Superconductivity and antiferromagnetism in quasi-one-dimensional organicconductors (Review Article)N. Dupuis, C. Bourbonnais, and J. C. Nickel

Citation: Low Temperature Physics 32, 380 (2006); doi: 10.1063/1.2199440 View online: http://dx.doi.org/10.1063/1.2199440 View Table of Contents: http://scitation.aip.org/content/aip/journal/ltp/32/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic-field-induced phase transitions in the quasi-one-dimensional organic conductor HMTSF–TCNQ Low Temp. Phys. 40, 371 (2014); 10.1063/1.4869591 Gigahertz-range synchronization at room temperature and other features of charge-density wave transport in thequasi-one-dimensional conductor NbS 3 Appl. Phys. Lett. 94, 152112 (2009); 10.1063/1.3111439 Variational cluster study of antiferromagnetism and dwave superconductivity in layered organic superconductors AIP Conf. Proc. 918, 322 (2007); 10.1063/1.2752003 Comparative Study of the Angular Magnetoresistance in QuasiOneDimensional Organic Conductors AIP Conf. Proc. 850, 1544 (2006); 10.1063/1.2355293 Coexistence of antiferromagnetism and triplet superconductivity J. Appl. Phys. 97, 10B108 (2005); 10.1063/1.1851922

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LOW TEMPERATURE PHYSICS VOLUME 32, NUMBER 4-5 APRIL-MAY 2006

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Superconductivity and antiferromagnetism in quasi-one-dimensional organicconductors „Review Article…

N. Dupuisa�

Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ United Kingdom;Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-Sud, 91405 Orsay, France

C. Bourbonnais

Regroupement Québecois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec,Canada J1K-2R1

J. C. Nickel

Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-Sud, 91405 Orsay, France;Regroupement Québecois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec,Canada J1K-2R1�Submitted September 15, 2005�Fiz. Nizk. Temp. 32, 505–520 �April–May 2006�

We review the current understanding of superconductivity in the quasi-one-dimensional organicconductors of the Bechgaard and Fabre salt families. We discuss the interplay between supercon-ductivity, antiferromagnetism, and charge-density-wave fluctuations. The connection to recentexperimental observations supporting unconventional pairing and the possibility of a triplet spinorder parameter for the superconducting phase is also presented. © 2006 American Institute ofPhysics. �DOI: 10.1063/1.2199440�

I. INTRODUCTION

Superconductivity in organic conductors was firstdiscovered1 in the ion radical salt �TMTSF�2PF6. Later on, itwas found in most Bechgaard and Fabre ��TMTTF�2X� andFabre ��TMTTF�2X� salts. These salts are based on the or-ganic molecules tetramethyltetraselenafulvalene �TMTSF�and tetramethyltetrathiafulvalene �TMTTF�. The monovalentanion X can be either a centrosymmetric �PF6, AsF6, etc.� ora non-centrosymmetric �ClO4, ReO4, NO3, FSO3, SCN, etc.�inorganic molecule. �See Refs. 2 and 3 for previous reviewson these compounds.� Although they are definitely not “high-Tc” superconductors—the transition temperature is of the or-der of 1 K—these quasi-one-dimensional �quasi-1D� con-ductors share several properties of high-Tc superconductorsand other strongly-correlated electron systems such as lay-ered organic superconductors4,5 or heavy-fermion materials.6

The metallic phase of all these conductors exhibits unusualproperties which cannot be explained within the frameworkof Landau’s Fermi liquid theory and remain to a large extentstill to be understood. The superconducting phase is uncon-ventional �not s-wave�. Magnetism is ubiquitous in these cor-related systems and might provide the key to the understand-ing of their behavior.

The quest for superconductivity in organic conductorswas originally motivated by Little’s proposal that highly po-larizable molecules could lead—via an excitonic pairingmechanism—to tremendously large transition temperatures.Early efforts towards the chemical synthesis of such com-pounds were not successful, as far as superconductivity isconcerned, but led to the realization of a 1D charge transfersalt �TTF-TCNQ� undergoing a Peierls instability at lowtemperatures.7 Attempts to suppress the Peierls state and sta-bilize a conducting �and possibly superconducting� state by

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increasing the 3D character of this 1D conductor proved tobe unsuccessful.

Organic superconductivity was eventually discovered inthe the Bechgaard salt �TMTSF�2PF6 under 9 kbar ofpressure.1 It was subsequently found in other members of the�TMTSF�2X series. Most of the Bechgaard salts are insulat-ing at ambient pressure and low temperatures,8 and it cameas a surprise that the insulating state of these materials is aspin-density-wave �SDW� rather than an ordinary Peierlsstate.2 The important part played by magnetism in thesecompounds was further revealed when it was found that theirphase diagram only shows a part of a larger sequence ofordered states, which includes the Néel and the spin-Peierlsphases of their sulfur analogs, the Fabre salts �TMTTF�2Xseries.9

The charge transfer from the organic molecules to theanions leads to a commensurate band filling 3/4 comingfrom the 2:1 stoichiometry. The metallic character of thesecompounds at high enough temperature is due to the delocal-ization of carriers via the overlap of �-orbitals betweenneighboring molecules along the stacking direction �a axis��Fig. 1�.2 The electronic dispersion relation obtained fromquantum chemistry calculations �extended Hückel method� iswell approximated by the following tight-binding form10–13

��k� = − 2ta cos�kaa/2� − 2t�b cos�kbb� − 2t�c cos�kcc�

� vF��ka� − kF� − 2t�b cos�kbb� − 2t�b� cos�kbb�

− 2t�c cos�kcc� + � , �1�

where it is assumed that the underlying lattice is orthorhom-bic. This expression is a simplification of the dispersionrelation—the actual crystal lattice symmetry is triclinic—butit retains the essential features. The conduction band along

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the chain direction has an overall width 4ta ranging between0.4 and 1.2 eV, depending on the organic molecule �TMTSFor TMTTF� and the anion. As the electronic overlaps in thetransverse b and c directions are much weaker than along theorganic stacks, the dispersion law is strongly anisotropic,t�b / ta�0.1 and t�c / t�b�0.03, and the Fermi surface con-sists of two open warped sheets �Fig. 1�. In the second line ofEq. �1�, the electronic dispersion is linearized around the two1D Fermi points ±kF, with vF the Fermi velocity along thechains �� is the chemical potential�. The next-nearest-chainhopping t�b� �2t�b

2 / ta+. . . is introduced in order to keep theshape of the Fermi surface unchanged despite the lineariza-tion. The anions located in centrosymmetric cavities lieslightly above or below the molecular planes. This structureleads to a dimerization of the organic stacks and a �weak�gap �D, thus making the hole-like band effectively half-filledat sufficiently low energy or temperature.14,15 �See Refs. 2, 7,and 9 for a detailed discussion of the structural properties ofquasi-1D organic conductors.� In the presence of interac-tions, commensurate band-filling introduces umklapp scatter-ing, which affects the nature of the possible phases in thesematerials.

What is remarkable about these electronic systems is thevariety of ground states that can be achieved either by chemi-cal means, namely substituting selenium by sulfur in the or-ganic molecule or changing the nature of the anion �its sizeor symmetry�, or applying pressure �Fig. 2�. At low pressure,members of the sulfur series are Mott insulators �MI� fromwhich either a lattice distorted spin-Peierls �SP� state—oftenpreceded by a charge-ordered �CO� state—or acommensurate-localized antiferromagnetic state �AF� can de-velop. On the other hand, itinerant antiferromagnetism �spin-density wave �SDW�� or superconductivity is found in theselenide series. Under pressure, the properties of the Fabresalts evolve towards those of the Bechgaard salts. The com-pound �TMTTF� PF spans the entire phase diagram as pres-

FIG. 1. A side view of the Bechgaard/Fabre salt crystal structure with theelectron orbitals of the organic stacks �courtesy of J. Ch. Ricquier� �a�.Electronic dispersion relation and projected 2D Fermi surface of�TMTTF�2Br �reprinted with permission from Ref. 13. Copyright 1994 byEDP Sciences� �b�.

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sure increases up to 50 kbar or so �Fig. 3�,16–18 thus showingthe universality of the phase diagram in Fig. 2.19

A large number of both theoretical and experimentalworks have been devoted to the understanding of the normalphase and the mechanisms leading to long-range order at lowtemperature. The presence of antiferromagnetism over alarge pressure range does indicate that repulsive interactionsamong carriers are important. The low dimensionality of thesystem is also expected to play a crucial role. On the onehand, in the presence of repulsive interactions a strongly an-isotropic Fermi surface with good nesting properties is pre-dominantly unstable against the formation of an SDW state,which is reinforced at low temperature by commensurateband filling. On the other hand, when the temperature ex-ceeds the transverse dispersion �t�b, 3D �or 2D� coherentelectronic motion is suppressed and the conductor behaves asif it were 1D; the Fermi liquid picture breaks down and thesystem becomes a Luttinger liquid.20,21 The relevance of 1D

FIG. 2. The generic phase diagram of the Bechgaard/Fabre salts as a func-tion of pressure or anion X substitution; Luttinger liquid �LL�, Mott insula-tor �MI�, charge order �CO�, spin-Peierls �SP�, antiferromagnetism �AF�,superconductivity �SC�, Fermi liquid �FL�.

FIG. 3. �P ,T� phase diagram of �TMTTF�2PF6. The shaded area above theSDW and SC phase indicates the region of the normal phase where spinfluctuations are significant. �Reprinted with permission from Ref. 17. Copy-right 2001 by EDP Sciences.�

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physics for the low-temperature properties �T� t�b�, as wellas a detailed description of the crossover from the Luttingerliquid to the Fermi liquid, is one of the most important issuesin the debate surrounding the theoretical description of thenormal state of these materials. As far as low-temperaturephases are concerned, a chief objective is to reach a gooddescription of the superconducting phase—the symmetry ofthe order parameter is still under debate—and the mecha-nisms leading to superconductivity. Owing to the close prox-imity of superconductivity and magnetism in the phase dia-gram of Fig. 2, it is essential to first discuss the origin ofantiferromagnetism in both series of compounds.

II. NÉEL ANTIFERROMAGNETISM AND SPIN-DENSITY WAVE

A. Fabre salts at ambient pressure: Mott-insulatorregime

The Fabre salts �TMTTF�2X at ambient pressure are lo-cated on the left of the phase diagram in Fig. 2. Both thenature of correlations and the mechanism of long-range orderat low temperature are now rather well understood. Belowthe temperature T��100 K �see Fig. 2�, the resistivity devel-ops a thermally activated behavior22 and the system enters aMott-insulator regime. The corresponding charge gap ��

��T� can be deduced from T� and turns out to be larger thanthe �bare� transverse bandwidth t�b, which in turn suppressesany possibility of transverse single-particle band motion andmakes the system essentially one-dimensional. The chargegap 2�� is also directly observed in the opticalconductivity.23 The members of the �TMTTF�2X series thusbehave as typical 1D Mott insulators below T� with the car-riers confined along the organic stacks—as a result of theumklapp scattering due to the commensurability of the elec-tronic density with the underlying lattice.14,15 This interpre-tation agrees with the absence of anomaly in the spin suscep-tibility at T�,24 in accordance with the spin-charge separationcharacteristic of 1D systems.21 It is further confirmed bymeasurements of the spin-lattice relaxation rate 1 /T1. TheLuttinger liquid theory predicts25,26

1

T1= C0T�s

2�T� + C1TK�, �2�

where C0 and C1 are temperature-independent constants.�s�T� is the uniform susceptibility and K� the Luttinger liq-uid charge stiffness parameter. The two contributions in �2�correspond to paramagnons or spinons �q�0� and AF spinfluctuations �q�2kF�. Both the temperature dependence of�s�T� and the presence of AF fluctuations lead to an enhance-ment of 1 /T1 with respect to the Korringa law �T1T�−1

=const that holds in higher-dimensional metals. In a 1D Mottinsulator K�=0, which leads to T1

−1=C0T�s2�T�+C1 in good

agreement with experimental measurements of T1 and �s.24

The low-energy excitations in the Mott-insulator regimeare 1D spin fluctuations. By lowering the temperature, thesefluctuations can propagate in the transverse direction andeventually drive an AF transition. This transition is not con-nected to Fermi-surface effects. The condition �� t�b pre-cludes a single-particle coherent motion in the transverse di-rection, and the concept of Fermi surface remains ill definedin the Fabre salts at ambient pressure. AF long-range ordercomes from interchain transfer of bound electron-hole pairs

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leading to a kinetic exchange interaction J� between spindensities on neighboring chains—much in analogy with theexchange interaction between localized spins in the Heisen-berg limit. An effective Hamiltonian can be derived from arenormalization group �RG� calculation,27,28

H� = J�� dxi,j�

Si�x� · S j�x� . J� ��

a

t�b*2

��

, �3�

where t�b* is the effective interchain hopping at the energy

scale �� and a is the lattice spacing along the chain. The sumin Eq. �3� is over nearest-neighbor chains. The naive valuet�b*2 /�� of the exchange interaction J� is enhanced by the

factor � /a, where �=vF /�� is the intrachain coherencelength induced by the Mott gap along which virtual inter-chain hoppings can take place. Within a mean-field treatmentof H�, the condition for the onset of long-range order isgiven by J��1D�2kF ,T�=1, where �1D�2kF ,T���T /���−1 isthe exact power-law form of the 1D AF spin susceptibility.This yields a Néel temperature

TN �t�b*2

��

. �4�

Since T� and �� decrease under pressure �Fig. 2�, Eq. �4�predicts an increase of TN with pressure—assuming a weakpressure dependence of tb

*—as observed experimentally �seeFig. 2�. The relation TNT�� t�b

*2 �const has been observed in�TMTTF�2Br.29

B. Bechgaard salts: itinerant magnetism

With increasing pressure, T� drops and finally mergeswith the AF transition line at Pm, beyond which there is nosign of a Mott gap in the normal phase. The Fabre salts thentend to behave similarly to the Bechgaard salts �Fig. 2�. Thechange of behavior at Pm is usually attributed to a deconfine-ment of carriers, i.e., a crossover from a Mott insulator toa—metallic—Luttinger liquid. At lower temperature, single-particle transverse hopping is expected to become relevantand induces a dimensional crossover at a temperature Tx

from the Luttinger liquid to a 2D or 3D metallic state. Withincreasing pressure, the AF transition becomes predomi-nantly driven by the instability of the whole warped Fermisurface due to the nesting mechanism. Although there is ageneral agreement on this scenario, there is considerable de-bate on how the dimensional crossover takes place and thenature of the low-temperature metallic state.

On the theoretical side, simple RG arguments indicatethat the crossover from the Luttinger liquid to the 2D regimetakes place at the temperature30

Tx �t�b

�� t�b

ta 1−K�/K�

, �5�

where K� is the Luttinger liquid parameter. For noninteract-ing electrons �K�=1�, Eq. �5� would give Tx� t�b: the 2DFermi surface is irrelevant when temperature is larger thanthe dispersion in the b direction. For interacting electrons�K��1�, the interchain hopping amplitude t�b is reduced toan effective value t�b

* and the dimensional crossover occursat a lower temperature Tx� t�b

* � t�b. A detailed theoreticalpicture of the dimensional crossover is still lacking. In par-

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ticular, whether it is a sharp crossover or rather extends overa wide temperature range—as shown by the shaded area inFig. 2—is still an open issue.

1. The strong-correlation picture

Some experiments seem to indicate that correlations stillplay an important role even in the low-temperature phase ofthe Bechgaard salts. For instance, a significant enhancementof 1 /T1T with respect to the Korringa law—although weakerthan in the Fabre salts at ambient pressure—is still present.24

This behavior has been explained in terms of 1D spin fluc-tuations persisting down to the dimensional crossover tem-perature Tx�10 K, below which the Korringa law isrecovered.24,26

The restoration of a plasma edge in the transverse b�direction at low temperature in �TMTSF�2PF6—absent in theFabre salts—suggests the gradual emergence of a coherentmotion in the �ab� planes below Tx�100 K.31,32 �b� is nor-mal to a and c in the �ab� plane. It differs from b due to thetriclinic structure.� However, the frequency dependence ofthe optical conductivity is inconsistent with a Drude-like me-tallic state.33,34,23 The low-energy peak carries only 1% of thetotal spectral weight and is too narrow to be interpreted as aDrude peak with a frequency-independent scattering time. Ithas been proposed that this peak is due to a collective modethat bears some similarities with the sliding of a charge-density wave—an interpretation supported by the new pho-non features that emerge at low temperature.33 Furthermore,99% of the total spectral weight is found in a finite energypeak around 200 cm−1. It has been suggested that this peak isa remnant of a �1/4�-filled Mott gap ��, observed in the lessmetallic Fabre salts at ambient pressure.35,23 In this picture,�TMTSF�2PF6 is close to the border between a Mott insula-tor and a Luttinger liquid, and the low-temperature metallicbehavior is made possible by the interchain coupling.23,36,37

A different interpretation has been proposed for the far-infrared spectrum in optical conductivity and is based on theweak half-filling character of the band for interactions in theHubbard limit.38

The longitudinal resistivity in �TMTSF�2PF6 is found tobe metallic, with a T2 law between the SDW transition and150 K, crossing over to a sublinear temperature dependenceabove 150 K with an exponent in the range 0.5–1.39,40 Whilethis observation would be consistent with a dimensionalcrossover to a low-temperature Fermi liquid regime takingplace at Tx�150 K, the transverse resistance �b along the baxis apparently fails to show the expected T2 behavior. Giventhe difficulties of a direct dc measurement, owing to nonuni-form current distributions between contacts, conflicting re-sults have been published in the literature.39,41 Nevertheless,below T�80 K �b can be deduced from �a�T2 and �c

�T1.5 using a tunneling argument, which yields �c

= ��a�b�1/2 and therefore �b�T. Moreover, contactless—microwave—transverse conductivity measurements in the�TMTSF�2PF6 salt fail to reveal the emergence of a Fermi-liquid T2 temperature dependence of the resistivity in the bdirection in this temperature range.42 As far as �c is con-cerned, a maximum around Tmax�80 K has been observed,with a metallic—though incoherent—behavior �c�T1.5 atlower temperature.43 T is highly sensitive to pressure,

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whereas the interchain hopping t�b is not. Therefore, Tmax

cannot be directly identified with t�b, but could be related toa—weakly—renormalized value t�b

* �Tx, in agreement withpredictions of the Luttinger liquid theory; see Eq. �5�. Thetransport measurements seem to be indicative of a gradualcrossover between a Luttinger liquid and a Fermi liquid oc-curring in the temperature range 40–80 K. The onset of 3Dcoherence and Fermi liquid behavior would then be relatedto the interplane coupling t�c between �a ,b� planes.43

The absence of Fermi liquid behavior down to very lowtemperatures in the Bechgaard salts seems to be further sup-ported by photoemission experiments. ARPES fails to detectquasiparticle features or the trace of a Fermi surface at150 K.44 Similar conclusions were deduced from integratedphotoemission at 50 K.45 However, photoemission results—e.g., the absence of dispersing structure and a power-lawfrequency dependence which is spread over a large energyscale of the order of 1 eV—do not conform with the predic-tions of the Luttinger theory and might be strongly influ-enced by surface effects.

The existence of strong correlations suggests that thekinetic interchain exchange J�, which drives the AF transi-tion in the sulfur series, still plays an important role in theBechgaard salts. In this picture, the decrease of TN with in-creasing pressure is due both to the decrease of J� and thedeterioration of the Fermi-surface nesting. This scenario issupported by RG calculations.28

All the experiments mentioned so far favor different—and sometimes incompatible—scenarios for the dimensionalcrossover. However, the high-temperature phase of the Bech-gaard salts is always analyzed on the basis of the Luttingerliquid theory. A consistent interpretation of the experimentalresults therefore requires one to find a common K� parameterand to determine the value of the remnant of the Mott gap��. NMR,24 dc transport,43,46 and optical measurements23,36

have been interpreted in terms of the Luttinger theory withK��0.23 and quarter-filled umklapp scattering.7,37 This in-terpretation, as well as the mere existence of strong correla-tions, is not without raising a number of unanswered ques-tions �see the next Section�. For instance, K��0.23 wouldlead according to �5� to Tx�10−3t�b a value much below theexperimental observations.

2. The weak-correlation picture

On the other hand, there are experiments pointing to theabsence of strong correlations in the Bechgaard salts. One ofthe most convincing arguments comes from the so-calledDanner-Chaikin oscillations.47 Resistance measurements of�TMTSF�2ClO4 in the c direction show pronounced reso-nances when an applied magnetic field is rotated in the �ac�plane at low temperature. The complete angular dependenceof the magnetoresistance can be reproduced within a semi-classical approach. The position of the resonance peaks isgiven by the zeros of the Bessel function J0��� evaluated at�=2t�bcBx /vFBz �c is the interchain spacing in the c direc-tion�. This enables a direct measure of the interchain hoppingamplitude in the b direction, yielding t�b�280 K above theanion ordering transition taking place at 24 K, in very goodagreement with values derived from band calculations.10–12

These results can hardly be reconciled with the existence of

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strong correlations. Sizeable 1D fluctuations should lead to astrong �k� ,w� dependence of the self-energy, and in turn to asignificant renormalization of k�-dependent quantities likethe interchain hopping amplitudes.28 This lends support tothe idea that the low-temperature phase of the Bechgaardsalts can be described as a weakly interacting Fermi liquidsubject to spin fluctuations induced by the nesting of theFermi surface.48,49

The weak-coupling approach has been particularly suc-cessful in the framework of the quantized nesting model.50–52

The latter explains the cascade of SDW phases induced by amagnetic field in �TMTSF�2PF6 and �TMTSF�2ClO4 andprovides a natural explanation for the quantization of theHall effect— xy =2Ne2 /h �N integer� per �ab� plane—observed in these phases. Furthermore, it reproduces the ex-perimental phase diagram only for interchain hopping ampli-tudes t�b , t�c, close to their unrenormalized values.

Despite the apparent success of the weak-coupling ap-proach, it has nevertheless become clear that the SDW phaseof the Bechgaard salts is not conventional. Recent experi-ments have shown that the 2kF SDW coexists with a 2kF anda—weaker—4kF charge-density wave �CDW� in�TMTSF�2PF6.53,54 Since there is no 2kF phonon softeningassociated to this transition, the emergence of this CDW statediffers from what is usually seen for an ordinary Peierlsstate. This unusual ground state can be explained on the basisof a quarter-filled 1D model with dimerization and onsite,nearest-neighbor and next-nearest-neighbor Coulombinteractions,55–59 but this explanation remains to be con-firmed.

3. The normal phase above the superconducting phase

It is remarkable that the superconducting phase lies nextto the SDW phase—which is actually a mixture SDW-CDW—and reaches its maximum transition temperature Tc

�1 K at the pressure Pc where TSDW and Tc join �see Figs. 2and 3�. In the normal phase above the SDW phase, the resis-tivity along the a axis decreases with temperature, reaches aminimum at Tmin, and then shows an upturn and a strongenhancement related to the proximity of the SDW phasetransition that occurs at TSDW�Tmin. The region of the nor-mal phase where strong AF fluctuations are present �TSDW

�T�Tmin� extends over the pressure range where theground state is superconducting �Fig. 3�. Its width in tem-perature decreases with increasing pressure, so that the su-perconducting transition temperature appears to be closelylinked to Tmin. These observations strongly suggest an inti-mate relationship between spin fluctuations and supercon-ductivity in the Bechgaard/Fabre salts.16,17 The importanceof spin fluctuations above the superconducting phase is fur-ther confirmed by the persistence of the enhancement of thespin-lattice relaxation rate 1 /T1 for P Pc.

24 Besides thepresence of spin fluctuations at low temperature, charge fluc-tuations have also been observed in the normal phase viaoptical conductivity measurements.33

III. SUPERCONDUCTIVITY

Some of the early experiments in the Bechgaard saltswere not in contradiction with a conventional BCS supercon-ducting state. For instance, the specific heat in

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�TMTSF�2ClO4 obeys the standard temperature dependenceC /T=�+�T2 above the superconducting transition, and thejump at the transition �C /�Tc�1.67 is close to the BCSvalue 1.43. The ratio 2��T=0� /Tc�3.33, obtained from thegap deduced from the thermodynamic critical field, is also inreasonable agreement with the prediction of the BCS theory�2� /Tc�3.52�.60,61 Early measurements of Hc2�T�, per-formed in the vicinity of the zero-field transition tempera-ture, were also interpreted on the basis of the BCStheory.62–65

Nevertheless, soon after the discovery of organic super-conductivity, the high-sensitivity of the superconductingstate to irradiation66,67 led Abrikosov68 to suggest the possi-bility of an unconventional—triplet—pairing, although thenonmagnetic nature of the induced defects is questionable.7

The sensitivity to nonmagnetic impurities, and thus the exis-tence of unconventional pairing, was later on clearly estab-lished by the suppression of the superconducting transitionupon alloying �TMTSF�2ClO4 with a very small concentra-tion of ReO4 anions.69,70 A recent study71 of the alloy�TMTSF�2�ClO4�x�ReO4�1−x with different cooling rates anddifferent values of x—has confirmed this in remarkable wayby showing that the transition temperature Tc is related to thescattering rate 1 /� by

ln�Tc0

Tc = ��1

2+

1

4��Tc − ��1

2 �6�

�Tc0 is the transition temperature of the pure system and �the digamma function�, as expected for an unconventionalsuperconductor in the presence of nonmagnetic impurities.72

Another indication of a possible unconventional pairingcame from the observation of Gor’kov and Jérome73 that theupper critical field Hc2�T�, extrapolated down to T=0, wouldexceed the Pauli limited field74,75 HP=1.84Tc0 /�B�2 T by afactor of 2. �The value of HP quoted here corresponds tos-wave pairing.� As the spin-orbit interaction is weak in thesesystems and cannot provide an explanation for such a largeHc2, it is tempting to again invoke triplet pairing. This issuehas been revived by recent measurements of the upper criti-cal field in �TMTSF�2PF6 with substantially improved accu-racy in angular alignment and lower temperatures. Lee etal.76,77 observed a pronounced upward curvature of Hc2�T�without saturation—down to T�Tc /60—for a field parallel

to the a or b� axis, with Hc2b��T� and Hc2

a �T� exceeding the

Pauli limited field HP by a factor of 4. Moreover, Hc2b��T�

becomes larger than Hc2a �T� at low temperatures. Similar re-

sults were obtained from simultaneous resistivity and torquemagnetization experiments in �TMTSF�2ClO4 �Fig. 4�.78 Theextrapolated value to zero temperature, Hc2�0��5 T, is atleast twice the Pauli limited field.

There are different mechanisms that can greatly increasethe orbital critical field Hc2

orb�T� in organic conductors. Super-conductivity in a weakly coupled plane system can survive ina strong parallel magnetic field if the interplane �zero-field�coherence length ��T� becomes smaller than the interplanespacing d at low temperature. Vortex cores, with size ��T��d, can then fit between planes without destroying the su-perconducting order in the planes, and lead to a Josephsonvortex lattice. In the Bechgaard salts, even for a field parallelto the b axis, the Josephson limit �T��d is, however,

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unlikely to be reached, since the interchain hopping ampli-tude t�c�5–10 K is larger than the transition temperatureTc�1.1 K. Nevertheless the orbital critical field can be en-hanced by a field-induced dimensional crossover.79–83 Amagnetic field parallel to the b� axis tends to localize thewave functions in the �ac� planes, which in turn weakens theorbital destruction of the superconducting order. When �c

=eHc� t�c �which corresponds to a field of a few tesla in theBechgaard salts�, the wave functions are essentially confinedin the �ac� planes, and the orbital effect of the field is com-pletely suppressed. The coexistence between SDW and su-perconductivity, as observed in a narrow pressure domain ofthe order of 0.8 kbar below the critical pressure Pc �Fig. 2�,can also lead to a large increase of the orbital upper criticalfield.84–88

Regardless of the origin of the large orbital critical field,another mechanism is required to exceed the Pauli limitedfield HP in the Bechgaard salts. For singlet spin pairing, thePauli limit may be overcome by a nonuniform Larkin–Ovchinnikov–Fulde–Ferrell �LOFF� state, where Cooperpairs form with a nonzero total momentum.89,90 This mecha-nism is particularly efficient in a 1D system,79,81,83,91 due tothe large phase space available for pairing at nonzero totalmomentum. For a linearized dispersion law, the mean-fieldupper critical field Hc

LOFF diverges as 1/T in a pure super-conductor. Lebed92 has argued that the quasi-1D anisotropyreduces Hc

LOFF below the experimental observations. Theonly possible explanation for a large upper critical fieldwould then be an equal-spin triplet pairing. A px-wave tripletstate with a d vector perpendicular to the b� axis wasproposed93 as a possible explanation of the experimental ob-servations reported in Refs. 76 and 77.

The triplet scenario in the Bechgaard salts is supportedby recent NMR Knight shift experiments �Fig. 5�.94,95 EarlyNMR experiments by Takigawa et al. already pointed to theunconventional nature of the superconducting state in�TMTSF�2ClO4.96 The proton spin-lattice relaxation rate1 /T1 does not exhibit a Hebel-Slichter peak. It decreasesrapidly just below T , in contrast to the typical BCS super-

FIG. 4. Resistivity �left scale� and torque magnetization �right� in�TMTSF�2ClO4 at 25 mK for H �b�. The dotted line and � symbols on thetorque curve represent a temperature-independent normal-state contribution.The onsets of diamagnetism and decreasing resistivity, upon decreasingfield, are indicated by the arrow near Hc2�5 T. Arrows in the low-fieldvortex state indicate field sweep directions. �Reprinted with permission fromRef. 78. Copyright 2004 by the American Physical Society.�

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conductor where it increases below Tc, reaching a maximumat T�0.9Tc. Furthermore, 1 /T1�T3 for Tc /2�T�Tc—as isthe case for most unconventional superconductors—suggesting zeros or lines of zeros in the excitation spectrum.Recent experiments by Lee et al. in �TMTSF�2PF6 show thatthe Knight shift, and therefore the spin susceptibility, re-mains unchanged at the superconducting transition.94,95 Thisindicates triplet spin pairing, since a singlet pairing wouldinevitably lead to a strong reduction of the spin susceptibility���T→0�→0�. It should, however, be noticed that the inter-pretation of the Knight shift results—due to a possible lackof sample thermalization during the time of theexperiment—has been questioned.7,97

In principle, the symmetry of the order parameter can bedetermined from tunneling spectroscopy. Sign changes of thepairing potential around the Fermi surface lead to zero-energy bound states in the superconducting gap.98 Thesestates manifest themselves as a zero-bias peak in the tunnel-ing conductance into the corresponding edge.99 More gener-ally, different pairing symmetries can be unambiguously dis-tinguished by tunneling spectroscopy in a magneticfield.100–102 In practice, however, the realization of tunneljunctions with the TMTSF salts appears to be very difficult.A large zero-bias conductance peak—suggesting p-wavesymmetry—across the junction between two organic super-conductors was observed.103 But the absence of temperaturebroadening could indicate that this peak is due to disorderrather than to a midgap state.104

Information about the symmetry of the order parametercan also be obtained from thermal conductivity measure-ments. The latter indicate the absence of nodes in the exci-tation spectrum of the superconducting state in�TMTSF� ClO ,105 thus suggesting a p -wave symmetry.

FIG. 5. 77Se NMR spectra Tc �0.81 K at 1.43 T�. Each trace is normalizedand offset for clarity. The temperatures shown in parentheses are the mea-sured equilibrium temperatures before the pulse. In the inset, the spin sus-ceptibility normalized by the normal-state � /�n from measured first mo-ments are compared with theoretical calculations98 for H /Hc2�0��0 �a� and0.63 �b�. Curves c and d are obtained from the ratio of applied field �1.43 T�to the measured upper critical field Hc2�T� at which the superconductingcriteria “onset” and “50 transition” have been used, respectively, to deter-mine Hc2�T�. �Reprinted with permission from Ref. 94. Copyright 2002 bythe American Physical Society.�

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However, because of the doubling of the Fermi surface in thepresence of anion ordering, a singlet d-or triplet f-wave orderphase would also be nodeless in �TMTSF�2ClO4 �see Fig. 6for the different gap symmetries in a quasi-1Dsuperconductor�.9,106

IV. MICROSCOPIC THEORIES OF THE SUPERCONDUCTINGSTATE

The phase diagram of the 1D electron gas within theg-ology framework108 is shown in Fig. 7. g1 and g2 denotethe backward and forward scattering amplitudes, respec-tively, and g3 the strength of the �half-filling� umklapp pro-cesses. Given the importance of spin fluctuations in thephase diagram of the Bechgaard/Fabre salts, as well as theexistence of AF ground states, the Bechgaard/Fabre saltsshould pertain to the upper right corner of the 1D phasediagram �g1 ,g20 and g1−2g2� �g3� where the umklappprocesses are relevant and the dominant fluctuations antifer-romagnetic. In the Fabre salts, the nonmagnetic insulatingphase observed below T��100 K indicates the importanceof umklapp scattering and suggests sizable values of g3 forthis series. Since the long-range Coulomb interaction favorsg1�g2, the Fabre salts are expected to lie to the right of thephase diagram, i.e. far away from the boundary g1−2g2

FIG. 6. Gap symmetries �r�k�� in a quasi-1D superconductor �after Ref.107, courtesy of Y. Suzumura�. r= + /− denotes the right/left sheet of theFermi surface. �Singlet pairing� s: const, dx2−y2: cos k�, dxy: r sin k�. �Tripletpairing� px: r, f: r cos k�, py: sin k�. Next-nearest-neighbor and longer-rangepairings are not considered.

FIG. 7. Phase diagram �leading fluctuations� of the 1D electron gas in pres-ence of umklapp scattering. SS �TS�: singlet �triplet� superconductivity. Agap develops in the charge sector �Mott insulating behavior� for g1−2g2

� �g �.

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= �g3�. Since the triplet superconducting phase is lying next tothe SDW phase �Fig. 7�, it is tempting to invoke a change ofthe couplings gi under pressure to argue in favor of apx-wave triplet superconducting state.68,109 Such a drasticchange of the couplings, which would explain why�TMTTF�2PF6 becomes superconducting above4.35 GPa,16–18 is, however, somewhat unrealistic and has notreceived any theoretical backing so far. The umklapp scatter-ing being much weaker in the Bechgaard salts, one cannotexclude that these compounds lie closer to the boundary be-tween the SDW and the triplet superconducting phase. Amoderate change of the couplings under pressure would thenbe sufficient to explain the superconducting phase of�TMTSF�2PF6 observed above 6 kbar or so. However, thedestruction of the superconducting phase by a weak mag-netic field and the observation of a cascade of SDW phasesfor slightly higher fields50–52 would imply that the interactionis strongly magnetic-field dependent—again a very unlikelyscenario.

In all probability, the very origin of the superconductinginstability lies in the 3D behavior of these quasi-1D conduc-tors. Thus the attractive interaction is a consequence of alow-energy mechanism that becomes more effective belowthe dimensional crossover temperature Tx. Transverse hop-ping makes retarded electron-phonon interactions more ef-fective, since it is easier for the electrons to avoid the Cou-lomb repulsion.109 By comparing the sulfur and selenideseries, it can, however, be argued that in the pressure rangewhere superconductivity is observed, the strength of theelectron-phonon interaction is too weak to explain the originof the attractive interaction. For narrow tight-binding bandsin the organics, the attraction is strongest for backscatteringprocesses in which 2kF phonons are exchanged.110,111 Ac-cording to the results of x-ray experiments performed on�TMTSF�2X, however, the electron-phonon vertex at thiswave vector does not undergo any significant increase in thenormal state �Fig. 8�. The amplitude of the�TMTSF�2PF6—which is directly involved in the strength ofthe phonon exchange—is weak. It is instructive to comparewith the sulfur analog compound �TMTTF�2PF6, for whichthe electron-phonon vertex at 2kF becomes singular, signal-ing a lattice instability towards a spin-Peierls distortion �Fig.8�. This instability produces a spin gap that is clearly visible

FIG. 8. Temperature dependence of the 2kF lattice susceptibility �I /T� as afunction of temperature in the normal phase of �TMTSF�2PF6 �top� and�TMTTF�2PF6 �bottom�. �Reprinted with permission from Ref. 53. Copy-right 1996 by EDP Sciences.�

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in the temperature dependence of the magnetic susceptibilityand nuclear relaxation rate.112,113 These effects are not seenin �TMTSF�2PF6 close to Pc. The persistent enhancement ofthese quantities indicates that interactions are dominantly re-pulsive �Sec. II B 3�, making the traditional phonon-mediated source of pairing inoperant.

Emery114 pointed out that near SDW instability, short-range AF spin fluctuations can give rise to anisotropic pair-ing and thus provide a possible explanation of the origin ofthe superconducting phase in the Bechgaard salts. Such fluc-tuations give rise to an oscillating potential that couples tothe electrons. Carriers can avoid the local Coulomb repulsionand take advantage of the attractive part of this potential bymoving on different chains. This mechanism, which can leadto superconductivity at low temperatures, is the spin analogof the so-called Kohn-Luttinger mechanism, which assumesthe pairing to originate in the exchange of charge-densityexcitations produced by Friedel oscillations.115 While mosttheoretical results on the spin-fluctuation-induced supercon-ductivity are based on RPA-like calculations,116,117,27,118–124

the existence of such an electronic pairing mechanism in aquasi-1D conductor has recently been confirmed by an RGapproach.125 Moreover, it has recently been realized thatCDW fluctuations can play an important role in stabilizing atriplet phase.126–128,107,129–131 Below we discuss in simpleterms the link between spin/charge fluctuations and uncon-ventional pairing,124 and present recent results obtained froma RG approach.129–131

A. Superconductivity from spin and charge fluctuations

Considering for the time being only intrachain interac-tions, the interacting part of the Hamiltonian within theg-ology framework108 reads

Hint = q

�gch��− q���q� + gspS�− q� · S�q�� �7�

�from now on we neglect the c axis and consider a 2Dmodel�, where �q and Sq are the charge- and spin-densityoperators in the Peierls channel �qx�2kF�, gch=g1−g2 /2 andgsp=−g2 /2. Starting from a half-filled extended Hubbardmodel, we obtaing g1=U−2V and g2=U+2V, where U is theon-site and V the nearest-neighbor lattice site �dimer� inter-action. For simplicity, we do not consider umklapp scattering�g3�, since it does not play an important role in the presentqualitative discussion. For repulsive interactions g1�g20,short-range spin fluctuations develop at low temperaturesdue to the nesting of the Fermi surface. They can be de-scribed by an effective Hamiltonian Hint

eff obtained from �7�by replacing the bare coupling constants by their �static� RPAvalues

gchRPA�q� =

gch

1 + gch�0�q�= gch − gch

2 �chRPA�q� ,

gspRPA�q� =

gsp

1 + gsp�0�q�= gsp − gsp

2 �spRPA�q� , �8�

where �RPA is the static ��=0� RPA susceptibility. The bareparticle-hole susceptibility diverges at low temperatures, i.e.,�0�Q�� ln�E0 /max�T , t�b� ��, due to the Q= �2kF ,�� nestingof the quasi-1D Fermi surface �� −��−� +��. �E is a

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high-energy cutoff of the order of the bandwidth.� The diver-gence is cut off by deviations from perfect nesting, charac-terized by the energy scale t�b� � Eq. �1��. In the Bechgaardsalts t�b� �10 K and varies with pressure.

When the nesting of the Fermi surface is good �smallt�b� �, the spin susceptibility �sp

RPA�Q� diverges at low tempera-tures, thus signaling the formation of a SDW. A larger valueof t�b� frustrates antiferromagnetism and, when exceeding athreshold value, eliminates the transition to the SDWphase.132,133 In that case, the �remaining� short-range spinfluctuations can lead to pairing between fermions. To seethis, we rewrite the effective Hamiltonian Hint

eff in the particle-particle �Cooper� channel

Hinteff =

k,k�

�gs�k,k��Os*�k�Os�k��

+ gt�k,k��Ot*�k� · Ot�k��� �9�

�we consider only Cooper pairs with zero total momentum�,where

gs�k,k�� = − 3gspRPA�k + k�� + gch

RPA�k + k�� ,

gt�k,k�� = − gspRPA�k + k�� − gch

RPA�k + k�� �10�

are the effective interactions in the singlet and triplet spinpairing channels �Fig. 9�. Os�k��Ot�k�� is the annihilationoperator of a pair �k ,−k� in a singlet �triplet� spin state, andOt= �Ot

1 ,Ot0 ,Ot

−1� denotes the three components Sz=1,0 ,−1of the triplet state �total spin S=1�.

On the basis of the effective Hamiltonian �9� the BCStheory predicts a superconducting transition whenever theeffective interaction gs,t turns out to be attractive in �at least�one pairing channel. A simple argument shows that this isindeed always the case in the presence of short-range spinfluctuations. The spin susceptibility �sp

RPA�k+k�� exhibits apronounced peak around k+k�=Q. Neglecting the unimpor-tant k� dependence, its Fourier series expansion reads

�spRPA�2kF,k� + k�� � =

n=0

�

an�− 1�n cos�n�k� + k�� ��

= n=0

�

an�− 1�n�cos nk� cos nk��

− sin nk� sin nk�� � , �11�

where an�0. Choosing an=a0, one obtains a diverging spinsusceptibility �sp

RPA�2kF ,k�+k�� ����k�+k�� −��. The condi-tion a0a1 . . . �0 gives a broadened peak around k�+k��=�. Equations �10� and �11� and �11� show that the effective

FIG. 9. Diagrammatic representation of the effective interaction gs,t in theCooper channel within the RPA.

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interaction in the singlet channel contains attractive interac-tions for any value of n. In real space, n corresponds to therange of the pairing interaction in the b direction. The domi-nant attractive interaction corresponds to nearest-neighbor-chain pairing �n=1� and a dx2−y2-wave-wave order parameter�r�k��cos k�� �r=sgn�kx��. The interaction is also attrac-tive in the triplet f-wave channel ��r�k���r cos k��. How-ever, all the three components of a �spin-one boson� SDWfluctuation contribute to the superconducting coupling in thesinglet channel—hence the factor of 3 in the first of equa-tions �10�. The latter therefore always dominates over thetriplet channel when charge fluctuations are not important.Note that the interaction is repulsive in the singlet dxy-wave��r�k���r sin k�� and the triplet py-wave �k�� channels.

Equations �10� show that CDW fluctuations tend to sup-press the singlet pairing but reinforce the triplet pairing. Inthe Bechgaard salts, the physical relevance of CDW fluctua-tions has been borne out by the puzzling observation of aCDW that coexists with the SDW �Sec. II B�.33,53,54 Withinthe framework of an extended anisotropic Hubbard model,recent RPA calculations have shown that the triplet f-wavepairing can overcome the singlet dx2−y2-wave pairing whenthe intrachain interactions are chosen such as to boost theCDW fluctuations with respect to the SDW ones.126–128 In ahalf-filled model, however, this requires the nearest-neighbor�intrachain� interaction V to exceed U /2. In a quarter-filledmodel—appropriate if one ignores the weak dimerizationalong the chains—the condition for f-wave superconductiv-ity becomes V2�U /2−V2 is the next-nearest-neighbor �in-trachain� Coulomb interaction—and appears even more un-realistic. Similar conclusions have been reached within a RGapproach.107

Given that electrons interact through the Coulomb inter-action, not only intrachain but also interchain interactions arepresent in practice. At large momentum transfer, the inter-chain interaction is well known to favor a CDW orderedstate.134–137 This mechanism is mostly responsible for CDWlong-range order observed in several organic and inorganiclow-dimensional solids �e.g. TTF-TCNQ�.138,139 In the Bech-gaard salts, both the interchain Coulomb interaction and thekinetic interchain coupling �t�b� are likely to be important inthe temperature range where superconductivity and SDW in-stability occur, and should be considered on equal footing. ARG approach has recently been used to determine the phasediagram of an extended quasi-1D electron gas model thatincludes interchain hopping, nesting deviations, and both in-trachain and interchain interactions.129–131 The intrachain in-teractions turn out to have a sizeable impact on the structureof the phase diagram. Unexpectedly, for reasonably smallvalues of the interchain interactions, the singlet dx2−y2-wavesuperconducting phase is destabilized to the benefit of thetriplet f-wave phase with a similar range of Tc. The SDWphase is also found to be close in stability to a CDW phase.Before presenting these results in more detail �Sec. IV B�, letus discuss in simple terms the role of interchain interactions.The interchain backward scattering amplitude g1

��0� con-tributes to the effective interaction in the Cooper channel:

gs�k�,k�� � → gs�k�,k�� � + 2g1��cos k� cos k��

− sin k� sin k�� � ,

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gt�k�,k�� � → gt�k�,k�� � + 2g1��− cos k� cos k��

+ sin k� sin k�� � . �12�

It thus tends to suppress singlet dx2−y2 pairing, but favorstriplet f-wave pairing. In addition to this “direct ” contribu-tion, g1

� reinforces CDW fluctuations,

gch�q�� → gch�q�� + 2g1� cos q� �13�

and therefore enhances the f-wave pairing over thedx2−y2-wave pairing via the mechanism of fluctuation ex-change �see Eqs. �10��. As to the interchain forward scatter-ing g2

�, its direct contribution to the DW channel is negli-gible, but it has a detrimental effect on both singlet andtriplet nearest-neighbor-chain pairings. This latter effect,which is neutralized by the umklapp scattering processes,can lead to next-nearest-neighbor-chain pairings when um-klapp processes are very weak.131

B. RG calculation of the phase diagram of quasi-1Dconductors

As a systematic and unbiased method with no a prioriassumption, the RG method is perfectly suited to study com-peting instabilities. The zero-temperature phase diagram ob-tained with this technique is shown in Fig. 10.130,131 In theabsence of interchain interactions �g1

�=g2�=0�, it confirms

the validity of the qualitative arguments given above. Whenthe nesting of the Fermi surface is nearly perfect �small t�� �the ground state is an SDW. Above a threshold value of t�� ,the low-temperature SDW instability is suppressed and theground state becomes a xx2−y2-wave superconducting �SCd�state with an order parameter �r�k���cos k�.125 In the pres-ence of interchain interactions �g1

�0�, the region of stabil-ity of the SCd phase shrinks, and a triplet superconductingf-wave �SCf� phase appears next to the d-wave phase forg1

�=g1� /�vF�0.1—obtained here for typical values of intra-

chain couplings and band parameters.130,131 For larger valuesof the interchain interactions, the SCd phase disappears andthe region of stability of the f-wave superconducting phasewidens. In addition, a CDW phase appears, thus giving thesequence of phase transitions SDW→CDW→SCf as afunction of t�� . For g̃1

��0.12, the SDW phase disappears.Note that for g̃1

��0.11, the region of stability of the CDWphase is very narrow, and there is essentially a direct transi-tion between the SDW and SCf phases.

FIG. 10. T=0 phase diagram as a function of t�� / t� and g̃1�. Circles: SDW,

squares: CDW, triangles: SCd ��r�k���cos k��, crosses: SCf ��r�k���r cos k��. The dashed lines indicate two �among many� possible pressureaxes, corresponding to transitions SDW→SCd and SDW→SCf .130,131

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The RG calculations yield Tc�30 K for the SDW phasein the case of perfect nesting and Tc�0.6–1.2 K for thesuperconducting phase, in reasonable agreement with the ex-perimental observations in the Bechgaard salts. Figure 11shows the transition temperature Tc as a function of t�� forthree different values of the interchain interactions, g̃1

�=0,0.11, and 0.14, corresponding to the three different se-quences of phase transitions as a function of t�� : SDW→SCd, SDW→ �CDW�→SCf , CD→WSCf . The phasediagram is unchanged when both g2

� and a weak umklappscattering amplitude g3 are included.130,131

The RG approach also provides important informationabout the fluctuations in the normal phase. The dominantfluctuations above the SCd phase are SDW fluctuations, asobserved experimentally �Sec. II B�. Although they saturatebelow T� t�� , where the SCd fluctuations become more andmore important, the latter dominate only in a very narrowtemperature range above the superconducting transition �Fig.12�. Above the SCf and CDW phases, one expects strongCDW fluctuations driven by g1

�. Figure 13 shows that forg1

��0.11-0.12, strong SDW and CDW fluctuations coexistabove the SCf phase. Remarkably, there are regions of thephase diagram where the SDW fluctuations remain the domi-nant ones in the normal phase above the SCf or CDW phase�Fig. 13b�.

A central result of the RG calculation is the close prox-imity of SDW, CDW, and SCf phases in the phase diagramof a quasi-1D conductor within a realistic range of values forthe repulsive intrachain and interchain interactions. Althoughthis proximity is found only in a small range of interchain

FIG. 11. Transition temperature as a function of t�� / t� and g̃1�=0, 0.11 and

0.14, corresponding to solid, dotted, and dashed lines, respectively.130,131

FIG. 12. Temperature dependence of the susceptibilities in the normal phaseabove the SCd phase �t�� =0.152t� and g̃1

�=0.08. The continuous, dotted,dashed, and dashed-dotted lines correspond to SDW, CDW, SCd and SCfcorrelations, respectively.130,131

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interactions, there are several features that suggest that thispart of the phase diagram is the relevant one for the Bech-gaard salts. i� SDW fluctuations remain important in the nor-mal phase throughout the whole phase diagram. They are thedominant fluctuations above the SCd phase, and remainstrong—being sometimes even dominant—above the SCfphase, where they coexist with strong CDW fluctuations, inaccordance with observations.24,33 ii� The SCf and CDWphases stand nearby in the theoretical phase diagram, theCDW phase always closely following the SCf phase whenthe interchain interactions increase. This agrees with the ex-perimental finding that both SDW and CDW coexist in theDW phase of the Bechgaard salts53,54 and the existence, be-sides SDW correlations, of CDW fluctuations in the normalstate above the superconducting phase.33 iii� Depending howone moves in practice in the phase diagram as a function ofpressure, these results are compatible with either a singletdx2−y2-wave or a triplet f-wave superconducting phase in theBechgaard salts �see the two pressure axes in Fig. 10�. More-over, one cannot rule out that both SCd and SCf phases existin these materials, with the sequence of phase transitionsSDW→SCd→SCf as a function of pressure. It is also pos-sible that the SCf phase is stabilized by a magnetic field,140

since an equal-spin pairing triplet phase is not sensitive tothe Pauli pair-breaking effect, contrary to the SCd phase.This would make possible the existence of large upper criti-cal fields exceeding the Pauli limit76,78 and would also pro-vide an explanation for the temperature independence of theNMR Knight shift in the superconducting phase.94

V. CONCLUSION

Notwithstanding the recent experimental progress, manyof the basic questions related to superconductivity in theBechgaard and Fabre salts remain largely open. The verynature of the superconducting state—the orbital symmetry of

FIG. 13. Temperature dependence of the susceptibilities in the normal phaseabove the SCf phase for g̃1

�=0.12, t�� =0.152t� �a� and t�� =0.176t� �b�.

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the order parameter and the singlet/triplet character of thepairing—is still not known without ambiguity even thoughrecent upper critical field measurements76–78 and NMRexperiments94,95 support a triplet pairing.

We have argued that the conventional electron-phononmechanism is unable to explain the origin of the supercon-ducting phase. On the other hand, the proximity of the SDWphase, as well as the observation of strong spin fluctuationsin the normal-state precursor to the superconductingphase,16,17,24 strongly suggest an intimate relationship be-tween antiferromagnetism and superconductivity in theBechgaard/Fabre salts. The scenario originally proposed byEmery,114 whereby short-range AF spin fluctuations can giverise to anisotropic pairing and thus stabilize a superconduct-ing phase, is so far the only one that is consistent with theexperimental observations and the repulsive nature of theelectron-electron interactions.

Within the framework of the RG approach, it has re-cently been shown that when spin and charge fluctuations aretaken into account on equal footing, both singlet dx2−y2- andtriplet f-wave superconducting phases can emerge at lowtemperatures whenever the nesting properties of the Fermisurface deteriorate under pressure.126–128,107,129,131 CDWfluctuations are enhanced by the long-range part of the Cou-lomb interaction. Remarkably, for a reasonably small valueof the interchain interactions, the singlet dx2−y2-wave phase isdestabilized to the benefit of a triplet f-wave with a similarrange of Tc.

130,131 The physical relevance of CDW fluctua-tions in the Bechgaard salts has been born out by the obser-vation of a CDW that actually coexists with the SDW.53,54

CDW fluctuations were also observed in the normal-stateprecursor to the superconducting state.33

As a systematic and unbiased method with no a prioriassumptions, the RG has proven to be a method of choice tostudy the physical properties of quasi-1D organic conductors.An important theoretical issue is now to go beyond the in-stabilities of the normal state. On the one hand, the RGanalysis should be extended to the low-temperature broken-symmetry phases in order to study the possible coexistenceof superconductivity and antiferromagnetism, as well asCDW and SDW, as observed in the Bechgaard salts.86,88,53,54

On the other hand, the RG technique might also enable totackle the unusual properties of the metallic phase. A recentRG analysis107 of the AF spin susceptibility in the normalphase has shown that below the dimensional crossover tem-perature, it differs significantly from the prediction of single-channel �RPA� theories. The interplay between the supercon-ducting and Peierls channels, which is at the origin of spin-fluctuation-induced superconductivity, might also beresponsible for the unusual properties of the metallic statebelow the dimensional crossover temperature.

a�E-mail: n.dupuis@imperial.ac.uk

1D. Jérome, A. Mazaud, M. Ribault, and K. Bechgaard, J. Phys. �France�Lett. 41, L95 �1980�.

2D. Jérome and H. J. Schulz, Adv. Phys. 31, 299 �1982�.3T. Ishiguro and K. Yamaji, Organic Superconductors, Vol. 88 of SpringerSeries in Solid-State Science, Springer-Verlag, Berlin, Heidelberg �1990�.

ticle is copyrighted as indicated in the article. Reuse of AIP content is subje

72.10.100.172 On: Sat, 0

4R. H. McKenzie, Comments Condens. Matter Phys. 18, 309 �1998�.5S. Lefebvre et al., Phys. Rev. Lett. 85, 5420 �2000�.6J. Flouquet, arXiv: cond-mat/0501602 �unpublished�.7D. Jérome, Chem. Rev. �Washington, D.C.� 104, 5565 �2004�.8K. Bechgaard et al., Solid State Commun. 33, 1119 �1980�.9C. Bourbonnais and D. Jérome, in Advances Synthetic Metals, TwentyYears of Progress in Science and Technology, P. Bernier, S. Lefrant, andG. Bidan �eds.�, Elsevier, New York �1999�, p. 206; arXiv: cond-mat/9903101.

10P M. Grant, J. Phys. �France� 44, 847 �1983�.11K. Yamaji, J. Phys. Soc. Jpn. 51, 2787 �1982�.12L. Ducasse et al., J. Phys. C 19, 3805 �1986�.13L. Balicas et al., J. Phys. �France� 4, 1539 �1994�.14V. J. Emery, R. Bruisma, and S. Barišić, Phys. Rev. Lett. 48, 1039 �1982�.15S. Barišić and S. Brazovskii, in Recent Developments in Condensed Mat-

ter Physics, J. T. Devreese �ed.�, Plenum, New York, �1981�, Vol. 1, p.327.

16D. Jaccard et al., J. Phys.: Condens. Matter 13, L89 �2001�.17H. Wilhelm et al., Eur. Phys. J. B 21, 175 �2001�.18T. Adachi et al., J. Am. Chem. Soc. 122, 3238 �2000�.19C. Bourbonnais and D. Jérome, Science 281, 1156 �1998�.20F. D. M. Haldane, J. Phys. C 14, 2585 �1981�.21T. Giamarchi, Quantum Physics in One Dimension, Oxford University

Press, Oxford �2004�.22C. Coulon et al., J. Phys. �France� 43, 1059 �1982�.23A. Schwartz et al., Phys. Rev. B 58, 1261 �1998�.24P. Wzietek et al., J. Phys. 3, 171 �1993�.25C. Bourbonnais et al., Phys. Rev. Lett. 62, 1532 �1989�.26C. Bourbonnais, J. Phys. 3, 143 �1993�.27C. Bourbonnais and L. G. Caron, Europhys. Lett. 5, 209 �1988�.28C. Bourbonnais and L. G. Caron, Int. J. Mod. Phys. B 5, 1033 �1991�.29S. E. Brown et al., Synth. Met. 86, 1937 �1997�.30C. Bourbonnais, F. Creuzet, D. Jérome, and K. Bechgaard, J. Phys.

�France� Lett. 45, L755 �1984�.31C. S. Jacobsen, D. B. Tanner, and K. Bechgaard, Phys. Rev. Lett. 46, 1142

�1981�.32C. S. Jacobsen, D. B. Tanner, and K. Bechgaard, Phys. Rev. B 28, 7019

�1983�.33N. Cao, T. Timusk, and K. Bechgaard, J. Phys. 6, 1719 �1996�.34M. Dressel, A. Schwartz, G. Grüner, and L. Degiorgi, Phys. Rev. Lett. 77,

398 �1996�.35T. Giamarchi, Physica B 230–232, 975 �1997�.36V. Vescoli et al., Science 281, 1181 �1998�.37T. Giamarchi, Chem. Rev. �Washington, D.C.� 104, 5037 �2004�.38J. Favand and F. Mila, Phys. Rev. B 54, 10425 �1996�.39J. Moser et al., Phys. Rev. Lett. 84, 2674 �2000�.40P. Auban-Senzier, D. Jérome, and J. Moser, in Physical Phenomena at

High Magnetic Fields, Z. Fisk, L. Gor’kov, and L. Schrieffer �eds.�, WorldScientic, Singapore �1999�.

41G. Mihály, I. Kézsmárki, F. Zámborszky, and L. Forro, Phys. Rev. Lett.84, 2670 �2000�.

42P. Fertey, M. Poirier, and P. Batail, Eur. Phys. J. B 10, 305 �1999�.43J. Moser et al., Eur. Phys. J. B 1, 39 �1998�.44F. Zwick et al., Phys. Rev. Lett. 79, 3982 �1997�.45B. Dardel et al., Europhys. Lett. 24, 687 �1993�.46A. Georges, T. Giamarchi, and N. Sandler, Phys. Rev. B 61, 16393

�2000�.47G. M. Danner, W. Kang, and P. M. Chaikin, Phys. Rev. Lett. 72, 3714

�1994�.48L. P. Gor’kov, J. Phys. �France� 6, 1697 �1996�.49A. T. Zheleznyak and V. M. Yakovenko, Eur. Phys. J. B 11, 385 �1999�.50P. M. Chaikin, J. Phys. �France� 6, 1875 �1996�.51P. Lederer, J. Phys. �France� 6, 1899 �1996�.52V. M. Yakovenko and H. S. Goan, J. Phys. 6, 1917 �1996�.53J. P. Pouget and S. Ravy, J. Phys. 6, 1501 �1996�.54S. Kagoshima et al., Solid State Commun. 110, 479 �1999�.55H. Seo and H. Fukuyama, J. Phys. Soc. Jpn. 66, 1249 �1997�.56N. Kobayashi, M. Ogata, and K. Yonemitsu, J. Phys. Soc. Jpn. 67, 1098

�1998�.57S. Mazumdar, S. Ramasesha, R. T. Clay, and D. K. Campbell, Phys. Rev.

Lett. 82, 1522 �1999�.58Y. Tomio and Y. Suzumura, J. Phys. Soc. Jpn. 69, 796 �2000�.59Y. Tomio and Y. Suzumura, J. Phys. Chem. Solids 62, 431 �2001�.60P. Garoche, R. Brusetti, and K. Bechgaard, Phys. Rev. Lett. 49, 1346

�1982�.61P. Garoche, R. Brusetti, and D. Jérome, J. Phys. �France� Lett. 43, L147

ct to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

3 May 2014 05:37:29

Low Temp. Phys. 32 �4-5�, April-May 2006 Dupuis et al. 391

This ar

�1982�.62D. L. Gubser et al., Phys. Rev. B 24, 478 �1981�.63K. Murata et al., Mol. Cryst. Liq. Cryst. 79, 639 �1982�.64R. L. Green, P. Haen, S. Z. Huang, and E. M. Engler, Mol. Cryst. Liq.

Cryst. Lett. 79, 183 �1982�.65R. Brusetti, M. Ribault, D. Jérome, and K. Bechgaard, J. Phys. �France�

43, 52 �1982�.66M. Y. Choi et al., Phys. Rev. B 25, 6208 �1985�.67S. Bouffard et al., J. Phys. C 15, 2951 �1982�.68A. A. Abrikosov, JETP Lett. 37, 503 �1983�.69C. Coulon et al., J. Phys. �France� 43, 1721 �1982�.70S. Tomic et al., J. Phys. Colloq. 44, C 3 �1983�.71N. Joo et al., Eur. Phys. J. B 40, 43 �2004�.72Q. Yuan et al., Phys. Rev. B 68, 174510 �2003�.73L. P. Gor’kov and D. Jérome, J. Phys. �France� Lett. 46, L643 �1985�.74A. M. Clogston, Phys. Rev. Lett. 9, 266 �1962�.75B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 �1962�.76I. J. Lee, M. J. Naughton, G. M. Danner, and P. M. Chaikin, Phys. Rev.

Lett. 78, 3555 �1997�.77I. J. Lee, P. M. Chaikin, and M. J. Naughton, Phys. Rev. B 62, R14669

�2000�.78J. I. Oh and M. J. Naughton, Phys. Rev. Lett. 92, 067001 �2004�.79A. G. Lebed, JETP Lett. 44, 114 �1986�.80L. I. Burlachkov, L. P. Gor’kov, and A. G. Lebed, Europhys. Lett. 4, 941

�1987�.81N. Dupuis, G. Montambaux, and C. A. R. Sá de Melo, Phys. Rev. Lett.

70, 2613 �1993�.82N. Dupuis and G. Montambaux, Phys. Rev. B 49, 8993 �1994�.83N. Dupuis, Phys. Rev. B 51, 9074 �1995�.84R. L. Greene and E. M. Engler, Phys. Rev. Lett. 45, 1587 �1980�.85R. Brusetti, M. Ribault, D. Jérome, and K. Bechgaard, J. Phys. �France�

43, 801 �1982�.86T. Vuletić et al., Eur. Phys. J. B 25, 319 �2002�.87I. J. Lee, P. M. Chaikin, and M. J. Naughton, Phys. Rev. Lett. 88, 207002

�2002�.88I. J. Lee et al., Phys. Rev. Lett. 94, 197001 �2005�.89A. I. Larkin and Y. N. Ovchinnikov, Sov. Phys. JETP 20, 762 �1965�.90P. Fulde and R. A. Ferrell, Physiol. Rev. 135, A 550 �1965�.91A. I. Buzdin and S. V. Polonskii, Sov. Phys. JETP 66, 422 �1983�.92A. G. Lebed, Phys. Rev. B 59, R721 �1999�.93A. G. Lebed, K. Machida, and M. Ozaki, Phys. Rev. B 62, R795 �2000�.94I. J. Lee et al., Phys. Rev. Lett. 88, 017004 �2002�.95I. J. Lee et al., Phys. Rev. B 68, 092510 �2003�.96M. Takigawa, H. Yasuoka, and G. Saito, J. Phys. Soc. Jpn. 56, 873 �1987�.97D. Jérome and C. R. Pasquier, in Superconductors, A. V. Narlikar �ed.�,

Springer-Verlag, Berlin �2005�.98P. Fulde and K. Maki, Phys. Rev. B 139, A 788 �1965�.99K. Sengupta et al., Phys. Rev. B 63, 144531 �2001�.100Y. Tanuma, K. Kuroki, Y. Tanaka, and S. Kashiwaya, Phys. Rev. B 64,

214510 �2001�.101Y. Tanuma et al., Phys. Rev. B 66, 094507 �2002�.102Y. Tanuma, K. Kuroki, Y. Tanaka, and S. Kashiwaya, Phys. Rev. B 68,

214513 �2003�.

ticle is copyrighted as indicated in the article. Reuse of AIP content is subje

72.10.100.172 On: Sat, 0

103H. I. Ha, J. I. Oh, J. Moser, and M. J. Naughton, Synth. Met. 137, 1215�2003�.

104M. Naughton, private communication �unpublished�.105S. Belin and K. Behnia, Phys. Rev. Lett. 79, 2125 �1997�.106H. Shimahara, Phys. Rev. B 61, R14936 �2000�.107Y. Fuseya and Y. Suzumura, J. Phys. Soc. Jpn. 74, 1264 �2005�.108J. Sólyom, Adv. Phys. 28, 201 �1979�.109V. J. Emery, J. Phys. Colloq. 44, C 3 �1983�.110 S. Barišić, J. Labbé, and J. Friedel, Phys. Rev. Lett. 25, 99 �1970�.111 W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698

�1979�.112 F. Creuzet et al., Synth. Met. 19, 289 �1987�.113 C. Bourbonnais and B. Dumoulin, J. Phys. 6, 1727 �1996�.114 V. J. Emery, Synth. Met. 13, 21 �1986�.115 W. Kohn and J. M. Luttinger, Phys. Rev. Lett. 15, 524 �1965�.116 M. T. Béal-Monod, C. Bourbonnais, and V. J. Emery, Phys. Rev. B 34,

7716 �1986�.117 L. G. Caron and C. Bourbonnais, Physica B 143, 453 �1986�.118 D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 �1986�.119 D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 35, 6694 �1987�.120K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554

�1986�.121H. Shimahara, J. Phys. Soc. Jpn. 58, 1735 �1988�.122H. Kino and H. Kontani, J. Low Temp. Phys. 117, 317 �1999�.123K. Kuroki and H. Aoki, Phys. Rev. B 60, 3060 �1999�.124D. J. Scalapino, Phys. Rep. 250, 329 �1995�.125R. Duprat and C. Bourbonnais, Eur. Phys. J. B 21, 219 �2001�.126K. Kuroki, R. Arita, and H. Aoki, Phys. Rev. B 63, 094509 �2001�.127S. Onari, R. Arita, K. Kuroki, and H. Aoki, Phys. Rev. B 70, 94523

�2004�.128Y. Tanaka and K. Kuroki, Phys. Rev. B 70, R060502 �2004�.129C. Bourbonnais and R. Duprat, Bull. Am. Phys. Soc. 49, 1:179 �2004�.130J. C. Nickel, R. Duprat, C. Bourbonnais, and N. Dupuis, Phys. Rev. Lett.

95, 247001 �2005�.131J. C. Nickel, R. Duprat, C. Bourbonnais, and N. Dupuis, arXiv: cond-

mat/0510744 �unpublished�.132Y. Hasegawa and H. Fukuyama, J. Phys. Soc. Jpn. 55, 3978 �1986�.133G. Montambaux, Phys. Rev. B 38, 4788 �1988�.134L. P. Gor’kov and I. E. Dzyaloshinskii, Sov. Phys. JETP 40, 198 �1974�.135L. Mihály and J. Sólyom, J. Low Temp. Phys. 24, 579 �1976�.136P. A. Lee, T. M. Rice, and R. A. Klemm, Phys. Rev. B 15, 2984 �1977�.137N. Menyhárd, Solid State Commun. 21, 495 �1977�.138S. Barišić and A. Bjeliš, in Theoretical Aspects of Band Structures and

Electronic Properties of Pseudo-One-Dimensional Solids, H. Kaminura�ed.�, D. Reidel, Dordrecht �1985�, p. 49.

139J. P. Pouget and R. Comes, in Charge Density Waves in Solids, L. P.Gor’kov and G. Grüner �eds.�, Elsevier Science, Amsterdam �1989�, p.85.

140H. Shimahara, J. Phys. Soc. Jpn. 69, 1966 �2000�.

This article was published in English in the original Russian journal. Repro-

duced here with stylistic changes by AIP.

ct to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

3 May 2014 05:37:29