# 1 Interacting electrons in quasi-one-dimensional organic superconductors

C. Bourbonnais and D. Jérome

## 1.1 Introduction

Superconductivity in organic materials has emerged in 1979 from an impor-
tant background of preexisting knowledge and experimental techniques. All
previous studies undertaken since 1973, which had been mostly performed on
the (TTF – TCNQ) series of charge transfer organic conductors had failed to
reveal superconductivity using chemistry and (or) pressure to suppress the
density-wave or the so-called Peierls instability inherent to one-dimensional
conductors. A breakthrough, which contributed to the discovery of organic
superconductivity, has been the synthesis of the molecule TMTSF by K. Bech-
gaard and co-workers [1].

Actually, in the early 70's leading ideas governing the search for new ma-
terials likely to exhibit good metallicity and possibly superconductivity were
driven by the possibility to minimize the role of electron-electron repulsions
and at the same time to increase the electron-phonon interaction, while keep-
ing the overlap between conducting stacks as large as possible. This led to the
synthesis of the new electron donating molecule TMTSF, presenting much
analogy with the previously known fulvalene donors in which the redox poten-
tial (ΔE)₁/₂ can be minimized, [2,3] by utilizing selenium instead of sulfur as
heteroatoms [4]. The next step was quite encouraging since the use of a high
pressure has allowed to remove the instability due to the divergence of the
Peierls channel down to the lowest temperatures in the two chain conductor
TMTSF – DMTCNQ [5]. A lucky situation has also been the synthesis of a
series of 1D organic salts based on the radical cationic molecule TMTTF (the
sulfur analog of the TMTSF molecule) and on a variety of inorganic monoan-
ions such as ClO₄, BF₄ or SCN [6,7]. A conducting character could thus be
anticipated from the intermolecular overlap of partly filled highest molecu-
lar orbitals (HOMO) of individual molecules. The compounds, (TMTTF)₂X,
were all insulating at ambient pressure but their crystal structure is the pro-
totype of the (TMTSF)₂X series in which superconductivity has been sub-
sequently discovered. In the rest of this article when we mention a (TM)₂X
compound this means that the organic molecule can be either TMTSF or
TMTTF. The structure exhibits a face to face packing of flat molecular units

<footnote n="1">¹ Réseau Québecois sur les Matériaux de Pointe (RQMP), Dpartement de physique, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K-2R1
cbourbon@physique.usherb.ca</footnote>

<footnote n="2">² Laboratoire de Physique des Solides, UMR 8502, Université de Paris-sud, 91405 Orsay, France jerome@lps.u-psud.fr</footnote>
<figure><img src="image_1.png" /><figcaption>**Fig. 1.1.** View of the cationic and anionic stacking perpendicular to the stacking axis, courtesy of J.Ch. Ricquier, IMN, Nantes</figcaption></figure>

along the *a* direction and the formation of molecular layers in the *a*–*b* planes separated by anions stacks (Fig. 1.1). The overall symmetry is triclinic, not too far from orthorhombic often taken as the approximate structure by theoreticians. In addition, the structure reveals an important peculiarity namely, the anions are located in centrosymmetrical cavities lie slightly above or below each molecular plane with a zigzag stacking of the molecules along the *a* direction. This structure leads in turn to a weak alternation of the interplanar distance (dimerization and a concomitant splitting of the HOMO conduction band into a filled lower sub-band separated from a half-filled upper (hole-like) band by a gap $\Delta_D$ at $\pm 2k_F$, called the dimerization gap in the new Brillouin zone. However, on account of the finite transverse dispersion, this dimerization gap does not lead to a genuine gap in the middle of the density of states as given from the extended-Hückel band calculation. The only claim that can be made is that these conductors show commensurate band filling (three-quarter filled with electrons or one quarter-empty with holes). This originates from the 2:1 stoichiometry. Consequently, according to a non-interacting particle band calculation, all compounds in the (TMTTF)₂X series should be found conducting.

<table><caption>**Table 1.1.** Calculated band parameters for three representative members of the (TM)₂X series according to the room temperature crystallographic data in reference [8]. The average intra and interstack interactions are given in lines 3 and 5, respectively. The bond dimerization is shown in line 4. All energies are in meV</caption><thead><tr><th></th><th>(TMTTF)₂PF₆</th><th>(TMTSF)₂PF₆</th><th>(TMTSF)₂ClO₄</th></tr></thead><tbody><tr><td>t₁</td><td>137</td><td>252</td><td>258</td></tr><tr><td>t₂</td><td>93</td><td>209</td><td>221</td></tr><tr><td>t</td><td>115</td><td>230</td><td>239</td></tr><tr><td>Δt/τ</td><td>0.38</td><td>0.187</td><td>0.155</td></tr><tr><td>t⊥</td><td>13</td><td>58</td><td>44</td></tr></tbody></table>
In Table (1.1) we report the band parameters of different members of the (TMTSF)₂X series as computed from crystallographic data [8]. The sulfur compounds exhibit bands that are significantly narrower and their crystallographic structure is more dimerized than those of the selenide compounds. (TMTTF)₂Br (not listed in Table (1.1)) is, however, an exception among the sulfur compounds with a dimerization of 0.13, which is smaller than the value calculated for (TMTSF)₂ClO₄. This might be due to the calculation of electronic bands based on rather old and less accurate crystallographic data than those used for the other compounds [9]. All (TM)₂X compounds with diversified anions can be gathered on a generic phase diagram displaying a wealth of different physical properties [10]. The gross features of the (TM)₂X phase diagram are shown in Fig. 1.2. Compounds on the left hand side of the phase diagram, such as (TMTTF)₂PF₆, are insulators below room temperature with a broad metal to insulator transition, while those on the right hand side of (TMTTF)₂Br exhibit an extended temperature regime with a metallic behavior and a sharp transition towards an insulating ground state. Therefore, the cause of the insulating nature of some members of the (TMTTF)₂X series should be determined in relation to the role of e-e repulsion and low dimensionality as we shall show later. Although the most extensive pressure studies have been performed on (TMTSF)₂PF₆ and (TMTTF)₂PF₆, recent studies of other compounds of the (TMTTF)₂X series with X= ReO₄, BF₄ and Br have shown that the main features observed under pressure in (TMTTF)₂PF₆ or in (TMTSF)₂PF₆ are also observed in other systems [11]. At this stage we may emphasize that the band filling of these materials is commensurate and in addition the existence of a dimerization in the crystal structure of the (TM)₂X series raises quite a challenging problem for the physics of one dimensional conductors since with the axial dimerization the conduction band becomes half-filled while it is originally quarter-filled from stoichiometry considerations. The commensurate band filling opens a new scattering channel for the carriers between both sides of the Fermi surface as then the total momentum transfer for two (four) electrons from one side of the 1-D Fermi surface to the other is equal to a reciprocal lattice vector (Umklapp scattering for half (quarter)-filled bands).

## 1.2 Elements of theory for interacting electrons in low dimension

In order to bring out the essential features behind the Luttinger liquid state we consider the electron gas model [167]. This model is of course a crude simplification of the interacting electron structure that takes place in actual organic compounds. Being defined with a small number of parameters, the model is nevertheless generic of a different kind of an electronic state that might be relevant to the description of organic conductors.
<figure><img src="image_2.png" /><figcaption>Fig. 1.2. Pressure dependence of transport activation in (TMTTF)₂PF₆. The activation for the c axis transport (Δc) although decreasing under pressure survives up to high pressures while the longitudinal transport (Δa) is no longer activated above 15 kbar when a dimensional cross-over occurs at a finite temperature [11]</figcaption></figure>

The model is based at the outset from the observation that a 1D non-interacting Fermi gas with a two-point Fermi surface at ±k₀² gives rise to two infrared logarithmic singularities in the response of the system to correlate pair of particles either in the electron-electron (Cooper) or in the 2k₀² electron-hole (Peierls) scattering channel. These are of the form

$$ \Pi^0 \sim 2N(E_F) \int_T^{E_F} \frac{d\epsilon}{\epsilon(k) - m\epsilon(k')} \\ \sim N(E_F) \ln \frac{E_F}{T}. \qquad (1.1) $$

The singularity, though similar for both types of pairing, results from a distinct symmetry property of the one-particle spectrum. Take for example the logarithmic integration of the Cooper channel where m = -1 and k' = -k; it express the formation of pairs of particles of total momentum zero for which ϵ(k) = ϵ(-k) is satisfied for the particles of each pair. In three dimensions, the slightest attraction between electrons in these time-reversed states inevitably leads to an instability of the metallic state towards BCS type of superconductivity. In one dimension, however, this singularity is not alone since the electron spectrum displays another type of symmetry property called 'nest-
ing' for which $m = 1$ and $k' = k - 2k_F^0$, where this time the energies of an electron (or a hole) state at $k$ and a hole (or an electron) state at $k - 2k_F^0$ are connected through the relation $\epsilon(k) = -\epsilon(k - 2k_F^0)$ – the 1D analog of eq. (??). The summation over a macroscopic number of intermediate states that are connected by nesting is responsible for a logarithmic integration of the form (1.1). It is the underlying mechanism of the Peierls lattice instability when $2k_F^0$ electron-hole pairs are coupled to low-frequency phonons [?].

What thus really makes one dimension so peculiar resides in the fact that the symmetry of the spectrum for the Cooper and Peierls instabilities refer to the same phase space of electronic states [166]. The two different kinds of pairing act as independent and simultaneous processes of the electron-electron scattering amplitude which interfere with and distort each other at all order of perturbation theory. What comes out of this interference is neither a BCS superconductor nor a Peierls/density-wave superstructure but a different instability of the Fermi liquid called a Luttinger liquid.

Following the example of a Fermi liquid, however, a selected emphasis is put on the electronic states close to the Fermi level where the argument $E_F/T$ of the logarithm goes to infinity at $T \to 0$. In the framework of the electron gas model, this amounts to write the one-particle kinetic term of the Hamiltonian in the form

$$H_0 = \sum_{p,k,\sigma} \epsilon_p(k) a_{p,k,\sigma}^{\dagger} a_{p,k,\sigma}, \quad (1.2)$$

where $\epsilon_p(k) \approx v_F(pk - k_F^0)$ is linearized around the two Fermi points $pk_F^0$ for right $p = +$ and left $p = -$ moving electrons (a similar continuum approximation neglecting the details at short distance has been made in the context of the Fermi liquid theory [cf. Eq.(??)]).

Accordingly, direct interactions between carriers in the model can be defined close to the Fermi points, a procedure called the 'g-ology' decomposition of the interaction [15,167]. For a rotationally invariant system, it allows to single out four different coupling constants: the backscattering and the forward scattering terms $g_1$ and $g_2$, for which two electrons near opposite Fermi points are coupled through a momentum transfer near $2k_F^0$ and zero, respectively; the $g_3$ coupling corresponds to Umklapp scattering where two electrons near $+k_F^0$ (resp. $-k_F^0$) are backscattered to $-k_F^0$ (resp. $+k_F^0$), a process that is made possible at half-filling if the reciprocal lattice vector $G = 4k_F^0 = 2\pi/a$ enters in the momentum conservation law; finally, one has the coupling $g_4$ by which two electrons near $k_F^0$ (resp. $-k_F^0$) experience a small momentum transfer which keeps them on the same branch [15] (Figure ??).

Focusing for the moment on the most important couplings $g_1$, $g_2$ and $g_3$, the interacting part $H_I$ of the total Hamiltonian can be written in the form

$$H_I = (2g_2 - g_1) \sum_{p,q} \rho_p(q) \rho_{-p}(-q) - g_1 \sum_{p,q} \mathbf{S}_p(q) \cdot \mathbf{S}_{-p}(-q)$$
$$ + \frac{1}{2L} \sum_{\{p,k,Q,\sigma\}} g_3 a_{p,k_1+pQ,\sigma}^{\dagger} a_{p,k_2-pQ+pG,\sigma'}^{\dagger} a_{-p,k_2,\sigma'} a_{-p,k_1,\sigma} \quad (1.3) $$

where

$$ \rho_p(q) = \frac{1}{2} (L)^{-\frac{1}{2}} \sum_{\{k,\alpha\}} a_{p,\alpha}^{\dagger}(k+q) a_{p,\alpha}(k), $$

and

$$ S_p(q) = \frac{1}{2} (L)^{-\frac{1}{2}} \sum_{\{k, \alpha \beta\}} a_{p, \alpha}^{\dagger}(k+q) \sigma^{\alpha \beta} a_{p, \beta}(k), $$

are respectively the long wavelength charge- and spin-density operators for right- or left-moving carriers.

$$ \begin{aligned} \tilde{g}'_1 &= -\tilde{g}_1^2 + \dots \\ (2\tilde{g}'_2 - \tilde{g}'_1) &= \tilde{g}_3^2 + \dots \\ \tilde{g}'_3 &= \tilde{g}_3(2\tilde{g}_2 - \tilde{g}_1) + \dots, \end{aligned} \quad (1.4) $$

where $\tilde{g}' = (d/d\ell)g$

The phase variable representation allows us to rewrite the Tomonaga-Luttinger Hamiltonian in the harmonic form

$$ H_{TL} = \sum_{\nu=\rho,\sigma} \frac{1}{2} \int \left[ \pi u_{\nu} K_{\nu} \Pi_{\nu}^2 + u_{\nu} (\pi K_{\nu})^{-1} \left( \frac{\partial \phi_{\nu}}{\partial x} \right)^2 \right] dx, \quad (1.5) $$

where $K_ν$ is the stiffness constants of acoustic excitations. The properties of the model are then entirely governed by the set of parameters $\{u_ν, K_ν\}$. These are functions of the microscopic coupling constants.

$$ H' = \frac{1}{L} \sum_{\{k,q,\alpha\}} g_1 a_{+,k_1+2k_F+q,\alpha}^{\dagger} a_{-,k_2-2k_F-q,\alpha'}^{\dagger} a_{+,k_2,\alpha'} a_{-,k_1,\alpha} + \frac{1}{2L} \sum_{\{p,k,q,\alpha\}} g_3 a_{p,k_1+p(2k_F+q),\alpha}^{\dagger} a_{p,k_2-p(2k_F+q)+pG,-\alpha}^{\dagger} a_{-p,k_2,-\alpha} a_{-p,k_1,-\alpha} \quad (1.6) $$

corresponds to the additional terms expressed in the fermion representation. In terms of phase variables, the part for antiparallel spins reads

$$ H' = \int dx \left\{ \frac{2g_1}{(2\pi\alpha_0)^2} \cos(\sqrt{8}\phi_\sigma) + \frac{2g_3}{(2\pi\alpha_0)^2} \cos(\sqrt{8}\phi_\rho) \right\}, \quad (1.7) $$

whereas the parallel part of $g_1$ goes into an additional renormalization of $u_σ$ and $K_ν$ in $H_{TL}$. From this expression, one first observes that $g_1$ ($g_3$) solely depends on the spin (charge) phase variable. Therefore the spin and charge parts of the total Hamiltonian $H = H_σ + H_ρ$ still commute and thus preserve spin-charge separation.
An exact solution of H cannot be found in the general case, except at a particular value of the coupling constants for each sector, corresponding to the Luther-Emery solutions [?, 18]. However, one can seek an approximate solution using scaling theory. In the framework of the Tomanaga-Luttinger model, we have already emphasized that anomalous dimensions found in single and pair correlation functions are hallmarks of scaling. In effect, a Luttinger liquid is a self-similar system when it is looked at different length x (and $u_ντ$) scales. We can profit by this property for the more general Hamiltonian by looking at the evolution or the flow of the couplings $g_{1,3}$ and Luttinger liquid parameters as a function of successive change of space and time scales. In practice, the corresponding space-time variations of the phase variables $\phi_ν$ are integrated out using $g_{1,3}$ as perturbations. Then by rescaling both the initial length ($x \to xe^{-l}$) and time ($u_ντ \to u_ντe^{-l}$) scales yields the renormalization group flow equations

$$ \begin{aligned} \frac{dK_ν}{dl} &= -\frac{1}{2}K_ν^2 g_ν^2 \\ \frac{dg_ν}{dl} &= g_ν(2 - 2K_ν) \end{aligned} \qquad (1.8) $$

for spin and charge parameters, where $g_σ \equiv g_1$ and $g_ρ \equiv g_3$. In the repulsive case, where $g_{i=1...4} > 0$, $g_1$ is marginally irrelevant in the spin sector, that is to say

What 1-D physics means is that instead of the usual description of low-lying excitations in terms of QP states in the Landau-Fermi liquid model, a collective mode description with decoupled spin and charge modes is a more appropriate starting point [12–14]. Such a model for 1D conductors has been proposed on the basis of a linearized energy spectrum for excitations close to the Fermi level adding the relevant Coulomb repulsions which are responsible for electron scattering with momentum transfer $2k_F$ and 0. This is the popular Tomonaga-Luttinger (TL) model for a 1-D conductor in which the spatial variation of all correlation functions (spin susceptibility at $2k_F$ or $4k_F$, CDW, superconductivity) exhibit a power law decay, characterized by the non-universal exponent $K_ρ$ (which is a function of microscopic coupling constant) [15]. When $K_ρ$ is larger than unity, interactions between carriers become attractive and favor superconducting correlations at a low temperature whereas, if $K_ρ$ is less than unity, repulsive interactions together with magnetic correlations are enhanced at a low temperature [16].

However, there exists an added peculiarity in these 1D conductors. The filling of the conduction band is commensurate. Stoichiometry imposes half a carrier (hole) per TM molecule, a concentration that cannot be modified by applying pressure. Consequently, uniformly spaced molecules along the stacking axis should lead to a situation where a unit cell contains 1/2 carriers, i.e., the conduction band is quarter-filled. However, the non-uniformity of molecular packing has been observed in early structural studies of the (TMTTF)₂X crystals [8]. The dimerization of the overlap between molecules
occurs along the stacks, a situation that is pronounced in the sulfur series, although it is also encountered in some members of the (TMTSF)₂X series (Table 1). The impact of such a dimerization on the electronic structure is generally quantified by a modulation of the intra-stack overlap integral, because both longitudinal and transverse molecular displacements can contribute to the intermolecular overlap and could make them half-filled band compounds. Commensurate band filling opens a scattering channel for the carriers between both sides of the Fermi surface so that the total momentum transfer for two (four) electrons from one side of the 1D Fermi surface to the other is equal to a reciprocal lattice vector (Umklapp scattering for half (quarter)-filled bands) [16]. Commensurability leads to important modifications in the model of the gapless Luttinger liquid, driving it towards a Mott-Hubbard type insulator with a gap in the charge sector; however, on account of the spin-charge separation of the 1D physics the spin sector remains gapless [16]. The amplitude of the charge gap strongly depends on the band filling and of the strength of the repulsive interactions. For half-filled band 1D conductors, the Mott Hubbard gap opens up as soon as the interactions between carriers become infinitesimally repulsive namely, $K_ρ < 1$. Quarter-filled band conductors, however, require more repulsive interactions before becoming 1D insulators. They become insulating when the condition $K_ρ < 0.25$ is fulfilled. Quite generally, the gap for conduction along the chains reads [16]

$$2\Delta_{\rho,a} \approx W\left(\frac{g_U}{W}\right)^{1/(2-2n^2K_\rho)}, \qquad (1.9)$$

where $g_U$ is the coupling constant $g_U = W(\frac{U}{W})^3$ and $n = 2$ (1) for the 1/4 (1/2)-filled situation in the Hubbard limit.

### 1.2.1 The role of interchain coupling

Electronic materials like organic conductors can be only considered as close realizations of 1D interacting fermion systems. In the solid state, molecular stacks are not completely independent, that is to say *interchain coupling*, though small, must be taken into account in their description. Two different kinds of interchain coupling are generally considered. First, we have potential coupling like Coulomb interaction which introduces scattering of particles on different stacks. In certain conditions, potential coupling may give rise to long-range (density-wave) order at finite temperature. Second, there is the kinetic coupling, commonly denoted $t_⊥$, which allows an electron to hop from one stack to another. In the following, we shall confine ourselves to the latter coupling, which is the most studied and by far the most complex. In effect, it is from $t_⊥$ that most instabilities of 1D quantum liquids discussed thus far can occur. These are also called *crossovers* which form the links between the physics in one and higher dimensions as either the restoration of a Fermi liquid component or the onset of long-range order. In the following, we will review
<figure><img src="image_3.png" /><figcaption>Fig. 1.3. The phase diagram of (TM)₂X</figcaption></figure>

the physics of both types of crossover induced by $t_⊥$ using the renormalization group method that was described earlier.

In the tight-binding picture of a linear array of $N_⊥$ chains we have

$$E_p(\mathbf{k}) = \epsilon_p(k) - 2t_{\perp b} \cos k_{\perp b}, \quad (1.10)$$

The reduction of the density of states along the chains also modifies the amplitude of interchain hopping. The crossover criteria mentioned above now becomes $(\xi/a)z(T)t_⊥ \sim E_F$, which leads to the usual downward renormalization of the one-particle crossover temperature: [17]

$$T_{x^1} \sim t_{\perp} \left( \frac{t_{\perp}}{E_F} \right)^{(1-\theta)/\theta} . \quad (1.11)$$

According to this expression, $T_{x^1}$ decreases when the interaction – which can be parametrized by $\theta$ – increases (Figure ??); it is non-zero as long as $\theta < 1$ for which $t_⊥$ remains a relevant variable. The system then undegoes a crossover to the formation of a Fermi liquid with quasi-particle weight $z(T_{x^1})$. For strong coupling, $T_{x^1}$ vanishes at the critical value $\theta_c = 1$ and becomes undefined for $\theta > 1$, $t_⊥$ being then marginal in the former case and irrelevant in the latter (Figure ??). Consequently, in the latter case, no transverse band motion is possible and the single-particle coherence is spatially confined along the stacks. These large values of $\theta$ cannot be attained from the above perturbative renormalization group. They are found on the
Luther-Emery line at half-filling or at quarter-filling in the presence of a charge gap; [18–20] in the gapless Tomanaga-Luttinger model for sufficiently strong coupling constants or when the range of interaction increases. [21]

The question now arises whether $t_{\perp}$ can promote interchain pair propa-
gation besides single-particle coherence. Actually, this possibility exists and
results from interchain pair-hopping processes, [28–30] a mechanism that
is not present in the Hamiltonian at the start but which emerges when
interactions along the stacks combine with $t_{\perp}$ in the one-dimensional re-
gion. The renormalization group approach proved to be particularly useful in
this respect making possible a unified description of both modes of propaga-
tion. [26, 28, 31–33]

In the one-dimensional regime, it is governed at the one-loop level by the
distinct flow equation

$$
\frac{d}{d\ell} \tilde{V}_{\perp} = f(\ell) + \tilde{V}_{\perp} \gamma(\ell) - \frac{1}{2} (\tilde{V}_{\perp})^2, \quad (1.12)
$$

where $\tilde{f}(\ell) \simeq -2[(\tilde{g}_2(\ell) + \tilde{g}_3(\ell))t_{\perp}/E_0]^2 e^{(2-2\theta(\ell))\ell}$. Here $\theta(\ell)$ and $\gamma(\ell)$ are the power law exponents of the one-particle propagator (Eqns. (??)) and (??)) and antiferromagnetic response (Eqn. (??)) respectively (these are scale dependent due to the presence of Umklapp scattering). One observes from the right-hand-side of the above equation that the seed term resulting from the perpendicular delocalization of the electron and hole within the pair competes with the second term due to antiferromagnetic correlations along the chains. The outcome of this competition will be determined by the sign of $2-2\theta(\ell)-\gamma(\ell)$. Regarding the last term, it is responsible for a simple pole singularity of $J_{\perp}$ at a non-zero $\ell_c = \ln(E_F/T_c)$, signaling the onset of long-range order at $T_c$. The temperature at which the change from the one-dimensional regime to the onset of transverse order occurs can be equated with a distinct dimensionality crossover denoted by $T_{x2} \simeq 2T_c$ for pair correlations. The latter makes sense as long as $zt_{\perp}$ is still a perturbation, that is to say for $T_{x2} > T_{x1}$, which defines the region of validity of (1.12).

By looking at the effective spectrum of the above model in which $t_{\perp} \to t_{\perp}^* = zt_{\perp}$ in (1.10), we observe that the whole spectrum obeys the relation $E_{\perp}^*(\mathbf{k}) = -E_{\perp}^*(\mathbf{k} + \mathbf{Q}_0)$, showing electron-hole symmetry or perfect nesting at $\mathbf{Q}_0 = (2k_F, \pi)$. The response of the free quasi-1D electron gas is still logarithmically singular $\chi^0(\mathbf{Q}_0, T) \sim \ln T_{x1}/T$ in the Peierls channel for $\mathbf{Q}_0$ electron-hole excitations within the energy shell $\sim T_{x1}$ above and below the coherent warped Fermi surface. This singularity is also to be found in the perturbation theory of the scattering amplitudes, and can therefore lead to an instability of the Fermi liquid. For repulsive interactions, the most favorable instability is the one that yields a spin-density-wave state. The temperature at which the SDW instability occurs can be readily obtained by extending the renormalization group method of § ?? below $T_{x1}$ (or $\ell > \ell_{x1} = \ln E_F/T_{x1}$). When perfect nesting prevails, a not too bad approximation consists of neglecting the interference between the Cooper and Peierls channels [22](we
shall revert to the problem of interference below $T_{x1}$ later in § ??). Thus by retaining the outer shell decomposition of $S_{I,2}$ in the latter channel only, one can write down a ladder flow equation

$$ \frac{d\tilde{J}}{d\ell} = \frac{1}{2}\tilde{J}^2 + \dots \qquad (1.13) $$

for an effective coupling constant $\tilde{J} = \tilde{g}_2 + \tilde{g}_3 - \tilde{V}_{\perp}$ ... that defines the net attraction between an electron and a hole separated by $Q_0$ (the origin of the exchange term $V_{\perp}$ will be discussed in § ??). This equation is integrated at once

$$ \tilde{J}(T) = \frac{\tilde{J}^*}{1 - \frac{1}{2}\tilde{J}^* \ln T_{x1}/T}, \qquad (1.14) $$

where $\tilde{J}^*$ is the effective SDW coupling at $T_{x1}$, resulting from the integration of 1D many-body effects at $\ell < \ell_{x1}$. The above expression leads to a simple pole singularity at the temperature scale

$$ T_c = T_{x1} e^{-2/J^*}, \qquad (1.15) $$

which corresponds to a BCS type of instability of the Fermi liquid towards a SDW state. As long as the nesting conditions are fulfilled, it invariably occurs for any non-zero interaction (dashed line of Fig. ??).

Nesting frustration is therefore required to suppress the transition. When nesting deviations are sufficiently strong, we will see, however, that the Fermi liquid is not stabilized after all. Actually, when interference between the Peierls and the Cooper channels is restored, the system turns out to become unstable to superconducting pairing, a mechanism akin to the Kohn-Luttinger mechanism for superconductivity in isotropic systems. [23–27] We shall return to this in § .

## 1.3 The Fabre salts series

### 1.3.1 The generic $(TM)_2X$ phase diagram

Although the $(TM)_2X$ generic phase diagram can be established by measuring the transport properties of various compounds, it has been most rewarding to use a single compound, namely $(TMTTF)_2PF_6$, and the help of a high pressure to span the generic diagram. The study of this strongly insulating system $(TMTTF)_2PF_6$ under high pressure has been very useful not only because it has led to the stabilization of superconductivity in a strongly insulating sulfur compound [34] but also because its location at the left end of the phase diagram has allowed several key properties of quasi 1-D conductors to be carefully monitored under pressure, for instance, the longitudinal or transverse transports activation and the one dimensional deconfinement
arising under pressure [11] which will be discussed later in Sec.1.3.3. The phase diagram in Fig. 1.2 where $(TMTTF)_2PF_6$ is the reference compound at ambient pressure has been obtained from the temperature dependences of $\rho_c(T)$ and $\rho_a(T)$ to be discussed below. In the low pressure region (see Fig. 1.4, $P<10$ kbar) $\rho_c(T)$ and $\rho_a(T)$ are both activated, with $\Delta_{\rho,c}$ being about 30% larger than $\Delta_{\rho,a}$. At higher pressures, the activation of $\rho_c$ follows a gentle decrease, while $\Delta_{\rho,a}$ collapses abruptly at a pressure of about 14 kbar, which also marks the onset of a non-monotonous temperature dependence of $\rho_c$ with a maximum arising at the temperature $T^*$. Above 14 kbar, $\rho_a(T)$ displays a metallic temperature dependence down to the sharp metal-insulator transition below 20 K, while $\rho_c(T)$ remains indicative of a weakly insulating state above the temperature $T^*$ increasing under pressure.

### 1.3.2 Longitudinal transport

A major problem encountered with the $(TM)_2X$ materials (as well as in most organic conductors) is the very strong pressure (or volume) dependence of their electronic properties, particularly the transport property [35–37]. This strong volume dependence is going along a particularly large thermal expansion. Hence, the only temperature dependence that can be compared with the prediction of the theory is the one measured at a constant volume. As all temperature dependences are obtained experimentally under constant pressure, a transformation to the constant volume $T$-dependence must be performed [36,37], in order to make a significant comparison with the theory. Fortunately this correction is not that relevant for the case of $(TMTTF)_2PF_6$ under high pressure since the compressibility of these materials is known to decrease under pressure [38]. This is no longer the case when the metallic phase is already stable at ambient pressure as we shall see later for the case of $(TMTSF)_2PF_6$ (§1.4.1).

The insulating character of these materials with partly filled conduction bands is not expected in the framework of non interacting electrons. The reason for this must be the existence of strong repulsive interactions between 1D carriers. One explanation for the different values and pressure dependences of the activation energies can be taken as an evidence for in-chain conduction made possible by thermally excited 1-D objects similar to solitons in conducting polymers [39], whereas transverse transport requires the excitation of real quasi-particles (QP’s) through a Mott-Hubbard gap larger than the soliton gap as we shall see in § ???. Contrasting with the strongly pressure dependence of the transport properties, the susceptibility is hardly sensitive to the location of a specific compound in the generic phase diagram. The spin susceptibility is dropping by about 40% between room and low temperature for all materials, although the actual magnitude and the low temperature behaviour depend on the compound (see Fig. 1.5).

Although the insulating behavior of $(TMTTF)_2X$ salts can be clearly ascribed to their commensurate band filling a closer examination is needed to
<figure><img src="image_4.png" /><figcaption>Fig. 1.4. (TMTTF)₂PF₆ longitudinal (left) and transverse (right) resistances versus temperature at different pressures [11]</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 1.5. Temperature dependence of the magnetic susceptibility of (TM)₂X compounds at ambient pressure</figcaption></figure>

determine which of the Umklapp scattering channels 1/2- or 1/4-filled is the most active. In the presence of such a gap, the transport is activated at low temperature $\rho_{\parallel} \propto e^{\Delta_{\rho,a}/T}$ but is expected to vary according to the power law

$$ \rho_{\parallel} \propto T^{4n^2 K_{\rho} - 3} \qquad (1.16) $$

in the high temperature regime, i.e., $T > \Delta_{\rho a}$ [16]. The material resembles a metal at high temperature along the longitudinal direction. In the high $T$ 1-D regime ($T > t_{\perp}$), the picture of non-coupled chains is approached. Therefore, the density of quasiparticle states should resemble the situation that prevails in a Luttinger liquid namely, $N(E) \sim |\omega|^{\alpha}$, where $\alpha$ is related
to $K_ρ$ through $α = \frac{1}{4}(K_ρ + 1/K_ρ - 2)$, forgetting about the influence of the Mott gap (supposedly smaller than the temperature).

### 1.3.3 Deconfinement and transverse transport

Under a pressure higher than 14 kbar the behavior of the $(TMTTF)_2PF_6$ resistance along the direction of the weakest coupling, i.e., along the c-axis, displays an insulating character with a maximum around 80-100 K and becomes metallic at lower temperatures, although remaining several orders of magnitude above the Mott-Ioffe critical value which is considered as the limit between metal and insulating-like transport [40]. Figure 1.6 displays the temperature dependence of $ρ_c$ in $(TMTTF)_2PF_6$, and also for other members of the $(TM)_2X$ family with different anions such as $(TMTTF)_2BF_4$ and $(TMTTF)_2Br$.

<figure><img src="image_6.png" /><figcaption>**Fig. 1.6.** Temperature dependence of the transverse transport along the c-axis in several compounds belonging to the $(TM)_2X$ series studied under pressure. The temperature of the maximum of resistivity shows the location of the dimensional crossover $T^*$ which is strongly pressure dependent for each compound. The resistivity upturn at low temperatures represents the stabilisation of the insulating SDW phase. In case of the selenium compound $TMTSF_2ClO_4$ the crossover lies around room temperature whereas for the sulfur compound $TMTTF_2Br$ a resistivity maximum is seen only after the constant volume correction is taken into account [41].</figcaption></figure>

The insulating character of the transverse transport has been interpreted as the signature of a non Fermi liquid behavior for carriers within planes (chains) [42]. When transverse transport along the c-direction is incoherent, transverse conductivity probes the physics of the a–b planes and conductivity in terms of the transverse coupling $t_{⊥c}$ is expressed in the tunneling approximation as
$$ \sigma_c(\omega, T) \propto t_{\perp c}^2 \int dx \int d\omega' A_{1D}(x, \omega') A_{1D}(x, \omega + \omega') \\ \times \frac{f(\omega') - f(\omega + \omega)}{\omega}, \qquad (1.17) $$

where $A_{1D}(x, \omega)$ is the one-electron spectral function of a single chain and $f(\omega)$ the Fermi-Dirac function. When the $a-b$ plane is mode of an array of weakly interacting Luttinger chains, Eq. (1.17) leads to a power law temperature dependence for the c-axis conduction. The temperature at which the c-axis transport switches from an insulating to a metallic temperature dependence corresponds to a crossover between two regimes; a high temperature regime with no QP weight at the Fermi energy (possibly a TL liquid in the 1D case) and another regime in which the QP weight increases with decreasing temperature. This interpretation does not imply that the transport along the c-direction must also become coherent below the cross-over. The c-axis transport may well remain incoherent with a Fermi liquid being established in the $a-b$ plane at temperatures below $T^*$. The temperature dependence of the resistivity along the least conducting direction is thus expressed as [43]:

$$ \rho_c(T) \propto T^{1-2\alpha}. \qquad (1.18) $$

Consequently, the temperature dependence of transport properties along the $a$ and $c$-axes above $T^*$ should possibly lead to a consistent determination of $K_\rho$.

Now, regarding $(TMTTF)_2PF_6$, we are facing a very interesting system, since the evolution from a Mott insulator to a metal can be carefully studied under pressure in a single sample and a decrease in compressibility under pressure makes constant volume correction less significant for temperature dependences measured at high pressures. Turning to the evaluation of the correlation coefficient from the temperature dependence of $\rho_c$ we end up fitting the data for $(TMTTF)_2PF_6$ in the pressure domain around 12 kbar, Fig. 1.4, with a very small value of $K_\rho$ (or large values of $\alpha$) which is not compatible with the value $K_\rho = 0.23$ derived from the far infrared (FIR, see below) and NMR data [44]. Consequently, the Mott gap seems to be important in this temperature regime governing the excitation for the motion of single particles along $c$. Tentatively, one can expect a transverse resistivity behaving according to [43],

$$ \rho_c(T) \propto T^{1-2\alpha} e^{\Delta_{\rho,c}/T}. \qquad (1.19) $$

Since the Mott-Hubbard gap varies exponentially with $K_\rho$ even a small variation in the ratio between the Coulomb interaction and the bandwidth under pressure can explain a significant decrease of all gaps moving from the left to the right in the generic phase diagram. An other signature of the Mott-Hubbard gap has been given by the frequency-dependent conductivity $\sigma(\omega)$ measured in various salts of the $(TM)_2X$ series exhibiting very different
<figure><img src="image_7.png" /><figcaption>**Fig. 1.7.** Frequency dependence of the optical conductivity of several (TM)₂X compounds for the light polarized along the *a* axis. After Dressel ??.</figcaption></figure>

values of the conductivity at room temperature [45] (see Fig. 1.7). The peak of the conductivity at a frequency $\omega_0$ correlates with the magnitude of the room temperature conductivity, namely both sulfur salts (TMTTF)₂PF₆ and (TMTTF)₂Br which are insulating exhibit a conductivity peak around $\omega_0 \approx 1000$ cm⁻¹ while the peak in (TMTSF)₂PF₆ occurs around 200 cm⁻¹ that is, very close to the zero frequency.

Both optical and transport data give $2\Delta_{\rho,a} = 800-1000$ K in (TMTTF)₂PF₆ at ambient pressure [11]. The difference between $K_{\rho}$ for selenium and sulfur compounds ($K_{\rho} = 0.18$ for the latter material) is a result of the difference between their bare bandwidths, since on-site repulsion, being a molecular property, is likely to be less sensitive to pressure than the intermolecular overlap along the stacking axis.

## Behind the Mott scale

- Absence of Plasma mode in the transverse direction ~ 1D effect

- Theory *vs* experiment

- NMR confirmation of $K_{\rho} = 0$ (Wzietek et al.)

- Prediction of the temperature dependence of resistivity at 1/4 filling and 1/2 filling and observation. The nuclear spin-lattice relaxation rate measured in NMR experiments is another quantity of practical importance if
one tries to gain information about spin correlations. It is given by the Moriya expression

$$T_1^{-1} = |A|^2 T \int \frac{\chi''(q, \omega)}{\omega} d^D q, \qquad (1.20)$$

which is taken in the zero Larmor frequency limit ($\omega \to 0$) and where A is proportional to the hyperfine matrix element. Thus the relaxation of nuclear spins gives in principle relevant information about the static, dynamics and dimensionality D of electronic spin correlations. This gives in turn a relatively easy access to the parameters $K_\rho$, $\bar{u}_\sigma(T)$ and $u_\sigma(T)$ of the electron gas. [46] According to Eqns. (??) and (??), the enhancements of $\chi''$ occur at $q \sim 0$ and in the interval $q \sim 2k_F \pm T/v_F$ close to $2k_F$. The integration is then readily done to give

$$T_1^{-1} \approx C_0(T)T\chi_{\sigma}^2(T) + C_1(T)T^{K_{\rho}}. \qquad (1.21)$$

Owing to the presence of $g_1$, we have $C_0(T) = C_0(u_\sigma(T)/\bar{u}_\sigma(T))^{1/2}$ and $C_1(T) = C_1(1 + \tilde{g}_1 \ln \omega_\sigma/T)^{1/2}$. As a function of temperature, two different behaviors can be singled out. At high temperature, where uniform spin correlations dominate and the $2k_F$ ones are small, the relaxation rate is then governed by the $C_0(T)T\chi_\sigma^2(T)$ term. In the low temperature domain, however, $2k_F$ spin correlations are singularly enhanced while uniform correlations remain finite so that $T_1^{-1} \sim C_1(T)T^{K_\rho}$.

### Nuclear magnetic resonance

Among other measurable quantities that are sensitive to one-dimensional physics, we have the temperature dependence of the nuclear spin-lattice relaxation rate $T_1^{-1}$. [44] Consider for example the insulating compounds (TMTTF)$_2$X. According to the scaling pictures of § ?? and § ??, the charge stiffness $K_\rho = 0$ vanishes in the presence of a gap below $T_\rho$, or for $\ell > \ell_\rho$. The resulting behavior for the relaxation rate is then

$$T_1^{-1} \sim C_1 + C_0 T \chi_{\sigma}^2. \qquad (1.22)$$

As shown in Fig. ??, this behavior is indeed found for (TMTTF)$_2$PF$_6$ salt when the relaxation rate data are combined with those of the spin susceptibility ($T\chi_\sigma^2$) in the normal phase above 40 K, namely above the onset of spin-Peierls fluctuations. [46, 47] A similar behavior is invariably found in all insulating materials down to low temperature where three-dimensional magnetic or lattice long-range order is stabilized. [36, 44, 48] Long-range order also prevents the observation of logarithmic corrections in $C_1$ which are expected to show up in the low temperature limit.
<figure><img src="image_8.png" /><figcaption>Fig. 1.8. After Ref. [46].</figcaption></figure>

Incursion of charge ordering

- Experiments

- Additional $4k_F$ instability for the sulfur compounds: Structurless anomaly in resistivity (Coulon et al., ~ 80's) NMR (Brown group); dielectric response (Nad et al.,

- Brief connection with the results electron gas model (Numerical results of Zotos et Mila and others) at 1/4 filling.

- Stability of CO under pressure effect (Brown): the case of $(\text{TMTTF})_2\text{AsF}_6$, $(\text{TMTTF})_2\text{BF}_4$ .... Additional figure on the "modification" of the phase diagram that would accomodate CO (?). [49]

1.3.4 The ordered states at low temperature

The spin-Peierls instability of the sulfur compounds

Among the members of the Fabre series, the two compounds $(\text{TMTTF})_2\text{PF}_6$ and $(\text{TMTTF})_2\text{AsF}_6$ develop an instability of the Mott insulating phase that involves both spins and lattice degrees of freedom. X-ray diffuse scattering measurements of Pouget et al., [50] on both compounds reveal 1D diffuse scattering lines at the wave vector $2k_F$ showing the onset of lattice softening below the temperature scale $T_{\text{SP}}^0 \approx 60$K for $(\text{TMTTF})_2\text{PF}_6$ and 45K for $(\text{TMTTF})_2\text{AsF}_6$. These lines condense into satellite reflections at the transition temperature $T_{\text{SP}} \approx 19$K for $(\text{TMTTF})_2\text{PF}_6$ salt and 15K, for $(\text{TMTTF})_2\text{AsF}_6$, where a static lattice distortion takes place at the wave vector $\mathbf{q}_0 = (2k_F, \pi, \pi)$. Recent elastic neutron scattering experiments did confirmed the existence of such a static distortion at $\mathbf{q}_0$ for $(\text{TMTTF})_2\text{PF}_6$ below $T_{\text{SP}}$ [51]. Since both the softening and the transition occurs in the Mott insulating state only spin excitations are gapless, one thus deals with a spin-Peierls (SP) transition with a pronounced quasi-1D character. The expected non magnetic nature of both SP fluctuations and long-range order
has been confirmed by the temperature dependence of the spin susceptibility and nuclear spin-lattice relaxation rate [44, 52]; these quantities are reduced in the fluctuation regime and show thermal activation below $T_{SP}$.

Following the example of the Peierls transition, the SP instability proceeds from the coupling of singular 1D bond-order-wave (BOW) electronic correlations to acoustic phonons at $2k_F$ [53–55]. Unlike the Peierls case, however, the coupling takes place within the Mott insulating state instead of a metal. The microscopic theory predicts a power law singularity in the 1D electronic BOW response below the Mott scale $T_ρ$, that is $χ_{BOW}(2k_F, T) ∼ (T/T_ρ)^{-γ}$. For weakly dimerized chains systems like (TMTTF)$_2$X, the power law exponent $γ = 1$ ($K_ρ = 0$), as for the antiferromagnetic spin response. The enhancement of the electron-phonon interaction at $2k_F$ by BOW correlations can be worked out by perturbation theory [53, 54]. In the random phase approximation, the static temperature electron-electron vertex part induced by the exchange of $2k_F$ phonons takes the form

$$ \Gamma_{\text{ph}}(2k_F, T) = \frac{g_{\text{ph}}^0}{1 - g_{\text{ph}}^0 \chi_{\text{BOW}}(2k_F, T)} \quad (1.23) $$

where $g_{\text{ph}}^0$ is the square of the bare $2k_F$ electron-phonon matrix element expressed in $πv_F$ units. A singularity of the temperature vertex part will then develop at the mean field temperature

$$ T_{\text{SP}}^0 = c g_{\text{ph}}^0 T_ρ, \quad (1.24) $$

where $c \gtrsim 1$. Since there is no phase transition in one dimension, $T_{\text{SP}}^0$ is not a true transition temperature but a temperature scale of lattice fluctuations that can be identified with the softening temperature born out by X-ray experiments. Although higher order fluctuations corrections will bring back the transition at $T = 0$, $T_{\text{SP}}^0$ remains the right temperature scale for the onset of short-range fluctuations [53].

As for long-range SP order, it is driven by the interchain interaction whose contributions combine Coulomb, interchain hopping and three-dimensional phonons [32, 54]. These coupling favor staggered bond order transversally to the chains at $q_0$. A molecular field treatment of interchain coupling, which takes into account one-dimensional fluctuations rigorously, leads to $T_{\text{SP}} \sim T_{\text{SP}}^0/3$, a standard result for most of the electronically driven quasi-1D structural transition in the adiabatic limit [?]. The observed values for the ratio $T_{\text{SP}}/T_{\text{SP}}^0$ in (TMTTF)$_2$PF$_6$ and (TMTTF)$_2$AsF$_6$ are compatible with this estimation.

The respective $T_{\text{SP}}$ of (TMTTF)$_2$AsF$_6$ and (TMTTF)$_2$PF$_6$ evolve differently under low pressure. For (TMTTF)$_2$AsF$_6$, $T_{\text{SP}}$ first increases under pressure and reaches a maximum when the charge ordering temperature $T_{CO}$ merges with $T_{\text{SP}}$ at 1.5 kbar, and finally decrease at higher pressure [57]. Charge order breaks the inversion symmetry in the unit cell and acts as a $4k_F$ charge-density-wave potential on molecular sites. The potential reduces
<figure><img src="image_9.png" /><figcaption>**Fig. 1.9.** The phase diagram of (TMTTF)₂PF₆ near the quantum critical point at *P*<sub>QCP</sub>. After Ref. [56].</figcaption></figure>

the strength of BOW correlations [58], and in turn the amplitude of both
$T_{\text{SP}}^0$ and $T_{\text{SP}}$. For (TMTTF)₂PF₆, however, $T_{\text{CO}}$ occurs at much lower tem-
perature [59], and $T_{\text{SP}}$ has a constant decrease under pressure. It finally goes
down rapidly, extrapolating to zero at $P_{\text{QCP}} \approx 9\text{kbar}$, where it competes
with a Néel state [47, 56, 60]. This critical pressure corresponding to the zero
temperature extrapolation of the critical lines can be put in the category of
a quantum critical point (Fig. 1.9) [56].

The decrease of $T_{\text{SP}}^0$ and $T_{\text{SP}}$ under pressure can be qualitatively under-
stood if one considers that the drop in $T_{\rho}$ weakens electronic BOW correla-
tions, which according to Eq. (1.24), reduces $T_{\text{SP}}^0$ (Figs. 1.9 and ??). As $T_{\text{SP}}^0$
carries on decreasing with pressure, it will reach values that become small
compared to the Debye energy $\omega_D$ of phonons ($\omega_D \sim 80\text{K in these materi-}
als$ [61]), where quantum effects enter into play [62–64]. In effect, the above
expression for $T_{\text{SP}}^0$ has been obtained for ‘static’ phonons, in the so-called adi-
abatic approximation where the molecules are supposed to have an infinite
mass. Adiabaticity is a reasonable assumption provided that $\omega_D/2\pi T_{\text{SP}}^0 \ll 1$.
As $T_{\text{SP}}^0$ decreases and $\omega_D$ increases under pressure, however, the adiabatic
condition will not be satisfied any more and the lattice softening will be re-
duced by the zero point motion of the molecular lattice. It has been shown
that a quantum-classical crossover is expected at $\omega_D/\pi T_{\text{SP}}^0(P) \approx 1$, where
quantum corrections completely suppress $T_{\text{SP}}^0$ and in turn $T_{\text{SP}}$ [62,63]. For a
system like (TMTTF)₂PF₆, this would take place after a reduction of $T_{\rho}$ by
a factor of two or so, which according to Fig. 1.3, corresponds to a pressure
of the order of $P_{\text{QCP}}$.

The Néel order

Sufficiently above $P_{\text{QCP}}$, antiferromagnetic correlations within the Mott state
are much less affected by lattice fluctuations that are sizably weaker if not ab-
sent in this pressure range. This is shown in the phase diagram by the absence
of a spin pseudo gap from the temperature dependent nuclear relaxation rate in (TMTTF)₂PF₆ at 15 kbar and in (TMTTF)₂Br at 1 bar [44]. This is also confirmed in the case of (TMTTF)₂Br by X-ray diffuse scattering experiments at ambient pressure [65]. The observation in these conditions of a temperature independent nuclear relaxation rate for both materials, indicates that the power law singularity of the AF response $\chi(2k_F, T) \sim (\pi v_F)^{-1}(T/\Delta_\rho)^{-\gamma}$ below the Mott scale $T_\rho$, albeit reduced, persists with the same exponent $\gamma = 1(K_\rho = 0)$. In the presence of a Mott gap, $\Delta_\rho > t_{\perp b}$, electron-hole bound pairs are formed and a coherent electron band motion in the transverse directions cannot take place. Propagation of order in the transverse directions leading to a Néel ordered state is nevertheless possible, but is provided by the antiferromagnetic interchain exchange

$$J_{\perp b,c} \approx \pi \frac{\xi_{\rho} t_{\perp b,c}^{*2}}{\alpha_{0} \Delta_{\rho}}. \qquad (1.25)$$

Here $\xi_\rho = v_F/\Delta_\rho$ is the size of electron-hole pairs within the Mott state and corresponds to the chain distance over which the exchange between chains takes place, $t_{\perp b,c}^*$ is the effective transverse hopping in the b and c directions, and $\alpha_0$ is the chain lattice constant [28, 32, 66]. Using the renormalization group method, a molecular field approach to the exchange coupling leads to a RPA form of the total magnetic susceptibility [26, 67], namely

$$\chi(\mathbf{q}_0, T) = \frac{\chi_{\text{AF}}(2k_F, T)}{1 - (J_{\perp b} + J_{\perp c})\chi_{\text{AF}}(2k_F, T)}, \qquad (1.26)$$

where $\mathbf{q}_0 = (2k_F, \pi, \pi)$ is the modulation wave vector of the Néel antiferromagnetic order. The temperature scale for Néel ordering is determined by the singularity of the susceptibility which leads

$$T_c \approx \frac{(t_{\perp b}^{*2} + t_{\perp c}^{*2})}{\Delta_{\rho}}. \qquad (1.27)$$

This result indicates that $T_c$ – essentially dominated by the exchange in the b direction – increases as the Mott gap $\Delta_\rho$ decreases, a feature commonly observed in the Fabre series (Figs. 1.3 and ??) [68, 69]. It is worth noting that in the quarter-filled Mott insulator compound (EDT-TTF-CONMe₂)₂AsF₆, the interchain exchange is also the driving force of antiferromagnetic long-range order. In these systems too, $T_c$ is found to increase as $\Delta_\rho$ is decreasing, a behavior that proved to be independent of the commensurability of Umklapp scattering processes behind the insulating gap (Fig. ).

When the pressure is further increased, the Mott insulating and Néel critical scales meet and then the spins order themselves directly from the metallic state. Antiferromagnetism becomes itinerant in character and corresponds to a SDW state. The interchain exchange enters in the weak coupling sector and continues to be active, albeit on a relatively small pressure range. Here the expression of the critical temperature modifies and reads
<figure><img src="image_10.png" /><figcaption>Fig. 1.10. After Ref. [68].</figcaption></figure>

$$T_c \approx g^{*2}t_{\perp}^* \qquad (1.28)$$

where $g^*$ is the normalized effective amplitude of electron-electron coupling around the transition. The calculations show that $T_c$ starts to decrease in this restricted pressure domain giving rise to a maximum in $T_c$, that is clearly seen in experiments for $(TMTTF)_2Br$ (Fig. 1.10) [68, 69], $(TMTTF)_2PF_6$ (Fig. ??, [34]) and mixed selenium-sulfur compounds $(TMDMTSF)_2PF_6$ [70, 71]. This weak coupling domain coupling quickly evolves to a regime where $t_{\perp}^*$ and then the single electron transverse coherence length are increasing rapidly under pressure signaling the beginning of a coherent band motion perpendicular to the chains. This yields the onset of electronic deconfinement and coherent nesting of the whole Fermi surface at $T^*$ [72]. Coherent nesting characterizes the mechanism of itinerant long-range order in the Fabre series at high pressure and in the Bechgaard salts at low pressure.

## Deconfinement and transport

- The cases of $(TMTTF)_2PF_6$ (Jaccard et al.,), $BF_4$ ...

- Evolution of the transverse and longitudinal transports: $T^*$, $\Delta_{\rho}(\|\text{and } \perp)$ (P. Auban et al.,); conjecture about the excitations contributing to transport.

- Restoration of the plasma edge along $b'$ (De Giorgi graph as a function of the anion X)
1.4 The Bechgaard salts

1.4.1 The metallic phase

The strongly metallic character of the (TMTSF)₂PF₆ salt has been one of the highlights in the search for organic conductors ? [73]. The temperature dependence of the longitudinal resistivity follows a power law T¹.⁴ from 300K down to about 100K. Below 35K, the resistivity of of (TMTSF)₂PF₆ or (TMTSF)₂AsF₆ is quadratic in temperature ρ_a(T) = ρ₀ + AT², which is valid down to the metal-insulator transition due to the onset of an itinerant anti-ferromagnetic state at 12K, Fig.1.11.

<figure><img src="image_11.png" /><figcaption>Fig. 1.11. Temperature dependence of the (TMTSF)₂PF₆ longitudinal resistance plotted versus T² [74]</figcaption></figure>

For high quality samples the resistance ratio ρ_a(300K)/ρ₀ can reach val-
ues as large as 800 [74], Fig.1.11. Furthermore, an interesting behaviour en-
countered in (TMTSF)₂PF₆ materials (and also in most organic conductors)
is the very strong pressure (or volume) dependence of their electronic prop-
erties, particularly the transport property [35–37]. In addition, the thermal
expansion of these materials is particularly large. Hence, the only tempera-
ture dependence that can be compared with the prediction of the theory is
the one measured at a constant volume. As all temperature dependences are
obtained under constant pressure, a constant volume transformation must
be performed. An example is given in Fig. 1.12 by the longitudinal trans-
port of (TMTSF)₂PF₆ behaving at high temperature similar to T² under
ambient pressure but varying sublinearly (\sim T¹.⁹³) from 300 to 150 K, once
the volume correction is taken into account [37]. The experimental power
law of longitudinal resistivity leads in turn to n²K_ρ=0.98 according to the
<figure><img src="image_12.png" /><figcaption>**Fig. 1.12.** Temperature dependence of the (TMTSF)₂PF₆ longitudinal resistance at constant volume showing quasi linear *T*-dependence, with the thermal dependence of the lattice parameter *a* displayed in the inset [37]</figcaption></figure>

theory of resistivity. Note that a similar power law for the temperature dependence of longitudinal transport ($\sim T^{0.93}$) can also be observed for two sulfur-compounds, BF₄ and PF₆ under high enough pressure when the correction to constant volume becomes negligible.

In the early days of the (TMTSF)₂X compound the lattice dimerization was believed to govern entirely the amplitude of the Mott-Hubbard gap [75,76]. When the half-filled scenario is privileged, namely ($n = 1$). Hence, $n^2K_ρ=0.98$ leads to $K_ρ$ is close to unity implying a weakly coupled electron gas. This situation of a very weak coupling is difficult to reconcile with an enhancement of the spin susceptibility and the characteristic enhancement of the nuclear spin relaxation rate [14, 44] but an additional argument against this weak coupling is made possible by the unusual behavior of transverse transport (Eq. (1.18)) [43]. The weak coupling value for $K_ρ \approx 1$ derived from the temperature dependence of longitudinal transport and the optical data (see below) would imply $\alpha \approx 0$ and consequently a metal-like temperature dependence for $ρ_c(T)$ which is *at variance* with the data.

More recently, an alternative interpretation based on new experimental results has been proposed assuming that the 1/4-filled scattering could justify the existence of the Mott gap in the entire (TM)₂X series [16]. With such a hypothesis ($n = 2$), the fit of the experimental data would thus lead to $K_ρ = 0.23$ and $\alpha = 0.64$ (see below, discussion on optical response) [77]. This value for $K_ρ$ agrees fairly well in the 1/4 filled scenario with band structure calculations and plasma edge measurements and $U/W = 0.7$ (which is in fair agreement with an increase in spin susceptibility at low temperatures) [14, 44]. Such a strong coupling value for $K_ρ$ implies that a system such as (TMTSF)₂PF₆ lies at the border between a 1D Mott insulator and a Luttinger liquid, though slightly on the insulating side. (TMTTF)₂Br is another particularly interesting system, in which the pressure coefficient of the
resistivity is very large. Hence, it is the correction to constant volume which makes the maximum in $\rho_c$ to appear around 150 K while this maximum is absent in the constant pressure runs, see (Fig. 1.6).

The other approach to the correlation coefficient is given by the far infrared optical studies of $(TMTSF)_2PF_6$ which have been very helpful for the determination of $K_\rho$ since the FIR gap of about $\Delta_{\rho,a} = 200 \text{ cm}^{-1}$ in $(TMTSF)_2PF_6$ has been attributed to the signature of the Mott-Hubbard gap [77]. Consequently, the frequency dependence of the conductivity above the Mott gap is closely linked to the dynamics of the excited carriers in the 1D regime. The theory predicts a power law dependence for the optical conductivity at frequencies larger than the Mott gap [16] namely,

$$ \sigma_a(\omega) \approx \omega^{4n^2K_\rho - 5} \qquad (1.29) $$

at $\omega > 2\Delta_\rho$ see Fig. (1.13).

<figure><img src="image_13.png" /><figcaption>Fig. 1.13. Far infra red data of $(TMTSF)_2PF_6$, experiment [77] (left) and theory [16] (right).</figcaption></figure>

According to the optical experimental data, see Fig.1.14 [77] $\sigma_a(\omega) \approx \omega^{-1.3}$ at high frequency leading to $n^2K_\rho=0.93$. This value for the correlation coefficient is fairly close to the one derived above from parrallel transport data but none of these experiments allow by themselves to discriminate between half or quarter filled Umklapp scattering.

### 1.4.2 Pseudogap and zero frequency mode in the metallic phase

It is also most illuminating to have a look at the conductivity in the far infrared regime. A large gap of order 1000K is observed in the frequency dependence of the FIR conductivity of sulfur compounds [78]. This is in line with the activation energy of the DC conductivity in those compounds. However, the surprise arose for selenium compounds which behave apparently like normal metals as far as DC transport is concerned in spite of the
<figure><img src="image_14.png" /><figcaption>**Fig. 1.14.** Optical conductivity above the Mott-Hubbard gap in several selenide conductors analyzed in terms of a power law $\omega_a^{-1.3}$.</figcaption></figure>

marked gap observed in the FIR regime at low temperature. The apparent normal behaviour of the resistivity varying quadratically in temperature for $(TMTSF)_2ClO_4$ or $(TMTSF)_2PF_6$ above the SDW transition could lead to the misleading conclusion of a 2 or 3-D Fermi gas in which the temperature dependence of the transport is governed by e-e scattering. However, the analysis of the conductivity in terms of the frequency reveals quite a striking breakdown of the Drude theory for single-particles. The inability of the Drude theory to describe the optical conductivity has been noticed by a number of experimentalists working on $(TMTTF)_2X$ with $X=ClO_4$, $PF_6$ or $SbF_6$ [79]. When the reflectance of $(TMTSF)_2ClO_4$ in the near-infrared is analyzed with the Drude model in the whole range of temperatures from 300 down to 30K the electron scattering rate is found to decrease gradually from $2.5 \times 10^{14} \text{ s}^{-1}$ at room temperature to $1.3 \times 10^{14} \text{ s}^{-1}$ at 30K [80]. Even if the RT value is not far from the value from DC conductivity, a drastic difference emerges at low temperature as $\sigma_{DC}$ increases by a factor about 100 between RT and 30K [81] as compared to the factor 2 for the optical lifetime.

An other striking feature of the optical conductivity has been noticed when the Kramers-Krönig transformation of the reflectance is performed in a broad frequency domain for $(TMTSF)_2ClO_4$ as well for all conducting materials at low temperature. Given the usual Drude relation $\sigma_{DC} = \omega_p^2\tau/4\pi$ between transport lifetime and plasma frequency data (the plasma frequency has been found nearly temperature independent [45, 80]) and the measured resistance ratio for $\rho_a$ of about 800 between RT and $2K$ obtained in good quality measurements, the Drude conductivity in the frequency range $\approx 40 \text{ cm}^{-1}$ should amount to at least $4000 \Omega \text{cm}^{-1}$ [82, 83]. The measured optical conductivity is at most of the order of $500 \Omega \text{cm}^{-1}$ [82]. Consequently, the rise in the conductivity as $\omega \to 0$ has been taken in $(TMTSF)_2ClO_4$ as well as in
the other salts with PF₆ or SbF₆ as an evidence for a hidden zero frequency mode. This mode is actually so narrow that it escapes a direct determination from K-K analysis of the reflectance which is limited to the frequency domain above 10 cm⁻¹. Estimates of the mode width have been obtained using the DC conductivity and the oscillator strength Ωₚ² of the mode with σDC = Ωₚ²τc/4π where Ωₚ is measured from the first zero crossing of the dielectric constant. This procedure gives a damping factor Γc = 0.005 cm⁻¹ and 0.09 cm⁻¹ at 2 and 25K respectively in (TMTSF)₂ClO₄ [83]. The con-

<figure><img src="image_15.png" /><figcaption>**Fig. 1.15.** Far infra red data of (TMTSF)₂ClO₄ from reference [83]. The dashed line is the Drude behaviour with 1/τ = 3.5 cm⁻¹ and ωp = 10⁴ cm⁻¹</figcaption></figure>

firmation of a very long scattering time for the DC conduction has also been brought by the rapid suppression of T_c by non-magnetic defects in the non conventional superconductor (TMTSF)₂ClO₄ leading to Γ_c = 0.56 cm⁻¹ at low temperature [84] meaning that the electron lifetime at low temperature is actually much longer than the value inferred from a Drude description.

There is now a wealth of experimental evidences showing the development of a narrow frequency mode in the Mott gap of (TMTSF)₂ClO₄ and related conducting compounds. From FIR data in (TMTSF)₂PF₆, it has also been shown that the narrow mode carries only a small fraction (a few percent) of the total spectral weight [45, 77] but it is this mode which explains the very large value of the DC conduction observed at low temperature [85].

### 1.4.3 Quarter-filled compounds

Even if the debate between 1/2 and 1/4 fillings may be relevant for (TM)₂X, this is no longer the case for new synthesized compounds in a family whose general structure precludes any dimerization. The structural peculiarity of
the salt (EDT – TTF – CONMe₂)₂AsF₆ is the absence of inversion center between adjacent molecules in stacks and instead the presence of a glide plane symmetry [86] (see Fig.1.16). The analysis of the transport data of (EDT – TTF – CONMe₂)₂AsF₆ has shown that in spite of the existence of a glide plane symmetry the carriers are localized, and even more localized than in the most insulating salts of the Fabre series known at present. Since the localization in this compound cannot be ascribed to a 1/2–Umklapp scattering or to the Anderson localization, 1/4-Umklapp scattering seems to be the only channel left to explain carrier localization in this commensurate 1D conductor. Under ambient pressure, given the total bandwidth deduced from quantum chemistry (W = Wtot(P = 1bar) = 0.350 eV (3850 K)) and the experimental Mott gap (2Δ<sub>ρ,a</sub> = 2700 K), the theory [16] gives, in the case of quarter filling stricto sensu: 2Δ<sub>ρ,a</sub> = W(U/W)<sup>2/(1-4K<sub>ρ</sub>)</sup>. This leads to K<sub>ρ</sub> = 0.1, with a reasonable U/W = 0.9.

The Mott gap of (EDT – TTF – CONMe₂)₂AsF₆ is much larger than the expected value of the bare interstack overlap $t_⊥$ (which is similar to that value of (TMTTF)₂Br, i.e., on the order of 100-200 K * ( check here anisotropy )* and even larger than that observed in the transverse coupling of (TMTSF)₂PF₆). Consequently, the large Mott gap makes the single particle hopping between neighboring stacks irrelevant, at least, in the pressure regime less than 20 kbar since the transverse hopping is renormalized to zero on account of a strong intrachain electron-hole interaction.

The interaction over bandwidth ratio $U/W \approx 0.9$ is also in fair agreement with the result of a crude analysis of the spin susceptibility of S-salts [14, 44] and indicates that these compounds lie in the strong coupling sector. With increasing pressure, the gap decreases steadily up to the pressure of 20 kbar above which it disappears sharply due to the competing transverse coupling. Below 15 kbar, the gap of (EDT – TTF – CONMe₂)₂AsF₆ (1350 K) is about equal to the gap of (TMTTF)₂X measured under ambient pressure [42]. Since the logarithmic pressure dependences of the gap seem to be similar for both compounds, we may say that (EDT – TTF – CONMe₂)₂AsF₆ can also be considered as part of the generic Fabre-Bechgaard salt diagram, provided the origin of the pressure axis is shifted to the left by 15 kbar. Hence, it is reasonable to expect that the TL parameter $K_ρ$ increases from left to right in the (TM)₂X diagram, since both optical and transport data suggest $K_ρ$= 0.23 in the Se compounds whilst it is only of the order of 0.1 in sulfur compounds. Let us add that the uniformly stacked 1-D conductor (DI – DCNQI)₂Ag, an other quarter filled compound reveals localization properties quite similar to those observed in the (TM)₂X series [87]. The normal state of this system is insulating at low pressure probably due to strong electron correlations but at pressures exceeding 15 kbar , the longitudinal resistivity is metallic above 100K with a quasi linear temperature dependence leading to $K_ρ$= 0.25 in the quarter-filled band hypothesis. This result shows once more that the conductor lies at the border with the quarter-filled Mott localized insulator.
<figure><img src="image_16.png" /><figcaption>Fig. 1.16. Structure of the quarter-filled compound (EDT – TTF – CONMe₂)₂AsF₆</figcaption></figure>

1.4.4 The spin-density-wave phase

At the core of the unity shown by the phase diagram of Fig. 1.3 is the shift on the pressure scale when selenium is substituted for the sulfur atom to give the Bechgaard salts series (TMTSF)₂X [73,88]. The selenium series was at the start considered more promising compared to previous organic compounds, mainly because the metal insulator transition at ambient pressure only occurs below 20K after high metallic conductivity have been attained [73].

In the two compounds of the series (TMTSF)₂PF₆ and (TMTSF)₂AsF₆, the transition occurs at $T_c \approx 12$K at 1 bar [73, 88, 89]. According to a variety of experimental probes, the transition shows all the characteristics of SDW long-range ordering [90, 91] ... (sliding, non linearity, fluctuations). Similar SDW is also found in (TMTTF)₂X but at much higher values on the pressure scale. In (TMTTF)₂PF₆ for example, about 40 kbar of pressure is needed to reach a $T_c \sim 10$K(Fig. ??) [34, 92], whereas for (TMTTF)₂Br [68, 93] and (TMTTF)₂BF₄ [94] about 10 kbar and 27 kbar must be applied respectively.

The emergence of a plasma edge in the b direction below 100K [95,96], and the recovery of transverse metallic and longitudinal Fermi liquid ($\sim T^2$) resistivity below $T^*$ for (TMTSF)₂PF₆ indicate that the transverse electron band motion has developed some coherence at the onset of the SDW transition. The mechanism of the instability will then naturally follow from the property of nesting of the Fermi surface based on the property $\epsilon_{-p}(\mathbf{k}) = -\epsilon_p(\mathbf{k}+\mathbf{q}_0)$ of the spectrum for a special wave vector $\mathbf{q}_0$, called the nesting vector. The relevance of the whole Fermi surface for the transition has been confirmed by the determination of the modulation wave vector of SDW by NMR [97], which coincides with the best nesting vector obtained by band calculations [8]. In the model spectrum of Eq. ?? for an orthorhombic lattice, perfect nesting is found at $\mathbf{q}_0 = (2k_F, \pi, \pi)$. Deviations with respect to this ideal situation, however, are likely to be found in practice. This amounts to use the spectra

$$ \epsilon_p(\mathbf{k}) = v(pk - k_F) - 2t_{\perp b}^* \cos k_{\perp b} - 2t_{\perp c}^* \cos k_{\perp c} - 2t'_{\perp b} \cos 2k_{\perp b}, \quad (1.30) $$
that includes small next-to-nearest-neighbor hopping $t'_{\perp b}$ along the b direction ($t'_{\perp b} \ll t_{\perp b}$). This leads to the modified nesting condition

$$ \epsilon_p(\mathbf{k}) = -\epsilon_{-p}(\mathbf{k} + \mathbf{q}_0) + 4t'_{\perp b} \cos 2k_{\perp b}, \quad (1.31) $$

with $k_{\perp b}$ dependent nesting frustration.

The determination of the temperature scale for the SDW instability follows from the Stoner form of the total susceptibility in weak coupling

$$ \chi(\mathbf{q}_0, T) \approx \chi_{1D}(2k_F, T_x) + \frac{\chi_0(\mathbf{q}_0, T)}{1 - \frac{1}{2}g^*\chi_0(\mathbf{q}_0, T)}, \quad (1.32) $$

where

$$ \chi_0(\mathbf{q}_0, T) = \ln \frac{T_x}{T} + \psi\left(\frac{1}{2}\right) - \Re\left\langle \psi\left(\frac{1}{2} + i \frac{t'_{\perp b} \cos 2k_{\perp b}}{\pi T}\right) \right\rangle_{k_{\perp b}} \quad (1.33) $$

is the bare susceptibility in the presence of nesting frustration $t'_{\perp b}$ ($\psi(x)$ is the digamma function and $\langle \dots \rangle_{k_{\perp b}}$ is an average over $k_{\perp b}$) [98]. The above expressions differ slightly from previous approaches [98–101] in that the contribution of intermediate electron-hole excitations to $\chi_0$ has been here restricted to an energy shell $\pm T_x$ (instead of $E_F$) around the Fermi level. More energetic excitations extending up to the Fermi energy are one-dimensional in character and do not contribute directly to the instability but to $\chi_{1D}$, the 1D background of the total staggered susceptibility, and to the renormalization of the interaction $g^*$ for SDW at $T_x$ [28, 31]. The above Stoner form also neglects the finite coupling between electron-hole and electron-electron pairings, an interference that persists even below $T_x$. This approximate weak coupling description of SDW remains qualitatively correct, however, as long as $t'_{\perp b}$ does not reach too large values, that is where the interference between density-wave and superconductivity can change the nature of the ground state (see § ?? ).

The susceptibility (1.32) develops a singularity at the critical temperature $T_c$ that depends on $t'_{\perp b}$, the main parameter that is standardly used to mimic the actual effect of pressure on the SDW state [27,98–100]. A finite $t'_{\perp b}$ will then reduce $T_c$ with respect to the BCS limiting value $T_c^0 = T_x e^{-2/g^*}$ at perfect nesting, a feature of the model that was soon linked with experiments done under pressure [88, 89, 102, 103] (inset of Fig. ?? ). Besides the monotonic increase of $T_x$ (Fig. ?? ) and the decrease of $g^*$ under pressure, the influence of $t'_{\perp b}$ on $T_c$ remains the most dominant effect. The expression (1.32) predicts that at the approach of the critical value $t'_{\perp b} \sim T_c^0$, $T_c$ rapidly goes down to zero – though the possibility of SDW at incommensurate $\mathbf{q}_0$ and very low temperature may introduce a change in the slope of $T_c$ near $P_c$ [104]. To $t'_{\perp b}$ will then correspond a critical pressure $P_c$ for the suppression of SDW; this variation of $T_c$ agrees with the characteristic pressure profile generally observed for (TMTSF)$_2$X at moderate pressure (Figs. ?? ) and for (TMTTF)$_2$X at higher pressure (Fig. ?? ). The fitness of the model to describe the $T_c$ of the SDW
state in (TMTSF)₂X can be further assessed if one considers the influence of
a transverse magnetic field on T_c. A perpendicular magnetic field H || c* tends
to confine the motion of electrons along the chain direction and gradually re-
stores nesting conditions. This in turn increases T_c with H [98], consistently
with early field dependent measurements of T_c in (TMTSF)₂PF₆ [105] and
(TMTSF)₂NO₃.

**Intrusion of a charge-density-wave order**

The reexamination of diffuse scattering X-ray patterns of (TMTSF)₂PF₆ by Pouget and Ravy [65, 106] revealed the emergence besides SDW of a charge-densit y-wave superstructure (CDW) at T_c; both have the same modulation vector q₀. These results were subsequently confirmed by Kagoshima et al., [107] who also found a similar superstucture in (TMTSF)₂AsF₆, but with a weaker amplitude. These results came as a surprise since at variance with ordinary Peierls phenomena, the CDW order is not preceded by any lattice softening in the normal state (the 2k_F diffuse scattering lines do exist at high temperature but their amplitudes become vanishingly small in the vicinity of T_c in the normal state [50, 106]). The CDW superstructure would then be entirely electronic in character with no lattice displacement involved.

In connection with these X-ray results, it is worth mentioning the earlier
optical conductivity measurements of Ng *et al.*, [108] on the isostructural
member of the series (TMTSF)₂SbF₆. The results show the growth in the
infrared of new phonon lines at *T*<sub>c</sub>, precisely those usually expected for the
excitations of a CDW superstructure; their temperature dependent intensity
follows roughly the one of the SDW order parameter below *T*<sub>c</sub>. CDW phonon
lines in the far infrared conductivity have also been found in the metallic
phase of (TMTSF)₂ClO₄ at low temperature [109], indicating that 2*k*<sub>*F*</sub>-CDW
and SDW correlations can coexist [17, 44].

On theoretical grounds the possibility for SDW and CDW to coexist has
been analyzed recently by considering the redistribution of charge and spin in
the unit cell when its internal – two-molecules – structure and the long-range
Coulomb interaction are taken into account. In the framework of extended
Hubbard model, numerical and mean-field approaches show that charge and
spin can be so rearranged in the unit cell that it can allow SDW, BOW and
CDW to coexist [55, 110–112]. Moreover, it was shown recently that when
interchain Coulomb interaction – even for small amplitude – is included, this
can favor BOW and CDW correlations while not affecting SDW [113].

1.4.5 Some features of the superconducting state

The superconducting transition

For all cases of superconductivity in the (TM)₂X series the first evidence has
been provided by a drop of the resistivity below the critical temperature and
the suppression of this drop under magnetic field. We shall focus the presentation on the two members of the (TMTSF)₂X series which have attracted most attention;

- i) (TMTSF)₂PF₆, because this has been the first organic superconductor to be found by transport measurements [88] and subsequently confirmed by magnetic shielding [114, 115] and also because the electronic properties of the 1-D electron gas on the organic stacks are only weakly (if at all) affected by the centrosymmetrical anions PF₆. The finding of a very small and still non-saturating resistivity under ambient pressure reaching the value of 10⁻⁵Ω⁻¹ cm⁻¹ verifier les unites at 12K triggered further pressure studies at a pressure of 9 kbar in a dilution refrigerator which led to the discovery of a zero resistance state below 1K see Fig. 1.17. The non saturation of the resistivity had been taken as a signature of superconducting precursor effects. We shall come again to this important question later.

<figure><img src="image_17.png" /><figcaption>**Fig. 1.17.** (TMTSF)₂PF₆, first observation of organic superconductivity under pressure [88].</figcaption></figure>

- ii) (TMTSF)₂ClO₄, because it is the only member of the (TM)₂X series displaying superconductivity at ambient pressure. However, the study of the superconducting state in (TMTSF)₂ClO₄ is meeting the problem of the ClO₄ anions ordering at 24K doubling the periodicity along the *b*-axis [116]. Consequently, great care must be taken to cool the sample slowly enough in order to reach a well anion- ordered state (R-state) at low temperature otherwise superconductivity is faced to its great sensitivity to disorder, a very important feature for organic superconductors which will be discussed more extensively below.
<figure><img src="image_18.png" /><figcaption>**Fig. 1.18.** (*TMTSF*₂ClO₄, first observation of organic superconductivity at ambient pressure [81]</figcaption></figure>

Since (*TMTSF*₂ClO₄ is an ambient pressure superconductor the thermodynamic evidence of the phase transition has been obtained from specific heat on single crystals [117] The electronic *Il s'agit ici de la chaleur speci-

<figure><img src="image_19.png" /><figcaption>**Fig. 1.19.** Electronic contribution to the specific heat of (*TMTSF*₂ClO₄, plotted as C<sub>e</sub>/T versus T [117]</figcaption></figure>

fique electronique contribution to specific heat of (*TMTSF*₂ClO₄ in a C<sub>e</sub>/T cl
versus *T* plot (Fig. 1.19), displays a very large anomaly around 1.2*K* [117]. Above 1.22*K*, the total specific heat obeys the classical relation in metals $C/T = \gamma + \beta T^2$ where the Sommerfeld constant for electrons $\gamma = 10.5$ mJ mol⁻¹K⁻² corresponding to a density of states at the Fermi level $N(E_F) = 2.1$ states eV⁻¹ mol⁻¹ for the two spin directions [117]. The specific heat jump at the transition amounts then to $\Delta C_e/\gamma T_c = 1.67$, i.e. only slightly larger than the BCS ratio. The behaviour of $C_e(T)$ in the superconducting state leads to the determination of the thermodynamical critical field $H_c = 44 \pm 2$ Oe and the single particle gap $2\Delta = 4$K. $T_c$ is depressed at a rate of 1.1 mK/Oe⁻¹ when a magnetic field is applied along the *c*\* axis [118]. Comparing the value of the density of states derived from the specific heat and the value of the Pauli susceptibility [119] lends support to a weak coupling Fermi liquid picture (at least in the low temperature range) [14].

## Critical fields

The anisotropic character of the electronic structure already known from the anisotropy of the optical data in the normal phase, is reflected in a severe anisotropy of the critical fields $H_{c2}$ measured along the three principal directions in $(TMTSF)_2ClO_4$ [?, 120–122]. Early data in $(TMTSF)_2ClO_4$ [?] are not in contradiction with the picture of singlet pairing but no data were given below 0.5K, the temperature domain where it would be most rewarding to see how $H_{c2}$ compares with the Pauli limit when *H* is perfectly aligned along the *a* and *b'* axes. This study has been revisited quite recently in perfectly aligned magnetic fields down to 0.2K [123].

<figure><img src="image_20.png" /><figcaption>Fig. 1.20. $(TMTSF)_2ClO_4$, critical fields along *a*, *b*′ and *c*\* [123]</figcaption></figure>

The linearity of the critical fields with temperature in the vicinity of the $T_c$ suggests an orbital limitation in the Ginzburg-Landau formalism for
the critical field and rules out a Pauli limitation which would favour a $(1 - T/T_c)^{1/2}$ dependence [121, 124].

Furthermore, in support to an early suggestion [124], the band structure parameters of $(TMTSF)_2ClO_4$ can explain the values and the anisotropies of the critical fields assuming the existence of nodes of the superconducting order parameter [123]. These results imply that critical fields values calculated without any contribution from the spin orbit coupling can overcome the Pauli limit at low temperature by factors of two or more [125, 126]. The situation for $(TMTSF)_2PF_6$ may, however, be quite different as the possibility of a non homogenous superconducting phase in the vicinity of the critical pressure opens an other possibility for an enhancement of the superconducting critical fields, *vide infra.refaire le graphique*

<figure><img src="image_21.png" /><figcaption>**Fig. 1.21.** $(TMTTF)_2PF_6$critical fields along $c^*$ under pressure in the SDW/SC coexistence regime [127]</figcaption></figure>

## Superconductivity and Pressure

The pressure dependence of $T_c$ is admittedly a remarkable feature for the $(TMTSF)_2X$ compounds since it is the pressure parameter which enabled organic superconductivity to be discovered. As far as $(TMTSF)_2PF_6$ is concerned, the strong pressure dependence observed above 10 kbar ($T_c$ is only 0.2K under 24 kbar [128]) leads to a Gruneisen constant $\delta \ln T_c/\delta \ln P = 11$ at 9 kbar [128] using the compressibility data at 16 kbar [38], $\delta \ln V/\delta P = 0.7\%$ kbar$^{-1} P$ *ou* $T$. This value of 11 is indeed larger than what has been derived for tin (7), the elemental superconductor with the strongest sensitivity to pressure. The pressure coefficient of superconductivity in $(TMTSF)_2ClO_4$ is even more dramatic since then $\delta \ln T_c/\delta \ln P = 36$ [129] using the compressibility of 1% kbar$^{-1}$ measured for $(TMTSF)_2PF_6$ at ambient pressure [38] P

cl

cl
ou $T?$. However, this remarkable sensitivity of $T_c$ in $(TMTSF)_2ClO_4$ might actually be related to the very specific problem of anion ordering in this compound as it has been suggested from the recent study on the sensitivity of $T_c$ against the presence of non-magnetic disorder [84]. Anion ordering reveals an uprise of the ordering temperature under pressure [129–131] which can be derived from the pressure dependence of a small kink in the temperature dependence of the resistivity, the signature of the ordering, moving from 24 up to 26.5K under 1.5 kbar and corroborated by studies at even higher pressures [132]. Together with this uprise there exists a slowing down in the dynamics of the anions needed for the ordering. Hence, high pressure studies require a special attention to the cooling rate which must be kept low enough to allow anion ordering at low temperature. This may be the explanation for the discrepancy between high pressure data showing the signature of anion ordering up to 8 kbar [131] and the absence of ordering claimed from the interpretation of magneto-angular oscillations [133].

## SDW-SC coexistence

A situation of non-homogenous superconductivity have been clearly identified near the border between SDW and Superconductivity in the $(TMTSF)_2X$ phase diagram [103]. At high pressure ($P > 9.4$ kbar) the superconducting phase emerges from a metallic state and can be reasonably thought of as homogeneous with a critical current density along the *a*-axis $J_c = 200$ A cm$^{-2}$, while below this critical pressure there exists a pressure domain for $(TMTSF)_2PF_6$ ($\approx 1$ kbar wide) in which a superconducting signature is observed at a nearly pressure independent temperature below the onset of a SDW instability where the critical current density is greatly reduced, $J_c \approx 10$ A cm$^{-2}$. This feature points in favour of a coexistence of SDW and SC macroscopic domains consisting of metallic (SC) tubes parallel to the *a* axis (Fig. 1.22). A related consequence of the existence of macroscopic insulating domains in the superconducting phase allowing a channeling of the lines of force in the material is the large increase of the upper critical field $H_{c2}$ [134] which had already been reported long time ago in $(TMTSF)_2PF_6$ for the first repetition *repetition? confirmation?* of organic superconductivity [135] and also in $(TMTSF)_2AsF_6$ [89].

Recent data obtained with $(TMTTF)_2PF_6$ under very high pressure where the coexistence regime is much broader than for $(TMTSF)_2PF_6$ have clearly shown that the critical field for $H||c^*$ is enhanced by a factor 10 at the border with the SDW phase where it can reach 1T, while it amounts to about 0.1 T at very high pressure when the superconducting state is homogeneous [127], a value similar to the observation in the R-state of $(TMTSF)_2ClO_4$ (Fig. (1.21) *Bonne figure?* It must be kept in mind that the slab formation in the vicinity of the SDW state with the formation of insulating domains is not related to the penetration of the magnetic field in a type II superconductor but the result from a competition between insulating and conducting phases. The
<figure><img src="image_22.png" /><figcaption>**Fig. 1.22.** Coexistence between SDW and Superconductivity in (TMTTF)₂PF₆ in the vicinity of the critical pressure for suppression of the SDW ground state [103].</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Fig. 1.23.** Sketch of the theory showing the coexistence between SDW and Superconductivity in the vicinity of the critical pressure [103]. $t'_b$ parametrizes the deviation from the perfect nesting of the Q-1-D Fermi surface.</figcaption></figure>
claim for the existence of tubular domains ||*a* is based on an analysis of the
resistance in the vicinity of the critical pressure at low temperature above *T*<sub>c</sub>
and of the critical currents in the superconducting state [103]. The theoreti-
cal approach relies on a variational model leading to an inhomogenous phase
with an energy lower than the energy of the homogenous states (Metallic or
Insulating SDW). Since it is the transverse *b* parameter which is expected
to govern the respective stability of the SDW and metallic phases [136] at
a pressure lower than the critical one but close enough to the homogenous
critical line the formation of macroscopic metallic domains with a smaller *b*
is energetically favoured in-between *b* expanded SDW domains. Similarly, at
pressures larger than the critical pressure insulating SDW domains should
be present between metallic regions (see Fig. 1.23). A model has been pro-
posed by Gorkov and Grigoriev [137] *bon journal ?* for the interpretation
of the SDW/SC coexistence regime in (TMTSF)<sub>2</sub>PF<sub>6</sub>. This model predicts
the existence of soliton domain walls in the SDW phase close to the critical
pressure between the uniformly gapped SDW phase and the SC phase. As
these domain walls should be perpendicular to the molecular stacking axis a
strong anomaly of the transport anisotropy could be anticipated related to
coexistence pressure regime. However, no such anomaly has been detected
experimentally [138].

c1

Superconductivity and non magnetic defects

It is the remarkable sensitivity of organic superconductivity to irradiation
[139,140] which led Abrikosov to suggest the possibility of triplet pairing in
these materials [141]. Although irradiation was recognized to be an excellent
method for the introduction of defects in a controlled way [142], defects thus
created can be magnetic [143] and the suppression of superconductivity by
irradiation induced defects as a signature of non-conventional pairing must
be taken with ‘a grain of salt’ since local magnetic moments can also act
as strong pair-breakers on s-wave superconductors. Several routes have been
followed to introduce an intrinsically non-magnetic perturbation modulating
the potential seen by the carriers on the organic stacks. Non magnetic disorder
has been achieved substituting TMTTF for TMTSF on the cationic stacks of
(TM)₂X salts with PF₆ [7] and ClO₄ salts [144]. However, in both situations
cationic alloying induces drastic modifications of the normal state electronic
properties since the SDW transition of (TMTSF)₂PF₆ is quickly broadened
and pushed towards higher temperature upon alloying [145].

Leaving the cation stack uniform, scattering centers can also be created
on the anion stacks with the solid solution (TMTSF)₂ClO₄(1-x)ReO₄xwhere
Tomić *et al.*, first mentioned the suppression of superconductivity upon alloy-
ing with a very small concentration of ReO₄ anions [146]*Bonne reference ?*.
In the case of a solid solution with tetrahedral anions such as ClO₄ or ReO₄
one is confronted to two potential sources of non-magnetic disorder which act
additively on the elastic electronic lifetime according to the Mathiessen's law.

c1
First the modulation due to the different chemical natures of the anions and second a disorder due to a progressive loss of long range ordering at $T_{AO}$ in the (TM)₂X solid solution although X-ray investigations have revealed that long-range order is preserved up to 3% ReO₄ with a correlation length $\xi_a > 200$Å [147]. Studies of superconductivity in (TMTSF)₂ClO₄ performed under extremely slow cooling conditions have shown that $T_c$ is a fast decreasing function of the non-magnetic disorder [84] where the residual resistivity along the c* axis has been used for the measure of the disorder in the alloys with different concentrations, Fig. 1.24. It must be emphasized that the residual resistivity is derived from a fit of the temperature dependence of the normal state resistivity according to a Fermi liquid model below the anion ordering temperature of 24K, namely $\rho(T) = \rho_0^{-} + AT^2$. This treatment of the resistivity below 10K or so allows to remove the influence of superconducting precursor effects above the ordering temperature.

<figure><img src="image_24.png" /><figcaption>**Fig. 1.24.** Phase diagram of (TM)₂X governed by non magnetic disorder. All open circles refer to the very slowly cooled samples in the R-state with different ReO₄⁻ contents. Open squares are data from the same samples corresponding to slightly larger cooling rates although keeping a metallic behaviour above $T_c$. A 10% sample with $\rho_0^{-} \approx 0.32\Omega.cm$ has provided four different $T_c$ depending on the cooling rate. One sample (8%) did not reveal any ordering down to 0.1K. These data show that the residual resistivity is a better characterization for the disorder than the nominal ReO₄⁻ concentration. Full dots (15 and 17%) are relaxed samples exhibiting a SDW ground state. The vertical bar is the error bar for a sample in which a maximum of the logarithmic derivative could not be clearly identified and therefore the actual SDW temperature should lie below 4K, the temperature of minimum resistivity. The full square is the Q-state of a 6% sample. The continuous line at the Metal-SC transition is the best fit of the data with the digamma function model providing $T_c^0 = 1.57$K. The dashed line at the Metal-SDW transition is only a guide for the eye. After Ref. [148]</figcaption></figure>
The suppression of $T_c$ is clearly related to the enhancement of the scattering rate in the solid solution. Since the additional scattering cannot be ascribed to magnetic scattering according to the EPR checks showing no additional traces of localized spins in the solid solution, the data in figure (1.24) cannot be reconciled with the picture of a superconducting gap keeping a constant sign over the whole ($\pm k_F$) Fermi surface. They require a picture of pair breaking in a superconductor with an unconventional gap symmetry. The conventional pair breaking theory for magnetic impurities in usual superconductors has been generalized to the case of non-magnetic impurities in unconventional materials and $T_c$ reads [149, 150],

$$ \ln\left(\frac{T_c^0}{T_c}\right) = \psi\left(\frac{1}{2} + \frac{\alpha T_c^0}{2\pi T_c}\right) - \psi\left(\frac{1}{2}\right) \quad (1.34) $$

with $\psi(x)$ being the digamma function, $\alpha = \hbar/2\tau k_B T_c^0$ the repairing parameter, $\tau$ the elastic scattering time and $T_c^0$ the limit of $T_c$ in the absence of any scattering. From the data in figure (1.24) the best fit leads to $T_c^0 = 1.57$K and a critical scattering for the suppression of superconductivity of $1/\tau_{cr} = 1.85$ cm$^{-1}$. Accordingly, $1/\tau$ amounts to $0.56$ cm$^{-1}$ in the pristine (TMTSF)$_2$ClO$_4$ sample. Such a value for the inverse carrier life time is admittedly significantly smaller than the predicted width at half height namely, $1/\tau \approx 2$ cm$^{-1}$ assuming a classical Drude behaviour involving the temperature dependence of the DC conductivity and the longitudinal plasma frequency [45]. The present derivation of the electron life time compares fairly well with far infrared optical measurements leading to a zero frequency conductivity peak with a width less than $2-4$ cm$^{-1}$ [79, 82]. Our results support the existence of a very narrow zero frequency peak carrying a minor fraction of the total spectral weight, which is probably the signature of a correlated low dimensional Fermionic gas.

The sensitivity of $T_c$ to non-magnetic disorder cannot be reconciled with a model of conventional superconductors. The gap must show regions of positive and negative signs on the Fermi surface which can be averaged out by a finite electron lifetime due to elastic scattering. As these defects are local the scattering momentum of order $2k_F$ can mix $+k_F$ and $-k_F$ states and therefore the sensitivity to non-magnetic scattering is still unable to tell the difference between $p_x$ or $d$ orbital symmetry for the superconducting wave function. A noticeable progress could be achieved paying attention to the spin part of the wave function. In the close vicinity of $T_c$ orbital limitation for the critical field is expected to prevail and therefore the analysis of the critical fields close to $T_c$ [124] does not imply a triplet pairing. When the magnetic field is oriented along the intermediate b-axis violations of the Pauli limitation have been claimed in (TMTSF)$_2$PF$_6$ [125] and recently in (TMTSF)$_2$ClO$_4$ [126] superconductors. However, it must be kept in mind that in all these experiments under transverse magnetic field along the b' axis the electronic structure is profoundly affected by the application of the field
which tends to localize the electrons as shown by the normal state crossing over from a metallic to an insulating state when investigated with a current along $c^*$ [123]. Furthermore it is still unclear whether the superconducting phase remains uniform under very strong transverse field [151, 152].

The nature of the superconducting coupling in (TM)₂X conductors is still intensively studied and debated. The absence of temperature dependence of the ¹³C Knight shift through the critical temperature at a pressure where (TMTSF)₂PF₆ is superconducting implies a triplet pairing [134]. However, the question of sample thermalization during the time of the NMR experiment has been questioned [153] and this result will have to be reconfirmed. The nature of the coupling in (TM)₂X superconductors has not yet reached a consensus. This is due in part to the lack of unambiguous experimental data for samples exhibiting superconductivity in the very low temperature region. This is *at variance* with the singlet coupling found in 2D organic superconductors with a $T_c$ in the 10K range as clearly indicated by the Knight shifts measurements in the superconducting state [154, 155]. It can be noticed that in spite of the established singlet coupling, the critical fields $H_{c2}$ of 2D superconductors can also greatly exceed the paramagnetic limit in the parallel geometry [156–158].

- µ-sr

- thermal conductivity

### An approach to the mechanism of superconductivity

Given the experimental uncertainty about the nature of superconductivity in the Bechgaard and Fabre salts, it is therefore still premature to favor the triplet scenario for pairing over the singlet one. In any case, however, superconductivity certainly differs from what is commonly seen in ordinary metals and this raises the question of the possible causes of pairing in these materials. This problem perplexed from the start every physicist and chemist of

<table><caption>**Table 1.2.** Please write your table caption here. ¹(TMTSF)₂PF₆: [134]ᵃ, [140]ᵇ, [159]ᶜ; ²(TMTSF)₂ClO₄: [160]ᵃ, [161]ᵇ, [126]ᶜ; ³(TMTSF)₂(ClO₄)₁⁻ₓ(ReO₄)ₓ: [84, 148]ᵃ.</caption><thead><tr><th>Exp./Δ(k<sub>⊥</sub>)</th><th>s</th><th>p</th><th>d</th><th>f</th></tr></thead><tbody><tr><td>ΔK</td><td>—</td><td>Const.<sup>(1a)</sup></td><td>—</td><td>Const.<sup>(1a)</sup></td></tr><tr><td>T<sub>1</sub><sup>-1</sup></td><td>—</td><td>—</td><td>T<sup>3(2a)</sup></td><td>T<sup>3(2a)</sup></td></tr><tr><td>κ(T)</td><td>T<sup>-2</sup>e<sup>-Δ/T(2b)</sup></td><td>T<sup>-2</sup>e<sup>-Δ/T(2b)</sup></td><td>—</td><td>—</td></tr><tr><td>H<sub>cr</sub></td><td>—</td><td>H<sub>c2</sub>(||<b>a</b>, <b>b</b>) > H<sub>P</sub><sup>(1c,2c)</sup></td><td>—</td><td>H<sub>c2</sub>(||<b>a</b>, <b>b</b>) > H<sub>P</sub><sup>(1b,2b)</sup></td></tr><tr><td>T<sub>c</sub>(τ<sub>0</sub>)</td><td>—</td><td>τ<sub>0</sub><sup>cr</sup>(1b,3a)</td><td>τ<sub>0</sub><sup>cr</sup>(1b,3a)</td><td>τ<sub>0</sub><sup>cr</sup>(1b,3a)</td></tr></tbody></table>
<figure><img src="image_25.png" /><figcaption>**Fig. 1.25.** Temperature dependence of the nuclear spin relaxation rate of (TMTSF)₂PF₆ at P=1bar (triangles), 5.5kbar (squares), 8kbar (crosses) and 10 kbar (full circles). After Ref. [47].</figcaption></figure>

the field, since the requirements for a traditional phonon-mediated mechanism for pairing are apparently not met [162]. Antiferromagnetism that completely surrounds superconductivity in the phase diagram represents the main obstacle for an effective attraction mediated by phonons takes place. Superconductivity is indeed invariably replaced by a SDW instability whether one moves backward on the pressure scale or whether at fixed pressure P > P_c, one moves along the magnetic field **H**(||*c**) axis where a cascade of field-induced SDW states is found. (TMTSF)₂ClO₄ is another example that illustrates how (TM)₂X are close to the threshold of a SDW instability. This compound, which presents an anion ordering on slow cooling, is already a superconductor at ambient pressure (*P*_c < 1 bar); it develops a SDW instability either beyond a critical alloying [146, 148], or by just cranking up the cooling rate [146, 163, 164]. Additional experimental weight in support of this view comes from the properties of the normal state at P > P_c. NMR experiments show indeed a strong enhancement of the spin-lattice relaxation rate T₁⁻¹ as a function of temperature, which reveals the existence of strong antiferromagnetic spin fluctuations in a very broad temperature domain of the metallic state above the superconducting instability (Fig. 1.25) [17, 44, 47, 165]. All this goes to show that even in the presence of superconductivity, the interactions
<figure><img src="image_26.png" /><figcaption>**Fig. 1.26.** Calculated phase transition temperature of the quasi-one-dimensional electron gas model as a function of the nesting deviation parameter $t'_{\perp b}$. Continuous line ($g_1^{\perp} = 0$), dotted line ($g_1^{\perp} = 0.11$,) and dashed line ($g_1^{\perp} = 0.14$.) After Nickel *et al.*, [113].</figcaption></figure>

in (TM)₂X at $P > P_c$ remain repulsive and favorable to a SDW state, and that only nesting conditions are apparently changing in each case.

According to the model described in §1.4.4, these fluctuations are made at the microscopic level of electron-hole pairs at $(\mathbf{k}, \mathbf{k} \pm \mathbf{q}_0)$, which then coexist with electron-electron (and hole-hole) pairing at $(\mathbf{k}, -\mathbf{k})$ responsible for the superconducting instability of the normal state at $P > P_c$. Since these two different pairings refer to the same electronic excitations around the Fermi surface, there will be some intrinsic dynamics or interference between them. We already came up against the problem of interfering pairing instabilities in the one-dimensional case (§ 1.2). For repulsive interactions and perfect nesting at $2k_F$, we have seen that interference between electron-hole and electron-electron pairing is maximum for a 1D – two points – Fermi surface and enters as a key ingredient in the formation of either a Luttinger liquid or a Mott insulating phase at commensurate filling [15, 166, 167]. At finite $t_{\perp b}$ and for temperature well below $T_x$, however, we will see that the outcome differs and may provide a logical link between SDW and superconductivity.

The connection between superconductivity and density-wave correlations goes back to the work of Kohn and Luttinger in the mid sixties [23]. They showed that the coupling between electron-hole (density-wave) and electron-electron correlations, albeit very small, is still present for a spherical Fermi surface. In this isotropic limit, $2k_F$ Friedel (charge) fluctuations act as an oscillating pairing potential for electrons giving rise to a purely electronic mechanism for superconductivity at large angular momentum. Emery suggested that this non-phonon mechanism should be working in the spin sector as well and that for the quasi-1D geometry, it can be boosted by the prox-
imity of a SDW state, which could yield experimentally reachable $T_c$ [24] – an effect that was early on confirmed in the framework of renormalized mean-field theory [25, 26, 67] and various RPA approaches [168–170]. However, these approaches amount to extract an effective superconducting coupling from short-range density-wave correlations, which in turn serves as the input of a ladder diagrammatic summation. The ladder theory, as a single channel approximation, neglects the quantum interference between the different kinds of pairings, and as such it cannot capture the dynamical emergence of superconductivity.

Because of the finite value of $t_⊥$, interference becomes non uniform along the Fermi surface which introduces a momentum dependence in the scattering amplitudes. This dependence can be parametrized by the set of transverse wavevectors for in going ($k_⊥1k_⊥2$) and outgoing ($k'_⊥1k'_⊥2$) electrons (here $k_⊥ \equiv k_⊥b$). The generalization of the 1D scaling equations Eqs ?? to now $k_⊥$-dependent interactions $g_{i=1,2,3}(k'_⊥1k'_⊥2k_⊥2k_⊥1)$ in the quasi-1D case, where both $t_⊥b$ and $t'_⊥b$ are present, has been worked out recently [27, 113, 171, 172]. The results can be put in the following schematic form:

$$ \partial_t g_i(k'_⊥1k'_⊥2k_⊥2k_⊥1) = \sum_{k_⊥} \sum_{n,n'=1}^{3} \left\{ \epsilon_{C,i}^{nn'} g_n(\{k_⊥\}) g_{n'}(\{k_⊥\}) \mathcal{L}_C(k_⊥, q_{C⊥}) + \epsilon_{P,i}^{nn'} g_n(\{k_⊥\}) g_{n'}(\{k_⊥\}) \mathcal{L}_P(k_⊥, q_{P⊥}, t'_{⊥b}) \right\}, \quad (1.35) $$

where $\mathcal{L}_{C,P}$ are the Cooper (electron-electron) and Peierls (electron-hole) loops, with $q_{C,P⊥}$ as their respective transverse momentum variables, and $\epsilon_{C,P,i}^{nn'} = \pm 1$, or 0. By integrating these flow equations, the interactions can develop a singularity that signals an instability of the normal state at a critical temperature $T_c$. The nature of ordering is determined by the profile of interactions in $\{k_⊥\}$ space, which corresponds to a divergence of a given order parameter susceptibility $\chi_μ$. Feeding these equations with a realistic set of bare parameters for the repulsive intrachain interactions $g_i$ and the band parameters $t_{⊥b}$ and $E_F$ in (TM)$_2$X, it is possible to follow the instabilities of the normal state as a function of $t'_{⊥b}$, which simulates the main influence of pressure in the model [27, 113, 171–173].

Thus at perfect nesting, when $t'_{⊥b} = 0$, the normal state develops a SDW instability at $T_c^0 \sim 20$K, which for small Umklapp scattering corresponds to the range of $T_c$ expected in most of (TMTSF)$_2$X at ambient pressure and (TMTTF)$_2$X at relatively high pressure. The range of $T_c$ roughly squares with the one obtained in the single channel approximation with no interference below $T_x$ (§1.4.4). As $t'_{⊥b}$ increases, $T_c$ is gradually decreasing until the critical range $t'_{⊥b} \simeq 0.6T_c^0$ is reached where the SDW is suppressed (full circles, Fig. 1.26). The metallic phase remains unstable at finite temperature, however, but the instability now takes place in the superconducting channel (open triangles, Fig. 1.26). The order parameter is of the form $\Delta(k_⊥) = \Delta \cos k_⊥$ and has nodes at $k_⊥ \pm \pi/2$; it corresponds to a interstack singlet or $d_{x^2-y^2}$-wave
pairing. It follows that for repulsive intrachain interactions, an attraction be-
tween electrons can be dynamically generated from the interference between
Cooper and Peierls scattering channels. The attraction between carriers on
neighboring chains can be seen as being mediated by the exchange of spin
fluctuations. Among the features that come out of the calculations, the fact
that SCd and SDW instability lines meet at the maximum of $T_c \sim$ 1K for
superconductivity, together with their respective $t'_{\perp b}$ dependence are worth
noticing in regard to the experimental phase diagram (Fig. 1.3).

Regarding the possible symmetries of the superconducting order param-
eter, an analysis of the momentum dependence of the scattering amplitude
gᵢ({k⊥}) reveals that for the electron gas model defined with only intrachain
repulsive interactions, the strongest superconducting instability is invariably
found in the singlet SCd-wave channel [27,173]. Triplet superconductivity in
the pₓ channel, which on phenomenological grounds has been proposed as a
possible candidate to describe superconductivity in the Bechgaard salts [174],
is strongly suppressed. In effect the triplet SCpₓ superconductivity, which has
a gap order parameter Δᵣ = rΔ with r = sign kₓ, is an intrachain pairing that
is subjected to the strongest repulsive part of the oscillating potential pro-
duced by SDW correlations [25]. More favorable conditions for triplet pairing
do exist but they take place at higher angular momentum, in the interchain
f-wave channel with an order parameter Δᵣ(k⊥) = rΔ cos k⊥, a possibility that
was shown to come out from the mean-field analysis [175]. For intrachain re-
pulsive couplings alone, however, the renormalization group results show that
the amplitude of triplet correlations are subordinate to those of the SCd-wave
channel in which the highest superconducting T_c is found [113,173].

Nevertheless following the Kohn-Luttinger picture, triplet pairing at high
angular momentum is connected to charge-density-wave fluctuations [23].
The presence of a CDW superstructure that coexists with a SDW state
in the Bechgaard salts (see § 1.4.4) has in this respect stressed their im-
portance in these salts close to P_c. Following the example of most quasi-
one-dimensional systems in which a CDW superstructure is found [176,177],
interchain Coulomb interaction is a physically relevant coupling that must
be taken into account in the presence of charge correlations. By including,
besides the g_i, interchain backward (g_1^⊥), forward (g_2^⊥) and Umklapp (g_3^⊥)
scattering amplitudes, one deals with the so-called the extended electron gas
model [178,179] for which realistic repulsive couplings allow us to expand the
range of possibilities of both superconducting and density-wave long-range
orders. The RG solution of Eqs. (1.35) shows that g_1^⊥ plays a key role in
the stability of SDW and SC_d orders and the emergence of triplet super-
conductivity and CDW order in the T - t'_{\perp b} phase diagram [113,171]. The
RG results depicted in Fig. 1.27 indeed show that beyond a relatively small
threshold in g_1^⊥, SC_d long-range order becomes unstable and an instability is
found in the interchain triplet f-wave channel above a critical t'_{\perp b}. The SC_f
T_c are comparable to SC_d but show stronger 'pressure' coefficient along the
<figure><img src="image_27.png" /><figcaption>**Fig. 1.27.** Calculated phase diagram of the extended quasi-one-dimensional electron gas model. After Nickel *et al.*, [113].</figcaption></figure>

$t'_{⊥b}$ (Fig. 1.26). The results also show that the strength of SDW correlations remain essentially unaffected and can still dominate the normal phase besides CDW correlations, whose amplitude grows with the strength interchain coupling. Although SDW order can precede the triplet $SC_f$ instability along the $t'_{⊥b}$ 'pressure' scale near the interchain coupling threshold, it is very close in stability with a CDW superstructure (Figs. 1.26 and 1.27).

Given the observation of a close proximity between SDW and CDW in the Bechgaard salts, it follows that not only singlet $SC_d$ but also $SC_f$ are serious candidates for the description of superconductivity in the Bechgaard salts. Since many parameters of the model are likely to change under pressure besides $t'_{⊥b}$, this puts some haziness about how one actually moves in the phase diagram of Fig. 1.27 and in turn on the most stable kind of superconductivity in these materials.

References

1. K. Bechgaard, D. O. Cowan, and A. N. Bloch, Journal of the Chemical Society Chemical Communications **22**, 937 (1974).

2. A. Garito and A. Heeger, Accounts of Chemical Research **7**, 232 (1974).

3. E. Engler *et al.*, Jour. Am. Chem. Soc. **99**, 5909 (1977).

4. J. Andersen *et al.*, Acta Cryst. B **34**, 1901 (1978).

5. A. Andrieux *et al.*, J. Physique. Paris **40**, 1199 (1979).

6. G. Brun *et al.*, C.R. Acad. Sc. (Paris) **284 C**, 211 (1977).

7. C. Coulon *et al.*, J. Phys. (Paris) **43**, 1059 (1982).

8. L. Ducasse *et al.*, J. Phys. C **39**, 3805 (1986).

9. K. Behnia *et al.*, Phys. Rev. Lett. **74**, 5272 (1995).

10. D. Jérome, Science **252**, 1509 (1991).
11. P. Auban, D. Jérome, C. Carcel, and J. Fabre, J. Phys. IV (France) **114**, 41 (2004), and to be published.

12. H. J. Schulz, Phys. Rev. Lett. **64**, 2831 (1990).

13. J. Voit, Rep. Prog. Phys. **58**, 977 (1995).

14. C. Bourbonnais and D. Jérome, in *Advances in Synthetic Metals, Twenty Years of Progress in Science and Technology*, edited by P. Bernier, S. Lefrant, and G. Bidan (Elsevier, New York, 1999), pp. 206–261, arXiv:cond-mat/9903101.

15. J. Solyom, Adv. Phys. **28**, 201 (1979).

16. T. Giamarchi, Physica **B230-232**, 975 (1997).

17. C. Bourbonnais et al., J. Phys. (Paris) Lett. **45**, L755 (1984).

18. V. J. Emery, in *Highly Conducting One-Dimensional Solids*, edited by J. T. Devreese, R. E. Evrard, and V. E. van Doren (Plenum Press, New York, 1979), p. 247.

19. J. Voit, Eur. Phys. J. B **5**, 505 (1998).

20. F. Mila and X. Zotos, Europhys. Lett. **24**, 133 (1993).

21. H. J. Schulz, J. Phys. C: Solid State Phys. **16**, 6769 (1983).

22. V. N. Prigodin and Y. A. Firsov, Sov. Phys. JETP **49**, 369 (1979).

23. W. Kohn and J. M. Luttinger, Phys. Rev. Lett. **15**, 524 (1965).

24. V. J. Emery, Synthetic Metals **13**, 21 (1986).

25. M. T. Béal-Monod, C. Bourbonnais, and V. J. Emery, Phys. Rev. B **34**, 7716 (1986).

26. C. Bourbonnais and L. G. Caron, Europhys. Lett. **5**, 209 (1988).

27. R. Duprat and C. Bourbonnais, Eur. Phys. J. B **21**, 219 (2001).

28. C. Bourbonnais and L. G. Caron, Physica **143B**, 450 (1986).

29. S. Brazovskii and Y. Yakovenko, J. Phys. (Paris) Lett. **46**, L (1985).

30. Y. A. Firsov, Y. N. Prigodin, and C. Seidel, Phys. Rep. **126**, 245 (1985).

31. C. Bourbonnais and L. G. Caron, Int. J. Mod. Phys. B **5**, 1033 (1991).

32. C. Bourbonnais, in *Les Houches, Session LVI (1991), Strongly interacting fermions and high-$T_c$ superconductivity*, edited by B. Doucot and J. Zinn-Justin (Elsevier Science, Amsterdam, 1995), p. 307.

33. C. Bourbonnais, in *Trends in High Magnetic Fields Science*, edited by L. P. L. C. Berthier, P. Boebinger and G. Martinez (Springer, Heidelberg, 2002), arXiv:cond-mat/0204345.

34. D. Jaccard et al., J. Phys.: Condens. Matter **13**, 89 (2001).

35. J. R. Cooper, Phys. Rev. B **19**, 2404 (1979).

36. D. Jérome, in *Organic Conductors: fundamentals and applications*, edited by J.-P. Farges (Dekker, New York, 1994), pp. 405–494.

37. P. Auban-Senzier, D. Jérome, and J. Moser, in *Physical Phenomena at High Magnetic Fields*, edited by R. Z. Fisk, L. Gorkov (World Scientific, Singapore, 1999), p. 211.

38. B. Gallois, Ph.D. thesis, Univ. Bordeaux I, 1987.

39. S. Brazovskii and S. Gordyunin, JETP Letters **31**, 371 (1980).

40. N. Mott, *Metal-Insulator Transitions* (Taylor and Francis, London, 1974).

41. P. Auban-Senzier (unpublished).

42. J. Moser et al., Eur. Phys. J. B **1**, 39 (1998).

43. A. Georges, T. Giamarchi, and N. Sandler, Phys. Rev. B **61**, 16393 (2000).

44. P. Wzietek et al., J. Phys. I (France) **3**, 171 (1993).

45. C. Jacobsen, D. Tanner, and K. Bechgaard, Phys. Rev. B **28**, 7019 (1983).
46. C. Bourbonnais et al., Phys. Rev. Lett. **62**, 1532 (1989).

47. F. Creuzet et al., Synthetic Metals **19**, 289 (1987).

48. B. Gotschy et al., J. Phys. I (France) **2**, 677 (1992).

49. This volume and to be published.

50. J. Pouget et al., Mol. Cryst. Liq. Cryst. **79**, 129 (1982).

51. P. Foury-Leylekian et al., Phys. Rev. B **70**, 180405(R) (2004).

52. M. Dumm et al., Phys. Rev. B **61**, 511 (2000).

53. C. Bourbonnais and B. Dumoulin, J. Phys. I (France) **6**, 1727 (1996).

54. L. G. Caron, C. Bourbonnais, F. Creuzet, and D. Jerome, Synthetic Metals **19**, 69 (1987).

55. J. Riera and D. Poilblanc, Phys. Rev. B **62**, R16243 (2000).

56. D. Chow et al., Phys. Rev. Lett. **81**, 3984 (1998).

57. F. Zamborszky et al., Phys. Rev. B **66**, 081103(R) (2002).

58. V. Ilakovac et al., Phys. Rev. B **50**, 7136 (1994).

59. D. S. Chow et al., Phys. Rev. Lett. **85**, 1698 (2000).

60. L. G. Caron et al., Synthetic Metals **27B**, 123 (1988).

61. J. P. Pouget et al., Phys. Rev. Lett. **37**, 437 (1976).

62. L. G. Caron and C. Bourbonnais, Phys. Rev. B **29**, 4230 (1984).

63. L. G. Caron and S. Moukouri, Phys. Rev. Lett **76**, 4050 (1996).

64. H. Bakrim and C. Bourbonnais (unpublished).

65. J. P. Pouget and S. Ravy, Synthetic Metals **85**, 1523 (1997).

66. S. Brazovskii and Y. Yakovenko, Sov. Phys. JETP **62**, 1340 (1985).

67. L. G. Caron and C. Bourbonnais, Physica **143B**, 453 (1986).

68. B. J. Klemme et al., Phys. Rev. Lett. **75**, 2408 (1995).

69. S. E. Brown et al., Synthetic Metals **86**, 1937 (1997).

70. P. Auban et al., J. Phys. (France) **50**, 2727 (1989).

71. P. Auban, D. Jérome, I. Johannsen, and K. Bechgaard, Synthetic Metals **41**, 2285 (1991).

72. C. Bourbonnais and D. Jérome, Science **281**, 1156 (1998).

73. K. Bechgaard et al., Solid State Comm. **33**, 1119 (1980).

74. S. Tomic et al., J. Phys. I (France) **1**, 1603 (1991).

75. S. Barisic and S. Brazovskii, in *Recent Developments in Condensed Matter Physics*, edited by J. T. Devreese (Plenum, New York, 1981), Vol. 1, p. 327.

76. V. J. Emery, R. Bruisma, and S. Barisic, Phys. Rev. Lett. **48**, 1039 (1982).

77. A. Schwartz et al., Phys. Rev. B **58**, 1261 (1998).

78. C. Jacobsen, in *Low Dimensional Conductors and Superconductors*, edited by D. Jérome and L. Caron (Plenum Press, New York, 1987), p. 253.

79. T. Timusk, in *Low Dimensional Conductors and Superconductors*, edited by D. Jérome and L. Caron (Plenum Press, New York, 1987), p. 275.

80. K. Kikuchi et al., Solid State Comm. **42**, 433 (1982).

81. K. Bechgaard, M. Carneiro, M. Olsen, and F. Rasmussen, Phys. Rev. Lett. **46**, 852 (1981).

82. H. Ng, T. Timusk, D. Jérome, and K. Bechgaard, Phys. Rev. B. **32**, 8041 (1985).

83. H. Ng, T. Timusk, and K. Bechgaard, J. Physique **44**, C3 (1983).

84. N. Joo et al., Eur. Phys. J. B **40**, 43 (2004).

85. For more references on the ground states of the insulating compounds see articles in [180].

86. K. Heuzé et al., Adv. Materials **15**, 1251 (2003).
87. T. Itou et al., Phys. Rev. Lett. 93, 216408 (2004).

88. D. Jérome, A. Mazaud, M. Ribault, and K. Bechgaard, J. Phys. (Paris) Lett. 41, L95 (1980).

89. R. Brusetti, M. Ribault, D. Jerome, and K. Bechgaard, J. Phys. (Paris) 43, 801 (1982).

90. K. Mortensen, Y. Tomkiewicz, T. D. Schultz, and E. M. Engler, Phys. Rev. Lett. 46, 1234 (1982).

91. A. Andrieux, D. Jérome, and K. Bechgaard, J. Phys. (Paris) Lett. 42, L87 (1981).

92. T. Adachi et al., J. Am. Chem. Soc. 122, 3238 (2000).

93. L. Balicas et al., J. Phys. I (France) 4, 1539 (1994).

94. P. Auban-Senzier et al., Synthetic Metals 133-134, 11 (2003).

95. C. S. Jacobsen, D. Tanner, and K. Bechgaard, Phys. Rev. Lett. 46, 1142 (1981).

96. V. Vescoli et al., Science 281, 1181 (1998).

97. T. Takahashi, Y. Maniwa, H. Kawamura, and G. Saito, Physica 143B, 417 (1986).

98. G. Montambaux, Phys. Rev. B 38, 4788 (1988).

99. B. Horowitz, H. Gutfreund, and M. Weger, Mol. Cryst. Liq. Cryst. 79, 591 (1982).

100. K. Yamaji, J. Phys. Soc. of Japan 51, 2787 (1982).

101. L. Gor'kov, J. Phys. I (France) 6, 1697 (1996).

102. D. Jérome and H. Schulz, Adv. in Physics 31, 299 (1982).

103. T. Vuletic et al., Eur. Phys. J. B 25, 319 (2002).

104. Y. Hasegawa and H. Fukuyama, J. Phys. Soc. Jpn. 55, 3978 (1986).

105. J. F. Kwak et al., Phys. Rev. Lett. 56, 972 (1986).

106. J. P. Pouget and S. Ravy, J. Phys. I (France) 6, 1501 (1996).

107. S. Kagoshima et al., Solid State Comm. 110, 479 (1999).

108. H. K. Ng, T. Timusk, and K. Bechgaard, Phys. Rev. B 30, 5842 (1984).

109. N. Cao, T. Timusk, and K. Bechgaard, J. Phys. I (France) 6, 1719 (1996).

110. S. Mazumdar, R. Ramasesha, R. T. Clay, and D. K. Campbell, Phys. Rev. Lett. 82, 1522 (1999).

111. M. Kobayashi, M. Ogata, and K. Yonemitsu, J. Phys. Soc. Jpn. 67, 1098 (1998).

112. H. Seo and H. Fukuyama, J. Phys. Soc. Jpn. 66, 1249 (1997).

113. J. C. Nickel, R. Duprat, C. Bourbonnais, and N. Dupuis, Phys. Rev. Lett. 95, 247001 (2005).

114. M. Ribault, G. Benedek, D. Jérome, and K. Bechgaard, J. Phys. Lett. Paris 41, L (1980).

115. K. Andres et al., Phys. Rev. Lett. 45, 1449 (1980).

116. J. Pouget, in Low Dimensional Conductors and Superconductors, edited by D. Jérome and L. Caron (Plenum Press, New York, 1987), p. 17.

117. P. Garoche, R. Brusetti, and K. Bechgaard, Phys. Rev. Lett. 49, 1346 (1982).

118. R. Brusetti, P. Garoche, and K. Bechgaard, J. Phys. C 16, 2495 (1983).

119. N. Miljak, J. R. Cooper, and K. Bechgaard, J. Phys. Coll. 44, 893 (1983).

120. D. U. Gubser et al., Mol. Cryst. Liq. Cryst. 79, 225 (1982).

121. R. L. Greene et al., Mol. Cryst. Liq. Cryst. 79, 183 (1982).

122. K. Murata et al., Mol. Cryst. Liq. Cryst. 79, 283 (1982).

123. N. Joo, to be published (2006).
124. L. P. Gorkov and D. Jérome, J. Phys. Lett. **46**, L643 (1985).

125. I. J. Lee, P. M. Chaikin, and M. J. Naughton, Phys. Rev. B **62**, R14669 (2000).

126. J. I. Oh and M. J. Naughton, Phys. Rev. Lett. **92**, 067001 (2004).

127. H. Wilhelm (2004), unpublished data and private communication.

128. H. J. Schulz et al., J. Physique. Lett **42**, L (1981).

129. D. Mailly, Ph.D. thesis, Thesis Université d'Orsay, 1983.

130. F. Creuzet, D. Jérome, and A. Moradpour, Mol. Cryst. Liq. Cryst. **119**, 297 (1985).

131. K. Murata et al., Mol. Cryst. Liq. Cryst. **119**, 245 (1985).

132. F. Guo et al., J. Phys. Soc. Jpn. **67**, 3000 (1998).

133. W. Kang, S. Hannahs, and P. Chaikin, Phys. Rev. Lett. **70**, 3091 (1993).

134. I. J. Lee et al., Phys. Rev. Lett. **88**, 17004 (2002).

135. R. L. Greene and E. M. Engler, Phys. Rev. Lett. **45**, 1587 (1980).

136. M. Héritier, G. Montambaux, and P. Lederer, J. Phys. (Paris) Lett. **45**, L943 (1984).

137. L. P. Gorkov and P. D. Grigoriev, Eur. Phys. J. B **71**, 425 (2005).

138. P. Auban-Senzier, (2006), to be published.

139. S. Bouffard et al., J. Phys. C **15**, 295 (1982).

140. M.-Y. Choi et al., Phys. Rev. B **25**, 6208 (1982).

141. A. A. Abrikosov, J. Low Temp. Phys **53**, 359 (1983).

142. L. Zuppirolli, in *Low Dimensional Conductors and Superconductors*, edited by D. Jérome and L. G. Caron (Plenum Press, New York, 1987), p. 307.

143. M. Sanquer and S. Bouffard, Mol. Cryst. Liq. Cryst. **119**, 147 (1985).

144. I. Johannsen et al., Mol. Cryst. Liq. Cryst. **119**, 277 (1985).

145. K. Mortensen and E. M. Engler, Phys. Rev. B **29**, 842 (1984).

146. S. Tomic, Jérome, P. Monod, and K. Bechgaard, J. de Phys. (Paris) Lett. **43**, L839 (1982).

147. S. Ravy, R. Moret, J. P. Pouget, and R. Comès, Physica **143 B**, 542 (1986).

148. N. Joo et al., Europhys. Lett. **72**, 645 (2005).

149. K. Maki, H. Won, and S. Haas, Phys. Rev. B **69**, 012502 (2004).

150. A. I. Larkin and Y. N. Ovchinikov, Sov. Phys. JETP **20**, 762 (1965).

151. A. G. Lebed, JETP Lett. **44**, 114 (1986).

152. N. Dupuis, G. Montambaux, and C. A. R. S. de Melo, Phys. Rev. Lett. **70**, 2613 (1993).

153. The claim for triplet superconducting pairing in (TM)₂X is based essentially on NMR investigations ($^{77}$Se Knight shift and spin-lattice relaxation time T₁) performed on (TMTSF)₂PF₆ under pressure [134]. The major problem regarding this experiment is the actual determination of the electron spins during the time of the NMR acquisition. The spin-echo relaxation time T₂ is of the order of 300 µs. After the sample is heated by the RF pulse the recovery of the base temperature is achieved with a thermal time constant of the order of 1 – 3 ms. Just after the RF pulse the temperature of the sample is not known because of the slow time response of the lock-in equipment measuring the sample resistance after the pulse. The T₁ measurement does not suffer from the same problem because it involves a time ≈ 200 – 500 ms which is large compared to the thermal recovery time. Hence, data published in reference [134] could support the existence of low energy quasi particles excitations (nodes in the supeconducting gap) but the claim for triplet paring which relies on the recording of the NMR signal immediately after the RF pulse should wait for a better control of the sample temperature at short times.
154. H. Mayaffre et al., Phys. Rev. Lett **75**, 4951 (1995).

155. K. Kanoda, K. Miyagawa, A. Kawamoto, and Y. Nakazawa, Phys. Rev. B **54**, 76 (1996).

156. S. Kamiya et al., Phys. Rev. B **65**, 134510 (2002).

157. F. Zuo et al., Phys. Rev. B **61**, 750 (2000).

158. D. Jérome and C. R. Pasquier, in *One dimensional organic Superconductors*, edited by A. V. Narlikar (Springer Verlag, Berlin, 2005).

159. I. J. Lee, M. J. Naughton, G. M. Danner, and P. M. Chaikin, Phys. Rev. Lett. **78**, 3555 (1997).

160. M. Takigawa, H. Yasuoka, and G. Saito, J. Phys. Soc. Jpn **56**, 873 (1987).

161. S. Belin and K. Behnia, Phys. Rev. Lett. **79**, 2125 (1997).

162. V. J. Emery, J. Phys. (Paris) Coll. **44**, 977 (1983).

163. T. Takahashi, Jérome, and K. Bechgaard, J. de Phys. (Paris) Lett. **43**, L565 (1982).

164. H. Schwenk, K. Andres, and F. Wudl, Phys. Rev. B **29**, 500 (1984).

165. W. Wu et al., Phys. Rev. Lett. **94**, 097004 (2005).

166. Y. A. Bychkov, L. P. Gorkov, and I. Dzyaloshinskii, Sov. Phys. JETP **23**, 489 (1966).

167. I. E. Dzyaloshinskii and A. I. Larkin, Sov. Phys. JETP **34**, 422 (1972).

168. D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B **34**, R8190 (1986).

169. H. Shimahara, J. Phys. Soc. Jpn. **58**, 1735 (1989).

170. H. Kino and H. Kontani, J. Phys. Soc. Jpn. **68**, 1481 (1999).

171. J. C. Nickel, R. Duprat, C. Bourbonnais, and N. Dupuis, Phys. Rev. B (2006), in press, arXiv:cond-mat/0510744.

172. C. Bourbonnais and R. Duprat, J. de Phys. IV **114**, 3 (2004).

173. Y. Fuseya and Y. Suzumura, J. Phys. Soc. Jpn. **74**, 1264 (2005).

174. A. G. Lebed, K. Machida, and M. Ozaki, Phys. Rev. B **62**, R795 (2000).

175. K. Kuroki, R. Arita, and H. Aoki, Phys. Rev. B **63**, 094509 (2001).

176. J. P. Pouget and R. Comes, in *Charge Density Waves in Solids*, edited by L. P. Gor'kov and G. Gruner (Elsevier Science, Amsterdam, 1989), p. 85.

177. S. Barisic and A. Bjelis, in *Theoretical Aspects of Band Structures and Electronic Properties of Pseudo-One-Dimensional Solids*, edited by H. Kaminura (D. Reidel, Dordrecht, 1985), p. 49.

178. L. P. Gor'kov and I. E. Dzyaloshinskii, Sov. Phys. JETP **40**, 198 (1974).

179. P. A. Lee, T. M. Rice, and R. A. Klemm, Phys. Rev. B **15**, 2984 (1977).

180. *Organic Conductors*, edited by J. Farges (Marcel Dekker, New York, 1994).