev. Lett.* **58** (1987) 2691.

[7] SCHWENK H., HESS E., ANDRES K., WUDL F. and AHARON-SHALOM E., *Phys. Lett.* **102A** (1984) 57.

[8] HEIDMANN C. P., ANDRES K. and SCHWEITZER D., *Physica* **143B** (1986) 357.

[9] ANDRES K., SCHWENK H. and VEITH H., *Physica* **143B** (1986) 334.

[10] OSHIMA K., URAYAMA H., YAMOCHI H. and SAITO G., *J. Phys. Soc. Jpn* **57** (1988) 730;
OSHIMA K., URAYAMA H., YAMOCHI H. and SAITO G., *Synth. Metals* **27** (1988) A473.

[11] TOKUMOTO M., KINOSHITA N., TANAKA Y. and ANZAI H., *J. Phys. Soc. Jpn* **60** (1991) 1426.

[12] SHIRBER J. E., OVERMYER D. L., CARLSON K. D., WILLIAMS J. M., KINI A. M., WANG H. H., CHARLIER H. A., LOVE B. J., WATKINS D. M. and YACONI G. A., *Phys. Rev. B* **44** (1991) 4666.

[13] ITO H., WATANABE M., NOGAMI Y., ISHIGURO T., KOMATSU T., SAITO G. and HOSOITO N., *J. Phys. Soc. Jpn* **60** (1991) 3230.

[14] QUIRION G., unpublished results; SCHWEITZER D., POLYCHRONIADIS K., KLUTZ T., KELLER H. J., HENNING I., HEINEN I., HAEBERLEN U., GOGU E. and GARTNER S., *Synth. Metals* **27** (1988) A465.

[15] RICE M. J. and LIPARI N. O., *Phys. Rev. Lett.* **38** (1977) 437;
YAMAJI K., *Solid State Commun.* **61** (1987) 413.

[16] CREUZET F., CREUZET G., JÉROME D., SCHWEITZER D. and KELLER H. J., *J. Phys. Lett. France* **46** (1985) L-1079.

[17] LARSEN J. and LENOIR C., *Synthesis* **2** (1989) 134.
[18] FERRARO J. R. and WILLIAMS J. M., Introduction to Synthetic Electrical Conductors (Acad. Press, 1987).

[19] SUGAI S. and SAITO G., *Solid State Commun.* **58** (1986) 759.

[20] KOZLOV M. E., POKHODNIA K. I. and YURCHENKO A. A., *Spectro. Acta* **43a** (1987) 323.

[21] KRIZA G., QUIRION G., TRAETTEBERG O., KANG W. and JÉROME D., *Phys. Rev. Lett.* **66** (1991) 1922.

[22] BULAEVSKII L. N., *Adv. Phys.* **37** (1988) 443.

[23] MENEGHETTI M., BOZIO R. and PECILE C., *J. Phys. France* **47** (1986) 1377.

[24] ENDRES H., KELLER H. J., SWIETLIK R., SCHWEITZER D., ANGERMUND K. and KRUGER C., *Z. Naturforsch.* **41a** (1986) 1319.

[25] BOZIO R., GIRLANDO A. and PECILE C., *Chem. Phys. Lett.* **52** (1977) 503;
BOZIO R., ZANON I., GIRLANDO A. and PECILE C., *J. Chem. Phys.* **71** (1979) 2282.

[26] WHANGBO M. H., WILLIAMS J. M., LEUNG P. C. W., BENO M. A., EMGE T. J., WANG H. H., CARLSON K. D. and CABTREE G. W., *J. Am. Chem. Soc.* **107** (1985) 5815.

[27] VARMA C., ZUANEN J. and RAGHAVACHARI K., *Science* **254** (1991) 989.

[28] ISHIGURO T. and YAMAJI K., Organic Superconductors (Springer Verlag, Heidelberg, 1990).

[29] SCALETTAR R. T., BICKERS N. E. and SCALAPINO D. J., *Phys. Rev. B* **40** (1989) 197;
See also MARSIGLIO F., *Phys. Rev. B* **42** (1990) 2416.

[30] SCHULTZ A. J., WANG H. H., WILLIAMS J. M. and FILHOL A., *J. Am. Chem. Soc.* **108** (1986) 7853.

[31] HOFFMANN R., *J. Chem. Phys.* **39** (1963) 1397;
WHANGBO M. H. and HOFFMANN R., *J. Am. Chem. Soc.* **100** (1978) 6093.

[32] AMMETER J. H., BÜRGI M. B., THIBEAULT J. and HOFFMANN R., *J. Am. Chem. Soc.* **100** (1978) 3686.

[33] HAWLEY M. E., GRAY K. E., TERRIS B. D., WANG H. H., CARLSON K. D. and WILLIAMS J. M., *Phys. Rev. Lett.* **57** (1986) 629.

[34] SKALSKI S., BETBEDER-MATIBET O., WEISS P. R., *Phys. Rev.* **136** (1964) A1500.

[35] CARLSON K. D., KINI A. M., KLEMM R. A., WANG H. H., WILLIAMS J. M., GEISER U., KUMAR S. K., FERRARO J. R., LYKKE K. R., WURZ P., FLESHLER S., DUDEK J. D., EASTMAN N. L., MOBLEY P. R., SEAMAN J. M., SUTIN J. D. B., YACONI G., PARKER D. H. and STOUT P., *Inorg. Chem.* **31** (1992) 3346.

[36] WILLIAMS J. M., SCHULTZ A. J., GEISER U., CARLSON K. D., KINI A. M., WANG H. H., KWOK W. K., WHANGBO M. H. and SHIRBER J. E., *Science* **252** (1991) 1501.