Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article)

Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article)

We review the current understanding of superconductivity in the quasi-one-dimensional organic conductors of the Bechgaard and Fabre salt families. We discuss the interplay between superconductivity, antiferromagnetism, and charge-density-wave fluctuations. The connection to recent experimental ob...

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Datum:2006
Hauptverfasser: Dupuis, N., Bourbonnais, C., Nickel, J.C.
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Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2006
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Spin Models
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spelling nasplib_isofts_kiev_ua-123456789-1201922025-02-09T16:32:06Z Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article) Dupuis, N. Bourbonnais, C. Nickel, J.C. Spin Models We review the current understanding of superconductivity in the quasi-one-dimensional organic conductors of the Bechgaard and Fabre salt families. We discuss the interplay between superconductivity, antiferromagnetism, and charge-density-wave fluctuations. The connection to recent experimental observations supporting unconventional pairing and the possibility of a triplet–spin order parameter for the superconducting phase is also presented. 2006 Article Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article) / N. Dupuis, C. Bourbonnais, J.C. Nickel // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 505–520. — Бібліогр.: 140 назв. — англ. 0132-6414 PACS: 74.70.Kn, 74.20.Mn, 75.30.Fv https://nasplib.isofts.kiev.ua/handle/123456789/120192 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Spin Models
Spin Models
spellingShingle Spin Models
Spin Models
Dupuis, N.
Bourbonnais, C.
Nickel, J.C.
Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article)
Физика низких температур
description We review the current understanding of superconductivity in the quasi-one-dimensional organic conductors of the Bechgaard and Fabre salt families. We discuss the interplay between superconductivity, antiferromagnetism, and charge-density-wave fluctuations. The connection to recent experimental observations supporting unconventional pairing and the possibility of a triplet–spin order parameter for the superconducting phase is also presented.
format Article
author Dupuis, N.
Bourbonnais, C.
Nickel, J.C.
author_facet Dupuis, N.
Bourbonnais, C.
Nickel, J.C.
author_sort Dupuis, N.
title Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article)
title_short Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article)
title_full Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article)
title_fullStr Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article)
title_full_unstemmed Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article)
title_sort superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (review article)
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2006
topic_facet Spin Models
url https://nasplib.isofts.kiev.ua/handle/123456789/120192
citation_txt Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article) / N. Dupuis, C. Bourbonnais, J.C. Nickel // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 505–520. — Бібліогр.: 140 назв. — англ.
series Физика низких температур
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AT bourbonnaisc superconductivityandantiferromagnetisminquasionedimensionalorganicconductorsreviewarticle
AT nickeljc superconductivityandantiferromagnetisminquasionedimensionalorganicconductorsreviewarticle
first_indexed 2025-11-27T23:55:35Z
last_indexed 2025-11-27T23:55:35Z
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fulltext Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5, p. 505–520 Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors (Review Article) N. Dupuis1,2, C. Bourbonnais3, and J.C. Nickel2,3 1Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ United Kingdom E-mail: n.dupuis@imperial.ac.uk 2Laboratoire de Physique des Solides, CNRS UMR 8502, Universit� Paris-Sud, 91405 Orsay, France 3Regroupement Qu�becois sur les Mat�riaux de Pointe, Universit� de Sherbrooke Sherbrooke, Qu�bec, Canada J1K-2R1 Received September 15, 2005 We review the current understanding of superconductivity in the quasi-one-dimensional or- ganic conductors of the Bechgaard and Fabre salt families. We discuss the interplay between su- perconductivity, antiferromagnetism, and charge-density-wave fluctuations. The connection to re- cent experimental observations supporting unconventional pairing and the possibility of a triplet–spin order parameter for the superconducting phase is also presented. PACS: 74.70.Kn, 74.20.Mn, 75.30.Fv Keywords: superconductivity, antiferromagnetism, unconventional pairing. 1. Introduction Superconductivity in organic conductors was first discovered in the ion radical salt (TMTSF)2PF6 [1]. Later on, it was found in most Bechgaard [(TMTSF)2X] and Fabre [(TMTTF)2 X] salts. These salts are based on the organic molecules tetra- methyltetraselenafulvalene (TMTSF) and tetra- methyltetrathiafulvalene (TMTTF). The monovalent anion X can be either a centrosymmetric (PF6, AsF6, etc.) or a non-centrosymmetric (ClO4, ReO4, NO3, FSO3, SCN, etc.) inorganic molecule. (See Refs. 2,3 for previous reviews on these compounds.) Although they are definitely not «high-Tc» superconductors — the transition temperature is of the order of 1 K –, these quasi-one-dimensional (quasi-1D) conductors share several properties of high-Tc superconductors and other strongly-correlated electron systems such as layered organic superconductors [4,5] or heavy-fer- mion materials [6]. The metallic phase of all these conductors exhibits unusual properties which cannot be explained within the framework of Landau’s Fermi liquid theory and remain to a large extent to be under- stood. The superconducting phase is unconventional (not s-wave). Magnetism is ubiquitous in these corre- lated systems and might provide the key to the under- standing of their behavior. The quest for superconductivity in organic conduc- tors was originally motivated by Little’s proposal that highly polarizable molecules could lead – via an excitonic pairing mechanism – to tremendously large transition temperatures. Early efforts towards the chemical synthesis of such compounds were not suc- cessful, as far as superconductivity is concerned, but led to the realization of a 1D charge transfer salt (TTF–TCNQ) undergoing a Peierls instability at low temperatures [7]. Attempts to suppress the Peierls state and stabilize a conducting (and possibly super- conducting) state by increasing the 3D character of this 1D conductor proved to be unsuccessful. Organic superconductivity was eventually discove- red in the the Bechgaard salt (TMTSF)2PF6 under 9 kbar of pressure [1]. It was subsequently found in © N. Dupuis, C. Bourbonnais, and J.C. Nickel, 2006 other members of the (TMTSF)2X series. Most of the Bechgaard salts are insulating at ambient pressure and low temperatures [8], and it came as a surprise that the insulating state of these materials is a spin-den- sity-wave (SDW) rather than an ordinary Peierls state [2]. The important part played by magnetism in these compounds was further revealed when it was found that their phase diagram only shows a part of a larger sequence of ordered states, which includes the N�el and the spin-Peierls phases of their sulfur ana- logs, the Fabre salts (TMTTF)2X series [9]. The charge transfer from the organic molecules to the anions leads to a commensurate band filling 3/4 coming from the 2:1 stoichiometry. The metallic char- acter of these compounds at high enough temperature is due to the delocalization of carriers via the overlap of �-orbitals between neighboring molecules along the stacking direction (a axis) (Fig. 1) [2]. The electronic dispersion relation obtained from quantum chemistry calculations (extended H�ckel method) is well ap- proximated by the following tight-binding form [10–13] �( ) cos( ) ( ) ( )k � � � �� �2 2 2 2t k a/ t k b t k ca a b b c ccos cos � � v k k t k b t k bF a F b b b b(| | ) cos( ) cos( )� � � � �� �2 2 � ��2t k cc ccos( ) ,� (1) where it is assumed that the underlying lattice is orthorhombic. This expression is a simplification of the dispersion relation — the actual crystal lattice symmetry is triclinic — but it retains the essential features. The conduction band along the chain direc- tion has an overall width 4ta ranging between 0.4 and 1.2 eV, depending on the organic molecule (TMTSF or TMTTF) and the anion. As the electronic overlaps in the transverse b and c directions are much weaker than along the organic stacks, the dispersion law is strongly anisotropic, t /tb a� � 01. and t /tc b� � � 0 03. , and the Fermi surface consists of two open warped sheets (Fig. 1). In the second line of Eq. (1), the electronic dispersion is linearized around the two 1D Fermi points �kF , with vF the Fermi ve- locity along the chains (� is the chemical potential). The next-nearest-chain hopping t t /tb b a� � �� � 2 2 � is introduced in order to keep the shape of the Fermi surface unchanged despite the linearization. The an- ions located in centrosymmetric cavities lie slightly above or below the molecular planes. This structure leads to a dimerization of the organic stacks and a (weak) gap D, thus making the hole-like band ef- fectively half-filled at sufficiently low energy or tem- perature [14,15]. (See Refs. 2,7,9 for a detailed dis- cussion of the structural properties of quasi-1D organic conductors.) In the presence of interactions, commensurate band-filling introduces Umklapp scat- tering, which affects the nature of the possible phases in these materials. What is remarkable about these electronic systems is the variety of ground states that can be achieved ei- ther by chemical means, namely substituting selenium by sulfur in the organic molecule or changing the na- ture of the anion (its size or symmetry), or applying pressure (Fig. 2). At low pressure, members of the sul- fur series are Mott insulators (MI) from which either a lattice distorted spin-Peierls (SP) state — often preceded by a charge ordered (CO) state — or a com- mensurate-localized antiferromagnetic state (AF) can develop. On the other hand, itinerant antiferro- magnetism (spin-density wave (SDW)) or supercon- ductivity is found in the selenide series. Under pres- sure, the properties of the Fabre salts evolve towards those of the Bechgaard salts. The compound (TMTTF)2PF6 spans the entire phase diagram as pres- sure increases up to 50 kbar or so (Fig. 3) [16–18], thus showing the universality of the phase diagram in Fig. 2 [19]. A large number of both theoretical and experimen- tal works have been devoted to the understanding of the normal phase and the mechanisms leading to long-range order at low temperature. The presence of antiferromagnetism over a large pressure range does 506 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 N. Dupuis, C. Bourbonnais, and J.C. Nickel –9.8 –10.0 –10.2 WII Wb ' X ZY X Y V E ,e V WI Wb � Wc Wc� Wa a b Fig. 1. A side view of the Bechgaard/Fabre salt crystal structure with the electron orbitals of the organic stacks (courtesy of J.Ch. Ricquier) (a). Electronic dispersion re- lation and projected 2D Fermi surface of (TMTTF)2Br (reprinted with permission from Ref. 13. Copyright 1994 by EDP Sciences) (b). indicate that repulsive interactions among carriers are important. The low-dimensionality of the system is also expected to play a crucial role. On the one hand, in the presence of repulsive interactions a strongly anisotropic Fermi surface with good nesting properties is predominantly unstable against the formation of an SDW state which is reinforced at low temperature by commensurate band filling. On the other hand, when the temperature exceeds the transverse dispersion �t b, 3D (or 2D) coherent electronic motion is sup- pressed and the conductor behaves as if it were 1D; the Fermi liquid picture breaks down and the system becomes a Luttinger liquid [20,21]. The relevance of 1D physics for the low-temperature properties (T t b� � ), as well as a detailed description of the cross- over from the Luttinger liquid to the Fermi liquid, is one of the most important issues in the debate sur- rounding the theoretical description of the normal state of these materials. As far as low-temperature phases are concerned, a chief objective is to reach a good description of the superconducting phase — the symmetry of the order parameter is still under debate — and the mechanisms leading to superconductivity. Owing to the close proximity of superconductivity and magnetism in the phase diagram of Fig. 2, it is es- sential to first discuss the origin of antiferromag- netism in both series of compounds. 2. N�el antiferromagnetism and spin-density wave 2.1. Fabre salts at ambient pressure: Mott-insulator regime The Fabre salts (TMTTF)2X at ambient pressure are located on the left of the phase diagram in Fig. 2. Both the nature of correlations and the mechanism of long-range order at low temperature are now rather well understood. Below the temperature T� � 100 K (see Fig. 2), the resistivity develops a thermally acti- vated behavior [22] and the system enters a Mott-insu- lator regime. The corresponding charge gap � ��� T can be deduced from T� and turns out to be larger than the (bare) transverse bandwidth t b� , which in turn suppresses any possibility of transverse single particle band motion and makes the system essentially one-di- mensional. The charge gap 2 � is also directly ob- served in the optical conductivity [23]. The members of the (TMTTF)2X series thus behave as typical 1D Mott insulators below T� with the carriers confined along the organic stacks — as a result of the Umklapp scattering due to the commensurability of the elec- tronic density with the underlying lattice [14,15]. This interpretation agrees with the absence of anomaly in the spin susceptibility at T� [24], in accordance with the spin-charge separation characteristic of 1D systems [21]. It is further confirmed by measurements of the spin-lattice relaxation rate 1 1/T . The Luttinger liquid theory predicts [25,26] 1 1 0 2 1T C T T C Ts K � �� �( ) , (2) where C0 and C1 are temperature independent con- stants. � s T( ) is the uniform susceptibility and K� the Luttinger liquid charge stiffness parameter. The two contributions in (2) correspond to paramagnons or spinons (q � 0) and AF spin fluctuations (q kF� 2 ). Both the temperature dependence of � s T( ) and the presence of AF fluctuations lead to an enhancement of 1 1/T with respect to the Korringa law ( )T T1 1� � const which holds in higher-dimensional metals. In a 1D Mott insulator K� � 0, which leads to T C T T Cs1 1 0 2 1 � � �� ( ) in good agreement with experi- mental measurements of T1 and � s [24]. The low-energy excitations in the Mott-insulator regime are 1D spin fluctuations. By lowering the tem- perature, these fluctuations can propagate in the transverse direction and eventually drive an AF tran- sition. This transition is not connected to Fermi sur- face effects. The condition � � �t b precludes a sin- gle-particle coherent motion in the transverse direction, and the concept of Fermi surface remains ill defined in the Fabre salts at ambient pressure. AF Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 507 T, K 100 10 1 10 20 30 50 P, kbar T� LL LL FL 2X Pm MI-CO SP SC Pc (TMTTF) AF 2X(TMTSF) X= P F 6 P F 6 B r C IO 4 Fig. 2. The generic phase diagram of the Bechgaard/Fabre salts as a function of pressure or anion X substitution; Luttinger liquid (LL), Mott insulator (MI), charge order (CO), spin-peierls (SP), antiferromagnetism (AF), super- conductivity (SC), Fermi liquid (FL). long-range order comes from interchain transfer of bound electron-hole pairs leading to a kinetic exchange interaction J� between spin densities on neighboring chains — much in analogy with the exchange interac- tion between localized spins in the Heisenberg limit. An effective Hamiltonian can be derived from a renormalization group (RG) calculation, [27,28] H J dx x x J a t i i j j b � � � � � � � �S S , ( ) ( ), � �� � 2 (3) where t b� * is the effective interchain hopping at the energy scale � and a the lattice spacing along the chain. The sum in Eq. (3) is over nearest-neighbor chains. The naive value t /b� *2 � of the exchange inter- action J� is enhanced by the factor ��/a where �� �� v /F is the intrachain coherence length in- duced by the Mott gap along which virtual interchain hoppings can take place. Within a mean-field treat- ment of H� , the condition for the onset of long-range order is given by J k TF� ��1D( , )2 1 where � �1D( , ) ( )2 1k T T/F � is the exact power law form of the 1D AF spin susceptibility. This yields a N�el temperature T t N b� � �2 � . (4) Since T� and � decrease under pressure (Fig. 2), Eq. (4) predicts an increase of TN with pressure — as- suming a weak pressure dependence of tb � — as observed experimentally (see Fig. 2). The relation T T tN b� � �� �2 const has been observed in (TMTTF)2Br [29]. 2.2. Bechgaard salts: itinerant magnetism With increasing pressure, T� drops and finally merges with the AF transition line at Pm , beyond which there is no sign of a Mott gap in the normal phase. The Fabre salts then tend to behave similarly to the Bechgaard salts (Fig. 2). The change of behavior at Pm is usually attributed to a deconfinement of carri- ers, i.e., a crossover from a Mott insulator to a — me- tallic — Luttinger liquid. At lower temperature, sin- gle-particle transverse hopping is expected to become relevant and induces a dimensional crossover at a tem- perature Tx from the Luttinger liquid to a 2D or 3D metallic state. With increasing pressure, the AF tran- sition becomes predominantly driven by the instabil- ity of the whole warped Fermi surface due to the nest- ing mechanism. Although there is a general agreement on this scenario, there is considerable debate on how the dimensional crossover takes place and the nature of the low-temperature metallic state. On the theoretical side, simple RG arguments indi- cate that the crossover from the Luttinger liquid to the 2D regime takes place at the temperature [30] T t t tx b b a K K� � � � � � �� � � ��� � � 1 , (5) where K� is the Luttinger liquid parameter. For non-interacting electrons (K� � 1), Eq. (5) would give T tx b� � : the 2D Fermi surface is irrelevant when tem- perature is larger than the dispersion in the b direction. For interacting electrons (K� � 1), the interchain hop- ping amplitude t b� is reduced to an effective value t b� * and the dimensional crossover occurs at a lower tem- perature T t tx b b� � � �� . A detailed theoretical picture of the dimensional crossover is still lacking. In particu- lar, whether it is a sharp crossover or rather extends over a wide temperature range — as shown by the shaded area in Fig. 2 — is still an open issue. 2.2.1. The strong-correlation picture Some experiments seem to indicate that correla- tions still play an important role even in the low-tem- perature phase of the Bechgaard salts. For instance, a significant enhancement of 1 1/T T with respect to the Korringa law — although weaker than 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 temperature Tx � 10 K, below which the Korringa law is recovered [24,26]. The restoration of a plasma edge in the transverse b� direction at low temperature in (TMTSF)2PF6 — ab- sent in the Fabre salts — suggests the gradual emer- gence of a coherent motion in the ( )ab planes below Tx � 100 K [31,32]. (b� is normal to a and c in the ( )ab plane. It differs from b due to the triclinic structure.) However, the frequency dependence of the optical conductivity is inconsistent with a Drude-like metal- lic state [33,34,23]. The low-energy peak carries only 1% of the total spectral weight and is too narrow to be interpreted as a Drude peak with a frequency-inde- pendent scattering time. It has been proposed that this peak is due to a collective mode that bears some simi- larities with the sliding of a charge-density wave — an interpretation supported by the new phonon features that emerge at low temperature [33]. Furthermore, 99% of the total spectral weight is found in a finite en- ergy peak around 200 cm �1. It has been suggested that this peak is a remnant of a (1 4/ )-filled Mott gap � , observed in the less metallic Fabre salts at ambient pressure [35,23]. In this picture, (TMTSF)2PF6 is close to the border between a Mott insulator and a Luttinger liquid, and the low-temperature metallic 508 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 N. Dupuis, C. Bourbonnais, and J.C. Nickel behavior is made possible by the interchain coupling [23,36,37]. A different interpretation has been pro- posed for the far infrared spectrum in optical conduc- tivity and is based on the weak half-filling character of the band for interactions in the Hubbard limit [38]. The longitudinal resistivity in (TMTSF)2PF6 is found to be metallic, with a T2 law between the SDW transition and 150 K, crossing over to a sublinear tem- perature dependence above 150 K with an exponent in the range 0.5–1 [39,40]. While this observation would be consistent with a dimensional crossover to a low-temperature Fermi liquid regime taking place at Tx � 150 K, the transverse resistance �b along the b axis apparently fails to show the expected T2 behav- ior. Given the difficulties of a direct dc measurement, owing to non-uniform current distributions between contacts, conflicting results have been published in the literature [39,41]. Nevertheless, below T � 80 K �b can be deduced from �a T� 2 and �c T� 15. 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 resis- tivity in the b direction in this temperature range [42]. As far as �c is concerned, a maximum around Tmax � 80 K has been observed, with a metallic — though incoherent — behavior �c T� 15. at lower tem- perature [43]. Tmax is highly sensitive to pressure, whereas the interchain hopping t b� is not. Therefore, Tmax cannot be directly identified with t b� , but could be related to a — weakly — renormalized value t Tb x� �* in agreement with predictions of the Luttinger liquid theory [see Eq. (5)]. The transport measurements seem to be indicative of a gradual cross- over between a Luttinger liquid and a Fermi liquid oc- curring in the temperature range 40–80 K. The onset of 3D coherence and Fermi liquid behavior would then be related to the interplane coupling t c� between ( , )a b planes [43]. The absence of Fermi liquid behavior down to very low temperatures in the Bechgaard salts seems to be further supported by photoemission experiments. ARPES fails to detect quasi-particle features or the trace of a Fermi surface at 150 K [44]. Similar conclu- sions were deduced from integrated photoemission at 50 K [45]. However, photoemission results — e.g. the absence of dispersing structure and a power-law fre- quency dependence which is spread over a large en- ergy scale of the order of 1 eV — do not conform with the predictions of the Luttinger theory and might be strongly influenced by surface effects. The existence of strong correlations suggests that the kinetic interchain exchange J� , which drives the AF transition in the sulfur series, still plays an impor- tant role in the Bechgaard salts. In this picture, the decrease of TN with increasing pressure is due both to the decrease of J� and the deterioration of the Fermi surface nesting. This scenario is supported by RG cal- culations [28]. All the experiments mentioned so far favor diffe- rent — and sometimes incompatible — scenarios for the dimensional crossover. However, the high-temper- ature phase of the Bechgaard salts is always analyzed on the basis of the Luttinger liquid theory. A consis- tent interpretation of the experimental results there- fore requires to find a common K� parameter and to determine the value of the remnant of the Mott gap � . NMR [24], dc transport [43,46], and optical mea- surements [23,36] have been interpreted in terms of the Luttinger theory with K� � 0 23. and quar- ter-filled Umklapp scattering [7,37]. This interpreta- tion, as well as the mere existence of strong correla- tions, is not without raising a number of unanswered questions (see the next section). For instance, K� � 0 23. would lead according to (5) to T tx b� � �10 3 , a value much below the experimental observations. 2.2.2. The weak-correlation picture On the other hand, there are experiments pointing to the absence of strong correlations in the Bechgaard salts. One of the most convincing arguments comes from the so-called Danner–Chaikin oscillations [47]. Resistance measurements of (TMTSF)2ClO4 in the c direction show pronounced resonances when an ap- plied magnetic field is rotated in the ( )ac plane at low temperature. The complete angular dependence of the magneto-resistance can be reproduced within a semiclassical approach. The position of the resonance peaks is given by the zeros of the Bessel function J0( )� evaluated at � � �2t cB /v Bb x F z (c is the interchain spacing in the c direction). This enables a direct mea- sure of the interchain hopping amplitude in the b di- rection, yielding t b� � 280 K above the anion order- ing transition taking place at 24 K, in very good agreement with values derived from band calculations [10–12]. These results can hardly be reconciled with the existence of strong correlations. Sizeable 1D fluc- tuations should lead to a strong (k w| | , ) dependence of the self-energy, and in turn to a significant renorma- lization of k� -dependent quantities like the interchain hopping amplitudes [28]. This lends support to the idea that the low-temperature phase of the Bechgaard salts can be described as a weakly interacting Fermi liquid subject to spin fluctuations induced by the nest- ing of the Fermi surface [48,49]. Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 509 The weak-coupling approach has been particularly successful in the framework of the Quantized Nesting Model [50–52]. The latter explains the cascade of SDW phases induced by a magnetic field in (TMTSF)2PF6 and (TMTSF)2ClO4, and provides a natural explanation for the quantization of the Hall effect — � xy Ne /h� 2 2 (N integer) per ( )ab plane — observed in these phases. Furthermore, it reproduces the experimental phase diagram only for interchain hopping amplitudes t tb c� �, close to their unre- normalized values. Despite the apparent success of the weak-coupling approach, it has nevertheless become clear that the SDW phase of the Bechgaard salts is not conven- tional. Recent experiments have shown that the 2kF SDW coexists with a 2kF and a — weaker — 4kF charge-density wave (CDW) in (TMTSF)2PF6 [53,54]. Since there is no 2kF phonon softening associ- ated to this transition, the emergence of this CDW state differs from what is usually seen for an ordinary Peierls state. This unusual ground-state can be ex- plained on the basis of a quarter-filled 1D model with dimerization and onsite, nearest-neighbor and next-nearest-neighbor Coulomb interactions [55–59], but this explanation remains to be confirmed. 2.2.3. The normal phase above the superconducting phase It is remarkable that the superconducting phase lies next to 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. 2 and 3). In the normal phase above the SDW phase, the resistivity along the a axis decreases with temperature, reaches a minimum at Tmin, and then shows an upturn and a strong enhancement related to the proximity of the SDW phase transition that occurs at T TSDW � min. The re- gion of the normal phase where strong AF fluctuations are present (T T TSDW � � min) extends over the pres- sure range where the ground state is superconducting (Fig. 3). Its width in temperature decreases with in- creasing pressure, so that the superconducting transi- tion temperature appears to be closely linked to Tmin. These observations strongly suggest an intimate rela- tionship between spin fluctuations and superconduc- tivity in the Bechgaard/Fabre salts [16,17]. The importance of spin fluctuations above the supercon- ducting phase is further confirmed by the persistence of the enhancement of the spin-lattice relaxation rate 1 1/T for P Pc� [24]. Besides the presence of spin fluc- tuations at low temperature, charge fluctuations have also been observed in the normal phase via optical conductivity measurements [33]. 3. Superconductivity Some of the early experiments in the Bechgaard salts were not in contradiction with a conventional BCS superconducting state. For instance, the specific heat in (TMTSF)2ClO4 obeys the standard tempera- ture dependence C/T T� �� � 2 above the supercon- ducting transition, and the jump at the transition C/ Tc� � 167. is close to the BCS value 1.43. The ratio 2 0 3 33 ( ) .T /Tc� � , obtained from the gap de- duced from the thermodynamical critical field, is also in reasonable agreement with the prediction of the BCS theory (2 3 52 /Tc � . ) [60,61]. Early measure- ments of H Tc2( ), performed in the vicinity of the zero-field transition temperature, were also interp- reted on the basis of the BCS theory [62–65]. Nevertheless, soon after the discovery of organic su- perconductivity, the high-sensitivity of the supercon- ducting state to irradiation [66,67] led Abrikosov [68] to suggest the possibility of an unconventional — trip- let — pairing, although the non-magnetic nature of the induced defects is questionable [7]. The sensitivity to non-magnetic impurities, and thus the existence of unconventional pairing, was later on clearly estab- lished by the suppression of the superconducting tran- sition upon alloying (TMTSF)2ClO4 with a very small concentration of ReO4 anions [69,70]. A recent study [71] of the alloy (TMTSF)2(ClO4)x(ReO4)1–x — with different cooling rates and different values of x — has confirmed this in remarkable way by showing 510 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 N. Dupuis, C. Bourbonnais, and J.C. Nickel 100 10 1 0 2 4 6 8 (TMTTF) PF2 6 M-HI SP M AF SDW SC T, K P, GPa Fig. 3. (P,T) phase diagram of (TMTTF)2PF6. The shaded area above the SDW and SC phase indicates the region of the normal phase where spin fluctuations are significant. (Reprinted with permission from Ref. 17. Copyright 2001 by EDP Sciences.) that the transition temperature Tc is related to the scattering rate 1/� by ln T T T c c c 0 1 2 1 4 1 2 � � �� � � �� � � � � �� � � �� � � � � � � �� � �� (6) (Tc0 is the transition temperature of the pure system and � the digamma function), as expected for an unconventional superconductor in the presence of non-magnetic impurities [72]. Another indication of a possible unconventional pairing came from the observation of Gor’kov and J�rome [73] that the upper critical field H Tc2( ), ex- trapolated down to T � 0, would exceed the Pauli lim- ited field [74,75] H T /P c B� �184 20. � T by a factor of 2. (The value of HP quoted here corresponds to s-wave pairing.) As spin-orbit interaction is weak in these systems and cannot provide an explanation for such a large Hc2, it is tempting to again invoke triplet pairing. This issue has been revived by recent measure- ments of the upper critical field in (TMTSF)2PF6 with substantially improved accuracy in angular alignment and lower temperatures. Lee et al. [76,77] observed a pronounced upward curvature of H Tc2( ) without saturation — down to T T /c� 60 — for a field parallel to the a or b� axis, with H Tc b 2 � ( ) and H Tc a 2( ) exceeding the Pauli limited field HP by a fac- tor of 4. Moreover, H Tc b 2 � ( ) becomes larger than H Tc a 2( ) at low temperatures. Similar results were ob- tained from simultaneous resistivity and torque mag- netization experiments in (TMTSF)2ClO4 (Fig. 4) [78]. The extrapolated value to zero temperature, Hc2 0 5( ) � T, is at least twice the Pauli limited field. There are different mechanisms that can greatly in- crease the orbital critical field H Tc2 orb ( ) in organic con- ductors. Superconductivity in a weakly-coupled plane system can survive in a strong parallel magnetic field if the interplane (zero-field) coherence length �� ( )T becomes smaller than the interplane spacing d at low temperature. Vortex cores, with size �� ( )T d� , can then fit between planes without destroying the super- conducting order in the planes, and lead to a Josephson vortex lattice. In the Bechgaard salts, even for a field parallel to the b� axis, the Josephson limit �� ( )T d� is however unlikely to be reached, since the interchain hopping amplitude t c� � 5–10 K is larger than the transition temperature Tc � 11. K. Neverthe- less the orbital critical field can be enhanced by a field-induced dimensional crossover [79–83]. A mag- netic field parallel to the b� axis tends to localize the wavefunctions in the ( )ac planes, which in turn weak- ens the orbital destruction of the superconducting or- der. When �c ceHc t� �� (which corresponds to a field of a few Tesla in the Bechgaard salts), the wave functions are essentially confined in the ( )ac planes and the orbital effect of the field is completely sup- pressed. The coexistence between SDW and supercon- ductivity, as observed in a narrow pressure domain of the order of 0.8 kbar below the critical pressure Pc (Fig. 2), can also lead to a large increase of the orbital upper critical field [84–88]. Regardless of the origin of the large orbital critical field, another mechanism is required to exceed the Pauli limited field HP in the Bechgaard salts. For sin- glet spin pairing, the Pauli limit may be overcome by a non-uniform Larkin–Ovchinnikov–Fulde–Ferrell (LOFF) state, where Cooper pairs form with a non- zero total momentum [89,90]. This mechanism is par- ticularly efficient in a 1D system [79,81,83,91], due to the large phase space available for pairing at nonzero total momentum. For a linearized dispersion law, the mean-field upper critical field Hc LOFF diverges as 1/T in a pure superconductor. Lebed [92] has argued that the quasi-1D anisotropy reduces Hc LOFF below the ex- perimental observations. The only possible explana- tion for a large upper critical field would then be an equal-spin triplet pairing. A px -wave triplet state with a d vector perpendicular to the b� axis was pro- posed [93] as a possible explanation of the experimen- tal observations reported in Refs. 76,77. The triplet scenario in the Bechgaard salts is sup- ported by recent NMR Knight shift experiments (Fig. 5) [94,95]. Early NMR experiments by Takiga- wa et al. already pointed to the unconventional nature of the superconducting state in (TMTSF)2ClO4 [96]. Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 511 T= 25 mK H // b� 0.3 0.2 0.1 0 0 2 4 6 8 10 �0H, T 2 0 –2 –4 –6 T o rq u e, 1 0 N m – 9 R e si st iv ity , cm � zz Fig. 4. Resistivity (left scale) and torque magnetization (right) in (TMTSF)2ClO4 at 25 mK for H b| | �. The dotted line and + symbols on the torque curve represent a tem- perature-independent normal state contribution. The onsets of diamagnetism and decreasing resistivity, upon decreasing field, are indicated by the arrow near Hc2 � 5 T. Arrows in the low field vortex state indicate field sweep direc- tions. (Reprinted with permission from Ref. 78. Copyright 2004 by the American Physical Society.) The proton spin lattice relaxation rate 1 1/T does not exhibit a Hebel–Slichter peak. It decreases rapidly just below Tc in contrast to the typical BCS supercon- ductor where it increases below Tc, reaching a maxi- mum at T Tc� 0 9. . Furthermore, 1 1 3/T T� for T / T Tc c2� � — as it is the case for most unconven- tional superconductors — suggesting zeros or lines of zeros in the excitation spectrum. Recent experiments by Lee et al. in (TMTSF)2PF6 show that the Knight shift, and therefore the spin susceptibility, remains unchanged at the superconducting transition [94,95]. This indicates triplet spin pairing, since a singlet pair- ing would inevitably lead to a strong reduction of the spin susceptibility ( ( ) )� T � �0 0 . It should however be noticed that the interpretation of the Knight shift results — due to a possible lack of sample therma- lization during the time of the experiment — has been questioned [7,97]. In principle, the symmetry of the order parameter can be determined from tunneling spectroscopy. Sign changes of the pairing potential around the Fermi sur- face lead to zero-energy bound states in the supercon- ducting gap [98]. These states manifest themselves as a zero-bias peak in the tunneling conductance into the corresponding edge [99]. More generally, different pairing symmetries can be unambiguously distin- guished by tunneling spectroscopy in a magnetic field [100–102]. In practice however, the realization of tun- nel junctions with the TMTSF salts appears to be very difficult. A large zero-bias conductance peak — sug- gesting p-wave symmetry — across the junction be- tween two organic superconductors was observed [103]. But the absence of temperature broadening could indicate that this peak is due to disorder rather than to a midgap state [104]. Information about the symmetry of the order param- eter can also be obtained from thermal conductivity measurements. The latter indicate the absence of nodes in the excitation spectrum of the superconducting state in (TMTSF)2ClO4 [105], thus suggesting a px -wave symmetry. However, because of the doubling of the Fermi surface in the presence of anion ordering, a sin- glet d- or triplet f-wave order phase would also be nodeless in (TMTSF)2ClO4 (see Fig. 6 for the different gap symmetries in a quasi-1D superconductor) [9,106]. 4. Microscopic theories of the superconducting state The phase diagram of the 1D electron gas within the g-ology framework [108] is shown in Fig. 7. g1 and g2 denote the backward and forward scattering amplitudes, respectively, and g3 the strength of the (half-filling) Umklapp processes. Given the impor- tance of spin fluctuations in the phase diagram of the Bechgaard/Fabre salts, as well as the existence of AF 512 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 N. Dupuis, C. Bourbonnais, and J.C. Nickel 8 7 6 5 4 3 2 1 0 7 7 S e sp e ct ra ,a rb . u n its –6000 –3000 0 60003000 Shift from < >� normal a bc d 1.70 K 1.40 K 0.90 K 0.83 K 0.76 K 0.56 K 0.51 K 0.40 K 0.32 K 0.80 K 0.50 K 0.65 K 0.35 K 0.20 K 0.09 K T/T (H)c 1.0 0.5 / n 0 0.3 0.6 0.9 1.2 Fig. 5. 77Se NMR spectra Tc (0.81 K at 1.43 T). Each trace is normalized and offset for clarity. The temperatures shown in parentheses are the measured equilibrium tem- peratures before the pulse. In the inset, the spin suscepti- bility normalized by the normal state � �/ n from measured first moments are compared with theoretical calculations [98] for H/Hc2 0 0( ) � (a) and 0.63 (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 super- conducting criteria «onset» and «50 transition» have been used, respectively, to determine Hc2( )T . (Reprinted with permission from Ref. 94. Copyright 2002 by the American Physical Society.) S in g le t T ri p le t + _ _ _ + _ + _ +_ + s p x p y d f x + + + + + _ __ _ + s d dxy + x – y2 2 Fig. 6. Gap symmetries �r k( )� in a quasi-1D superconduc- tor (after Ref. 107, courtesy of Y. Suzumura). r = + /– denotes the right/left sheet of the Fermi surface. (Singlet pairing) s: const, d x y2 2 � : cos k� , dxy: r ksin � . (Triplet pairing) px: r, f: r kcos � , py: sin k� . Next-nearest-neigh- bor and longer-range pairings are not considered. ground states, the Bechgaard/Fabre salts should per- tain to the upper right corner of the 1D phase diagram (g g1 2 0, � and g g g1 2 32� � | |) where the Umklapp processes are relevant and the dominant fluctuations antiferromagnetic. In the Fabre salts, the non-mag- netic insulating phase observed below T� � 100 K indicates the importance of Umklapp scattering and suggests sizable values of g3 for this series. Since the long-range Coulomb interaction favors g g1 2� , the Fabre salts are expected to lie to the right of the phase diagram, i.e. far away from the boundary g g g1 2 32� � | |. Since the triplet superconducting phase is lying next to the SDW phase (Fig. 7), it is tempting to invoke a change of the couplings gi under pressure to argue in favor of a px -wave triplet superconducting state [68,109]. Such a drastic change of the couplings, which would explain why (TMTTF)2PF6 becomes superconducting above 4.35 GPa [16–18], is however somewhat unrealistic and has not received any theo- retical backing so far. The Umklapp scattering being much weaker in the Bechgaard salts, one cannot ex- clude that these compounds lie closer to the boundary between the SDW and the triplet superconducting phase. A moderate change of the couplings under pres- sure would then be sufficient to explain the supercon- ducting phase of (TMTSF)2PF6 observed above 6 kbar or so. However, the destruction of the super- conducting phase by a weak magnetic field and the ob- servation of a cascade of SDW phases for slightly higher fields [50–52] would imply that the interaction is strongly magnetic-field dependent — again a very unlikely scenario. In all probability, the very origin of the supercon- ducting instability lies in the 3D behavior of these quasi-1D conductors. Thus the attractive interaction is a consequence of a low-energy mechanism that be- comes more effective below the dimensional crossover temperature Tx . Transverse hopping makes retarded electron-phonon interactions more effective, since it is easier for the electrons to avoid the Coulomb repul- sion [109]. By comparing the sulfur and selenide series, it can however be argued that, in the pressure range where superconductivity is observed, the strength of the electron–phonon interaction is too weak to explain the origin of the attractive interac- tion. For narrow tight-binding bands in the organics, the attraction is strongest for backscattering processes 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 ver- tex at this wave vector does not undergo any signifi- cant increase in the normal state (Fig. 8). The ampli- tude of the 2kF lattice susceptibility in (TMTSF)2PF6 — which is directly involved in the strength of the phonon exchange — is weak. It is instructive to com- pare with the sulfur analog compound (TMTTF)2PF6, for which the electron-phonon vertex at 2kF becomes singular, signaling a lattice instability towards a spin-Peierls distortion (Fig. 8). This instability pro- duces a spin gap that is clearly visible in the tempera- ture dependence of the magnetic susceptibility and nu- clear relaxation rate [112,113]. These effects are not seen in (TMTSF)2PF6 close to Pc. The persistent en- hancement of these quantities indicates that interac- tions are dominantly repulsive (Sec. 2.2.3.), making the traditional phonon-mediated source of pairing inoperant. Emery [114] pointed out that near an SDW insta- bility, short-range AF spin fluctuations can give rise to anisotropic pairing and thus provide a possible Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 513 0 SDWTS CDWSS g g g – 2g =|g | 1 2 31 2 Fig. 7. Phase diagram (leading fluctuations) of the 1D electron gas in presence of Umklapp scattering. SS (TS): singlet (triplet) superconductivity. A gap develops in the charge sector (Mott insulating behavior) for g g g1 2 32� � | | . T, K 0 (TMTTF) 2PF6 (TMTSF)2PF6 50 0 100 150 200 I/ T ,a rb .u n its Fig. 8. Temperature dependence of the 2kF lattice suscep- tibility (I/T) as a function of temperature in the normal phase of (TMTSF)2PF6 (top) and (TMTTF)2PF6 (bot- tom). (Reprinted with permission from Ref. 53. Copyright 1996 by EDP Sciences.) explanation of the origin of the superconducting phase in the Bechgaard salts. Such fluctuations give rise to an oscillating potential that couples to the electrons. Carriers can avoid the local Coulomb repulsion and take advantage of the attractive part of this potential by moving on different chains. This mechanism, which can lead to superconductivity at low temperatures, is the spin-analog of the so-called Kohn–Luttinger mechanism which assumes the pairing to originate in the exchange of charge-density excitations pro- duced by Friedel oscillations [115]. While most theo- retical results on the spin-fluctuation-induced super- conductivity are based on RPA-like calculations [116,117,27,118–124], the existence of such an elec- tronic pairing mechanism in a quasi-1D conductor has been recently confirmed by an RG approach [125]. Moreover, it has been recently realized that CDW fluc- tuations can play an important role in stabilizing a trip- let phase [126–128,107,129–131]. Below we discuss in simple terms the link between spin/charge fluctuations and unconventional pairing [124], and present recent re- sults obtained from an RG approach [129–131]. 4.1. Superconductivity from spin and charge fluctuations Considering for the time being only intrachain in- teractions, the interacting part of the Hamiltonian within the g-ology framework [108] reads H g g q int ch sp� � � � �[ ( ) ( ) ( ) ( )]� �q q S q S q (7) (from now on we neglect the c axis and consider a 2D-model), where �q and Sq are the charge- and spin-density operators in the Peierls channel ( )q kx F� 2 , g g g /ch � �1 2 2 and g g /sp � � 2 2. Starting from a half-filled extended Hubbard model, we obtain g U V1 2� � and g U V2 2� � , where U is the onsite and V the nearest-neighbor lattice site (dimer) interaction. For simplicity, we do not con- sider Umklapp scattering (g3), since it does not play an important role in the present qualitative discus- sion. For repulsive interactions g g1 2 0� � , short-range spin fluctuations develop at low tempera- tures due to the nesting of the Fermi surface. They can be described by an effective Hamiltonian Hint eff ob- tained from (7) by replacing the bare coupling con- stants by their (static) RPA values g g g g gRPA RPA ch ch ch ch ch ch( ) ( ) ( ),q q q� � � � 1 0 2 � � g g g g gRPA RPA sp sp sp sp sp sp( ) ( ) ( ),q q q� � � � 1 0 2 � � (8) where �RPA is the static (� � 0) RPA susceptibility. The bare particle-hole susceptibility diverges at low temperatures, i.e. �0 0( ) ( ( , ))Q � ��ln maxE / T t b , due to the Q � ( , )2kF � nesting of the quasi-1D Fermi sur- face ( � �k k Q� � ��� ). (E0 is a high-energy cut- off of the order of the bandwidth.) The divergence is cut off by deviations from perfect nesting, character- ized by the energy scale ��t b [Eq. (1)]. In the Bechgaard salts �� �t b 10 K and varies with pressure. When the nesting of the Fermi surface is good (small ��t b), the spin susceptibility � sp RPA( )Q diverges at low temperatures, thus signaling the formation of an SDW. A larger value of ��t b frustrates antiferro- magnetism and, when exceeding a threshold value, eliminates the transition to the SDW phase [132,133]. In that case, the (remaining) short-range spin fluctua- tions can lead to pairing between fermions. To see this, we rewrite the effective Hamiltonian Hint eff in the particle-particle (Cooper) channel H g O Os s sint eff � � � � � ��[ ( , ) ( ) ( ) k,k k k k k � � ��g O Ot t t( , ) ( ) ( )]k k k k (9) (we consider only Cooper pairs with zero total mo- mentum), where g g gs RPA RPA( , ) ( ) ( ),k k k k k k� � � � � � � �3 sp ch g g gt RPA RPA( , ) ( ) ( )k k k k k k� � � � � � � �sp ch (10) are the effective interactions in the singlet and triplet spin pairing channels (Fig. 9). Os ( )k (O kt ( )), is the annihilation operator of a pair ( , )k k� in a singlet (triplet) spin state, and Ot t t tO O O� �( , , )1 0 1 denotes the three components Sz � �1 0 1, , of the triplet state (total spin S � 1). On the basis of the effective Hamiltonian (9) the BCS theory predicts a superconducting transition whenever the effective interaction g s t, turns out to be attractive in (at least) one pairing channel. A simple argument shows that this is indeed always the case in the presence of short-range spin fluctuations. The spin susceptibility � sp RPA( )k k� � exhibits a pronounced peak around k k Q� � � . Neglecting the unimportant k| | dependence, its Fourier series expansion reads � sp RPA F n n nk k k a n k k( , ) ( ) cos[ ( )]2 1 0 � � � � � �� � � � � � �� = a nk nk nk nkn n n � � � � � �� � � � � 0 1( ) [cos cos sin sin ], (11) 514 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 N. Dupuis, C. Bourbonnais, and J.C. Nickel where an ! 0. Choosing a an � 0, one obtains a diver- ging spin susceptibility � sp RPA Fk k k( , )2 � �� � � � � � �� �" �( )k k . The condition a a0 1 0� � !... gives a broadened peak around k k� �� � �. Eqs. (10, 11) show that the effective interaction in the singlet channel contains attractive interactions for any value of n. In real space, n corresponds to the range of the pairing interaction in the b direction. The dominant attractive interaction corresponds to nearest-neigh- bor-chain pairing (n � 1) and a d x y2 2 � -wave order pa- rameter r k k( ) cos� �� (r kx� sgn( )). The interac- tion is also attractive in the triplet f-wave channel ( ( ) cos ) r k r k� �� . However, all the three components of a (spin-one boson) SDW fluctuation contribute to the superconducting coupling in the sin- glet channel — hence the factor of 3 in the first of equations (10). The latter therefore always dominates over the triplet one when charge fluctuations are not important. Note that the interaction is repulsive in the singlet dxy -wave ( ( ) sin ) r k r k� �� and the trip- let py -wave (sink� ) channels. Equations (10) show that CDW fluctuations tend to suppress the singlet pairing, but reinforce the trip- let one. In the Bechgaard salts, the physical relevance of CDW fluctuations has been borne out by the puz- zling observation of a CDW that coexists with the SDW (Sec. 2.2.) [33,53,54]. Within the framework of an extended anisotropic Hubbard model, recent RPA calculations have shown that the triplet f-wave pair- ing can overcome the singlet d x y2 2 � -wave pairing when the intrachain interactions are chosen such as to boost the CDW fluctuations with respect to the SDW ones [126–128]. In a half-filled model, this however requires the nearest-neighbor (intrachain) interaction V to exceed U/2. In a quarter-filled model — appro- priate if one ignores the weak dimerization along the chains — the condition for f-wave superconductivity becomes V U/2 2! — V2 is the next-nearest-neighbor (intrachain) Coulomb interaction — and appears even more unrealistic. Similar conclusions were reached within an RG approach [107]. Given that electrons interact through the Coulomb interaction, not only intrachain but also interchain in- teractions are present in practice. At large momentum transfer, the interchain interaction is well known to favor a CDW ordered state [134–137]. This mecha- nism is mostly responsible for CDW long-range order observed in several organic and inorganic low-dimen- sional solids (e.g. TTF-TCNQ) [138,139]. In the Bechgaard salts, both the interchain Coulomb interac- tion and the kinetic interchain coupling (t b� ) are likely to be important in the temperature range where superconductivity and SDW instability occur, and should be considered on equal footing. An RG ap- proach has recently been used to determine the phase diagram of an extended quasi-1D electron gas model that includes interchain hopping, nesting deviations and both intrachain and interchain interactions [129–131]. The intrachain interactions turn out to have a sizeable impact on the structure of the phase di- agram. Unexpectedly, for reasonably small values of the interchain interactions, the singlet dx y2 2 � -wave superconducting phase is destabilized to the benefit of the triplet f-wave phase with a similar range of Tc. The SDW phase is also found to be close in stability to a CDW phase. Before presenting these results in more detail (Sec. 4.2.), let us discuss in simple terms the role of interchain interactions. The interchain back- ward scattering amplitude g1 � (� 0) contributes to the effective interaction in the Cooper channel, g k k g k ks s( , ) ( , )� � � �� � � � � � � �� � � � �2 1g k k k k[cos cos sin sin ], g k k g k kt t( , ) ( , )� � � �� � � � � � � � �� � � � �2 1g k k k k[ cos cos sin sin ]. (12) It thus tends to suppress singlet d x y2 2 � pairing, but favors triplet f-wave pairing. In addition to this «di- rect» contribution, g1 � reinforces CDW fluctuations, g q g q g qch ch( ) ( ) cos� � � �� � 2 1 (13) and therefore enhances the f-wave pairing over the d x y2 2 � -wave pairing via the mechanism of fluctuation exchange [see Eqs. (10)]. As for the interchain for- ward scattering g2 � , its direct contribution to the DW channel is negligible, but it has a detrimental effect on both singlet and triplet nearest-neighbor-chain pairings. This latter effect, which is neutralized by the Umklapp scattering processes, can lead to next-nearest-neighbor-chain pairings when Umklapp processes are very weak [131]. Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 515 g =s,t + + + … Fig. 9. Diagrammatic representation of the effective inter- action gs t, in the Cooper channel within the RPA. 4.2. RG calculation of the phase diagram of quasi-1D conductors As a systematic and unbiased method with no a pri- ori assumption, the RG method is perfectly suited to study competing instabilities. The zero-temperature phase diagram obtained with this technique is shown in Fig. 10 [130,131]. In the absence of interchain interactions (g g1 2 0� �� � ), it confirms the validity of the qualitative arguments given above. When the 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 the ground state becomes a dx y2 2 � -wave superconducting (SCd) state with an or- der parameter r k k( ) cos� �� [125]. In the presence of interchain interactions (g1 0� � ), the region of sta- bility of the SCd phase shrinks, and a triplet supercon- ducting f-wave (SCf) phase appears next to the d-wave phase for ~ .g g / vF1 1 01� �� � � — obtained here for typical values of intrachain couplings and band pa- rameters [130,131]. For larger values of the interchain interactions, the SCd phase disappears and the region of stability of the f-wave superconducting phase wid- ens. In addition a CDW phase appears, thus giving the sequence of phase transitions SDW�CDW�SCf as a function of ��t . For ~ .g1 012�� , the SDW phase dis- appears. Note that for ~ .g1 011� � , the region of stabil- ity of the CDW phase is very narrow, and there is es- sentially a direct transition between the SDW and SCf phases. The RG calculations yield Tc � 30 K for the SDW phase in the case of perfect nesting and Tc � 0.6–1.2 K for the superconducting phase, in reasonable agree- ment with the experimental observations in the Bechgaard salts. Fig. 11 shows the transition tempera- ture Tc as a function of ��t for three different values of the interchain interactions, ~g1 0� � , 0.11 and 0.14, corresponding to the three different sequences of phase transitions as a function of ��t : SDW�SCd, SDW�(CDW)�SCf and CD�WSCf . The phase di- agram is unchanged when both g2 � and a weak Umklapp scattering amplitude g3 are included [130,131]. The RG approach also provides important informa- tion about the fluctuations in the normal phase. The dominant fluctuations above the SCd phase are SDW fluctuations as observed experimentally (Sec. 2.2.). Although they saturate below T t� �� where the SCd fluctuations become more and more important, the latter dominate only in a very narrow temperature range above the superconducting transition (Fig. 12). Above the SCf and CDW phases, one expects strong CDW fluctuations driven by g1 � . Fig. 13 shows that for ~g1 � � 0.11–0.12, strong SDW and CDW fluctua- tions coexist above the SCf phase. Remarkably, there are regions of the phase diagram where the SDW fluc- tuations remain the dominant ones in the normal phase above the SCf or CDW phase (Fig. 13,b). A central result of the RG calculation is the close proximity of SDW, CDW and SCf phases in the phase diagram of a quasi-1D conductor within a real- istic range of values for the repulsive intrachain and interchain interactions. Although this proximity is found only in a small range of interchain interactions, there are several features that suggest that this part of the phase diagram is the relevant one for the Bechgaard salts. i) SDW fluctuations remain impor- tant in the normal phase throughout the whole phase diagram. They are the dominant fluctuations above the SCd phase, and remain strong — being sometimes even dominant — above the SCf phase where they co- exist with strong CDW fluctuations, in accordance with observations [24,33]. ii) The SCf and CDW phases stand nearby in the theoretical phase diagram, the CDW phase always closely following the SCf phase when the interchain interactions increase. This agrees with the experimental finding that both SDW 516 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 N. Dupuis, C. Bourbonnais, and J.C. Nickel 0 0.04 0.08 0.12 0 0.04 0.08 0.12 0.16 0.2 t /t g 1 ~ SDW CDW SCd SCf � � � � Fig. 10. T = 0 phase diagram as a function of t t�� �/ and ~g1 � . Circles: SDW, squares: CDW, triangles: SCd ( ( ) cos )�r k k� �� , crosses: SCf (�r k r k( ) cos� �� ). The dashed lines indicate two (among many) possible pressure axes, corresponding to transitions SDW�SCd and SDW�SCf [130,131]. 0.0001 0.001 0.01 0.1 0 0.04 0.08 0.12 0.16 0.2 t /t� � � T /tc � Fig. 11. Transition temperature as a function of �� �t /t for ~g1 0� � , 0.11 and 0.14, corresponding to solid, dotted, and dashed lines, respectively [130,131]. and CDW coexist in the DW phase of the Bechgaard salts [53,54] and the existence, besides SDW correla- tions, of CDW fluctuations in the normal state above the superconducting phase [33]. iii) Depending how one moves in practice in the phase diagram as a func- tion of pressure, these results are compatible with ei- ther a singlet dx y2 2 � -wave or a triplet f-wave super- conducting phase in the Bechgaard salts (see the two pressure axes in Fig. 10). Moreover, one cannot ex- clude that both SCd and SCf phases exist in these ma- terials, with the sequence of phase transitions SDW�SCd�SCf as a function of pressure. It is also possible that the SCf phase is stabilized by a magnetic field [140], since an equal-spin pairing triplet phase is not sensitive to the Pauli pair breaking effect contrary to the SCd phase. This would make possible the exis- tence of large upper critical fields exceeding the Pauli limit [76,78], and would also provide an explanation for the temperature independence of the NMR Knight shift in the superconducting phase [94]. 5. Conclusion Notwithstanding the recent experimental pro- gresses, many of the basic questions related to super- conductivity in the Bechgaard and Fabre salts remain largely open. The very nature of the superconducting state — the orbital symmetry of the order parameter and the singlet/triplet character of the pairing — is still not known without ambiguity even though recent upper critical field measurements [76–78] and NMR experiments [94,95] support a triplet pairing. We argued that the conventional electron–phonon mechanism is unable to explain the origin of the super- conducting phase. On the other hand, the proximity of the SDW phase, as well as the observation of strong spin fluctuations in the normal state precursor to the superconducting phase [16,17,24], strongly suggest an intimate relationship between antiferromagnetism and superconductivity in the Bechgaard/Fabre salts. The scenario originally proposed by Emery [114], whereby short-range AF spin fluctuations can give rise to anisotropic pairing and thus stabilize a superconduct- ing phase, is so far the only one that is consistent with the experimental observations and the repulsive na- ture of the electron-electron interactions. Within the framework of the RG approach, it has recently been shown that when spin and charge fluctu- ations are taken into account on equal footing, both singlet dx y2 2 � - and triplet f-wave superconducting phases can emerge at low temperatures whenever the nesting properties of the Fermi surface deteriorate un- der pressure [126–128,107,129–131]. CDW fluctua- tions are enhanced by the long-range part of the Cou- lomb interaction. Remarkably, for a reasonably small value of the interchain interactions, the singlet dx y2 2 � -wave phase is destabilized to the benefit of a triplet f-wave with a similar range of Tc [130,131]. The physical relevance of CDW fluctuations in the Bechgaard salts has been born out by the observation of a CDW that actually coexists with the SDW [53,54]. CDW fluctuations were also observed in the normal state precursor to the superconducting state [33]. As a systematic and unbiased method with no a pri- ori assumptions, the RG has proven to be a method of choice to study the physical properties of quasi-1D or- ganic conductors. An important theoretical issue is Superconductivity and antiferromagnetism in quasi-one-dimensional organic conductors Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 517 0.1 1 10 100 1000 0.001 0.01 0.1 1 10 100 T/t � � v F 0.0001 Fig. 12. Temperature dependence of the susceptibilities in the normal phase above the SCd phase ( � �� �t t0152. and ~ .g1 008� � ). The continuous, dotted, dashed, and dashed-dotted lines correspond to SDW, CDW, SCd and SCf correlations, respectively [130,131]. 0.1 1 10 100 1000 0.001 0.01 0.1 1 10 100 0.1 1 10 100 1000 0.001 0.01 0.1 1 10 100 T/t � � v F T/t � � v F a b 0.0001 Fig. 13. Temperature dependence of the susceptibilities in the normal phase above the SCf phase for ~ .g1 012� � , � �� �t t0152. (a) and � �� �t t0176. (b). now to go beyond the instabilities of the normal state. On the one hand, the RG analysis should be extended to the low-temperature broken-symmetry phases in or- der to study the possible coexistence of superconduc- tivity and antiferromagnetism, as well as CDW and SDW, as observed in the Bechgaard salts [86,88,53,54]. On the other hand, the RG technique might also enable to tackle the unusual properties of the metallic phase. 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