# Organic Superconductivity

Denis Jérome

► To cite this version:

Denis Jérome. Organic Superconductivity. Superconductivity in New Materials., pp.149 - 216, 2011, 10.1016/s1572-0934(11)04005-4. hal-04997218

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# Organic Superconductivity: A mouse may be of service to a lion

Denis Jérôme

Laboratoire de Physique des Solides, UMR 8502, Université Paris-Sud, 91405, Orsay, France

## Abstract

The research on the possibility of superconductivity in organic matter has been boosted by the quest for new materials becoming superconductors at critical temperatures higher than those available in the seventies. After a few years of research which have led to the discovery of the first metal-like organic conductors in quasi one dimensional (Q1D) charge transfer compounds, superconductivity was finally discovered in the Q1D Bechgaard salt, $(TMTSF)_2PF_6$ in 1979-80. The intense work which has followed the discovery has shown that this Bechgaard salt is a member of an extended isostructural family of compounds forming the generic $(TM)_2X$ phase diagram and exhibiting a broad variety of physical properties controllable by chemistry or by the application of a high pressure. On the low pressure side of the diagram, all compounds behave as insulators (1D localization for 1/2-filled bands or charge ordered insulator for 1/4-filled bands). Chemistry or high pressure promotes a two dimensional deconfinement of the carriers becoming in turn metallic and even three dimensional at low temperature with the stabilization first of an itinerant antiferromagnet (the spin density wave insulator) and under higher pressure of superconductivity around 1.4K. The experimental investigation of these superconductors has shown that superconductivity arises in a background of antiferromagnetic fluctuations which are also responsible for a non Fermi liquid linear term $AT$ in the normal state resistivity at low temperature above the critical pressure needed to suppress the spin-density-wave state. The correlation is such that $A \to 0$ as $T_c \to 0$. These correlations suggest that anomalous scattering and Cooper pairing have a common origin, both rooted in the low-frequency antiferromagnetic fluctuations measured by NMR. Several experimental investigations of superconductivity under magnetic field or in alloyed compounds reveal a pairing which must be singlet with nodes in the gap. A recent development of a renormalization group approach of the Peierls and Cooper divergences treated on equal ground in Q1D conductors is supporting all experimental observations very successfully. Even if Q1D organic superconductors cannot compete with inorganic materials which exhibit superconductivity at about ten times higher critical temperatures they represent very clean and flexible systems on which universal properties of correlated Fermions can be studied aiming to enhance the specifications of high $T_c$ superconductors. These materials are model superconductors where practically most theoretical aspects are under control.

**Key words:** Quasi one dimensional organic superconductors, non-Fermi liquid behaviour, antiferromagnetic fluctuations

**PACS:** 74.70.Kn

## 1. Introduction

Superconductivity is a domain of condensed matter physics and materials science nearly hundred years old and yet still keeping busy an impressive number of physicists and materials scientists. It is interesting to notice that two major achievements in superconductivity have been obtained by chance as this is often inherent to scientific discoveries.

The first one is the discovery of the initial phenomenon of superconductivity by Gilles Holst in 1911 at Leiden by chance when he was involved in the

measurement of the resistance of high purity metals at low temperature in the laboratory of Heike Kamerlingh Onnes who had just succeeded three years before liquifying helium.

The second one is the discovery of superconductivity above 30K by G. Bednorz and A. Müller in 1986 in copper oxides, also an unexpected discovery since oxides were studied at that time for their insulating and magnetic properties rather than for their conductivity.

However, it is the prospect of using superconductivity in power transmission, computer circuits, small magnetic fields detectors, etc., which maintained a very
<figure><img src="image_2.png" /><figcaption>Figure 1: The figure displays the evolution of $T_c$ in materials according to the date of the discovery of their superconducting properties. It clearly shows a renewed effort in superconductivity at the beginning of the eighties.</figcaption></figure>

strong pressure on both theoreticians and materials scientists keeping in mind the synthesis of new conducting compounds in which superconductivity could be stabilized as close as possible to room temperature for an easier and cheaper application of the phenomenon.

With this idea in mind an international conference on Organic Superconductors was organized by W.A. Little at Hawaii in 1969 i.e ten years before the discovery of this phenomenon in organic conductors and at a time where conducting organic crystals were still unheard of. It is fair to recognize that the interest manifested by the whole physics community after the meeting has stimulated thinking and the research of new materials.

## 2. Superconductivity in metals and alloys (1911 → 1973)

### 2.1. The early days

Superconductivity (a new state of matter) is one of the major discovery of the twentieth century in physics. From an experimental point of view the finding by G. Holst in the laboratory of H. Kamerlingh Onnes[1] in 1911 of a current travelling without resistance through a metal cooled at very low temperature paved the way to a very large number of industrial applications. And this, long before the proposal of a satisfactory theoretical framework by Bardeen, Cooper and Schrieffer (BCS)[2] in 1957 explaining both the zero resistance state and the

magnetic flux expulsion (the Meissner effect). Furthermore, the BCS theory has also been the paradigm of modern physical theories based on the importance of the quantum nature at low temperature since it relies on the establishment of a new long range ordered state described by the BCS wave function. BCS emphasized the existence of a two body attractive interaction between charge carriers (either electrons or holes) being a prerequisite for a Bose condensation of electron pairs into the superconducting state. This net attractive coupling in spite of the Coulomb repulsion between carriers of the same sign relies closely on an attraction between electrons mediated by their interaction with excitations of the lattice namely, the phonons.

Consequently, a major achievement of the BCS theory has been understanding the ionic mass dependence of the critical temperature i.e. $(T_c \propto M^{-1/2})$ also called, the isotope effect. Although Fröhlich had proposed in 1954 a model for superconductivity based on the involvement of the lattice [3], this theory was unable to account for the superconductivity of metals but turned out to be relevant subsequently for the interpretation of some of the transport properties of one dimensional organic compounds to be encountered later in this presentation, *vide infra*.

### 2.2. Looking for high $T_c$

At the beginning of the 60's, searching for new materials exhibiting the highest possible values of superconducting $T_c$ was already a strong motivation in materials sciences and the term *high temperature superconductor* was already commonly used refering to the intermetallic compounds of the A15 structure, namely, $(Nb_3Sn$ or $V_3Si$)[4]. The hidden 1D nature of the A15 structure provides an enhancement of the density of states at the Fermi level lying close to the van-Hove singularity of the density of states at the band edges of the 1D d-band. Within the BCS formalism large $T_c$ could be expected. They were actually observed (17-23K) but an upper limit was found to the increase of $T_c$ since the large value of $N(E_F)$ makes also the structure unstable against a cubic to tetragonal Jahn-Teller band distortion[5, 6]. The theory[7] showed that $T_c$ is maximized in the compounds $Nb_3Sn$ and $V_3Si$. In this context at the beginning of the 80's some papers based on metallurgical considerations, regarded 25-30K as the highest possible value for $T_c$ [8]. However, at that time, attempts to increase $T_c$ were still based on the phonon mediated BCS theory and its strong coupling extension[9, 10].

Expending the successful idea of the isotope effect of the BCS theory other models were proposed in which excitations of the lattice responsible for the electron
pairing had been replaced by higher energy excitations
namely, electronic excitations, with the hope of new ma-
terials with T_c higher than those explained by the BCS
theory. Consequently, the small electronic mass m_e of
the polarizable medium would lead to an enhancement
of T_c of the order of (M/m_e)1/2 times the value which is
observed in a conventional superconductor, admittedly
a huge factor. V.L.Ginzburg[11, 12] considered in 1964
the possibility for the paring of electrons in metal lay-
ers sandwiched between polarizable dielectrics through
virtual excitations at high energy. W.A.Little made an
important suggestion[13, 14]: a new mechanism for su-
perconductivity expected to lead to a drastic enhance-
ment of the superconducting T_c to be observed in espe-
cially designed macromolecules. The idea of Little was
indeed strongly rooted in the extension of the isotope
effect proposed by BCS. However a prerequisite to the

<figure><img src="image_3.png" /><figcaption>Figure 2: Schematic picture of Little's suggestion. Charge carriers *a* and *b* moving along a conducting spine are bound *via* a virtual electronic excitation of polarizable side groups.</figcaption></figure>

model of Little was the achievement of conduction in
molecular crystals namely, new types of conductors in
which the conduction would proceed through the trans-
port of charge between π -like molecular orbitals rather
than atomic orbitals. Actually, the concept of a syn-
thetic metal had already been launched by McCoy and
Moore[15] when they proposed to prepare composite
metallic substances from non-metallic constituent ele-
ments. Regarding to the possibility of superconductivity
in materials other than metals, Fritz London in 1937[16]
has been the first to suggest that compounds with aro-
matic rings such as anthracene, naphtalene,... might ex-
hibit a current running freely around the rings under
a magnetic field. The first successful attempt to pro-
mote metal-like conduction between open shell molec-
ular species came out in 1954 with the synthesis of the
molecular salt of perylene oxidized with bromine [17]
although this salt was rather unstable. Coming back
to 1D materials, the model of Little was based on the
use of a long conjugated polymer such as a polyacety-
lene molecule grafted by polarizable side groups[18]

(see figure 2). Admittedly, this formidable task in syn-
thetic chemistry did not reach its initial goal but the idea
to link organic metallicity and one dimensionality was
launched and turned out to be a very strong stimulant
for the development of organic superconductors. Need-
less to say that a lot of basic physical problems had
been overlooked in the seminal paper of Little[13] still
based on the popular Fermi liquid BCS theory. The first
point is that one dimensional conductors had been re-
garded previously as textbook examples by Peierls[19],
who established that a gap should spontaneously open
in the conduction band at the Fermi level concommi-
tantly with a lattice distortion and lead in turn to the sta-
bilization of a dielectric insulating state at low tempera-
ture long before this has been established in the platino-
cyanate chains[20]. Bychkov et-al[21] criticized the
model of Little regarding its potentiality to lead to high
temperature superconductivity. Following Bychkov et-
al, the one-dimensional character of the model system
proposed by Little makes it a unique problem in which
there exists a built-in coupling between superconduct-
ing and dielectric instabilities. It follows that each of
these instabilities cannot be considered separately in the
mechanism proposed by Little in one dimension. Last
but not least, as shown by Landau[22, 23], fluctuations
should be very efficient in a 1D conductor shifting any
long range ordering towards very low temperature.

3. Organic crystals becoming conducting (1974 → 1979)

3.1. *1D Charge transfer compounds*

A major step has been accomplished in 1970 towards
the discovery of new materials for superconductivity
with the synthesis by F.Wudl[24] of the new molecule
tetrathiafulvalene, (TTF). This molecule containing
four sulfur heteroatoms in the fulvalene skeleton can
easily donate electrons when it is combined to elec-
tron accepting species. This discovery has deeply in-
fluenced the subsequent evolution of the chemistry of
organic conductors. The discovery of the molecule
TTF triggered the synthesis of the first stable organic
metal, a charge transfer complex, TTF – TCNQ see
fig.3. The system is made up of two kinds of flat
molecules each forming segregated parallel conducting
stacks. as the orbitals involved in the conduction (π-
HOMO and π-LUMO for TTF and TCNQ respectively)
are associated with the molecule as a whole rather than
with a particular atom with carriers in each stacks pro-
vided by an interstack charge transfer *at variance* with
<figure><img src="image_4.png" /><figcaption>Figure 3: An artist view of the TTF – TCNQ stacked structure according to reference[25].</figcaption></figure>

other organic conductors such as the doped conjugated polymers which came out a little bit later. The overlap of the molecular orbitals being largest along the stack direction and much weaker between them makes the electron dispersion one dimensional (1D). In a first approximation the energy of this 1D conductor depends only on the electron wave vector along the *b*\* direction in the reciprocal space and due to the inverted band structure coming from TTF and TCNQ respectively the 1D bands have to intersect the Fermi energy at Fermi wave-vectors $\pm k_F$. In turn, the band filling is not determined by stoichiometric considerations but rather by a subtle chemical balance. Hence, the band filling does not need to be commensurate with the underlying lattice.

## 3.2. Peierls transition and Fröhlich conductivity

The announcement of a large and metal-like conduction in TTF – TCNQ was made in 1973, simultaneously by D. Cowan and colleagues at Baltimore[26] and A. Heeger and colleagues at Pennsylvania[27]. The Pennsylvania group made a stimulating and somewhat provocative claim announcing a giant conductivity peak of the order of $10^5 (\Omega cm)^{-1}$ at 60 K arising just above a very sharp transition towards an insulating ground state at low temperature.

This conductivity peak was attributed by their authors to precursor signs of an incipient superconductor. The claims for superconductivity were made on the basis of 1D superconducting fluctuations enhanced by the softening of the $2k_F$ phonon mode coupled to the electrons. This was still traditional thinking since it was believed from the study of the A15 superconductors that soft phonons modes were responsible for the high values of their $T_c$ [8]. Unfortunately, a conductivity peak

with such a giant amplitude could never be reproduced by other groups although the metallic character of this novel molecular material was not disputed[28].

<figure><img src="image_5.png" /><figcaption>Figure 4: STM data from reference[29] showing the charge density wave at 36K on TCNQ stacks with longitudinal $\lambda_b = 3.39b$ and commensurate transverse ordering, $\lambda_a = 4a$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: Pressure dependence of the conductivity of TTF – TCNQ showing the drop of metallic conductivity in the commensurability domain. The transverse conduction (right) remains unaffected by commensurability[30].</figcaption></figure>

Quite important has been the finding by the Orsay group from X-ray scattering studies of Bragg satellites allowing to link the metal-insulator transition to the onset of the instability predicted by Peierls[31] in a conducting chain. The Peierls transition leading to a charge density wave structure has been reexamined more recently in the direct space by STM techniques[29], (see, figure 4).

High pressure measurements expected to enable the stabilization of superconductivity[32, 33] have played for the first time a decisive role in the study of organics as shown by the study of the temperature-pressure phase diagram of TTF – TCNQ. As displayed on fig. 5
left, there is a slow increase of the Peierls temperature under pressure to which a sharp peak around 19 kbar is superimposed. This peak has been attributed to a third order commensurability pinning enhancing the stability of the Peierls state with a concomitant increase of the transition temperature. The wave length of the Peierls distorsion becomes commensurate with the underlying lattice in the vicinity of 19 kbar ($\lambda_{\parallel} = 3b$). This has also been confirmed by direct neutron scattering data[34].

Taking advantage of a commensurability attainable under pressure, the possibility to perform systematic studies of the conductivity in the metallic temperature domain by sweeping the helium gas pressure at fixed temperatures did show a drop of longitudinal conductivity closely related to commensurability conditions and particularly pronounced at low temperature[35] see fig.5 right, where Fröhlich fluctuations are pinned to the lattice and no longer contribute to the conduction. This behaviour brought the experimental confirmation of collective Fröhlich fluctuations in a 1D temperature regime above a Peierls transition and, to a large extent, contributed to the understanding of the conduction peak announced in 1974. Furthermore, the role of 1D electron-electron repulsive interactions has been recognized for the first time in organics by the 4k<sub>F</sub> signature in X-ray diffuse scattering experiments[36]. While high

<figure><img src="image_7.png" /><figcaption>Figure 6: Schematic picture of the conductivity of TTF-TCNQ showing the contributions related to Fröhlich fluctuations and single particles according to [30].</figcaption></figure>

pressure appeared to be a parameter far more influential for organic conductors than for regular metals, it failed to stabilize superconductivity in TTF-TCNQ. However very high pressure studies[37] have succeeded recently making TTF-TCNQ sufficiently three dimen-

sional preventing in turn the onset of a Peierls instability at high temperature *albeit* without the existence of superconductivity[33].

When the study of TTF-TCNQ was going on it was recognized that electron polarizability is an important factor to reduce the screened on-site e-e repulsion. Therefore, the redox potential ($\Delta E$)$_{1/2}$ should be minimized, [38, 39] in order to fulfill this goal and obtain potentially conducting materials. Hence, new charge transfer compounds with the acceptor TCNQ were synthesized using other heteroatoms besides sulfur for the donor molecule, i.e. substituting sulfur for selenium in the TTF skeleton thus leading to the TSF molecule.

Much attention was put on the tetramethylated derivative of the TSF molecule which, when combined to the dimethylated TCNQ gave rise to TMTSF-DMTCNQ. The outcome of this study has been truly decisive for the quest of organic superconductivity [40] since for the first time a metallic state could be stabilized under pressure in an organic compound as the ground state replacing the usual Peierls ground state observed in all other compounds pertaining to the TTF-TCNQ series[41].

## 4. Organic superconductivity and phase diagrams (1980 → 2004)

### 4.1. $(TMTSF)_2PF_6$: The prototype 1D superconductor

At the same time, the Copenhagen group led by Klaus Bechgaard succeeded in the synthesis of a new series of 1D conducting salts all based on the TMTSF cation radical molecule, see fig.7, namely, $(TMTSF)_2X$ where X is an inorganic mono-anion with various possible symetry, spherical ($PF_6$, $AsF_6$, $SbF_6$, $TaF_6$), tetrahedral ($BF_4$, $ClO_4$, $ReO_4$) or triangular ($NO_3$)[42]. These or-

<figure><img src="image_8.png" /><figcaption>Figure 7: Side view of the molecule (TMTSF).</figcaption></figure>
ganic molecules form stacks along the *a* axis in the crystalline states, fig. 8, and give rise to conductors for which the intermolecular coupling is much stronger along the stacking axis than along any of the transverse directions. They are one dimensional (1D) conductors, fig. 9. All these compounds but the one with X= ClO₄ did reveal an insulating ground state under ambient pressure. What is special with (TMTSF)₂PF₆, the prototype of the so-called Bechgaard salts, unlike the previously investigated TTF – TCNQ, is the magnetic origin of the ambient pressure insulating state related to the onset of itinerant antiferromagnetism opening a gap at Fermi level over the quasi-planar (1D) Fermi surface.

<figure><img src="image_9.png" /><figcaption>Figure 8: Side view of (TM)₂X conductors along the stacking *a* axis.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9: View of (TM)₂X conductors along the stacking *a* axis [43].</figcaption></figure>

In this latter compound the conductivity reached the value of 10⁵ (Ωcm)⁻¹ at 12K with still a very strong temperature dependence. This behaviour for the transport properties together with absence of any lattice modulation [44, 45] as precursors to the metal-insulator transition were new features in this field still dominated by the CDW philosophy stimulated further investigations under pressure and allowed the stabilization of a metallic state down to liquid helium temperature at a

pressure of about 9 kbar leading for the first time to the stabilization of a highly conducting metallic state at 1.2 K, see fig. 10. The finding of a very small and still non-saturating resistivity at 1.3K extrapolating linearly to a nearly zero value at *T* = 0K was a strong enough motivation to trigger further studies under pressure in a dilution refrigerator which rapidly led to the discovery of a zero resistance state below 1K. As this zero resistance state was easily suppressed by a small magnetic field transverse to the most conducting direction, superconductivity was claimed [46]. The discovery of superconductivity in the (TMTSF)₂X family was a very exciting phenomenon since it was the first time such an instability could be stabilized in an organic compound. This happened about 15 years after the publication of Little's suggestion [13].

However, as we shall see in the following of this article, superconductivity turned out to be only the tip of the iceberg for the properties of these materials. There is much more physics to be studied besides superconductivity in the (TM)₂X series. We cannot understand the existence of superconductivity without touching all particular features revealed by the phase diagram of (TM)₂X. This is reason why a large part of the following sections are devoted to the investigation of the chemical phase diagram of the (TM)₂X series varying the nature of the monoanion X and also to evolution of the physical properties of (TMTTF)₂X and (TMTSF)₂X or under pressure.

<figure><img src="image_11.png" /><figcaption>Figure 10: Left: Picture of the tube containing single crystals of (TMTSF)₂PF₆ from the batch KS 284 for the first pressure runs in which organic superconductivity has been discovered, "Runs at 12kbar (or below)". Right: Warning at the back of the tube, "Keep results confidential".</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 11: (TMTSF)₂PF₆: first observation of organic superconductivity (under a pressure of 9 kbar). Resistivity data normalized to the 4.2K value are shown for two samples. (Reprinted with permission from ref[?]. Copyright 1980 EDP Sciences.).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 12: Generic temperature-pressure phase diagram of the (TM)₂X family of organic conductors. The diagram is drawn for the compound (TMTTF)₂PF₆ taken for the origin of the pressure scale. The location of the compounds (TMTSF)₂PF₆ and (TMTSF)₂ClO₄ under ambient pressure are ≈ 37 and 47 kbar respectively in this diagram.</figcaption></figure>

## 4.2. The 1D generic phase diagram for (TM)₂X materials

In the mid-1980's, the isostructural family comprising the sulfur molecule with the same series of monovalents was investigated under pressure and it was realized that (TMTTF)₂X and (TMTSF)₂X salts both belong to a unique class of materials forming the generic (TM)₂X phase diagram[47], fig.12. In particular, (TMTTF)₂PF₆, although the most insulating compound of the phase diagram can be made superconducting at low temperatures under a pressure of 45 kbar[48–50]. Because of their particular crystal structures, such materials can be considered at first glance as prototypes for 1D physics[51].

What 1D physics means in a nutshell, is the replacement of Landau-Fermi quasiparticles low lying excitations by decoupled spin and charge collective modes [52, 53]. The model for 1D conductors starting from a linearized energy spectrum for excitations close to the Fermi level and adding the relevant Coulomb repulsions which are responsible for electron scattering with momentum transfer 0 and $2k_F$ is known as the popular Tomonaga-Luttinger (TL) model. In this model the spatial variation of all correlation functions (spin susceptibility at $2k_F$ or $4k_F$, CDW, Superconductivity) exhibit a power law decay at large distance, characterized by a non-universal exponent $K_ρ$ (which is a function of the microscopic coupling constants) [54].

However an important peculiarity of (TM)₂X materials make them different from the usual picture of TL conductors. Unlike two-stacks TTF – TCNQ materials, (TM)₂X conductors exhibit a band filing which is commensurate with the underlying 1D lattice due to the 2:1 stoichiometry imposing half a carrier (hole) per TM molecule. Consequently, given a uniform spacing of the molecules along the stacking axis the unit cell contains 1/2 carrier, i.e. the conduction band is quarter-filled. However, non-uniformity of the intermolecular spacing had been noticed from the early structural studies of (TMTTF)₂X crystals[55] due to the periodicity of the anion packing twice the periodicity of the molecule packing. This non-uniformity is at the origin of a dimerization of the intrastack overlap between molecules which is important in the sulfur series (prominent 1/2 Umklapp scattering) but still present although less developed in some members of the (TMTSF)₂X series. A dramatic consequence of this commensurate situation for (TM)₂X materials is the existence of localization channels due to electron-electron Umklapp scatterings with momentum transfer $4k_F$ (two particles scattering) or $8k_F$ (four particles scattering) corresponding to half or quarter-filled bands respectively[56, 57]. This localization is a typical outcome of 1D physics in the presence of repulsive interactions and leads to a charge gap $Δ_ρ$ in the quasiparticle spectrum although no ordering is expected at any finite temperature for a purely 1D system. The spin sector remains gap-less on account of the
separation between spin and charge degrees of freedom in 1D conductors. Fortunately, most features of the 1D localization and the physical properties of this electron gas can be studied by transport and optical properties of the materials. The most salient effect is the existence of a minimum of resistance at $T_ρ$ below which the resistance becomes activated. According to the theory of the commensurate 1D electron gas worked out by a number of authors, using the extended Hubbard, DMRG and bosonisation models, (see reference[58] for further reading) it as been shown that both Umklapp mechanisms compete for the establishment of the charge gap, 1/4 and 1/2 Umklapp scatterings leading to charge ordered and Mott insulators respectively[59]. The experimental data from transport are unable to discriminate which of the two localizing channels is the dominant one at low temperature. In spite of the charge gap a metal-like behaviour of the longitudinal resistance can still be observed at $T$ larger than $Δ_ρ$ leading to a resistance displaying a power law $ρ(T) ≈ T^θ$ ($θ > 0$). Experimental studies have revealed such a metallic behaviour for the resistance either in (TMTTF)₂X under pressure[60, 61] or in (TMTSF)₂X even at ambient pressure[30]. What has been found is a sublinear exponent, namely $θ = 0.93$ for the constant volume temperature dependence of the (TMTSF)₂PF₆ resistance[60, 61]. Similarly, the optical conductivity reveals a power law frequency dependence at high frequency above a charge gap, $σ(ω) ≈ ω^{-1.3}$[62]. Transport and optical data experiments can both be explained within the theoretical framework of commensurate conductors with commensurability one or two. However, in the case of half-filling (commensurability unity) the Luttinger parameter $K_ρ$ should be very close to unity (very weak repulsive interactions) since $θ = 4K_ρ - 3$. A situation which implies weak coupling $K_ρ ≈ 1$ would be hard to reconcile with the exchange enhancement of the spin susceptibility[63]. On the other hand, when quarter-filled Umklapp scatterings prevail at high temperature $θ = 16K_ρ - 3$ and consequently $K_ρ = 0.23$ according to the data of (TMTSF)₂PF₆.

### 4.3. Transverse transport and deconfinement

The temperature dependence of the transverse conduction along the least conducting direction also provides additional arguments in favour of a predominance of the quarter-filled Umklapp above $T_ρ$. There is little doubt that the transport along $c^*$ is essentially diffusive over a broad temperature regime since it is only around helium temperature that a weak plasma edge can be detected in optical data for light polarized along $c^*$ in (TMTSF)₂ClO₄ [64]. The behavior of the (TMTSF)₂PF₆ resistance along the c direction

displays an insulating behaviour below room temperature going through a maximum around $T^* ≈ 80 - 100$ K and becomes metallic at lower temperatures, although remaining several orders of magnitude above the Mott-Ioffe critical value which is considered as the limit between metal and insulating-like transport [65]. The results of the extensive investigation performed in (TMTTF)₂PF₆ are displayed on fig.13,[60].

<figure><img src="image_14.png" /><figcaption>Figure 13: (TMTTF)₂PF₆, longitudinal (left) and transverse (right) resistances versus temperature at different pressures[60].</figcaption></figure>

Transverse transport requires the tunneling of Fermions (at variance with the longitudinal transport which is related to 1D collective modes) between neighbouring chains and therefore probes the amount of quasi particles (QP) existing close to Fermi level. The insulating character of the transverse transport at high temperature can thus be interpreted as the signature of a non Fermi-Landau behavior [66]. When transport along the c-direction is incoherent, transverse conductivity probes the physics of the a-b planes namely, the physics of the weakly interacting Luttinger chains in the a-b planes. The resistivity along the least conducting direction depends on the one-electron spectral function of a single chain and reads $ρ_c(T) ≈ T^{1-2α}$[67] where α is related to $K_ρ$, ($α = \frac{1}{4}(K_ρ + 1/K_ρ - 2)$.

The temperature $T^*$ at which the c-axis transport switches from an insulating to a metallic temperature dependence corresponds to a cross-over between two regimes (a deconfinement) ; a high temperature regime with no QP weight at Fermi energy (possibly a TL liquid in the 1D case) and another regime in which the QP weight increases with decreasing temperature. This picture does not necessarily imply that the transport along the c-direction must also become coherent below the cross-over since the c-axis transport may remain incoherent with a progressive establishment of a Fermi liquid in a-b planes at temperatures below $T^*$. Consequently, the temperature dependence of transport properties along a and c-axes in the 1D regime above $T^*$ leads to a determination of $K_ρ$ of order 0.23 in
(TMTSF)₂PF₆ and (TMTTF)₂PF₆ under a pressure of 12-15 kbar respectively. The Luttinger parameter is much smaller in (TMTTF)₂PF₆ at ambient pressure[60], (Kρ=0.18) due to the enhancement of correlations in the left region of the phase diagram, fig.14. Optical data measured throughout the entire generic diagram reach fairly similar conclusions[68].

The decrease of the charge gap under pressure, fig.14 can be ascribed to a weakening of intra chain correlations moving from left to right in the diagram. Carriers are confined on the 1D chains and the transverse coupling $t_{⊥b}$ renormalized by intrachain interactions is not pertinent (the effective coupling = 0 K)[69, 70]. This is the 1D confined regime. However, when this gap $Δ_a$ decreasing under pressure becomes of the order of the bare transverse coupling $t_{⊥b} ≈ 200$ K, a close interplay between this gap and the transverse coupling occurs giving rise to the sharp suppression of the localization observed around 15 kbar in (TMTTF)₂PF₆ and in turn to a sharp rise of the effective transverse coupling becoming pertinent at higher pressures approaching therefore the bare coupling $t_{⊥b}$. The deconfinement of the carriers is observed around $T^*$.

<figure><img src="image_15.png" /><figcaption>Figure 14: Pressure dependence of transport activation in (TMTTF)₂PF₆. The activation for the *c* axis transport ($Δ_c$) although decreasing under pressure survives up to high pressures while the longitudinal transport ($Δ_a$) is no longer activated above 15 kbar when a dimensional cross-over (1D to 2D) occurs at a finite temperature [60].</figcaption></figure>

Below this deconfinement temperature charge excitations lose their 1D character and resemble more and more what is expected in Fermi liquids (quasiparticles), leading in turn to a quadratic temperature dependence for the longitudinal resistivity observed in (TMTTF)₂X compounds under high pressure[71] or even at ambient pressure in (TMTSF)₂PF₆ [72], *vide infra*. However, electron excitations of this "Fermi liquid" retain a low energy gap in the far infra-red spectrum in which most of the oscillator strength is carried by states

above the gap coexisting with a very narrow and intense zero frequency peak in the conductivity[62, 73].

A phase transition towards a long range ordered insulating phase has been observed recently in the charge localized temperature domain. The phase at low temperature has been ascribed, according to NMR data to the onset of a charge disproportionation between molecules on the molecular chains[74]. Since the charge of this low temperature phase (see, the CO state on the left side of the generic (TMTTF)₂PF₆ diagram, figure (12), is no longer uniform some ferroelectricity can be expected as shown by a signature in dielectric measurements[75]. The stability of the CO state (often called a Wigner state) is a direct consequence of the long range nature of the Coulomb repulsion which, in terms of the extended Hubbard model, amounts to finite on-site U and second-neighbours V repulsions. An other example of the charge ordering in 1D conductors will be given in the following section.

### 4.4. A textbook example for the Wigner crystal, 1/4 filled compounds

<figure><img src="image_16.png" /><figcaption>Figure 15: Generic (*P/T*) phase diagram of [EDT-TTF-CONMe₂]₂X with open and close symbols for EDT₂AsF₆ and EDT₂Br respectively. The origin of pressure is taken for EDT₂Br. High temperature triangles provide a determination of *T*CO from the *P*-dependence of the longitudinal conductivity at room temperature and a BCS relation between the gap and the transition temperature. *T*CO is also deduced from the kink on the *P*-dependence of the conductivity at room temperature (triangle at 4 kbar). The minimum of the longitudinal resistivity (*T*ρ) in the 1D regime is reported when it is observed below room temperature (at 5.6 kbar). Circles (cross (x) indicate the antiferromagnetic ground state observed by *H* and ¹³C NMR. Crosses at 4 K and 7 - 9 kbar mean the lowest temperature reached without any magnetic ordering. Squares indicate the metal to insulator transition derived from transverse and longitudinal resistivity. The cross at (3K, 12.5 kbar) marks the onset of a weak localisation also present at higher pressure.</figcaption></figure>
The details of the *T/P* phase diagram for the TMTTF compounds described previously clearly involve a competition between 1/2 and 1/4 Umklapp scatterings. The dimerization in the g-ology approach to the physics of 1D conductors [56] results in a localization of charges on the bonds between adjacent molecules due to 1/2-filled Umklapp scattering[58]. The ground state is referred to as a dimer-Mott insulator [76]; in the real materials, spin-Peierls (SP) or antiferromagnetic (AF) ordering occurs at low temperature[63]. On the other hand, the next bear neighbour Coulomb interaction competing with the on-site interaction is known to lead to charge ordering[77] which tends to favour an AF ground state, but here the exchange integrals, and consequently the AF wavevector, are controlled by the CO order parameter. The opportunity to study the progression from insulator→metal in a truly 1/4-filled quasi-one dimensional compound (i.e., without dimerization) presents itself with the crystallization of [EDT-TTF-CONMe₂]₂X, for which there is no center of symmetry between the molecules along the stack, at variance with the (TM)₂X series and their inherent dimers, and where extensive refinement of X-ray synchrotron data indicate uniform distribution of the molecular units, yielding a band quarter-filled with holes [78].

The *T/P* phase diagram shown in Fig.15, drawn from a combination of transport and NMR measurements, is simpler than for the TMTTF systems because of the absence of dimerized stacks. At ambient pressure and *T*=300K, both compounds are in a CO (Wigner) insulating state and cooling results in antiferromagnetically ordered ground states. The application of pressure suppresses the CO phase in favor of a quasi-one dimensional conductor that, upon cooling, undergoes a transition towards which is likely to be a Peierls state since no sign of magnetic ordering could be detected in NMR experiments. Still higher pressures suppress the Peierls transition, presumably as the nesting condition responsible for this Peierls transition is weakened. Similar properties were observed for both compounds once the pressure is shifted for the AsF₆ material by -7 kbar relative to EDT₂Br and for this reason the phase diagram as presented on fig.15 applies to both compounds, EDT₂Br and EDT₂AsF₆, *albeit* with the appropriate shift in pressure. Experiments on another structurally more complex 1/4-filled system, (DI-DCNQI)₂X [79, 80] have not yet clarified the existence of an intermediate density wave insulator.

### 5. Organic superconductivity and recent developments, (2005 →)

In this Section we intend to survey some aspects related to the experimental features of the condensed superconducting state as well as those of the metallic background leading to a superconducting ground state when stability conditions are fulfilled.

#### 5.1. *Experimental investigations of the SC phase*

##### 5.1.1. *The SC phase from transport measurements*

As mentioned above the first evidence for superconductivity in (TMTSF)₂PF₆ came from a study of the electronic transport under pressure[46], fig.11. This discovery has been confirmed in the same material a few months later by the IBM group[81] which found a decrease of *T*<sub>c</sub> under pressure attributed to "a hardening of the lattice modes". Recent and extensive studies revealed that this early conclusion was probably naive. Thanks to some additional chemistry, superconductivity could be observed without the need of pressure in (TMTSF)₂ClO₄. This compound is the only member of the (TM)₂X series displaying superconductivity under ambient pressure.

<figure><img src="image_17.png" /><figcaption>Figure 16: (TMTSF)₂ClO₄, first observation of organic superconductivity at ambient pressure. After Ref. [82]</figcaption></figure>

However, the study of the superconducting state in (TMTSF)₂ClO₄ is meeting the problem of the non-centrosymmetrical ClO₄ anions ordering at 24K and doubling the periodicity along the *b*-axis [83]. Consequently, great care must be taken to cool the sample
<figure><img src="image_18.png" /><figcaption>Figure 17: Diamagnetic shielding of (TMTSF)₂ClO₄ at T=005K for magnetic fields oriented along the three crystallographic axes, from ref.[87].</figcaption></figure>

slowly enough in order to reach a well anion-ordered state (R-state) at low temperature; otherwise superconductivity is faced to its great sensitivity to disorder, a very important feature for organic superconductors which will be discussed more extensively below, see Sec.5.3.1.

### 5.1.2. Evidence for SC from magnetic data

Soon after the discovery of organic superconductivity in (TMTSF)₂PF₆ using transport measurements AC susceptibility measurements performed on the same material detected an anomaly at $T_c$ indicative of a transition into a diamagnetic state[84]. A confirmation of the bulk nature of organic superconductivity has been given subsequently by the observation of shielding and Meissner signals in (TMTSF)₂PF₆ [85]. Similar effects have been observed in (TMTSF)₂ClO₄ [86, 87] with a detailed analysis of the anisotropic Meissner effect carried on the case of (TMTSF)₂ClO₄ [87], fig.17. The first critical fields obtained from the Meissner magnetization curves at low temperature read, 0.2,1 and 10 Oe along axes a, b' and c' respectively. Considering the values for the second critical fields $H_{c2}$ derived either from the Meissner experiments plus the knowledge of the thermodynamical field[87] or from a direct measurements of transport, superconductivity has to be considered in the extreme type II limit. The Ginzburg-Landau parameter $\kappa$ can even overcome 1000 when the field is along a due to the weak interchain coupling in these Q1D conductors making the field penetration very easy for

the parallel configuration of the external field. Historically, the critical fields of Q-1-D superconductors have first been considered in the framework of the dirty limit of type II superconductors. This approach led to the conclusion that the band structure anisotropy is unable to explain the anisotropy of $H_{c2}$[30]. However, the recent extensive study of the suppression of superconductivity by non-magnetic defects[88] cannot support the dirty limit approach for these materials. An interpretation for critical fields assuming the pure limit of type II superconductors has been suggested in 1985[89] see also Section 5.3.1. This proposal was based on the calculation of the microscopic expressions for the effective mass tensor in the Ginzburg-Landau equation near $T_c$[90]. As we shall see, this interpretation provides a reasonable interpretation of the data.

With an orthorhombic tight binding model for (TMTSF)₂ClO₄ taking into account the doubling of the b parameter due to the anion ordering below 24K and also the $k_\perp$ dependence of the gap over the Fermi surface leading to $<|\Psi|^2>=1/2$ for a cosine $k_\perp$ dependence over the Fermi surface which is expected for a singlet d-wave pairing, see Section 5.3.1 critical fields can be written in a practical way as,

$$H_{c2}^a = \frac{96.8 \times 10^3}{t_b t_c} T_c(T_c - T) \quad (1)$$

$$H_{c2}^b = \frac{193.6 \times 10^3}{t_a t_c} T_c(T_c - T) \quad (2)$$

$$H_{c2}^c = \frac{386 \times 10^3}{t_a t_b} T_c(T_c - T) \quad (3)$$

where the magnetic field is in kOe and $t_c$ in Kelvin.

Using recent data of field derivatives, $dH_{c2}^a/dT$,

(a) 67.5

(b') 36.3

(c*) 1.39

<caption>Table 1: Critical fields derivatives $dH_{c2}^a/dT$ near $T_c$ in kOe/K according to reference [91] along a, b' and c' axes.</caption>

data near $T_c$ obtained with a perfectly aligned magnetic field[91], Eqs.(1-3) lead to band parameters $t_a : t_b : t_c = 1200, 300$ and 6K respectively which compare favourably with realistic parameters for the band
structure[93]. Therefore, orbital limitation is governing
critical fields close to T_c along the three directions.

When the magnetic field is oriented along partic-
ular crystals axes namely the intermediate *b*-axis or
along the chain axis, violations of the Pauli limita-
tion have been claimed in (TMTSF)₂PF₆ [94] and re-
cently in (TMTSF)₂ClO₄ superconductors[95]. How-
ever, it must be kept in mind that none of these ex-
periments involve any thermodynamic determination of
the superconducting state such as specific heat mea-
surements. Furthermore, in all these experiments un-
der transverse magnetic field along the *b'* axis, the elec-
tronic structure is profoundly affected by the application
of the field which tends to localize the electrons along
the *c*\* direction as shown by the normal state trans-
port ρ_c crossing over from a metallic to an insulating
behaviour when it is investigated under magnetic field
aligned along *b*'[96]. It is still unclear whether the su-
perconducting phase remains uniform under very strong
transverse field[97, 98]. In terms of lower critical fields
The determination of the thermodynamical critical field
H_c = 44 ± 2 Oe can be obtained from electronic specific
heat measurements, C_e(T) *vide infra*.

5.1.3. *The SC phase from calorimetric measurements*

Since (TMTSF)₂ClO₄ is an ambient pressure super-
conductor the thermodynamic evidence of the phase
transition has been obtained from specific heat on single
crystals[99]. The electronic contribution to the specific
heat of (TMTSF)₂ClO₄ in a Cₑ/T vs T plot (Fig. 19),
dipslays a very large anomaly around 1.2K[99]. Above
1.22K, the total specific heat obeys the classical relation
in metals C/T = γ + βT², where the Sommerfeld con-
stant for electrons γ = 10.5 mJ mol⁻¹K⁻², correspond-

<figure><img src="image_19.png" /><figcaption>Figure 18: Critical fields of (TMTSF)₂ClO₄ determined from T_c,onset for the three principal axes with an indication for the Pauli limit at low temperature (after [91, 92]).</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Figure 19: Electronic contribution to the specific heat of (TMTSF)₂ClO₄, plotted as Cₑ/T versus T. After [99]</figcaption></figure>

ing to a density of states at the Fermi level N(E<sub>F</sub>) =
2.1 states eV<sup>-1</sup> mol<sup>-1</sup> for the two spin directions[99].
The specific heat jump at the transition amounts then
to ΔC<sub>e</sub>/γT<sub>c</sub> = 1.67, i.e. only slightly larger than
the BCS ratio for a s-wave superconductor. There-
fore, the specific heat data and the comparison between
the value of the density of states derived specific heat
Pauli susceptibility[100] lend support to a weak cou-
pling Fermi liquid picture (at least in the low temper-
ature range)[63]. Specific heat has also been used to
investigate the pair breaking effect of disorder on SC
in (TMTSF)<sub>2</sub>ClO<sub>4</sub> in fast cooled samples[101], see also
Sec.5.3.1.

5.2. *The SC phase diagram under pressure*

5.2.1. *The SDW-SC coexistence regime*

From the early days a situation of non-homogenous
superconductivity have been clearly identified near the
border between SDW and Superconductivity in the
(TMTSF)₂X phase diagram[81, 102–104]. At high
pressure (P > 9.4 kbar), the superconducting phase
emerges from a metallic state and can be reasonably
thought of as homogeneous with a critical current den-
sity along the *a*-axis J_c = 200 Å cm⁻²[103]. Below
this critical pressure, there exists a pressure domain for
(TMTSF)₂PF₆ (≈ 1 kbar wide) in which a superconduct-
ing signature detected from the ρ_a data is observed at
a nearly pressure independent temperature below the
onset of a SDW instability, where the critical current
density is greatly reduced, J_c ≈ 10 Å cm⁻². This fea-
ture points in favor of a coexistence between SDW and
SC macroscopic domains. Reference[103] suggested
metallic (SC) tubes parallel to the *a* axis (Fig. 20) with a
sudden vanishing of the SC signal around 8.6 kbar. The
<figure><img src="image_21.png" /><figcaption>Figure 20: First study of the coexistence between SDW and Superconductivity in (TMTSF)₂PF₆ in the vicinity of the critical pressure for suppression of the SDW ground state using the longitudinal resistivity. After Ref. [103]</figcaption></figure>

existence of coexisting macroscopic regions of SDW and SC order is also supported by a recent NMR investigation performed at a pressure slightly lower than the critical pressure for the establishment of the homogeneous state[105]. This latter study has enabled a quantification of the relative volume fractions in the SDW-metallic regime using the proton NMR linewidth as the local probe. A related consequence of the existence of macroscopic insulating domains in the superconducting phase allowing a channeling of the lines of force in the material is the large increase of the upper critical field $H_{c2}$[106], which had already been reported long time ago in (TMTSF)₂PF₆ when organic superconductivity received its confirmation[81], and also subsequently in (TMTSF)₂AsF₆ [107]. It must be kept in mind that the slab formation in the vicinity of the SDW state with the formation of insulating domains is not related to the SC state itself namely, the penetration of the magnetic field in a type II superconductor, but is the result from a competition existing between insulating and conducting phases. The claim for the existence of tubular domains $||a$ is based on an analysis of the resistance and of the critical currents in the superconducting state in the vicinity of the critical pressure at low temperature[103].

A theoretical approach predicting a phase separation relies on a variational model leading to an inhomogeneous phase with an energy lower than the energy of the homogeneous states (Metallic or Insulating SDW). Since it is the transverse *b* parameter that is expected to govern the balance between SDW and metallic phases at a pressure lower than the critical one but close enough to the

homogenous critical line the formation of macroscopic metallic domains with a smaller *b* is energetically favored in-between *b* expanded SDW domains[108]. Similarly, at pressures larger than the critical pressure, insulating SDW domains should be present between metallic regions (see Fig. 20). This interpretation is *at variance* with a model based on similar experimental data for the critical fields in the coexistence regime claiming the formation of thin superconducting slabs perpendicular to *a* sandwiched between SDW insulating domains is the result of a self organization process taking advantage of the largest field penetration length perpendicular to the direction of the field[106].

An other approach has been taken by Gorkov and Grigoriev for the interpretation of the SDW/SC coexistence regime in (TMTSF)₂PF₆ [109]. This model predicts the existence of soliton domain walls along the *b* direction in the SDW phase close to the critical pressure between the uniformly gapped SDW phase and the SC phase. As these domain walls should be perpendicular to the molecular stacking axis, a strong anomaly of the transport anisotropy could be anticipated related to the formation of conducting slabs perpendicular to the *a* axis in the coexistence pressure regime. However, no such anomaly has been reported.

Recent data obtained with (TMTTF)₂PF₆ do show that such a coexistence regime exists in this compound although under a much higher pressure[49, 50, 110]. Hence, due to a smaller compressibility, the coexistence regime is much broader in pressure than for (TMTSF)₂PF₆, about 10kbar instead of 1kbar. Similarly the coexistence is accompanied by an enhancement of the critical field for $H||c^*$ by a factor 10 at the border with the SDW phase reaching 1T as compared to only 0.1 T at very high pressure when the superconducting state is homogeneous[111], a value similar to the observation in the R-state of (TMTSF)₂ClO₄. Finally, an experimental study performed in (TMTSF)₂ReO₄ under pressure has revealed the existence of conducting filaments parallel to *a* in the pressure domain close to 10 kbar when metallic and insulating domains coexist at low temperature as a consequence of two coexisting anion orders in this material[112].

The coexistence regime has recently been revisited in (TMTSF)₂PF₆ performing transport measurements along the three principal axes leading to the first comprehensive investigation of this organic superconductor in the vicinity of the endpoint of the spin density wave - metal phase transition[113]. At low temperature, the transition of metallic domains towards superconductivity is used to reveal the various textures in the inho-
<figure><img src="image_22.png" /><figcaption>Figure 21: Phase diagram of (TMTSF)₂PF₆ as determined from resistivity measurements along the three axes (circles: ρ_a; squares: ρ_b; triangles: ρ_c) after ref[113]. The filled (open) symbols correspond to the transition towards SC (SDW) respectively. The contrast of colors between P_c2 and P_c is illustrating the increase in SC volume fraction moving from P_c2 up to P_c corresponding to the three different regimes explained in the text. Based on the knowledge of T_SDW and T_c, the pressure of 5.5 kbar in ref[105] would amount to 8.8 kbar in the present phase diagram.</figcaption></figure>

mogenous state. The existence of metallic walls perpendicular to the *a*-axis is confirmed. These walls first appear as filaments along the *c*-axis and evolve under pressure towards slabs parallel to the *b'*-planes. The stability of a soliton phase in this pressure regime might be a good starting point for understanding the physics of the coexistence regime. Figure.21 displays the details of the phase coexistence between metal and insulator in (TMTSF)₂PF₆ close to the critical pressure.

### 5.2.2. *The SC phase under pressure*

The pressure stability of SC is admittedly a remarkable feature of the (TM)₂X compounds. Pressure has enabled organic superconductivity to be discovered but is also the parameter which controls the SC phase very efficiently.

As far as (TMTSF)₂PF₆ is concerned, in the vicinity of the critical pressure the pressure coefficient amounts to $\delta \ln T_c / \delta P = -11\% \text{ kbar}^{-1}$ leading in turn to a Gruneisen constant for superconductivity $\delta \ln T_c / \delta \ln V = 18$ at 9 kbar[118], with the compressibility data measured under pressure (at 16 kbar $\delta \ln V / \delta P = -0.7\% \text{ kbar}^{-1}$)[119] in the same compound. This value is indeed sizably larger than 7, the value which is obtained in tin, the elemental superconductor exhibiting the strongest sensitivity to pressure[120]. Although the SC phase diagram has been established from the early investigations[30], a detailed determination has been performed only recently[121], see fig.22 top. A look at the (TMTSF)₂PF₆ phase dia-

<figure><img src="image_23.png" /><figcaption>Figure 22: Top: (P, T) phase diagram of (TMTSF)₂PF₆ 6 deduced from resistivity measurements. The data points below 11kbar are deduced from measurements along the three crystallographic axes: down triangles for ρ_a, squares for ρ_b, hexagons for ρ_c, empty symbols for T_SDW and full symbols for T_SC (refs[113, 114]). Above the critical pressure (9.4 kbar), only longitudinal resistivity data are plotted: (red) circles from this work performed on one single crystal measured under eight successive pressure runs and (green) triangles from reference[115]. Bottom: Pressure dependence of the superconducting transition of (TMTSF)₂ClO₄ deduced from longitudinal resistivity measurements: (red) circles from this work (the star at 1 bar is deduced from a ρ_c measurement[116]) and (green) triangles from ref[117]. Dashed and dotted horizontal lines indicate the lowest temperature reached without occurrence of superconductivity in both studies. The (red) continuous line is a linear fit of our data including the 1 bar data.</figcaption></figure>

gram shows that the strong pressure dependence of T_c is however restricted to the close vicinity of the border with the SDW phase. It tends to level off at very high pressure.

The pressure coefficient of superconductivity in (TMTSF)₂ClO₄, see fig.22 (bottom), resembles that of (TMTSF)₂PF₆ and is even more dramatic since $\delta \ln T_c / \delta \ln V = 36$[122] using the compressibility of 1% kbar⁻¹ (this is the value measured for (TMTSF)₂PF₆ at ambient pressure [119], as to the best of our knowledge compressibility data for (TMTSF)₂ClO₄ are still missing). However, it becomes clear comparing recent and old data[123, 124] that the stability of the SC state of (TMTSF)₂ClO₄ under pres-
sure is also determined by an additional factor namely, the residual disorder of the $ClO_4$ anions which interferes with the suppression of the SC state[121] leading to the absence of SC above 10 kbar at variance with the very high pressure behaviour of $(TMTSF)_2PF_6$ for which SC remains at $T_c$ = 0.2 K even 15 kbar above the critical pressure. The remarkable sensitivity of $T_c$ in $(TMTSF)_2ClO_4$ might actually be related to the very specific problem of anion ordering in this compound as it has been suggested from the recent study on the sensitivity of $T_c$ against the presence of non-magnetic disorder[125]. Anion ordering reveals an uprise of the ordering temperature under pressure[122, 126, 127], which can be derived from the pressure dependence of a small kink in the temperature dependence of the resistivity, the signature of the ordering, increasing from 24 to 26.5K under the first 1.5 kbar and leveling off at higher pressures, corroborating studies at performed at very high pressures[128]. Accompanying this rise under pressure, a slowing down in the dynamics of the anions is expected. Hence, high pressure studies require a special attention to the cooling rate which must be kept low enough to allow a proper anion ordering at low temperature. This may be the explanation for the discrepancy between high pressure data showing the signature of anion ordering up to 8 kbar [127], and the absence of ordering claimed from the interpretation of magnetomechanical oscillations[129] or also an explanation for the differences between old and new data for the phase diagram of $(TMTSF)_2ClO_4$ displayed in fig.22.

## 5.3. Nature of SC

### 5.3.1. Role of non-magnetic defects on SC

From the very beginning of organic superconductivity, the finding of a common border between an insulating SDW phase and the SC state[30] and even of a coexistence pressure domain for both instabilities[103] raised a great and controversial interest concerning the nature of the SC pairing.

It is the remarkable sensitivity of organic superconductivity to irradiation[130, 131] that first led Abrikosov to suggest the possibility of triplet pairing in these materials[132]. Although irradiation was recognized to be an excellent method for the introduction of defects in a controlled way[133], defects thus created can be magnetic [134], and the suppression of superconductivity by irradiation induced defects as a signature of non-conventional pairing must be taken with 'a grain of salt' since local magnetic moments can also act as strong pair-breakers on s-wave superconductors. Several routes have been followed to introduce an intrinsically

non-magnetic perturbation modulating the potential seen by the carriers on the organic stacks.

Non magnetic disorder has been achieved substituting TMTTF for TMTSF on the cationic stacks of $(TMTSF)_2(ClO_4)_{(1-x)}(ReO_4)_x$ salts with $PF_6$[135] and $ClO_4$ salts [136]. However, in both situations cationic alloying induces drastic modifications of the normal state electronic properties since the SDW transition of $(TMTSF)_2PF_6$ is quickly broadened and pushed towards higher temperature upon alloying[137]. An influence of the sample purity on the superconducting $T_c$ of $(TMTSF)_2ClO_4$ had been noticed by Gubser et al[138] who found that purer crystals have higher $T_c$ and that even samples coming from the same batch could show different $T_c$. Leaving the cation stack uniform, scattering centers can also be created on the anion stacks with the solid solution[139] $(TMTSF)_2(ClO_4)_{(1-x)}(ReO_4)_x$, where Tomić et al.,[140] first mentioned the suppression of superconductivity upon alloying with a very small concentration of $ReO_4$ anions. In the case of a solid solution with tetrahedral anions such as $ClO_4$ or $ReO_4$, one is confronted to two potential sources of non-magnetic disorder that act additively on the elastic electronic lifetime according to the Mathiessen's law. First the modulation due to the different chemical natures of the anions and second a disorder due to a progressive loss of long-range ordering at $T_{AO}$ in the $(TMTSF)_2(ClO_4)_{(1-x)}(ReO_4)_x$ solid solution. However, X-ray investigations have revealed that long-range order is preserved up to 3% $ReO_4$ with a correlation length $\xi_a$ > 20nm[141]. Studies of superconductivity in $(TMTSF)_2ClO_4$ performed under extremely slow cooling conditions have shown that $T_c$ is a fast decreasing function of the non-magnetic disorder[125], where the residual resistivity along the c* axis has been used for the measure of the disorder in the alloys with different concentrations (Fig. 23). It must be emphasized that the residual resistivity is derived following a linear extrapolation of the normal state resistivity according to the analysis of the normal state transport discussed in Section 5.4.2. The suppression of $T_c$ is clearly related to the enhancement of the scattering rate in the solid solution. Since the additional scattering cannot be ascribed to magnetic scattering according to the EPR checks showing no visible traces of localized spins in the solid solution[125], the data in figure (23) cannot be reconciled with the picture of a superconducting gap keeping a constant sign over the whole $(\pm k_F)$ Fermi surface. They require a picture of pair breaking in a superconductor with an unconventional gap symmetry.

The conventional pair breaking theory for magnetic
<figure><img src="image_24.png" /><figcaption>Figure 23: Phase diagram of (TMTSF)₂(ClO₄)₃(ReO₄)₃ governed by non magnetic disorder according to reference [125]. All data point refer either to very slowly cooled samples in the R-state with different ReO₄ contents (x% R) or to slightly quenched samples (x% Q run number).</figcaption></figure>

impurities in usual superconductors has been generalized to the case of non-magnetic impurities in unconventional materials and the correction to $T_c$ obeys the following relation [142, 143],

$$ \ln\left(\frac{T_c^0}{T_c}\right) = \psi\left(\frac{1}{2} + \frac{\alpha T_c^0}{2\pi T_c}\right) - \psi\left(\frac{1}{2}\right), \quad (4) $$

with $\psi(x)$ being the Digamma function, $\alpha = \hbar/2\tau k_B T_c^0$ the depairing parameter, $\tau$ the elastic scattering time and $T_c^0$ the limit of $T_c$ in the absence of any scattering (obviously a virtual sample). A major interest of the study of SC on alloys is to provide a direct access to the electron lifetime at low temperature which is not accessible from conductivity or optical data, vide infra.

From the data in figure (23), the best fit of the data with the Digamma function leads to $T_c^0 = 1.20K$ and a critical scattering for the suppression of superconductivity of $1/\tau_{cr} = 1.5 \text{ cm}^{-1}$ (2.14K). According to the experimental dependence of $T_c$ versus the residual resistivity in the pristine (TMTSF)₂ClO₄ sample shown on fig.23, $1/\tau_0$ becomes $0.15 \text{ cm}^{-1}$ (0.22K), i.e, ($\tau_0 = 3.10^{-11} s$) in a sample with $T_c = 1.2K$. Such a long elastic lifetime would lead to a mean free path of $\lambda = 550 \text{ nm}$ ($v_F = 1.8 \times 10^5 \text{ m/s}$)[144] and is in a very good agreement with the model of clean limit superconductivity and $\xi_a \approx 60 \text{ nm}$ along the a axis of pure samples[145].

Since the residual resistivity of (TMTSF)₂ClO₄ is $\approx \rho_0 = 2.5 \mu\Omega cm$, the previous results suggest that superconductivity cannot be stabilized in samples exhibiting too large values of the residual longitudinal resistivity namely, $\rho_0 > 25 - 30 \mu\Omega cm$.

There are no similar data available for $T_c$ in (TMTSF)₂PF₆ using the alloying facility but an investigation has been undertaken measuring $T_c$ in irradiated samples[130, 131] leading to an upper limit of the residual resistivity above which superconductivity is no longer stable of $\approx 80 \mu\Omega cm$. Considering that this latter value is probably overestimated due to possible cracks inherent to the experiments performed in 1981 this value provides a similar order of magnitude for the enhanced residual scattering needed to destroy superconductivity. However, in case of irradiation damages it was recognized that impurity centers carry a spin and therefore it is not possible to distinguish between the different phenomena responsible for the pair breaking. The data obtained on the alloyed-(TMTSF)₂ClO₄ series are therefore the only ones leading to the conclusion that organic superconductivity is suppressed by non magnetic impurities.

The carrier lifetime in (TMTSF)₂ClO₄ is admittedly about 10-20 times larger than what a classical Drude behaviour would predict according to the measured temperature dependence of transport and the room temperature lifetime $\tau_0 = 3.10^{-15} s$[146]. The inverse lifetime is also significantly smaller than the predicted width at half height namely, $1/\tau \approx 2 \text{ cm}^{-1}$. assuming a classical Drude behaviour and derived from the temperature dependence of the DC conductivity and the longitudinal plasma frequency[147].

These transport data support the existence of a very narrow peak at zero frequency carrying a minor fraction of the total spectral weight, in fair agreement with far infrared optical measurements leading to a zero frequency conductivity peak of width less than $2-4 \text{ cm}^{-1}$[148, 149].

The sensitivity of $T_c$ to non-magnetic disorder cannot be reconciled with a model of conventional superconductors. The gap must show regions of positive and negative signs on the Fermi surface, which can be averaged out by a finite electron lifetime due to elastic scattering. As these defects are local, the scattering momentum of order $2k_F$ can mix $+k_F$ and $-k_F$ states and therefore the sensitivity to non-magnetic scattering is still unable to tell the difference between $p_x$ and $d$ orbital symmetries for the superconducting wave function. The only way to discriminate between the two scenarios is the determination of the spin part of the wave function.

### 5.3.2. Spin part of the SC pairing

While NMR studies concluded in the early days in favour of nodes in the SC gap[150], specific heat data performed down to 0.4K[101] and thermal conductivity data supported the model of a s-type BCS gap covering
the entire Fermi surface[151]. Regarding the spin part of the SC wave function, a triplet pairing was claimed from a divergence of the critical field $H_{c2}(T)$ measured by transport and exceeding the Pauli limiting value reported at low temperature in (TMTSF)₂PF₆ under pressure when $H$ is applied along the $b'$ or $a$ axes [152] and also from the absence of a change in the ⁷⁷Se Knight shift at $T_c$ [153], While, the recent finding of

<figure><img src="image_25.png" /><figcaption>Figure 24: ⁷⁷Se relative shifts vs temperature for $H \parallel b'$, $a$, respectively $b$ and $a$. At $H = 40$kOe, there is no observable change from the normal state shifts at the lowest temperatures measured (100mK). Zero shift is arbitrarily set to the normal state first moment, according to reference[154].</figcaption></figure>

a $T_c$ determined by the value of the electron lifetime in the presence of non magnetic lattice defects has conclusively ruled out the s-wave hypothesis and suggested the existence of nodes with zeros of the SC gap on the Fermi surface[88] favouring singlet-d, triplet-p or even f pairings[155] new NMR experiments[154] have brought another scenario.

A clear change of the ⁷⁷Se Knight, see fig.24, observed at $T_c$ in the compound (TMTSF)₂ClO₄ at fields lower than those used in the previous (TMTSF)₂PF₆ studies[153] provided a conclusive evidence in favour of singlet pairing at least in low magnetic fields up to ≈2.5 Tesla and reported the possibility of a transition from low to high field superconductivity at a field of the order of the Pauli paramagnetic limit[154]. Furthermore, muon spectroscopy performed in the SC state of (TMTSF)₂ClO₄ [156] has concluded to the absence of spontaneous magnetic fields i.e, the time reversal symmetry is likely not broken in fair agreement with the NMR data[154].

### 5.3.3. Beyond Pauli limit in low dimensional superconductors

A recent transport study of the anisotropic superconductivity of (TMTSF)₂ClO₄ with the possibility of a fine angular control of the magnetic field has confirmed the survival of superconductivity far above $H_P$ when the magnetic field is lying along $a$ or $b'$-axes[91, 92], see Fig.18 in Sec.5.1.2. These findings are corroborated by an NMR investigation which shows 1) that $(T_1T)^{-1}$ recovers its value of the normal phase at a field which is not far from the value of the Pauli limit ($H_P \approx 2.4T$), fig.25, while 2) the resistivity reaches the value of the normal state at a field which about twice larger. An ongoing specific heat study of the SC state of (TMTSF)₂ClO₄ [157] is confirming that the recovery of the normal phase specific heat occurs at the Pauli field when $H \parallel a$. Hence, the $a$ direction is a very special direction for superconductivity in (TMTSF)₂ClO₄. Along this latter direction a (Josephson) vortex state can survive above the Pauli field but is strongly suppressed at small tilt angles in the $a-c^*$ plane although the density of quasiparticles at Fermi level of this high field superconducting phase is essentially that of the normal state. The high field behaviour is probably related to a modulated SC phase such as a FFLO phase[143, 158, 159]. The modulated state represents an energy comprise in which less condensation energy is gained but also less Zeeman energy is lost entering the superconducting state of a fully singlet paired state. The singlet Cooper pairs of such a state acquire a finite momentum $q_{FFLO}$ which amounts for a Q1D Fermi surface to the nesting vector between $k_\uparrow$ and $k_\downarrow$ Fermi sheets. When the field is along $b'$ the planes become decoupled at fields exceeding the value of the transverse coupling along $c^*[96, 160]$ and a vortex phase can also survive up to fields larger than the Pauli limit[97, 98, 161].

## 5.4. The metallic phase of the (TM)₂X diagram, control parameters of Q1D organic superconductivity

### 5.4.1. Contribution of transport towards the investigation of the metallic phase

The anisotropic character of the electronic structure and the shape of the (TM)₂X single crystals make the study of the various components of the resistivity both informative and possible. Two directions are able to provide reliable conductivity data using the four contact technique, measurements along $a$ or $c^*$. The four contact technique is very delicate for the $b'$ axis as the results are very dependent on the distance between current and potential contacts in the $a-c$ planes. Hence, only $\rho_a$ and $\rho_c$ will be considered in this section.
<figure><img src="image_26.png" /><figcaption>Figure 25: $^{77}$Se $(T_1T)^{-1}$ versus H at T = 100mK (left axis). Also shown is the interlayer resistance $R_{zz}$(H/a) over a similar range of fields (right-hand axis) after reference[154].</figcaption></figure>

The task is to clarify the various regimes for the physics of the (TM)₂X diagram. First, looking at the transport along $c^*$ we can be confident that no coherence between adjacent a-b planes can be established at high temperature. Since this question does not seem to have received yet much theoretical attention, we can use as a rule of thumb an approximate definition proposed long time ago for transport between chains[30, 162] and adapt it to transport between planes. Therefore if the transport within a-b planes is coherent and characterized by a scattering time $\tau_{//}$, interplane coherence is established when the lifetime of a carrier on a plane(determined by the interplane coupling $t_c$) is shorter than its intra plane scattering time. Therefore, coherence establishes along the least conducting direction when the inequality,

$$ \hbar/\tau_{//} < t_c \qquad (5) $$

is fulfilled.

Considering the small value of $t_c$ such a condition cannot be fulfilled until the low temperature domain is reached when the resistivity is small enough. Furthermore, within the high temperature transverse incoherent domain, $\hbar/\tau_{//} > t_c$, two regimes can be distinguished for the a-b plane. First, a high T regime where $\rho_a$ and $\rho_c$ display opposite temperature coefficients. As discussed in Sec.4.3, this is the 1D regime where $\rho_c$ probes the density of quasiparticle states at Fermi level and $\rho_a$ is related to the conduction process of the 1D regime. The maximum of $\rho_c$ is a signature for a cross over from the 1D to a 2D regime. Below $T^*$, additional density of quasiparticle is recovered and $\rho_c$ begins to follow the same temperature dependence as $\rho_a$ although the transverse motion along $c^*$ remains a diffusive one.

As Section.4.3 has already been devoted to the physics at high temperature, between 300K and $T^*$, the followings sections intend to highlight the temperature regime say, below $T^*$.

Early studies of the longitudinal transport have revealed a $T^2$ law obeyed below 100 K down to the onset of the SDW transition at 12 K in (TMTSF)₂PF₆ [72, 163] and down to 10K in (TMTSF)₂ClO₄ [164, 165] with a coefficient B of this $T^2$ contribution found to be pressure dependent in (TMTSF)₂PF₆ [66]. The resistance along $c^*$ has also been extensively studied due to the possibility to obtain data of high quality[66, 166], i.e, making the comparison between different pressure runs possible keeping the same amount of impurities for one run to the other.

However, approaching the low temperature region, the resistivity turns to a linear temperature dependence and becomes even sublinear in the very vicinity of the SC critical temperature when pressure is close to the critical pressure for the suppression of the SDW phase. Actually, the sublinear behaviour noticed in the first investigation of (TMTSF)₂PF₆ under pressure down to the temperature of pumped helium is the unusual feature acted as a strong stimulant for the research at even lower temperatures using a pressure cell embedded in a dilution refrigerator which led to the discovery of superconductivity at 0.9K[46].

Finding of a strongly temperature dependent resistance prior to the SC transition around 1K was unusual as the resistance of regular metals is expected to saturate at low temperature towards a residual value following a power law with an exponent larger than unity.

The fact that a strong temperature dependence of the resistivity prevails at low temperature has first been taken as a piece of evidence for the existence of paraconductivity in a quasi-one dimensional conductor above $T_c$ [118]. The calculation of Azlamazov-Larkin diagrams based on a time dependent Ginzburg-Landau theory provides, (i), a 3D domain restricted to the very vicinity of $T_c$ where the transverse coherence extends over several interchain distances d, i.e, $\xi_{\perp}/d > 1$ with $\rho_{\parallel}^{3D} \propto T^{-1/2}(\ln T/T_c)^{1/2}$ leading in turn to the classical $1/(T - T_c)^{1/2}$ divergence of the paraconductivity and (ii), a 1D regime where $\xi_{\perp}/d < 1$ with $\rho_{\parallel}^{1D} \propto T^{1/2}(\ln T/T_o)^{-3/2}$ where $T_o$ stands for a renormalized mean field temperature, about one third of the actual mean field temperature [30].

The fluctuation approach was conceivable in the context of the eighties[167], since long time before the actual discovery of organic superconductivity it was believed that mean free paths exceeding 300nm (needed
to explain the conduction in terms of a single particle model) were unlikely in organic conductors thought to be dirty materials. It was considered instead that a large value of the conductivity in a 1D organic conductor following the *TTF*–*TCNQ* story should be the signature of a fluctuating collective transport [168, 169]. The fluctuation approach to organic SC has however been disputed by other groups [170].

While the comparison between experiments and the theory for paraconducting fluctuations performed in ref [118] provided a reasonable agreement in the 3D regime where a square root temperature divergence of the conductivity is expected from the contribution of the superconducting fluctuations, such a contribution to the transport is far less obvious at higher temperature as far as one dimensional fluctuations are concerned. 1D superconducting fluctuations could provide the right order of magnitude for the resistivity with a reasonable value of the renormalized mean field temperature $T_o$ but they fail to reproduce the correct linear temperature dependence characterizing the range of 10 K [118].

Even more important is the question of the purity needed to explain the conductivity at low temperature in these organic superconductors.

After twenty five years of intense research on organic superconductors it has emerged that excellent purity is indeed attainable in these organic materials which are grown by very selective electrocrystallisation routes. Moreover, high purity seems to be the rule for (TM)₂X superconductors rather than the exception. It is a prerequisite for the stability of the superconducting phase as shown in the study of the superconducting temperature varying the elastic lifetime of the carriers via the introduction of a controlled amount of non magnetic defects [125] see Section 5.3.1. According to this latter study and the longitudinal coherence length $\xi_\parallel(0) \approx 60$nm known from critical fields determination [171] the elastic mean free path should reach about 1000nm in pristine (TMTSF)₂ClO₄ samples i.e more than 2800 intermolecular distances. Consequently, there is an agreement to consider superconductivity in these materials in the clean limit of type II superconductors [89].

High purity is also a prerequisite for the stabilization of spin density wave phases induced under high magnetic field first in (TMTSF)₂ClO₄ [172, 173] and later in (TMTSF)₂PF₆ [174, 175] known to be easily suppressed by non magnetic impurities [176, 177]. The physics of field-induced spin density wave phases is not discussed in this article.

It is now agreed from recent data that a value of longitudinal conductivity of order $10^5 (\Omega cm)^{-1}$ is attainable at helium temperature with a single particle mechanism

without addressing the need for a dominant superconducting collective transport.

The accumulation of data over the last 25 years did suggest the need for a complete and careful reinvestigation of the metallic phase of (TM)₂X superconductors and for significant progresses in the theory. This new visit will be surveyed in the rest of this article.

### 5.4.2. Longitudinal transport

The new visit to the metallic region of the (TM)₂X phase diagram consisted in a systematic investigation of the resistivity under 7-8 pressure performed on a unique single crystal in order to minimize effects of impurities inherent to the comparison between different samples. What is clear at first sight on fig.26 showing a typical behaviour of the resistance at low temperature in (TMTSF)₂PF₆ and (TMTSF)₂ClO₄, is the linear behaviour of $\rho_a$ in (TMTSF)₂PF₆ below 10K and also the for (TMTSF)₂ClO₄ although at a somewhat lower temperature below 4K or so.

The study has been conducted on (TMTSF)₂PF₆ and on (TMTSF)₂ClO₄ on the longitudinal and the transverse conductivity $\rho_c$ in a wide pressure domain.

Timusk et al have the first to pay attention to this linear law in (TMTSF)₂ClO₄ pointing to a non-phonon scattering similar to what is seen in cuprates mechanism [73].

The detailed analysis of the transport [178] in (TMTSF)₂PF₆ amounts to fitting the experimental data by a second order polynomial form $\rho(T) = \rho_0 + AT + BT^2$. Below 8 K, $\rho_a(T)$ is strictly linear in temperature, with $\rho(T) = \rho_0 + AT$, i.e, $B=0$. Above 8 K, $\rho(T)$ acquires a curvature, which can be captured by the quadratic component, keeping the same value for $\rho_0$ as for the low temperature regime.

For the analysis of the (TMTSF)₂ClO₄ data the temperature domain was restricted from 0.1 and 17K since the actual temperature dependence above 17 K can be affected by the extrinsic influence of anion ordering occuring around 25K. The data for (TMTSF)₂ClO₄ at P = 4.9 kbar on fig.26 also reveal a resistivity which retains a finite temperature dependence approaching 0 K and follows a quadratic dependence above 12 K or so. What can be learnt from both systems on fig.26 is that the linear contribution to the resistivity cannot be strong at high temperature while the quadratic one cannot be dominant at low temperature. It is now becoming clear that a second order polynomial form of the resistivity such as $\rho(T) = \rho_0 + AT + BT^2$ keeping both factors A and B constant over the whole temperature range 0-20K
<figure><img src="image_27.png" /></figure>

Figure 26: Temperature dependence of the longitudinal resistivity below 20K, at zero field and under $H = 0.05$ T along $c^*$ in order to suppress SC. Second order polynomial fits $\rho(T) = \rho_0 + AT + BT^2$, according to the sliding fit procedure described in the text are shown for the $T$ intervals 2–6K and $T$ interval 18–22K in blue and red respectively. (TMTSF)₂PF₆ and (TMTSF)₂ClO₄ are displayed at the top and bottom respectively. For both compounds, top left insets show the temperature dependence of $A$ and $B$ coefficients and right insets the pressure dependence of the residual resistivity deduced from the low temperature fit at different pressures. See[178].

cannot afford for the experimental data.

An other approach can be taken instead in both compounds using a fitting procedure based on the second order polynomial fit such as $\rho(T) = \rho_0 + A(T)T + B(T)T^2$ for which both A and B prefactors are now temperature dependent with the value of $\rho_0$ dependent on pressure only. For the determination of $A(T)$ and $B(T)$ a sliding fit procedure of the experimental data has been applied using a window of 4K centered on $T$. Applying this fitting procedure to all successive pressure runs up to 21 and 17 kbar for (TMTSF)₂PF₆ and (TMTSF)₂ClO₄ respectively, the results for the $T$ dependences of $A$ and $B$ at different pressures can be extracted and displayed on fig.27. The data for (TMTSF)₂PF₆ are particularly interesting. They are also the more reliable data as the temperature dependence of the resistivity of (TMTSF)₂ClO₄ may be affected by the progressive orientation of anions below the anion ordering temperature. What is shown on the

<figure><img src="image_28.png" /></figure>

Figure 27: Top: Temperature dependence of $A$ (left) and $B$ (right) coefficients of the polynomial fit, $\rho(T) = \rho_0 + AT + BT^2$ at different pressures for (TMTSF)₂PF₆. Bottom: Temperature dependence of $A$ (left) and $B$ (right) coefficients of the polynomial fit, $\rho(T) = \rho_0 + AT + BT^2$ at different pressures for (TMTSF)₂ClO₄. Each temperature point corresponds to the center of a 4K window used for the fit. The lines for (TMTSF)₂PF₆ are guides for the eyes showing a crossing regime for $B$ at various pressures around 10K.

upper part of fig.27 is a decrease of $A$ and a concomitant decrease of $B$ versus $T$ on warming. However, it must be noticed that $B$ at low temperature is increasing under pressure *at variance* with the $B$ at high temperature which is decreasing, all curves crossing in the region of 10K, see fig.27. This behaviour could be related to the opening of a pseudogap in the density of states at Fermi level which will be discussed again in Sec.6.1.

Comparing the pressure dependence of $A$ determined in the window 2K-4K and $T_c$ is particularly enlightening. As far as (TMTSF)₂PF₆ is concerned, it is remarkable that both quantities follow parallel behaviours under pressure, fig.28. $A$ and $T_c$ are related in both compounds as it can be seen on fig.28. Using the values of $A$ determined from the linear regime at low temperature and $T_c$ from the onset temperature of SC one can see that $A$ correlates fairly well with the power law of $T_c$ and quite remarkably, that both vanish when $T_c \to 0$ with the exception for (TMTSF)₂ClO₄ when $T_c$ is becoming small, Fig.29. The power law in fig.29 is $A \propto T_c^{0.63}$ with an exponent which depends slightly on the position of the window over which the fit is performed. But in all cases, crucially, the existence of the linear term in the resistivity matches with the existence of SC. In other words, a pure $T^2$ Fermi liquid resistivity is only seen when $T_c$ has been completely suppressed with increas-
<figure><img src="image_29.png" /></figure>

Figure 28: Pressure dependence of $T_c$ onset and A coefficient, obtained from the low temperature ($2 < T < 6$K) polynomial fit and the sliding fit procedure described in the text, for $(TMTSF)_2PF_6$ (top) and $(TMTSF)_2ClO_4$ (bottom). The horizontal dashed lines indicate our lowest reached temperature, 0.08 K. Below this temperature, the vertical lines are A values for $(TMTSF)_2ClO_4$ obtained at 10.4 and 17 kbar without superconductivity. The dotted line is a linear fit of $T_c$ onset data points for $(TMTSF)_2ClO_4$.

ing pressure.

The case of the relation between A and $T_c$ for $(TMTSF)_2ClO_4$ is somewhat different from $(TMTSF)_2PF_6$. They both exhibit similar pressure dependence as long as $T_c$ is not becoming too small, i.e, at pressures smaller than 10 kbar or so, a finite value for A persists even in the absence of SC as shown by the pressure runs at 10.4 and 17 kbar where no SC can be observed above 0.1K, fig.28.

Such a difference between pressure dependences in these two organic superconductors can be explained by the particular role played by the anions in case of $(TMTSF)_2ClO_4$ as seen in Section.5.3.1 since the effect of a frozen anion weak disorder is to affect the stability of the superconducting phase (pair breaking effect) but not the inelastic contribution to the transport. A clear illustration for the different response of pair breaking and A to the presence of non magnetic impurities is given on fig.30 which display the SC transition seen from $\rho_c$ in $(TMTSF)_2ClO_4$ with various doping levels of $ReO_4$[88, 179]. Unlike A, no direct correlation can be established between B and $T_c$. The quadratic law

<figure><img src="image_30.png" /></figure>

Figure 29: A coefficient, obtained from the low temperature ($2 < T < 6$K) polynomial fit and the sliding fit procedure described in the text at different pressures, against $T_c$ onset for $(TMTSF)_2PF_6$. The line is a power fit through the origin $A = 0.379 T_c^{0.63}$.

<figure><img src="image_31.png" /></figure>

Figure 30: $(TMTSF)_2(ClO_4)_{1-x}(ReO_4)_x$ transverse resistivity showing the paraconducting behaviour even though $T_c$ is strongly affected by pair breaking[179].

is a well established behaviour in $(TMTSF)_2PF_6$ below 100K or so[72, 180]. This law was noticed at the beginning of the $(TMTSF)_2ClO_4$ study[164] and in the early studies of $(TMTSF)_2AsF_6$ under ambient pressure [72] and is still present under 21.8 kbar in $(TMTSF)_2PF_6$. The same is true for $(TMTSF)_2ClO_4$ even under 10.4 and 17 kbar when no SC can be observed.

Furthermore, for both compounds an experimental correlation can be established between the coefficient B at high temperature, say 20K, and the electronic spin susceptibility. As shown in fig.31, the prefactor B of the $T^2$ law taken at high temperature and the square of the electronic spin susceptibility given by the NMR data at high temperature ($\chi^2(q=0) \propto 1/T_1T$)[180, 183] follow the same pressure dependence. This behaviour is
<figure><img src="image_32.png" /></figure>

Figure 31: Pressure dependence of the B coefficient, obtained from the high temperature (26 < T < 30K) polynomial fit, and of the spin susceptibility measured under pressure via NMR relaxation experiments[180, 181]. The ambient pressure point is deduced from a-axis resistivity data in ref[72]. The relation B ∝ χ² is indicative of the Kadowaki-Woods relation encountered in strongly correlated metals[182].

actually reminiscent of the Kadowaki-Woods law ob-
served in various strongly correlated metals[182, 184].
Very much like A, the B contribution is not affected
by non magnetic impurities created by alloying or by
a frozen disorder of the anions[121]. B is controlled
by pressure only. An important behaviour is emerg-
ing from the analysis of the temperature dependence
of B displayed on fig.27. The prefactor B is re-
lated to the density of states at the Fermi level. Its
drop below 15K in (TMTSF)₂PF₆ and below 10K in
(TMTSF)₂ClO₄ suggests a concomitant drop of the den-
sity of states i.e, the occurence of a pseudo-gap at Fermi
level. However, the absence of any sign of pseudo-
gap in the static spin susceptibility[185] suggests that
the spin sector is not affected by the opening of this
pseudo-gap which is therefore restricted to the charge
sector only.

5.4.3. *Transverse transport*

The transverse transport along *c* corroborates most of
the results already obtained with *ρ*<sub>*a*</sub> in the low temper-
ature domain say, below, 15-20 K. At higher tempera-
tures the temperature dependence of *ρ*<sub>*c*</sub> deviates from
that of *ρ*<sub>*a*</sub>. This behaviour is ascribed to the cross over
from coherent to diffusive transport already mentioned
above and occuring along the direction of weakest cou-
pling in the temperature domain above 15-20 K. Since
we do not expect large effects on the transport proper-
ties crossing *T*<sub>cross</sub>, the cross over cannot be identified
with a great accuracy.

The theoretical treatment of the transverse resistiv-
ity ρ_c leads to a similar temperature dependence for
diffusive or coherent 3D anisotropic limits[162]. In

both cases the resistivity is proportional to 1/τ∥, hence,
ρ_c(T) ≈ ρ_a(T). However, the difference between
these extreme regimes is given by the anisotropy of
the resistance which reads ρ_c/ρ_a ≈ (t_a/t_c)²(a/c)² and
(t_a/t_c)(a/c)² in the limit of diffusive and coherent mo-
tion respectively[118]. There is a change of the ρ_c/ρ_a
anisotropy by a factor ≈ t_a/t_c = 750 going from one
extreme situation to the other (assuming for t_a and t_c
=1500 and 2 K respectively[93]. This factor can also
be 240 using the band parameters derived from the SC
critical fields in (TMTSF)₂ClO₄.

The cross over between these two extreme situations
is determined by the coupling along *c* and the longitudi-
nal (or in-plane) scattering rate *τ*∥(*T*). when the follow-
ing equation. For a diffusive to coherent transition to
take place at *T*<sub>cross</sub> the condition *ħ*/*τ*∥(*T*<sub>cross</sub>) ≈ *t*<sub>c</sub> must
be fulfilled crudely speaking. This means that the tem-
perature dependence of *ħ*/*τ*∥ has to be known. Hope-
fully, such an information is accessible from the knowl-
edge of the residual scattering rate, 1/*τ*<sub>0</sub> and the temper-
ature dependence of ρ<sub>a</sub> assuming that the temperature
dependence of ρ<sub>a</sub> is due to the scattering rate only.

We first consider the case of (TMTSF)₂ClO₄ in which
the residual scattering rate is known from the evolu-
tion of T_c in ReO₄ doped (TMTSF)₂ClO₄ samples, see
Section.5.3.1, 1/τ₀=0.2K ( 0.15 cm⁻¹).This determina-
tion of 1/τ₀ enables in turn the calibration of the resis-
tivity in units of degrees Kelvin.

With the previous determination of $\hbar/\tau_0 = 0.2K$ in a
pristine sample $\hbar/\tau_\parallel(T)$ reaches $t_c = 2K$ when $\rho_a(T_{cross})$
$= 10 \times \rho_0$ namely, a temperature $T_{cross} = 15K$ at low
pressure, rising up to 30K under 17 kbar according to
the measured temperature dependence of $\rho_a$, fig.16. If a
value of 5K is taken for $t_c$ according to the critical fields
data, Sec.5.1.2, $T_{cross}$ would rise up to 25 K and 65K
under ambient and 17 kbar pressure respectively. The
main factor increasing $T_{cross}$ under pressure is the de-
crease of B assuming a much smaller effect of pressure
on $t_c$.

This crude estimate of Tcross hints that in most of
the low temperature regime investigated in Section5.4.2,
transverse coherence prevails. The data of ρ_c(T) con-
firm a temperature dependence analogous to the tem-
perature dependence of ρ_a reported in Section5.4.2.
The c/a anisotropy measured by the anisotropy of B
amounts to about 4000 under 1.5kbar with only a weak
decrease under pressure. Moving to (TMTSF)₂PF₆ and
following under pressure the amplitude of the quadratic
law obeyed by ρ_c in the temperature domain 100-
20K reveals a drastic pressure dependence of the
<figure><img src="image_33.png" /></figure>

Figure 32: Pressure dependence of the resistivity anisotropy in (TMTSF)₂PF₆ derived either from the linear or from the quadratic contribution to ρ_c(T) [186]. The insert shows the high pressure domain where the pressure dependence of (TMTSF)₂PF₆ is smaller. Data for (TMTSF)₂ClO₄ given by the ratio of the B coefficients are also displayed with a shift of about 10 kbar for the upper pressure scale. The very small pressure dependence of the anisotropy suggests that (TMTSF)₂ClO₄ is already located well inside the coherent regime under ambient pressure [187].

anisotropy between 1 bar and 10 kbar, see, fig.32 and a much smoother one above. Crudely speaking the c/a anisotropy evolves from 400,000 at low pressure to 120,000 under 4.75 kbar and 6000 under 10 kbar and 4800 under 22 kbar. This is very likely the signature of a transverse transport evolving from diffusive at low pressure above 20K to coherent below 20K under 10 kbar. As shown by the data on fig.32 the large decrease of ρ_c between 1 bar and 10 kbar can be ascribed to the pressure dependence of the c/a anisotropy accompanying the cross over since the prefactor B_a from ρ_a does not reveal such a dramatic pressure dependence in particular between 1 bar and 10 kbar. The c axis data are in striking contrast with the smooth pressure dependence of B_a from the longitudinal resistivity displayed on fig.31. The evolution of the c/a anisotropy by a factor 100 is actually not that far from the value 750-200 expected above between the two extreme situations depending on the choice of t_c. The data of anisotropy are in fair agreement with the existence of a c-transverse cross over regime displayed on fig.33. A deviation at 35K from the Kohler's rule followed by the quadratic resistivity of (TMTSF)₂PF₆ under ambient pressure has been attributed to the onset of the coherent-diffusive transition [163, 188].

<figure><img src="image_34.png" /></figure>

Figure 33: Generic (TM)₂X phase diagram on which the c* cross-over regime has been added.

# 6. Discussion

## 6.1. A charge pseudo-gap?

Given the new data showing a temperature dependence of the B prefactor for the quadratic law of the resistivity It is worth recalling some old experiments performed on (TMTSF)₂PF₆ and (TMTSF)₂ClO₄ which used to be controversial and never led to clear conclusions. A first case is given with the study of (TMTSF)₂ClO₄ down to 1.2K by far infrared spectroscopy which has shown the onset of a strong magneto-absorption signal in FIR at 3.8 meV [189] which had been associated with an energy gap in the electronic density of states although not related to neither superconductivity nor to a SDW state according to the authors. A second case comes from quasi particle tunneling experiments of S/I/S junctions on either (TMTSF)₂ClO₄ [190]. The tunneling spectra of (TMTSF)₂ClO₄/insulator/lead did show a pseudo-gap in the density of states of 2Δ=3.4 meV at 1.2K which remained present even under a magnetic field strong enough to suppress the superconductivity of lead. This result was objected by the data of an other tunneling experiment [191] which failed to detect such a pronounced anomaly in the density of states of the organics. The discrepancies between the two studies could be due to the quality of the junctions since the authors of reference [190] emphasized that they were able to detect the structure due to electron-phonon interactions of superconducting lead on the wings of the tunneling characteristics, a proof of high quality tunnel junctions. A similar gap 2Δ=3.6 meV in a
(TMTSF)₂PF₆ Schottky tunneling device has been reported under 11 kbar[192] at a temperature of 50 mK. Since such a large value cannot be attributed to the quasiparticle gap of superconducting (TMTSF)₂PF₆, which cannot be much larger than 2Δ =0.6 meV or to other superconducting materials in such a Schottky barrier it can be related to some intrinsic property of the organic electrode. This gap survives the suppression of the (TMTSF)₂PF₆ superconductivity up to at least 10K. The reports of tunneling experiments on (TMTSF)₂ClO₄/insulator/ Lead junctions showing structures due to the superconductivity of the organic and lead with a shallow depression of the density of states persistis even if superconductivity of lead is destroyed under magnetic field[190, 191].

Finally, there are two contradictory reports for the spin susceptibility of (TMTSF)₂ClO₄ at low temperature. The measurement of the ESR susceptibility[193] does reveal a significant drop below 30K or so whereas the measurement of the susceptibility with a Faraday set up reveals a nearly temperature independent susceptibility below 30K[100, 185]. Such an apparent flatness of χ<sub>Faraday</sub> might possibly be ascribed to some compensation coming from a low temperature Curie tail[193] but since the ESR line becomes Dysonian at low temperature because of the high conductivity of the material, the ESR data may not be as reliable as the static measurements and the static susceptibility data should the one to take into consideration.

In conclusion, the existence of a pseudo-gap in the density of quasi particle states below 15K with a width 2Δ of about 36-40 K seems to be supported by several experimental results. It is probably *not* due to 1D superconducting fluctuations as believed earlier but may be related to the decrease of the coefficient *B* below the temperature Δ ≈ 15K and to the theory which will be sketched in the following section.

## 6.2. Antiferromagnetic fluctuations

Transport has firmly established a close connection between the existence of the linear in temperature resistivity and superconductivity. What about AF fluctuations which have been detected in the early NMR studies of these materials[69].

We have access to the contribution to the relaxation provided by AF fluctuations subtracting from the experimental data the normal contribution fluctuations (q = 0) following a Korriga law. When such a subtraction is performed, (assuming a regular Korringa relaxation at low temperature extrapolated from the high temperature limit), the additional relaxation can be compared to the amplitude of the anomalous scattering rate in transport.

Such a comparison is shown in figure.34 in the case of (TMTSF)₂PF₆ and (TMTSF)₂ClO₄ where the additional contribution to the (q = 0) Korringa relaxation, Δ(1/T₁)/T and A, are both seen to decrease at increasing temperature. Since transport and NMR experiments in (TMTSF)₂PF₆ have not been conducted at exactly the same pressure a second best approach consists in using the transport data of the measurements at 11.8 kbar and the NMR data available at 11 kbar in the literature[181] (inset, fig.34). The comparison between temperature

<figure><img src="image_35.png" /></figure>

**Figure 34:** Top: Temperature dependence of the fluctuation-induced relaxation Δ(1/T₁)/T at 11 kbar deduced from NMR data in ref[181] and of the coefficient of the linear resistivity (A) at 11.8 kbar for (TMTSF)₂PF₆. The insert shows the ⁷⁷Se relaxation data at four different pressures where the bump of extra relaxation coming from AF fluctuations is clearly seen[181]. Similar results have been obtained in ref[194]. Bottom: Temperature dependence of the fluctuation-induced relaxation Δ(1/T₁)/T at 5 kbar at room temperature from ref[195], estimated to be around 1.5 kbar at low temperature, and of the coefficient of the linear resistivity (A) at 1.5 kbar for (TMTSF)₂ClO₄.

dependencies of A and Δ(1/T₁T) should not be taken at face value. It is only indicative of a common origin for the linear law in transport and the enhancement of relaxation, the onset of AF fluctuations at low temperature.

## 6.3. What about paraconductive fluctuations?

After the recent reinvestigation of the transport properties in the metallic phase it is essential to return to the problem which made transport of
(TMTSF)₂X anomalous at low temperature namely, the paraconductivity[118, 196]. As shown on fig.35 a clear downward curvature is observed in *R* vs *T* below 3K when the pressure is close to the critical pressure for the suppression of the SDW phase. Such a behaviour cannot be ascribed to some inelastic scattering, it must be the signature of a paraconductive contribution adding to the normal component of the conductivity. The paraconductive contribution reads $\Delta\sigma = (\rho_n - \rho_{ex})/\rho_n\rho_{ex}$ where $\rho_{ex}$ is the measured resistivity and $\rho_n$ is the normal contribution provided from the fit of the experimental data in the low temperature regime above 4K. In the case of (TMTSF)₂PF₆ under 10 kbar displayed on fig.35 the normal resistivity is linear between 4 and 8K. The paraconducting contribution derived on the bottom fig.35 shows a power law dependence as $\epsilon^{-\alpha}$ with $\alpha$ evolving from 1/2 to 3/2 at increasing temperatures. Above 3K or so the determination of $\Delta\sigma$ is no longer reliable because it becomes more sensitive to the procedure used to determine the normal contribution. Paraconductivity can be detected on both *c* and *a* components of the resistivity with the same percentage although it may be easier to see on $\rho_c$ than $\rho_a$ for a question of signal to noise ratio since the transverse signal is much larger than the longitudinal one. Figure.30 provides some illustration for the paraconducting behaviour in the vicinity of $T_c$ in some samples of (TMTSF)₂(ClO₄)ₐ⁻(ReO₄)ₓ. The downward curvature is quite visible approaching $T_c$ even in samples where SC is depressed by the pair breaking effect of non magnetic impurities.

## 6.4. Some theory

This article is not proposing an exhaustive survey of all 1D theories proposed over the years to understand the main features of the (TM)₂X phase diagram including the onset of superconductivity. The state of the art can be found in reviews[197, 198] or in the comprehensive textbook by Giamarchi[58]. However, as the field of conducting organic crystals has benefited a lot at its beginning from the somewhat provocative promises made by Little of a superconductivity at high temperature in 1D organic conductors mediated by a non phonon pairing mechanism it is interesting to see after 30 years of work on the subject what the situation is in terms of superconducting mechanism.

The one dimensional electronic structure and the dominant role of magnetism in organic conductors are inherent to the physical properties of a large number of organic salts pertaining to the first generation of conductors[199, 200]. Furthermore, the dominant role

<figure><img src="image_36.png" /><figcaption>Figure 35: (Top), Resistivity of (TMTSF)₂PF₆ along the *c* axis under 10 kbar together with the fit for the normal component in the low temperature domain. (Bottom), Excess conductivity plotted in a log-log scale versus $\epsilon = (T - T_c)/T_c$ and $T_c$ (midpoint)=1.05 K.. The different lines illustrate the power law divergence with the exponents 1/2, 1 and 3/2 corresponding to the AL paraconductivity in 3, 2 and 1D. The lower insert displays the excess conductivity versus *T*, *i.e.*, $T_c=0$ K which emphasizes the 1D regime.</figcaption></figure>

of on-site Coulomb repulsion received a brilliant confirmation on the first 1D metal-like charge transfer compounds TTF – TCNQ with the observation in diffuse X-ray experiments of a $4k_F$ scattering besides the regular $2k_F$ Peierls one[36].

As to the single chain commensurate organic conductors (TM)₂X, the existence of repulsive interactions is visible everywhere in the generic phase diagram. It is responsible for the localization which takes place at high temperature for compounds located in the upper left side of the diagram due to the existence of an Umklapp scattering term in these half field conductors. It is also visible through the stabilization of an antiferromagnetic (insulating) ground state once the carriers become deconfined over *a* – *b* planes with a Q1D Fermi surface either chemically, the case of (TMTSF)₂ClO₄ or under pressure for (TMTSF)₂PF₆. Furthermore, even when the alteration of the Fermi surface nesting prevents the establishment of a SDW long range order strong anti-
<figure><img src="image_37.png" /><figcaption>Figure 36: (a)Temperature variation in the normalized static susceptibility in the SDW and SCd channels at various transverse coupling $t'_{\perp b}$ on either side of the threshold value $t'^*_{\perp b}$. (b) RG phase diagram of the Q1D electron gas model. The dashed line for the pressure dependent $\Theta$ stands as the CW scale in the SC sector. The horizontal dotted line defines the temperature domain for the CW behaviour. From reference[203]</figcaption></figure>

ferromagnetic fluctuations are still visible at low temperature through the enhancement of the nuclear relaxation above the regular Korringa contribution. In the context of strong electron-electron repulsions it is hard to imagine how a pairing could take place to provide the necessary attraction between quasiparticles to form a Cooper pair. However, a possibility to overcome this repulsion has been investigated in the sixties by Kohn and Luttinger[201] who suggested the possibility of a non phonon attractive pairing through the mediation of $2k_F$ charge fluctuations around the electrons. Emery in 1983 considered the formation of the pairs between electrons on neighbouring stacks[202] leading to a gap function with nodes on the Fermi surface but did not keep this option on the basis of the specific heatjump data at $T_c$. The Sherbrooke group then proposed an attractive pairing mechanism based on the exchange of spin fluctuations between neighbouring stacks[181].

There is little doubt that in the deconfined regime of the (TM)₂X diagram a weak coupling approach is valid. This hypothesis is justified by the treatment of the SDW ground state which is the stabilization of an itinerant an-

tiferromagnet by nesting of the Q1D Fermi surface. An other justification came from the treatment of the field-induced SDW phases for which a weak coupling explanation has been highly successful[108, 204]. Hence, it is in the weak-coupling limit that a the theory of the metallic phase of (TM)₂X compounds has been developed with the help of the weak coupling renormalization group in the framework of the repulsive Q1D electron gas by the Sherbrooke group. Hence, a solution to the problem of the interplay between antiferromagnetism and superconductivity in the Bechgaard salts has been worked out[155, 205, 206]. The calculated phase diagram as shown on fig.36 captures the essential features of the experimentally-determined phase diagram of (TMTSF)₂PF₆ [203]. The superconducting state below $T_c$ has d-wave symmetry [154], with pairing coming from antiferromagnetic spin fluctuations. The tuning parameter in this theory which mimics the effect of pressure is the nesting frustration of the Fermi surface,i.e, the second near neighbour interaction between adjacent stacks in the $b'$ direction. As long as $P < P_c$ namely, $t'_{\perp b} < t'^*_{\perp b}$ on fig.36 (25.6 K with parameters of the theory[203] the susceptibility $\chi_{SDW}$ goes like $(T-T_{SDW})$ signaling the onset of the long range AF state at $T_{SDW}$. When the transverse coupling approaches the critical value $T_{SDW}$ drops suddenly although the phase coexistence regime, see Section.5.2.1, is not accounted by this theory. However, when $T_{SDW}$ reaches a non zero minimum value at $t'_{\perp b} = t'^*_{\perp b}$ the d-superconductivity channel takes over at the temperature $T_c$ continuously decreasing at increasing values of $t'_{\perp b}$.

The normal state above $T_c$ is characterized by the constructive interference of antiferromagnetic and pairing correlations, which enhances the amplitude of spin fluctuations [203].

The antiferromagnetic correlation length $\xi(T)$ increases according to $\xi = c(T + \Theta)^{-1/2}$ as $T \to T_c$, This correlation length is expected to impart an anomalous temperature dependence to any quantity that depends on spin fluctuations. For instance, it was shown [203] to account in detail for the deviation from the Fermi-liquid behavior in the NMR relaxation rate measured in the Bechgaard salts [154, 183, 210]. In this regime $\chi_{SDW}$ exhibits a Curie-Weiss behaviour for a 2D antiferromagnet[211, 212] at a temperature $T < \Theta$ where $\Theta$ is a positive temperature scale increasing very sharply under pressure[203]. The low temperature CW regime is characterized by a very specific behaviour of the relaxation time $T_1T$ going like $C(T + \Theta)$ which has been first noticed by Wu et-al[209] in (TMTSF)₂PF₆ under pressure and sub-
<figure><img src="image_38.png" /><figcaption>Figure 37: Temperature dependence of $^{77}$Se $T_1T$ in (TMTSF)₂PF₆ under 9 kbar Θ= 1.5 K, under 11 kbar, Θ= 10 K from the data in [207, 208] and (TMTSF)₂ClO₄ under ambient pressure (circles) from the data in [69] Θ=1.5K. A value of Θ=11 K was found for (TMTSF)₂PF₆ under 9 kbar [209]. The differences between experiments may be due to differences in pressure calibrations (ΔP ≈ 2 kbar).</figcaption></figure>

sequently by Shinagawa et-al for the relaxation of (TMTSF)₂ClO₄ studied at low temperature[154]. Data for (TMTSF)₂ClO₄ and (TMTSF)₂PF₆ under pressure obtained from refs[69, 183] are displayed on fig.37. This figure suggests that (TMTSF)₂ClO₄ under ambient pressure and (TMTSF)₂PF₆ under 11 kbar are located very close to the quantum critical point at Θ=0 K. It also illustrates the fast increase of Θ under pressure.

Through Umklapp scattering, antiferromagnetic spin fluctuations will also convey an anomalous temperature dependence to the quasi-particle scattering rate $\tau^{-1}$, in addition to the regular Fermi-liquid component which goes as $T^2$. Evaluation of the imaginary part of the one-particle self-energy yields $\tau^{-1} = aT\xi + bT^2$, where a and b are constants. It is then natural to expect the resistivity to contain a linear term $aT$ (in the limit $T \ll \Theta$), whose magnitude would presumably be correlated with $T_c$, as both scattering and pairing are caused by the same antiferromagnetic correlations. The combined effect of pairing and antiferromagnetic correlations conspires to produce the remarkably linear resistivity observed in (TMTSF)₂PF₆ on the border of SDW order. However, the theory will have to be improved to account for the possibility of a pseudo-gap opening in the charge sector below 15K without any effect on the spin sector.

## 7. Conclusion

This short presentation of superconductivity in organic matter started with an historical approach because it can be instructive to see how the whole story has developed from quite provocative theoretical

statements[13] and subsequently led to the discovery of the first organic superconductor thanks to a very cooperative and productive work between Chemistry and Physics communities over the past 35 years. It is worth citing V.L.Ginzburg in this conclusion[213]: *I believe that it was undoubtedly the discussion of the possible exciton mechanism of superconductivity that stimulated the search for such superconductors and studies of them (organic superconductors and intercalated layered superconductors).* In addition, an other statement of Ginzburg: *the organic superconductors are clearly interesting by themselves or, to be more precise, irrespective of the high $T_c$ problem cannot be more appropriate when one consider the development of this field.* While the actual mechanism behind organic superconductivity is still not fully settled we can be confident in saying that the model proposed by Little in 1964 is probably not the one most appropriate to these materials.

Thanks to the determining role of high pressure governing their physical properties, organic conductors (and superconductors) have shown their superiority compared to high $T_c$ cuprates in terms of purity and in the variety of phenomena which can be studied in a single system keeping both structure and chemical purity constant. Recently, it was even believed that the entire physics of organics including superconductivity could be reproduced without the burden of high pressure techniques tuning the carrier concentration with a field effect technique. Unfortunately, the numerous articles published on this subject cannot be cited anymore but it is an interesting aspect of superconductivity at the interfaces between two insulators which is now emerging on far more solid grounds[214].

A perfect illustration for the scientific interest of organic conductors is provided by the amount of knowledge and new physical phenomena which came out of the studies of organic superconductors. To summarize, the optimized location for the superconductivity $T_c$ close to the Mott localization resulting from a compromise between the interchain coupling and the strength of the correlations and the likeliness of an unconventional pairing mechanism which remains a challenging topic and awaits new experimental evidences. It is also important to mention the magnetic field confinement discovered in the 1D's leading to the phenomenon of field induced spin density wave phases[172, 174, 202] and the quantization of the Hall effect in these phases[108, 174, 175]. Furthermore, the angular dependent magnetoresistance specific to these anisotropic conductors[215, 216], the so-called Lebed oscillations in (TMTSF)₂X provide a nice illustration
for the new features of the quasi-1D conductors[217]. All these fascinating phenomena are beyond the scope of this survey but are well documented in recent review books[197, 198].

It is worth concluding with the most recent developments. There exists in the low-temperature metallic phase of the salt (TMTSF)₂PF₆ close to the SDW order a correlation between SC and two properties of the metallic phase: (i) the non-Fermi liquid linear-T resistivity as $T \to 0$ and (ii) the contribution to the spin-lattice relaxation coming from the presence of antiferromagnetic fluctuations close to the SDW instability. Away from the SDW phase at elevated temperatures, the resistivity acquires a $T^2$ curvature whose pressure dependence appear to correlate well with the Korringa law observed in NMR experiments when $T_c \to 0$. *The correlation between non-Fermi-liquid resistivity and superconducting $T_c$ further suggests that anomalous scattering and pairing have a common origin* [203, 218, 219].

A similar correlation has been observed in (TMTSF)₂ClO₄ under pressure[220] where NMR studies have also been performed. Furthermore, the study of (TMTSF)₂ClO₄ has shown that organic superconductivity is governed by two control parameters: (i) the intrinsic control parameter monitored by the intrinsic strength of interactions (pressure) and (ii) the pair breaking mechanism governed by the elastic electron life time which is quite influential in these spin-singlet superconductors[154] with line nodes in the gap[125].

We also note that the phase diagram of the iron-pnictide superconductors Ba(Fe₁-xCox)₂As₂, with its adjacent semi-metallic SDW and superconducting phases[221–223], closely resembles that of (TM)₂X. The present data analysis supports the suggestion made in ref[218] regarding a similar interpretation holding for both families of materials.

Among the work and those remaining for the future of Q1D organic superconductors we can make a few suggestions:

1. A reinvestigation of the FIR properties of (TMTSF)₂ClO₄ at low temperature and very low frequency.

2. Accurate measurements of the nuclear relaxation at low temperature in the metallic state of (TMTSF)₂PF₆ and (TMTSF)₂ClO₄ versus pressure.

3. The measurements of the transport properties and NMR relaxation in alloyed-(TMTSF)₂ClO₄ under pressure at low temperature.

4. The studies of the superconducting phase of

(TMTSF)₂ClO₄ via specific heat and transport techniques under aligned fields.

Thinking at the present situation of these Q1D organic superconductors and how they could be helpful for understanding the physics of other correlated superconductors, HTSC, pnictides, heavy Fermions, it is appropriate to say that *a mouse may be of service to a lion*!

This survey of organic superconductivity is certainly incomplete without mentioning the other class of organic superconductors based on the molecule BEDT-TTF making two-dimensional conducting 2:1 salts, (ET)₂X, with various polymeric monoanions[224, 225]. The 2D compounds have provided not only higher values of $T_c$ with evidences for d-like pairing[226–228] but also textbook examples for the two dimensional fermiology and a playground for the study of the 2D Mott transition with a common border with superconductivity and a "bad" metal at high temperature plus a spin pseudo-gap[229].

What about higher $T_c$'s in Q1D compounds? A recent study of an organic salt of the BEDT-TTF molecule, the polytype $\beta'$(BEDT-TTF)₂ICl₂ under a very high pressure has revealed a superconducting transition at 14 K at 8.2 GPa[230]. This compound is claimed to be a 1D Mott insulator at ambient pressure with a T/P phase diagram analogous to the generic (TM)₂X diagram. If the open nature (1D) of the Fermi surface under pressure is confirmed, this compound would thus be the organic conductor with the highest $T_c$. It would also suggest that high $T_c$'s are not restricted to the higher dimensionality (2D or 3D) compounds.

## 8. Acknowledgments

At the end of this historical perspective of organic superconductors I wish to recall the contribution of several colleagues, in particular my close colleagues who have played a pioneering role in this research. J.Friedel, 40 years ago, pointed out the relevance of low dimensionality (1 and 2D) in superconductivity[7] and who has continously supported the activity in one dimensional conductors at Orsay, M.Weger who brought the first TTF-TCNQ samples to Orsay in 1974 for physical investigations under pressure[32], K.Bechgaard with whom the cooperation established in the seventies is still very much active, M.Ribault who understood the importance of linking high pressure and very low temperature measurements and played a crucial role in the initial discovery. A lot of experiments and data processing would not have been performed without the contri-
bution of P. Auban-Senzier, P. Wzietek and C.Pasquier at Orsay and P. Batail at Angers. S.Brown has contributed to solve the controversy about the parity of organic superconductivity. I wish to mention the important contribution made by S. Yonezawa in the laboratory of Y. Maeno taking advantage of very low temperature facilities in accurately aligned fields to study the non conventional properties of the superconducting phase of (TMTSF)₂ClO₄. The theoreticians T.Giamarchi and the late H.J.Schulz are very close colleagues who made very significant contributions to the one dimensional theory and who have been deeply involved in the tight cooperation with experimentalists. Last but not least, I wish to acknowledge the Sherbrooke group, C. Bourbonnais and L. Caron who have developed an original and remarkably fruitful theoretical approach to 1D conductors in close cooperation with experimentalists at Orsay over the last 25 years and L.Taillefer and N. Doiron-Leyraud who made recently a remarkable contribution connecting organics to other strongly correlated superconductors.

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