Nature of superconductivity in quasi-one-dimensionalorganic Bechgaard superconductors

Denis Jeromea,∗, Shingo Yonezawab

aLaboratoire de Physique des Solides, CNRS UMR 8502, Universite Paris-Sud, F-91405 Orsay, FrancebDepartment of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

Abstract

It is the saturation of the transition temperature Tc in the range of 24 K for knownmaterials in the late sixties which triggered the search for additional materials be-ing able to offer new coupling mechanisms leading in turn to higher Tc’s. As a re-sult of this stimulation, superconductivity in organic matter has been discovered intetramethyl-tetraselenafulvalene-hexafluorophosphate, (TMTSF)2PF6, in 1979, in thelaboratory founded at Orsay by Professor Friedel and his colleagues in 1962. Althoughthis conductor is a prototype example for low dimensional physics, we intend to re-strict ourselves in this article to the superconducting phase of the ambient pressuresuperconductor (TMTSF)2ClO4, in which most studies of the superconducting phasehave been performed. We shall present a series of experimental results supportingnodal d-wave symmetry for the superconducting gap in these prototypical quasi-one-dimensional conductors.

Keywords: One dimensional conductors, Organic superconductivity,

Bechgaard salts, (TMTSF)2ClO4

1. Introduction — Historical overview

Searching for new materials exhibiting the highest possible values for the super-conducting critical temperature Tc was a strong motivation in materials sciences in theearly 70’s, and the term high temperature superconductor was already commonly usedreferring to the intermetallic compounds of the A15 structure, namely materials suchas Nb3Sn or V3Si [1].

Expending the very successful explanation of the isotope effect in the Bardeen-Cooper-Schrieffer (BCS) theory, other models were proposed in which excitations ofthe lattice responsible for the electron pairing had been replaced by higher-energy ex-citations, namely, electronic excitations, with the hope of finding new materials withTc higher than those explained by the BCS theory. Consequently, the small electronicmass me of the polarizable medium would lead to an enhancement of Tc of the or-der of (M/me)1/2 times the value which is observed in a conventional superconductor

∗corresponding author

Preprint submitted to Elsevier August 27, 2019

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where M is an atomic mass. This is admittedly a huge factor. V. L. Ginzburg [2, 3]considered in 1964 the possibility for the pairing of electrons in metal layers sand-wiched between polarizable dielectrics through virtual excitations at high energy. But,the most provocative suggestion came from W. A. Little in 1964 [4, 5], who predictedroom-temperature superconductivity with a new pairing mechanism leading to a drasticenhancement of the superconducting Tc.

The idea of Little was rooted in the extension of the isotope effect proposed byBCS, replacing the mediating phonon by an electronic excitation in especially designedquasi-one-dimensional (Q1D) macromolecules. However a prerequisite to the model ofLittle was the achievement of metallic conduction in organic molecular crystals. Thiswas not a trivial problem in the sixties.

A short time later, the synthesis of the first stable organic compound displayingmetallic conduction below room temperature, the charge transfer complex TTF-TCNQcame out. This compound is made up of two kinds of flat molecules each forming seg-regated parallel conducting stacks. It fulfills the conditions for an organic conductor asthe orbitals involved in the conduction (π-HOMO, highest occupied molecular orbitaland π-LUMO, lowest unoccupied molecular orbitals for TTF and TCNQ respectively)are associated with the molecule as a whole rather than with a particular atom. Freecarriers within each stacks are given by an interstack charge transfer at variance withother organic conductors known at that time such as the conducting polymers, in whichcharges are provided by doping [6]. However, the conducting behaviour in TTF-TCNQis stopped at low temperature by a metal-insulator transition accompanying a Peierlsdistortion [7]. The Peierls ground state turned out to be very robust despite numerousattempts to suppress it under high pressure making the 1D conductor in turn more twodimensional [8, 9]. After more than thirty years, the insulating state is finally found tobe almost suppressed at pressure as high as 8 GPa [10].

The clue to overcome the natural tendency for a 1D conductor to undergo a Peierlstransition towards an insulating ground state came after a fair amount of experimentalworks in physics together with chemistry using the newly discovered organic donortetramethyl-tetraselena-fulvalene TMTSF [11].

The Copenhagen group led by Klaus Bechgaard, very experienced with the chem-istry of selenium succeeded in the synthesis of a new series of conducting salts allbased on the TMTSF molecule with the stoichiometry 2:1 namely, (TMTSF)2X, whereX is an inorganic mono-anion with various possible symmetry, octahedral (PF6, AsF6,SbF6, TaF6), tetrahedral (BF4, ClO4, ReO4) or triangular (NO3) [11]. All these com-pounds but the one with X = ClO4 did reveal an insulating ground state under ambientpressure.

What is so special with (TMTSF)2PF6 , the prototype of the so-called Bechgaardsalts, unlike previously investigated TTF-TCNQ, is the magnetic origin of the ambient-pressure insulating state [12] contrasting with the Peierls-like ground states discoveredpreviously in charge transfer compounds. The ground state of (TMTSF)2PF6 turnedout to be a spin density wave (SDW) state as shown in Fig. 1, similar to the predic-tions made by Lomer [13] in 1962 and by Overhauser [14] for metals. However, theSDW has been suppressed under an hydrostatic pressure of about 9 kbar enabling thestabilization of a metal-like conduction down to liquid helium temperature, and finallythe stabilization of superconductivity below 1K found back in December 1979 [15], as

2

presented in Fig. 2.It is now established that the Q1D electron gas model with weak-coupling limit

explains fairly well the properties of the SDW phases in (TMTSF)2X materials, both thesuppression of the SDW phase under pressure [16, 17] and the stabilization of magneticfield-induced SDW phases [18, 19]. The non-interacting part of the Q1D electrongas model is defined in terms of a strongly anisotropic electron spectrum yielding anorthorhombic variant of the real open Fermi surface in the ab plane of the Bechgaardsalts. The spectrum E(k) = vF(|k| − kF) − 2tb cos kb − 2t′b cos 2kb as a function of themomentum k = (k, kb) is characterized by an intrachain or longitudinal Fermi energyEF = vFkF, which revolves around 3000 K in (TMTSF)2X [20, 21]; here vF and kF

are the longitudinal Fermi velocity and wave vector. This energy is much larger thanthe interchain hopping integra l tb (≈ 200 K), in turn much bigger than the second-nearest neighbor transverse hopping amplitude t′b. The latter stands as the antinestingparameter of the spectrum, which simulates the main influence of pressure in the model.

The unnesting parameters of the band structure, t′

b and similarly t′

c for the c∗ di-rection both play an important role in the T − P and T − P − H phase diagrams of(TMTSF)2X. When t

′

b exceeds a critical unnesting band integral of the order of theSDW transition temperature (≈ 15 − 30 K) in case of complete nesting [16, 17], theSDW ground state is suppressed in favour of a metallic phase with the possibility ofrestoration of SDW phases under magnetic field along the c∗ axis [22].

The close proximity between antiferromagnetism and superconducting groundstates of (TM)2X superconductors and the deviation of the metallic phase from thetraditional Fermi liquid behaviour have been recognized as early as the beginning ofthe eighties. The possibility for a pairing mechanism involving carriers on neighbour-ing chains in these quasi 1D conductors avoiding the Coulomb repulsion has beenproposed by V. Emery in the context of the exchange phonon mechanism [23]. Soonafter, Emery and coworkers introduced the possibility that antiferromagnetic fluctua-tions play a role in the paring mechanism [24, 25] but concluded that superconductivitycould not emerge from pairing on the same organic chain. The exchange of spin fluc-tuations between carriers on neighbouring chains was thus proposed [24] to providethe necessary glue for pairing in analogy with the exchange of charge density wavesproposed by Kohn and Luttinger [26] in the context of a new pairing mechanism in lowdimensional conductors.

In the context of superconductivity in heavy fermions metals discovered the sameyear as organic superconductivity [27], J. Hirsch performed a Monte Carlo simulationof the Hubbard model showing an enhancement of anisotropic singlet pairing correla-tions due to the on site Coulomb repulsion leading eventually to an anisotropic singletsuperconducting state [28].

One year later, L. Caron and C. Bourbonnais [29, 30] extending their theory for thegeneric (TM)2X phase diagram to the metallic domain made the proposal for a gapequation with singlet superconductivity based on an interchain magnetic coupling withan attraction deriving from an interchain exchange interaction overcoming the on-stackCoulomb repulsion. Taking into account the interference between the diverging Cooperand Peierls channels, the renormalization treatment of Q1D conductors received morerecently a significant improvement [31]. An overview of the theory of 1D conductorscan be found in the textbook by T. Giamarchi [32].

3

c

a

(a)

Figure 1: (a) Side view of the TMTSF molecule (yellow and red dots are selenium and carbon atoms re-spectively, hydrogens not shown) and (TMTSF)2PF6 Q1D structure seen along the b axis, courtesy of J.Ch.Ricquier, IMN, Nantes. The yellow and green clouds around the atoms schematically present real-spacedistribution of molecular orbitals responsible for the electronic conduction. (b) Generic phase diagram forthe (TM)2X family [33] based on experiments on the sulfur compound (TMTTF)2SbF6 under ambientpressure taken as the origin for the pressure scale. All colored phases are long-range ordered. The curvebetween the 1D Metal and charge localization marks the onset of 1D charge localization, which ends around15 kbar, slightly above (TMTTF)2Br . The 1D to 2D deconfinement occurs at the continuous line in thehigher-pressure regime. The line between 2D and 3D regimes defines the upper limit for low temperature3D coherent domain. There exists a small pressure window around 45 kbar in this generic diagram whereSC coexists with SDW according to reference [34, 35, 36]. (TMTSF)2ClO4 is the only compound to exhibitsuperconductivity under ambient pressure.

For several experimental reasons we are now entitled to attribute the pairing inorganic superconductivity to a mechanism which may differ from the regular electron-phonon driven pairing in traditional superconductors. First, superconductivity of Q1DBechgaard salts shares a common border with magnetism as displayed on the genericdiagram in Fig. 1. Second, strong antiferromagnetic fluctuations in the normal stateabove Tc in the vicinity of the SDW phase provide the dominant contribution to the nu-clear hyperfine relaxation and also controls the linear temperature dependence of elec-tronic transport. Some experimental results point to the existence of a non-conventionalpairing mechanism. They will be summarized below.

2. Basic properties of superconductivity

Although superconductivity in organic conductors has first been stabilized underpressure [15] (see Fig. 2), more detailed investigations of this phenomenon have beenconducted in (TMTSF)2ClO4 for experimental reasons since it is the only compoundof the Q1D-Bechgaard salts series which exhibits superconductivity at 1.2 K underambient pressure. Additional evidences for superconductivity in (TMTSF)2X con-ductors came out from (TMTSF)2ClO4 transport studies [37], specific heat measure-ments [38, 39] and Meissner flux expulsion (Fig. 3) [40]. More recent specific-heatdata are presented in Sec. 3.1.

4

Figure 2: First observation of superconductivity in (TMTSF)2PF6 under a pressure of 9 kbar [15]. Theresistance of two samples is normalized to its value at 4.5 K.

Regarding evidences from the Meissner expulsion, the lower critical field Hc1 ob-tained from the Meissner magnetization curves at low temperature, 0.2, 1, and 10 Oealong axes a, b′ and c∗, respectively. Following the values for the upper critical fieldsHc2 derived either from the Meissner experiments and the knowledge of the thermody-namical field [40] or from a direct measurements of transport, superconductivity is inthe extreme type II limit. The Ginzburg-Landau parameter κ can even overcome 1000when the field is along the a axis due to the weak interchain coupling in these Q1Dconductors making the field penetration very easy for this external-field configuration.An interpretation for the critical fields assuming the clean limit has been suggested in1985 [41]. According to this theory, the slopes of Hc2(T ) near H = 0 should be givenby:

Hc2 ‖ a(T ) =98.7 × 103

tb′ tc∗Tc0(Tc0 − T ) , (1)

Hc2 ‖ b′ (T ) =199 × 103

tc∗ taTc0(Tc0 − T ) , (2)

Hc2 ‖ c∗ (T ) =365 × 103

tatb′Tc0(Tc0 − T ) , (3)

where Hc2 is given in unit of kOe and the hopping integrals in K [42]. This proposalwas based on the calculation of the microscopic expressions for the effective masstensor in the Ginzburg-Landau equation near Tc [43].

Given the experimental determination of the critical field derivatives near Tcof (TMTSF)2ClO4 from transport studies [44, 42], dHc2 ‖ a/dT = −67 kOe/K,dHc2 ‖ b′/dT = −36 kOe/K, and dHc2 ‖ c∗/dT = −1.5 kOe/K, Eqs. (1-3) lead to bandparameters ta : tb′ : tc∗ = 1200, 310, and 7 K, respectively. If one use slopes from athermodynamic study [45], dHc2 ‖ a/dT = −81 kOe/K, dHc2 ‖ b′/dT = −23 kOe/K, anddHc2 ‖ c∗/dT = −1.1 kOe/K, we obtain ta : tb′ : tc∗ = 1800, 250, and 6 K. These valuesare in reasonable agreement with the realistic band parameters [22]

5

Figure 3: Diamagnetic shielding of (TMTSF)2ClO4 at T = 0.05 K for magnetic fields oriented along thethree crystallographic axes, from Ref. [40].

Figure 4: Critical fields of (TMTSF)2ClO4 determined from the onset temperature of the c∗-axis resistanceT onset

c for fields along the three principal axes with an indication for the Pauli limit at low temperature.After [44, 42].

From the slopes of Hc2(T ), one can also deduce the superconducting coherencelengths ξi (i = a, b′, c∗), by using formulae Horb

c2 = −0.73Tc0 dHc2(T )/dT |T=Tc0 [46]and Horb

c2 ‖ i = Φ0/(2πξ jξk), where Φ0 is the flux quantum. We obtain (ξa, ξb′ , ξc∗ ) =

(620 Å, 330 Å, 14 Å) from the transport phase diagram [42]1, and (ξa, ξb′ , ξc∗ ) =

(1100 Å, 300 Å, 14 Å) from thermodynamic phase diagram. The obtained coherencelengths are quite anisotropic reflecting the Q1D nature of (TMTSF)2ClO4. Also noticethat the coherence lengths are much larger than the mean free path along the a axis,la ∼ 1.6 µm [42]. Thus, this system is well within the clean limit ξ l.

1The coherence length values in Ref. [42] should be multiplied by ∼ 1.4 because of a trivial calculationerror.

6

3. Non-s-wave superconducting nature in (TMTSF)2X

Since the superconducting phase of (TMTSF)2ClO4 is located next to the spin-density wave (SDW) phase as shown in Fig. 1(b), possibility of non-s-wave pairingmediated by spin fluctuation has been proposed. Experimentally, early evidences fornon-s-wave pairing in (TMTSF)2ClO4 have been obtained with the spin-lattice re-laxation rate 1/T1 measurement with the nuclear magnetic resonance (NMR) tech-nique, which revealed absence of the coherence peak just below Tc as well as thepower-low behavior at lower temperatures [47]. This behavior is theoretically in-terpreted as a consequence of non-s-wave pairing [48]. Strong suppression of su-perconductivity by non-magnetic impurities was revealed by using alloyed samples(TMTSF)2(ClO4)1−x(ReO4)x [49, 50], as described in detail in Sec. 3.2. Observationof a

√H dependence of the low-temperature specific heat [45], as well as the tempera-

ture dependence of the specific heat in zero field described below, also provide strongevidence for nodal SC state. Furthermore, the in-plane field-angle dependence of thespecific heat provides information on the location of nodes, as explained in Sec. 3.4.We note that several experiments claim fully gapped states: the thermal conductiv-ity [51] reveals electronic thermal conductivity vanishes exponentially below Tc aftersubtraction of phonon contribution; in-field muon spin rotation (µSR) [52] revealed thetemperature dependence of the penetration depth suggesting a fully gapped states butonly in magnetic fields. Nevertheless, we believe that so far nodal superconductingscenario has been accumulating more direct evidences. We also note that zero-fieldµSR measurement [52] could not detect spontaneous time-reversal symmetry breaking(i.e. spontaneous magnetization) in the superconducting state, excluding possibility of“chiral” superconducting state. Experiments on superconductivity in (TMTSF)2X arealso reviewed excellently in Refs. [53, 54, 55, 56].

Theoretically, as already explained, spin-fluctuation pairing mechanism in(TMTSF)2X has been proposed as early as 1986 [24]. Because of the simplicity ofthe Q1D electronic structure in (TMTSF)2X, as well as stimulation by interesting ex-periments, a tremendous amount of theories have been proposed. Microscopic theoriesconsidering spin and/or charge fluctuations have proposed unconventional supercon-ducting state, not only spin-singlet d-wave-like states, but also spin-triplet p-wave-likeor f -wave-like states, based on methods such as random phase approximation (RPA)or fluctuation exchange (FLEX) theories [57, 58, 59, 60, 61, 62, 63], quantum MonteCarlo method [64, 65, 66], perturbation theory [67], and RG theory [68, 69, 70, 71].Considering only the one pair of Fermi surface sheets, which is the electronic bandstructure for (TMTSF)2PF6 , s or p-wave-like states can be fully gapped, whereas d orf -wave-like states should have nodes on the Fermi surface. In case of (TMTSF)2ClO4, the Fermi surface consists of two pairs of sheets at low temperatures because of theband folding due to the anion ordering below TAO = 24 K [21]. For such “folded”Fermi surfaces, it has been pointed out that a fully-gapped d-wave-like state is alsopossible [72]. For more details of theories, see review articles such as Refs. [73, 74, 54]

3.1. Specific heat data

Recent new data of the temperature dependence of the specific heat Cp is presentedin Fig. 5. This data is obtained by the ac technique [75] and using only one single

7

crystal. To improve the accuracy of the obtained data, we measured the heater-currentfrequency f response in the amplitude of the temperature oscillation Tac and we fit-ted the Tac( f ) data with the theoretical function Tac( f ) = P/(8π fCp)[1 + (4π f τ1)2 +

(4π f τ2)−2]−1/2 to obtain Cp, where τ1 and τ2 are the external and internal relaxationrates, respectively. More details will be published elsewhere. In Fig. 5, we com-pare results for different samples. Both samples exhibits sharp anomaly at aroundTc ∼ 1.2 K, indicating bulk superconductivity. The electronic specific-heat coefficientis found to be γe = 10.6–10.8 mJ/K2mol, in good agreement with the previous works(γe = 10.5 mJ/K2mol) [38, 39], although the phononic specific heat coefficient exhibits∼ 20% variation depending on samples, but still comparable to βp = 11.4 mJ/K4molreported in Ref. [39]. In addition, it can be checked from the data in Fig. 5(b) that theentropy of the superconducting state at Tc equals that of the normal state at the sametemperature within ∼ 13% for both samples.

0

10

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5

(TMTSF)2ClO4(a)

c p / T

(m

J/K2 m

ol)

T (K)

Sample #1

Sample #2

0

2

4

6

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0 0.5 1 1.5 2 2.5

(TMTSF)2ClO4(b)

c el /

T (

mJ/

K2 m

ol)

T (K)

Sample #1

Sample #2

Figure 5: (a) Temperature dependence of the specific heat of (TMTSF)2ClO4 . We present data for twodifferent single crystals, Sample #1 (0.257 mg; blue circles) and Sample #2 (0.364 mg; red squares). Thebroken curves are fitting results with the Sommerfeld-Debye formula Cp/T = γe + βpT 2 to the normal statedata (T > 1.3 K). Resulting fitting parameters are γe = 10.8 ± 0.2 mJ/K2mol and βp = 12.6 ± 0.1 mJ/K4molfor Sample #1, and γe = 10.6 ± 0.4 mJ/K2mol and βp = 9.8 ± 0.2 mJ/K4mol for Sample #2. (b) Electronicspecific heat Cel/T of the two samples.

Figure 5(b) displays several features supporting non-s-wave pairing state. Firstly,the height of the specific-heat jump at Tc, ∆C, nearly equals to γeTc. This is notablysmaller than the expectation of the BCS theory, in which ∆C/γeTc is expected to be1.43. Instead, it is known that ∆C/γeTc can be smaller than 1.43 if the SC gap hassubstantial anisotropy. In particular, if the gap has line nodes the ratio can be evensmaller than 1.0 [76, 23]. Secondly, Cel/T exhibits linear temperature dependence ina wide temperature range below ∼ 0.7 K. Such linear behavior also evidences nodalsuperconducting gap. The finite intercept for the linear extrapolation of Cel/T to zerotemperature is expected in case of a finite elastic scattering time [77]. Using the datafor sample # 1, the residual density of states amounts to about 18.5% the value of thenormal state according to Fig. 5(b). This will be commented further in the next section.

3.2. Non magnetic defects

A basic property of the BCS superconducting s-wave function is the isotropic (k-independent) gapping on the Fermi surface. Hence, no pair breaking is expected from

8

the scattering of electrons against spinless impurities [78], since such scatterings es-sentially just mixes and averages gaps at different k positions. Experimentally, thisproperty has been verified in non-magnetic dilute alloys of s-wave superconductors andprovided a strong support to the BCS model of conventional s-wave superconductors.However, the condition for isotropic gap is no longer met for the case of non-s-wavepairing, in which the average of the gap ∆(k) over the Fermi surface vanishes due tosign changes in ∆(k), i.e.

∑FS ∆(k) ∼ 0. Consequently, Tc for these superconductors

should be strongly affected by any non-magnetic scattering, cancelling out positive andnegative parts of the gap. Theories on effects of non-magnetic impurities on Tc in suchsuperconductors have been deduced by generalizing conventional pair breaking theoryfor magnetic impurities in s-wave superconductors. Then the famous relation,

ln(

T 0c

Tc

)= ψ

(12

+αT 0

c

2πTc

)− ψ

(12

), (4)

with ψ(x) being the Digamma function and α = ~/2τkBT 0c the depairing parameter

related to the elastic scattering time τ is obtained [79, 80]. Experimentally, it hasbeen found that this relation holds for non-s-wave superconductors such as Sr2RuO4(Tc = 1.5 K; most likely a p-wave spin-triplet superconductor) [81].

It is also the remarkable sensitivity of organic superconductivity to irradiation de-tected in the early years [82, 83], which led Abrikosov to suggest the possibility oftriplet pairing in these materials [84]. A more recent investigation of the influence ofnon magnetic defects on organic superconductivity has been conducted following aprocedure which rules out the addition of possible magnetic impurities, which is thecase for X-ray irradiated samples [85]. Attempts to synthesize non stoichiometric com-pounds have not been successful for these organic salts. However, what turned out to befeasible is an iso-electronic anion solid solution keeping the charge transfer constant.One attempt has been to create non-magnetic disorder through the synthesis solid so-lutions with centrosymetrical anions such as AsF6 and SbF6. This attempt turned outto be unsuccessful as the effect of disorder happened to be very limited with only aminute effect on Tc [86].

An other scheme with which non-magnetic defects can be introduced in a controlledway for non-centro-symetrical anions in the (TMTSF)2X series is either by fast coolingpreventing the complete ordering of the tetrahedral ClO4 anions or by introducing ReO4anions to the ClO4 site by making the solid solution (TMTSF)2(ClO4)(1−x)(ReO4)x. Asdisplayed on Fig. 6, Tc in the solid solution is suppressed and the suppression is clearlyrelated to the residual resistivity, the enhancement of the elastic scattering in the normalstate. The data on Fig.6 show that the relation Tc versus ρ0 follows Eq. (4) with goodaccuracy with Tc

0 = 1.23 K.It has been checked that the additional scattering cannot be ascribed to magnetic

scattering with the electron paramagnetic resonance (EPR) technique, which shows noadditional traces of localized spins in the solid solution. Thus, the data in Fig. 6 cannotbe reconciled with the picture of a superconducting gap keeping a constant sign over thewhole (±kF) Fermi surface. They require a picture of pair breaking in a superconductorwith an anisotropic gap symmetry.

It is interesting to compare the residual density of states predicted by theories with

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0.00 0.05 0.10 0.15 0.20 0.25

0.0

0.2

0.4

0.6

0.8

1.0

1.2

10R

!

7R

6Q2

6Q1

6R

5Q1

5R

4Q1

4R

3R

2R

1R

Critical tem

peratu

re (K

)

Residual resistivity (!.cm)

0.0 0.2 0.4 0.6 0.8 1.0

(TMTSF)2

ClO4x

ReO4(1-x)

Figure 6: Phase diagram of (TMTSF)2(ClO4)(1−x)(ReO4)x , governed by non magnetic disorder accordingto reference [50]. All dots refer to very slowly cooled samples in the R-state (the so-called relaxed state)with different ReO−4 contents. A sample with ρ0 = 0.27(Ωcm)−1 i.e, beyond the critical defect concentration,is metallic down to the lowest temperature of the experiment. The continuous line is a fit of Eq. (4) to thedata with Tc

0 = 1.23 K.

experimental data. Figure 6 shows that the depairing parameter of the pristine sam-ple amounts to about 6.25% the critical value for the suppression of superconductivity.Given the ratio Γ/Γ0=0.65 for the pristine sample where Γ is the scattering rate, thecalculation of Sun and Maki [77] leads in turn to a residual density of states N(0)=0.26N0 which is fairly close to the residual density of states derived from our specificheat experiments, see the previous subsection. In the NMR data in Ref. [87], the spinlattice relaxation rate 1/T1 below 0.2 K is revealed to amount ∼ 25% of that in thenormal state. Since, 1/T1 is proportional to the square of the density of states, the ob-served residual value of 1/T1 corresponds to ∼ 50% of the density of states remainingin the superconducting state. Such a residual density of states from NMR comparesvery favorably with the value 18.5% provided by the measurement of the electronicspecific heat. The larger value found by NMR can be attributed to the field dependenceof the density of states, as reported in Ref. [45], since NMR data have been taken undermagnetic field µ0H = 0.96 T along the b′ axis or 1.3 T along the a axis.

The influence of non magnetic impurities on the superconducting phase implies theexistence of positive as well as negative values for the superconducting order parameter.It precludes the usual case of s-symmetry but is still unable to discriminate between twopossible options namely, singlet-d (g) or triplet-p (f) [69], see Fig. 8.

3.3. Spin susceptibility in the superconducting phaseThe detailed study of the behaviour of spin static and dynamic properties has been

undertaken via 77Se Knight shift and 1/T1 measurements across Tc in the compound(TMTSF)2ClO4 [87].

These data have provided a solid evidence in favour of singlet pairing, as presentedin Fig. 7(a). Furthermore, the temperature dependence of the relaxation rate shown inFig. 7(b) does not display the exponential behaviour expected in a regular fully gappeds-wave superconductor but instead a power law dependence below Tc with a linearregime establishing below 0.2K showing that there is a non zero density of states at

10

the Fermi level. Because 1/T1 is proportional to the square of the density of states,the observed residual 1/T1 value amounting 25-30% of the density in the normal stateindicates that the density of states is recovered by 50% at µ0H = 0.96 T for H ‖ b′ andµ0H = 1.3–1.4 T for H ‖ a.

Moreover, a steep increase of the spin-lattice relaxation rate versus magnetic fieldfor both field orientations ‖ a and b′ has provided the evidence for a sharp cross-overor even a phase transition occurring at low temperature under magnetic field between 1and 2 Tesla from the low field d-wave singlet phase and a high field regime exceedingthe paramagnetic limit Hp being either a triplet-paired state [88, 89] or an inhomoge-neous Fulde-Ferrell-Larkin-Ovchinnikov state [90, 91]. The nature of this high-fieldphase is further discussed in Sec. 4.

Figure 7: 77Se Knight shift and 1/T1 vs T for (TMTSF)2ClO4 , for H//b′ and a, according to reference [87].The sign of the variation of Knight shift at Tc depends on the sign of the hyperfine field. A linear temperaturedependence of the relaxation rate is recovered at very low temperature signaling the existence of unpairedcarriers at the Fermi level.

The conclusions of the (TMTSF)2ClO4 NMR data are at variance with the previousones obtained in (TMTSF)2PF6 regarding the spin part of the SC wave function. Atriplet pairing was claimed in (TMTSF)2PF6 [92] from the absence of any change in the77Se Knight shift at Tc together with the divergence of the critical field Hc2 exceedingthe Pauli limiting value when H is applied along the b′ or a axes [93]. The origin ofthe discrepancy between the data in the two organic superconducting brothers is notfully clarified but one may consider that the experimental situation is more delicatein (TMTSF)2PF6 since high pressure is required. Consequently, phase coexistence(SC/SDW) may be present and the actual sample temperature may not be known withenough accuracy in (TMTSF)2PF6.

3.4. Magneto calorimetric studies

In addition to the spin state, the orbital gap structure of the SC state is anotherfundamentally important information. As we explained in the preceding sections, ev-

11

Figure 8: Possible gap symmetries agreeing with the different experimental results. The spin-singlet d-wave(or g-wave) symmetry is the only symmetry agreeing with all experiments, (yellow column on line).

idence for non-s-wave pairing state in (TMTSF)2X had been accumulated. To revealmore precise gap structures, one of the common techniques is measurement of field-angle-dependent quasiparticle excitations. As first proposed by Volovik [94], super-conductors with nodes (or zeros) exhibit field dependent quasiparticle excitation withmomentum close to the nodal position, induced by the energy shift δω caused by the su-percurrent surrounding vortices penetrating the sample. Such field-induced excitationsis now called the Volovik effect. This quasiparticle excitation is also field-direction de-pendent [95], because δω is proportional to the inner product of the Fermi velocity vFat the node and the superfluid velocity vs, which is in turn perpendicular to the appliedfield. Thus, if one rotates the magnetic field within a certain plane, it is expected that thequasiparticle density of states oscillates as a function of the field angle. Such oscilla-tion can be detected by measuring e.g. the specific heat or thermal conductivity whilerotating the magnetic field within the conducting plane. Indeed, such studies havebeen widely performed in 3D or Q2D tetragonal systems such as CeCoIn5 [96, 97],YNi2B2C [98], Sr2RuO4 [99, 100], and many other materials [101].

In contrast to Q2D systems, the story for Q1D systems is not so simple, becausesubstantial in-plane anisotropy of Hc2 leads to pronounced specific-heat oscillation,even concealing the oscillation originating from the gap anisotropy. In addition, inQ1D systems, generally vF is not parallel to the Fermi wavenumber kF. Therefore, evenif the specific-heat oscillation originating from the gap is observed, the field directionwhere the specific heat exhibits anomaly does not have any direct relation to the gapnode position in the k space: one can only obtain the direction of the Fermi velocityat nodes from the field-angle dependent quasiparticle excitation. To reveal the gapstructure in k-space, one should know the band structure of the material investigated.For these reasons, the gap structure investigation of Q1D superconductors by the field-angle-induced quasiparticle excitation had not been explored.

Actual experiments have been reported by Yonezawa et al. [45, 102] who devel-oped a highly sensitive calorimeter based on the “bath modulating method” [103] and

12

vF

vF

kF

H

Γ

vF

vF

kF

HΓ

quasiparticleexcitation

Δ(k)

(a) (b) (c)

H // vortex

vs ⊥ vortex

Figure 9: Schematic description of the Volovik effect in a superconductor with gap nodes or zeros. (a) Super-current flowing around magnetic vortices. Supercurrent velocity vs is perpendicular to the vortex direction,namely the magnetic field direction. (b) Quasiparticle excitation around gap nodes excited by the Volovikeffect. (c) Quasiparticle excitation when the field is parallel to the Fermi velocity at a node. In such situation,the excitation at this node is zero, since vs · vF = 0 at this node.

measured the field-strength and field-angle dependence of the heat capacity of one sin-gle crystal of (TMTSF)2ClO4 . The in-plane field-angle φ dependence presented inFig. 10 is of particular interest. In addition to the large oscillation in the heat capac-ity originating from the in-plane anisotropy of Hc2, additional kink structures in C(φ)curves are observed. The kinks are located φ = ±10; i.e., for fields ±10 away fromthe crystalline a axis within the ab plane.

(h)

(i)

(k)

(j)

Figure 10: Observed in-plane field-angle dependent quasiparticle excitation in (TMTSF)2ClO4 [45].

By comparing the experimental data with a simple simulation shown in Fig. 10(g)and (j), it is suggested that±10 is the direction of vF at the gap nodes. Nagai et al. [104]calculated the field-angle dependence of the specific heat based on the quasiclassicalframework together with the first-principles band calculations, and they deduced simi-

13

lar conclusion. Thus, it is now clarified that the Fermi-surface positions at which vF ispointing ±10 away from the a axis are candidate nodal positions in the k space.

Based on the Fermi surface obtained by the tight-binding band calculation [21],Yonezawa et al. proposed that the d-wave-like state with nodes at ky = ±0.25/b∗ bestmatches with experiment [45]. Here, ky is the wavevector perpendicular to the a axisand b∗ = π/b is the size of the first Brillouin zone along the ky direction. Consideringthe nesting vectors of the Fermi surface of (TMTSF)2ClO4, this state is likely to berealized if the intra-band nesting plays the dominant role for Cooper pairing [102].

One comment that should be made here; There is actually a still some debateconcerning the detailed Fermi-surface shape of (TMTSF)2ClO4. The conclusion ofthe nodal position in the k space strongly depends on the assumption of the Fermi-surface shape, which is affected by the anion gap ∆a. A value of ∆a ∼ 100 meVhas been used in the tight-binding-model calculation [21], and for the analysis of thespecific heat data [45]. On the other hand, Nagai et al. [104] recently performed first-principles band calculation for the anion-ordered low-temperature crystal structure of(TMTSF)2ClO4, and evaluate ∆a as nearly zero. As a result, the calculated Fermisurfaces nearly touches each other. Another first-principles calculation by Alemanyet al. [105] revealed small but sizable anion-order effect with ∆a ∼ 14 meV, accompa-nied by a weak anti-crossing between the split bands. Thus the resultant Fermi surfacesare well separated in the k space. More recent calculation by Aizawa et al. obtained asimilar gap value ∆a ∼ 8.7 meV [106]. Experimentally, ∆a should be finite but seems tobe no more than 25 meV [107, 108]. A value around 14 meV is confirmed by a recentanalysis of magnetoresistance oscillations in (TMTSF)2ClO4 by G. Montambaux andD. Jerome [109].

Returning back to the nodal SC gap structure, the d-wave-like state with nodes atky = ±0.25b∗ remains a candidate structure even with ∆a = 0 meV according to thedetailed analysis [102], at least within the tight-binding model. Experimental deter-mination of ∆a and analyses based on the relevant band structure are still necessary toresolve the nodal structure. Microscopic theories on the gap structure based on realisticband structure is also important to finally settle this issue.

4. High-field superconducting state

As already mentioned in previous sections, (TMTSF)2X salts have been knownto exhibit divergent behavior in Hc2 determined from resistivity as the temperaturedecreases. The origin of this behavior has been attributed to spin-triplet pairing or tothe Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states [110, 111, 112, 113, 114, 115,116, 117, 118, 88, 119, 120, 61, 121, 122]. In case of (TMTSF)2ClO4 , the former isexcluded, since clear decrease in the spin susceptibility is observed [87] as describedin Sec. 3.3. In addition, sudden increase of the nuclear-lattice relaxation rate 1/T1observed above around 2 T [87] was considered as a consequence of formation ofunusual high-field SC phases.

The FFLO state [90, 91] can be realized when spin-singlet Cooper pairs are formedamong Zeeman-split Fermi surfaces in high magnetic fields [123]. Because of the Zee-man split, the Fermi wavenumber for the up-spin electron kF↑ and that for the down-spin electron kF↓ are not equal. Thus, when a Cooper pair is formed between kF↑ and

14

−kF↓ electrons as presented in Fig. 11, the pair acquires the non-zero center-of-massmomentum qFFLO = kF↑ − kF↓. This momentum results in the spatial oscillation of theSC order parameter. This means that the FFLO state is accompanied by the transla-tional symmetry breaking. In particular, for Q1D systems, qFFLO = kF↑ − kF↓ shouldbe nearly fixed to the a axis, since the number of pairs can be maximized if qFFLOmatches with the nesting vector between the spin-up and spin-down Fermi surfaces,which is nearly parallel to the a axis, as schematically shown in Fig. 11(b). Indeed, itis theoretically shown that the FFLO state with qFFLO ‖ a generally acquires high Tc ina Q1D system [124]. Observation of unusual phenomena resulting from the symmetrybreaking with such a fixed qFFLO can be a hallmark of the Q1D FFLO state.

↑↓ ↑ ↓

(k’, ↓)–k

qFFLO

(k, ↑)

(k’, ↓)qFFLO(k, ↑)

k’ = –k + qFFLO

(k’’, ↑)

–k’’

(a) 3D / 2D (b) Q1D

↓↑

Figure 11: Schematic comparison between FFLO pair formations for (a) 3D or 2D Fermi surfaces and (b)Q1D Fermi surfaces.

There are only a few candidate materials for the FFLO state. The heavy Fermioncompound CeCoIn5 clearly exhibits an unusual high-field phase [125, 126]. How-ever, this phase may not be a textbook-like FFLO state, since the phase is revealedto be accompanied by antiferromagnetic ordering [127]. Other leading candidatesare the two-dimensional organic superconductors κ-(BEDT-TTF)2Cu(NCS)2 and λ-(BETS)2FeCl4. In the former, the existence of additional high-field SC phase has beenconfirmed by magnetic and thermodynamic measurements [128, 129], as well as by anNMR study [130]. More recently, substantial increase of 1/T1 attributable to the An-dreev reflections originating from the order-parameter modulation is observed [131].In the latter compound, oscillatory behavior in the electric resistivity due to the vortexflow is observed [132]. This behavior is believed to be a consequence of the “lock-ing” effect between vortices and order-parameter modulation. Its sister compound λ-(BETS)2GaCl4 also exhibits a signature of the FFLO state [133].

For (TMTSF)2ClO4, only the unusual divergent-like behavior of Hc2(T ) for H ‖ b′

had been known for the high-field state [134, 135]. In 2008, Yonezawa et al. inves-tigated the in-plane field-angle dependence of the onset temperature of superconduc-tivity, T onset

c , based on the c∗-axis resistance measurements of (TMTSF)2ClO4 singlecrystals [44, 42]. They made use of the anisotropy of Hc2 to accurately deduce T onset

c :they compare the resistance in fields exactly parallel to the ab plane, as well as infields tilted away from the ab plane only by a few degrees. The c∗ axis component ofthe field induced by the tilting destroys the superconductivity, allowing one to extract

15

contribution of superconductivity by comparing resistances for the two field directions.It is revealed that, not only for H ‖ b′, but also for H ‖ a, T onset

c remains finite up to5 T, the maximal field achieved in this study, as shown in Fig. 4. In particular, the onsetcurve for H ‖ a exhibits a peculiar “S” shape, a limited behavior at around 0.8 K andan increase again below 0.3 K. The behavior for H ‖ a resembles that observed in thepressure-induced superconductivity in (TMTSF)2PF6 [93], and is recently theoreticallytreated within the FFLO scenario [122, 124]. What is more, they observed unusualmodulation in T onset

c (φ) above 3.0 T. In particular, the maxima of the T onsetc (φ) curve,

which is located at φ = 0 (H ‖ a) and φ = 90 (H ‖ b′) at low fields, the latter is foundto shift away from the crystalline axis at high fields. This is in some sense a (quasi)field-induced breaking of symmetry. 2 To the best of our knowledge, such a modulationin Tc has never been reported in any other FFLO candidates.

(a) (b)

Figure 12: (a) Polar plot of the φ dependence of T onsetc at several magnetic fields. The red line indicates the

new principal axis emerging above 3 T [44]. (b) Comparison of T onsetc (φ) for different samples [42]. The

blue and red points indicate T onsetc of Sample #1 (very clean) and Sample #2 (moderately clean), respectively.

Substantial difference is seen for |φ| > 19, whereas the sample dependence is rather small for smaller fieldangles. This difference is attributed to the fact that the FFLO state, as well the field-induced 2D confinementfor H ‖ b′, is very sensitive to impurity scatterings.

This unusual phenomena is interpreted as a consequence of the formation of FFLOstates. In FFLO states, the modulation vector qFFLO of the SC order parameter breaksthe translational symmetry of the SC state, and may lead to unusual field-angle de-pendence of T onset

c . Such interpretation have been indeed supported by recent theories.Croitoru and Buzdin [136, 137] found that Tc(φ) exhibit unusual φ dependence oncethe system is in the FFLO state, by solving linearized Eilenberger equations for s-wave

2Since the crystal structure of (TMTSF)2ClO4 belongs to the triclinic space group, any spatial symmetryexcept for the inversion symmetry is already broken by the lattice. Therefore, strictly speaking, it is noaccurate to say “symmetry breaking by the magnetic field” in the present case. Nevertheless, T onset

c (φ)exhibits nearly a mirror symmetry with respect to the a and b′ axes in low field.

16

superconductivity in an highly anisotropic quasi-two-dimensional model. More re-cently, they revealed similar results by a quasiclassical formalism for an s-wave modelwith a more-realistic Q1D band as well [138].

It is then natural that much effort has been devoted to search for thermodynamicevidence of the realization of the high-field FFLO phase. Interestingly, a specific-heatstudy with accurately aligned magnetic fields revealed that anomaly in the specific heatat T onset

c cannot be detected [45]: For field directions along the three principal axes,only detected anomaly is located close to the curve below which resistivity is zero asshown in Fig. 14. (This may be just coincident, since in some experiments the zero-resistance state is observed up to around 3 T for H ‖ a and H ‖ b′ [135, 87]. Alsosee Fig. 13.) The field at which the specific heat anomaly is detected and the specificheat recovers its normal state value should be assigned as the thermodynamic Hc2, andabove this field superconductivity has a density of states nearly equal to that in thenormal state. Nevertheless, the resistivity anomaly observed above the thermodynamicHc2 is quite robust and reproduced by several groups [134, 135, 44].

One possible explanation is that the density of states in the high-field FFLO stateis nearly equal to that in the normal state, because of the zero-gap region in real spaceoriginating from the order-parameter modulation [90]. This is reasonable, but exper-imental efforts to reveal the thermodynamic phase boundary between the FFLO stateand normal state is highly required to support the scenario. Another explanation is thatthe high-field FFLO region intrinsically acquires fluctuating nature, probably assistedby the low-dimensional electronic state in this material. We emphasize here that, evenwith fluctuating superconductivity, the observed anomalous behavior in T onset

c (φ) israther difficult to explain without “quasi symmetry breaking” in the underlying pairingchannel.

Figure 13: Field dependence of 1/(T1T ) for H ‖a and H ‖ b′ obtained by the NMR study on(TMTSF)2ClO4 [87].

Figure 14: Superconducting phase diagram of(TMTSF)2ClO4 obtained by the specific heat andresistivity measurements [44, 45].

17

-10 -5 0 5 10 15 20

0.0

0.1

0.2

0.3

0.4

0.5

(TMTSF)2

ClO4

1 bar

(TMTSF)2

PF6

11 kbar

T1

T (s.K

)

T (K)

9 kbar

Figure 15: Plot of the nuclear relaxation versus temperature according to the data of Ref. [139]. A Korringaregime, T1T = const is observed down to 25 K. The 2D AF regime is observed below ≈ 15 K and the smallCurie-Weiss temperature of the 9 kbar run is the signature of the contribution of quantum critical fluctuationsto the nuclear relaxation. The Curie-Weiss temperature becomes zero at the QCP. These data show that theQCP should be slightly below 9 kbar with the present pressure scale. The inset shows that the organicsuperconductor (TMTSF)2ClO4 at ambient pressure is very close to fulfill quantum critical conditions.

5. Metallic state above Tc: antiferromagnetic fluctuation and its relation to super-conductivity

Interestingly, the metallic phase of (TMTSF)2PF6 in the 3D coherent regime whenpressure is in the neighbourhood of the critical pressure Pc behaves in way far fromwhat is expected for a Fermi liquid. This behavior indicates dominance of quantumcritical fluctuations near Pc. What is more, close relation between the non-Fermi-liquid behavior and superconductivity has been recently revealed both experimentallyand theoretically, as described in detail below.

Experimentally, NMR measurements of 1/T1 have probed antiferromagnetic fluc-tuations. The canonical Korringa law, 1/T1T ∝ χ2(q = 0,T ), is well obeyed at hightemperature, say, above 25 K, but the low temperature behaviour deviates strongly fromthe standard relaxation in paramagnetic metals. As shown in Fig. 15, an additional con-tribution to the relaxation rate emerges on top of the usual Korringa relaxation. Thisadditional contribution rising at low temperature has been attributed to the onset of an-tiferromagnetic fluctuations in the vicinity of Pc [140, 141, 142]. On the other hand, inthe low temperature regime, the relaxation rate follows a law such as T1T = C(T + Θ)as shown in Fig. 15. This is the Curie-Weiss behaviour for the relaxation which is to beobserved in a 2D fluctuating antiferromagnet [143, 144, 145, 146]. Similar behavior isalso found in a 13C NMR study [147].

The positive Curie-Weiss temperature Θ, which provides the energy scale of thefluctuations, becomes zero when pressure is equal to Pc (the quantum critical condi-tions). When Θ becomes large comparable to T , the standard relaxation mechanismis expected to recover down to low temperatures, in agreement with the observation atvery high pressures [148].

The existence of fluctuations is also observed as anomalous behavior in transport.

18

T

T2

Figure 16: A log-log plot of the inelastic longitudinal resistivity of (TMTSF)2PF6 below 20 K, according toRef. [140].

At P = Pc, the inelastic scattering in transport reveals at once a strong linear term atlow temperature, as presented in a log-log plot of the resistivity versus T , Fig. 16. Thisstrongly linear behavior evolves to quadratic behavior in the high temperature regime.As pressure is increased away from Pc, the resistivity exhibits a general tendency tobecome quadratic at all temperatures [140], see Figs. 16 and 1. The existence of a lineartemperature dependence of the resistivity is at variance with the sole T 2 dependenceexpected from the ordinary electron-electron scattering in a conventional Fermi liquid,indicating dominant scattering involving spin fluctuations.

Figure 17: Coefficient A of linear resistivity as a function of Tc plotted versus Tc/Tc0 for (TMTSF)2PF6. Tc is defined as the midpoint of the transition and the error bars come from the 10 % and 90 % pointswith Tc0 = 1.23K under the pressure of 8 kbar which provides the maximum Tc in the SDW/SC coexistenceregime. The dashed line is a linear fit to all data points except that at Tc = 0.87 K, according to Ref. [140].

Furthermore, the investigation of both transport and superconductivity under pres-sure in (TMTSF)2PF6 has established a correlation between the amplitude of the lineartemperature dependence of the resistivity and the value of Tc, as displayed in Fig. 17.This correlation suggests a common origin for the inelastic scattering of the metallicphase and pairing in the SC phase (TMTSF)2PF6 [140], as discussed in the rest of thissection.

19

Within the framework of a weak-coupling limit, the problem of the interplay be-tween antiferromagnetism and superconductivity in the Bechgaard salts has been theo-retically worked out using the renormalization group (RG) approach [146, 69] as sum-marized below. The theories take into account only the 2D problem. The RG integra-tion of high-energy electronic degrees of freedom was carried out down to the Fermilevel, and leads to a renormalization of the couplings at the temperature T [31, 70, 146].The RG flow superimposes the 2kF electron-hole (density-wave) and Cooper pairingmany-body processes, which combine and interfere at every order of perturbation. Asa function of the ‘pressure’ parameter t′b, i.e the unnesting interchain coupling, a sin-gularity in the scattering amplitudes signals an instability of the metallic state towardthe formation of an ordered state at some characteristic temperature scale. At low t′b,nesting is sufficiently strong to induce a SDW instability in the temperature range ofexperimentally observed TSDW ∼ 10-20 K. When the antinesting parameter approachesthe threshold t′∗b from below (t′∗b ≈ 25.4 K using the above parameters), TSDW sharplydecreas es and a s a result of interference, SDW correlations ensure Cooper pairing at-traction in the superconducting d-wave (SCd) channel. This gives rise to an instabilityof the normal state SCd order at the temperature Tc with pairing coming from antifer-romagnetic spin fluctuations between carriers of neighbouring chains. Such a pairingmodel actually supports the conjecture of interchain pairing in order for the electronsto avoid the Coulomb repulsion made by V. Emery in 1983 and 1986 [23, 24].

≈Pressure"

Figure 18: Calculated phase diagram of the quasi-onedimensionalelectron gas model from the renormaliza-tion group method at the one-loop level [146]. Θ and the dash-dotted line defines the temperature region ofthe Curie-Weiss behavior for the inverse normalized SDW response function.

The calculated phase diagram shown in Fig. 18 with reasonable parameters g1 =

g2/2 ≈ 0.32 for the backward and forward scattering amplitudes respectively and g3 ≈

0.02 for the longitudinal Umklapp scattering term [146, 142] captures the essentialfeatures of the experimentally-determined phase diagram of (TMTSF)2PF6 presentedin Fig. 1.

Sedeki et al. [149] have proceeded to an evaluation of the imaginary part of theone-particle self-energy. In addition to the regular Fermi-liquid component, whichgoes as T 2, low-frequency spin fluctuations yield τ−1 = aTξ, where a is a constant and

20

the antiferromagnetic correlation length ξ(T ) increases according to ξ = c(T + Θ)−1/2

as T → Tc, where Θ is the temperature scale for spin fluctuations [149]. It is thennatural to expect the Umklapp resistivity to contain (in the limit T Θ) a linear termAT , whose magnitude would presumably be correlated with Tc, as both scattering andpairing are caused by the same antiferromagnetic correlations. The observation of a T -linear law for the resistivity up to 8 K in (TMTSF)2PF6 under a pressure of 11.8 kbaras displayed in Fig. 16 is therefore consistent with the value of Θ = 8 K determinedfrom NMR relaxation a t 11 kba r displayed in Fig. 15.

We add one comment that, in (TMTSF)2X, the existence of the quantum criticalpoint is actually not trivial because the boundary between the SDW and SC phases isa first-order phase transition within the pressure-temperature phase diagram, in con-trast to ordinary theories on quantum criticality assuming a second-order transition.However, it has been recently revealed that other typical “quantum critical” materialssuch as iron pnictides [150] indeed exhibit first-order-like behavior in the vicinity ofthe quantum critical point, evidenced by phase separation between magnetically or-dered and paramagnetic phases detected by µSR studies [151]. Thus, it is now gettingclearer that the quantum criticality near a first-order transition observed in (TMTSF)2Xprobably shares general and important physics with a broad class of materials.

6. Conclusion

Both experimental and theoretical views point the contribution of electron correla-tions to the superconducting pairing problem. The extensive experimental evidence infavor of the emergence of superconductivity in the (TM)2X family next to the stabilitypressure threshold for antiferromagnetism has shown the need for a unified descrip-tion of all electronic excitations that lies at the core of both density-wave and super-conducting correlations. In this matter, the recent progresses of the renormalizationgroup method for the 1D-2D electron gas model have resulted in predictions about thepossible symmetries of the superconducting order parameter when a purely electronicmechanism is involved, predictions that often differ from phenomenologically basedapproaches to superconductivity but are in fair agreement with recent experimentalfindings.

To summarize, firstly, the SC order parameter is displaying lines of nodes whichare governing the stability against impurity and thermodynamics of the SC phase andhave been located by the field angular dependence of the specific heat. Secondly, theelectron scattering in the metallic phase above Tc suggests the existence of strong an-tiferromagnetic fluctuations leading to the possibility of a spin mediated pairing in theSC phase. The pairing mechanism behind organic superconductivity is likely differentfrom the proposal made by Little but it is nevertheless a phonon-less mechanism, atleast in (TM)2X superconductors.

What is also emerging from the work on these prototype 1D organic superconduc-tors is their very simple electronic nature with only a single band at Fermi level, noprominent spin orbit coupling and extremely high chemical purity and stability.

They should be considered in several respects as model systems to inspire thephysics of the more complex high Tc superconductors, especially for pnictides andelectron-doped cuprates. Most concepts discovered in these simple low dimensional

21

conductors may also become of interest for the study of other 1D or Q1D systems suchas carbon nanotubes, the purple bronze superconductor Li0.9Mo6O17 [152, 153], thenewly-discovered telluride superconductor Ta4Pd3Te16 [154], and artificial 1D struc-tures.

This article shows that we still have plenty of food for thought in the field of organicsuperconductors.

Acknowledgements

We are grateful to Professor Jacques Friedel who has welcomed and strongly sup-ported the research activity on low dimensional conductors at Orsay from its verybeginning. He has contributed through continuous encouragements and numerousdiscussions. D.J. wishes to acknowledge the remarkably fruitful cooperation withKlaus Bechgaard who provided the samples for the experiments performed in Kyoto,Patrick Batail for the chemistry of various 1D and 2D conductors, with late HeinzSchulz, Thierry Giamarchi and Claude Bourbonnais for the theory, with the groupof Louis Taillefer at Sherbrooke for recent experimental work, with Stuart Brown atUCLA and with our Orsay colleagues C. Pasquier, N. Joo and P. Senzier. S.Y. ac-knowledges Y. Maeno, K. Ishida, H. Aizawa, K. Kuroki for useful discussions, andT. Kajikawa, S. Kusaba, for technical assistance. This work was supported in France byCNRS and in Japan by Grants-in-Aids for Scientific Research (KAKENHI 22103002,23540407, 23110715, and 26287078) from Ministry of Education, Culture, Sports,Science and Technology (MEXT) of Japan and from Japan Society for the Promotionof Science (JSPS).

22

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