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# Organic superconductors: the TMTSF2X family

Jacques Friedel, Denis Jerome

► To cite this version:

Jacques Friedel, Denis Jerome. Organic superconductors: the TMTSF2X family. Contemporary Physics, 1982. [arXiv:0409102v1]

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# Organic superconductors: the (TMTSF)₂X family

J. FRIEDEL and D. JÉROME

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

**ABSTRACT.** This review article is intended to give an elementary introduction to the basic theoretical understanding of the Quasi-One-Dimensional electron gas in organic conductors. It provides a simple theoretical picture of the electron gas instabilities arising at low temperature. Furthermore it gives a survey of the salient experimental results related on the one hand to the Peierls (insulating) instability of TTF-TCNQ and on the other hand to the superconducting instability recently discovered in the family of (TMTSF)₂X conducting salts.

Special emphasis is put on the consequence of One-Dimensionality for the existence of precursor regimes of the low temperature phase transitions up to temperatures much higher than the phase transition temperatures.

## 1. Introduction

This article is concerned first with the influence of one-dimensionality on electrical conductivity. It is also concerned with the manifestation of one-dimensional characteristics in real solids. Solids of chainlike structure show one-dimensional characteristics but, of course, the chains do interact laterally, though this interaction may be small compared to the longitudinal interaction. Thus we can expect that at high temperature when $k_B T$ is large compared to the lateral interaction, the solid will show its predominantly one-dimensional character: at low enough temperatures, the three-dimensional character will ultimately show up.

Thus the discussion is couched in terms of two temperatures: $T_1$ the temperature below which one-dimensional behaviour predominates; $T_3$ the temperature below which three-dimensional ordering occurs.

An important characteristic of one-dimensional systems is the very marked influence of fluctuations on their properties. For example, a one-dimensional system cannot sustain long-range order above the absolute zero and such a system has its own characteristic instabilities. The Peierls instability is one such: if the one-dimensional system is metallic then it is possible to lower the energy of the electrons by a lattice modulation of such a wavelength that it produces energy gaps at the Fermi level. The appropriate wave number in this case is just $2k_M$, twice the Fermi wave vector of the electrons in the chain. In general this modulation will not be commensurate with the lattice periodicity. If it is not commensurate, then although there is a gap at the Fermi level, it is still possible to drive the whole system, electrons and lattice modulation, with an electric field to produce a current. This is termed Fröhlich conductivity.

Another possible instability, though not necessarily confined to one-dimensional systems, is a static spin modulation without a lattice modulation. In this the electron spin density varies periodically along the chain; an antiferromagnetic arrangement of spins is an extreme example of this. Another is the formation of superconducting electron pairs; ultimately this will give rise to superconductivity with the possibility not only of singlet pairing (with opposite spin) but also of triplet pairing (parallel spin). In one dimension, long-lived pairs may persist in substantial concentrations well above
the superconducting transition temperature, giving rise to a much higher than normal conductivity termed paraconductivity.

In what follows, evidence for these and other effects is presented and discussed, particularly in relation to the organic materials (tetramethyltetraselenafulvalene)₂X, (TMTSF)₂X.

## 2. Historical perspective and basic theoretical models

The discovery of antiferromagnetism and superconductivity in organic chain conductors (Bechgaard et al. 1980, Jérôme et al. 1980) is the most recent development of a concerted study of this type of solids, which originated with a proposal by Little (1964) and has already been reviewed in its earlier stages by Berlinsky (1976).

It is of interest to recall this proposal and to sketch the main points of the later experimental and theoretical developments of the field, before describing in more detail the properties of the family of compounds (called (TMTSF)₂X) where these new properties have been found.

### 2.1. Little's proposal of superconducting organic chains and its problems

In a search for high temperature superconductivity, Little proposed to use chain conductors in a polarizable medium: a large polarization could induce, over a wide spectrum of frequencies, a large effective attraction between conducting electrons which could lead to a strong BCS superconductivity. The general idea was not new and indeed this mechanism might be responsible for the fairly high critical temperatures observed in agglomerated and partly oxidized fine metallic powders. Little proposed that polarization could be obtained by attaching lateral polarizable organic radicals with double bonds to a conducting chain; he pointed out that the chain itself could be made of organic material and need not conduct at zero frequency and zero temperature: it could have an (electron) energy gap, as in (CH)₃ or carotenoid chains, if that gap were small compared with the possible superconducting gap. Thus arose the possibility of purely organic superconductors.

Little's proposal had at the time a very large impact, and was the starting point of many researches, despite the fact that his patent, a very thick folio, did not provide any experimental confirmation. It raised questions about a number of interconnected aspects of the problem which were soon pointed out and are worth listing:

* superconducting fluctuations?

* nature of the conducting chains?

* nature of the interactions (electron-electron or electron-lattice)?

### 2.2. One-dimensional fluctuations

The phase transition aspects are suitably discussed by analogy with magnetic transition models and we note first that a purely one-dimensional order is not possible at finite temperatures (Steiner et al. 1976). This is well known and easy to understand for a ferromagnetic Ising lattice where only one component of spin is allowed and the order parameter has only one component. Imagine that a reversal of direction of magnetization is produced at points along the chain with a (small) concentration $c=a/\xi$ where a is the intersite distance along the chain and $\xi$ is the average distance between points. The change in free energy is:

$$ \Delta F = \alpha_{\parallel} c + k_B T c \ln c $$

where $\alpha_{\parallel}(>0)$ is the internal energy due to the rupture of order along the chain and
$-k_B c \ln c$ the entropy of position of the points. Minimizing with respect to $c$ gives a finite coherence length:

$$ \xi(T) = a \exp \left( 1 + \frac{\alpha_{\parallel}}{k_B T} \right) \qquad (1) $$

Superconductivity has actually an order parameter with two continuous degrees of freedom (amplitude and phase). Because a change of phase costs less energy than a change of amplitude, it is then equivalent at low temperature to a Heisenberg two-dimensional (XY) ferromagnet since the latter has a two-component order parameter. The analysis is valid up to a mean field temperature $T_1$ of the order of the gap observed at 0 K. The analogy with the XY ferromagnet gives a more complex variation of $\xi(T)$ due to the more progressive rupture of order by its continuous variation in space. At high temperature (i.e. not too much below $T_1$):

$$ \xi(T) \propto a \frac{\alpha_{\parallel}}{k_B T} \qquad (2) $$

with lower temperature corrections.

To obtain a finite order for macroscopic chains, there are then two solutions (fig. 1):

(1) Build a three-dimensional network of size $l$. The critical temperature $T_3$ for three-dimensional order will then be given by (fig. 1(a)):

$$ \xi(T_3) \approx l \qquad (3) $$

(2) Pack the chains close to each other into a parallel array, and allow a (small) coupling $\Delta_{\perp}$ between them (Dupas and Renard 1978) (fig. 1(b)). If each chain has $z$ neighbouring chains, a mean field treatment of the thermal disorder of ordered segments of size $\xi$ along each chain gives:

$$ k_B T_3 \approx z \Delta_{\perp} \xi(T_3) \qquad (4) $$

Thus for the superconductive case:

$$ k_B T_3 = (z \alpha_{\parallel} \alpha_{\perp})^{1/2} \qquad (5) $$

<figure><img src="image_3.png" /><figcaption>Fig. 1. Networks of chain conductors. (a) Three-dimensional network; (b) three dimensional piling up of parallel chains; (c) localized couplings of three-dimensional piling up of parallel chains.</figcaption></figure>
while for the Ising lattice one would have:

$$k_B T_3 \approx \frac{\alpha_{\parallel}}{\ln \frac{k_B T_3}{ez\alpha_{\perp}}} = \frac{\alpha_{\parallel}}{\ln \frac{\alpha_{\parallel}}{ez\alpha_{\perp}} \ln \frac{\alpha_{\parallel}}{ez\alpha_{\perp} \ln \dots}} \approx \frac{\alpha_{\parallel}}{\ln \frac{\alpha_{\parallel}}{ez\alpha_{\perp}}} \quad (6)$$

In practice, most experimental studies refer so far to close-packed lattices of conducting chains, where (5) applies. It has however been recently suggested by Schulz (1982) that randomly distributed and strong but localized interchain couplings between parallel chains as pictured in fig. 1 (c) could raise $T_3$ to the value (3); thus $k_B T_3 \approx \alpha_{\parallel} a/l$.

## 2.3. (High temperature) chain conductors

The chains could be metallic, as indeed are the Pt chains in KCP, the first linear-chain compound studied in detail, by Zeller (1973), for its conductivity. Gaps in carotenoids are too large to fit Little's prescription. But large n or p dopings have produced metallic conduction in organic chains such as $(CH)_x$, in the same way as in highly-doped semi-conductors.

But much of the effort has concentrated so far on compounds with one or two types of chain, each made up of a piling of flat organic molecules with suitable bonds.

Charge transfer produces excess electrons or excess holes in these chains, and the intermolecular electron transfer along each chain is large enough for a coherent band picture to apply to their motion along the chains. This is the case for the TTF-TCNQ (two-chain) compounds first studied, and also the $(TMTSF)_2X$ family discussed here.

If each chain is first considered separately, the conductivity along it can be ascribed to a number of electrons (or holes) partly filling a narrow one-dimensional band, with a width defined by a 'parallel' intermolecular transfer energy $t_{\parallel}$. This is taken as a positive quantity; and in all the compounds considered, this is of the order of 1 eV (fig. 2).

Because of electron-electron, electron-phonon, electron-defect collisions, the Fermi electrons have, in their flow along a chain, a finite lifetime $\tau_{\parallel}$. This can in turn be split into a part which transfers a Fermi electron backwards from $k_M$ to $-k_M$, thus transferring a momentum $-2k_M$ (or vice versa), and a part which keeps the Fermi electron forward on the same branch of $E(k_z)$, thus transferring a momentum very near to zero.

The part of $\tau_{\parallel}$ due to static defects would tend to localize the electrons of this perfectly one dimensional problem (Mott and Twose 1961, Thouless 1973). But that part of $\tau_{\parallel}$ due to forward phonon scattering delocalizes the electrons again at high

<figure><img src="image_4.png" /><figcaption>Fig. 2. $E(k)$ for an isolated (non-modulated) chain.</figcaption></figure>
enough temperatures, by strongly enough coupling electron states localized in different parts of the chain (Gogolin et al. 1976, Kaveh et al. 1978 b). And indeed all the chain compounds referred to above show metallic conductivity $\sigma_{\parallel}$ along the chains over large ranges of temperatures (fig. 3). With a Fermi velocity $v_M$ and $1/bb'$ chains per unit area (fig. 1(b)), the Fermi shift $\delta k = \epsilon\epsilon\tau_{\parallel}/\hbar$ due to an electric field $\epsilon$ gives then:

$$ \sigma_{\parallel} = 4e^2 v_M \tau_{\parallel} / \hbar b b' \quad (7) $$

Normal electron transfer, between neighbouring chains, is not completely negligible, as shown by the fact that the conductivity $\sigma_{\perp}$ normal to the chains varies with Tin a way very similar to $\sigma_{\parallel}(T)$, although it is usually several orders of magnitude smaller.

As first pointed out by Weger (1978), three different regimes can occur.

(1) Coherent three dimensional couplings. The extreme case occurs for very strong interchain couplings, due to very large interchain transfer integrals $t_{\perp}$ (again taken as positive), of the order of $t_{\parallel}$. This produces a two- or three-dimensional conductor, with a closed Fermi surface, with a form that can be deduced from de Haas-van Alphen or cyclotron resonance experiments (fig. 4(a)). The 'chains' lose most of their physical identity and produce only some (small) anisotropy of the conductivity.

(2) Coherent one-dimensional couplings. This case occurs for more moderate interchain couplings. $t_{\perp}$ is definitely less than $t_{\parallel}$. It warps the planar Fermi surfaces of independent chains, over regions of $k_{\parallel}$ of order $|t_{\perp}|(\partial E / \partial k_{\parallel})_{E_M}| \approx t_{\perp}/2t_{\parallel}a$ (fig. 4(b)). These warped planes correspond to two separate and infinite sheets of the Fermi surface, which cannot give rise to cyclotron resonance, except for H very near to the chain axis. This also corresponds to highly anisotropic electrical conductivity. This regime occurs for:

$$ \hbar\tau_{\parallel}^{-1} \ll t_{\perp} \ll t_{\parallel} \quad (8) $$

where $\hbar\tau_{\parallel}^{-1}$ is the Fermi energy broadening due to $\tau_{\parallel}$.

<figure><img src="image_5.png" /><figcaption>Fig. 3. Temperature variation of the parallel resistivity in a quasi-one-dimensional electron gas (schematic). (a) Low temperature insulator; (b) low temperature superconductor.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. Effect of a finite transverse coupling $t_{\perp}$. (a) Strong coupling (coherent three-dimensional coupling); (b) moderate coupling (coherent one-dimensional coupling); (c) weak coupling (incoherent one-dimensional coupling).</figcaption></figure>

(3) Incoherent one-dimensional coupling. This case occurs for weak interchain couplings. The condition is correlative:

$$t_{\perp} \ll \hbar\tau_{\parallel}^{-1} \qquad (9)$$

The Fermi surface is then made of two diffuse planes normal to $k_z$, as for independent chains (fig. 4(c)).

This seems to be the case of KCP and of most of the compounds of the TTF-TCNQ and of the (TMTSF)₂X families, at least at high temperatures ($T > T_1$, fig. 3).

If then an electron travels initially along chain 1 (fig. 1(b)), $t_{\perp}$ will produce a progressive transfer to a neighbouring chain such as 2. For small values of the time $t$, the electron wave function reads:

$$|\psi\rangle = \cos\frac{1}{\hbar}t_{\perp}t|\psi_1\rangle + \sin\frac{1}{\hbar}t_{\perp}t|\psi_2\rangle$$

At time $t = \tau_{\parallel}$ ($\ll 1/\hbar t_{\perp}$), the electron wave function loses its coherence. The electron has acquired a small probability of being on chain 2, equal to $\sin^2 \tau_{\parallel} t_{\perp} \approx t_{\perp}^2 \tau_{\parallel}^2 / \hbar^2$. Its rate of transfer per unit time is then:

$$\frac{1}{\tau_{\perp}} \approx \frac{1}{\tau_{\parallel}} \left( \frac{\tau_{\parallel}^2 t_{\perp}^2}{\hbar^2} \right) = \frac{1}{\hbar^2} |t_{\perp}|^2 \tau_{\parallel} \qquad (10)$$

This is obviously a diffusive process, with successive incoherent interchain hoppings.
The normal conductivity then reads, according to the Einstein relation:

$$ \sigma_{\perp} = n(E_M)k_B T D_{\perp} \frac{e^2}{k_B T} $$

where $n(E_M)=4/\hbar v_M b b'$ is the density of states per unit energy and per unit volume; thus, $n(E_M)k_B T$ is the number of thermally excited carriers, e their charge, and the diffusion coefficient is:

$$ D_{\perp} = b^2 \tau_{\perp}^{-1} $$

Therefore:

$$ \sigma_{\perp} \approx n(E_M) e^2 b^2 \tau_{\perp}^{-1} $$

and, with $\hbar v_M \approx -2at_{\parallel} \sin k_M a \approx at_{\parallel}$

$$ \frac{\sigma_{\perp}}{\sigma_{\parallel}} \approx \left( \frac{t_{\perp}}{t_{\parallel}} \frac{b}{a} \right)^2 \qquad (11) $$

As indeed observed in the compounds referred to above, $\sigma_{\perp}/\sigma_{\parallel}$ remains constant at high temperatures (Cohen et al. 1974), with values generally of the order of $10^{-3}$.

A more direct proof of this behaviour is offered by nuclear magnetic resonance measurements (Soda et al. 1977). Over a large range of magnetic fields, and thus of electron frequencies, $\omega_e$, the spin-lattice relaxation time $T_1$ (not to be confused with the temperature $T_1$ in fig. 3) at temperature $T$ varies as:

$$ (T_1 T)^{-1} = A + B \omega_e^{-1/2} $$

a law typical of one-dimensional electronic motion. More exactly, the A term is due to backward scatterings and is fairly insensitive to $\omega_e$. The B term is due to forward scatterings, involving spin excitations at small $q$s. The corresponding autocorrelation function reads:

$$ \phi_q(t) \approx \exp(-D_{\parallel}q^2t) $$

where $D_{\parallel}$ is the diffusion constant along the chain. Thus:

$$ \phi = \sum_q \phi_q(t) \propto (D_{\parallel}t)^{-1/2} $$

and the contribution to $(T_1 T)^{-1}$ reads

$$ \int_{0}^{\infty} \cos \omega_e t \phi(t) dt \propto \omega_e^{-1/2}. $$

$T_1 T^{-1}$ departs from this linear law as a function of $\omega_e^{-1/2}$ for small enough $\omega_e$s for which it saturates, as expected in a three-dimensional motion of the electrons. The cross-over value of $\omega_e$ between the two regimes gives a direct measurement of $\tau_{\perp}^{-1}$. One finds typically for TTF-TCNQ at room temperatures (fig. 5):

$$ \tau_{\parallel} \approx 5 \times 10^{-15} \text{s} \quad \text{and} \quad \tau_{\perp} \approx 5 \times 10^{-12} \text{s} $$

These values are consistent with the condition $\hbar^{-1}t_{\perp}\tau_{\parallel}=(\tau_{\parallel}/\tau_{\perp})^{1/2} \ll 1$.

As the temperature is lowered, $\tau_{\parallel}$ might increase so much that it reaches a cross-over regime to $\hbar^{-1}t_{\perp}\tau_{\parallel} \gg 1$. The intermediate regime ($T_3 < T < T_1$) pictured in fig. 3, where $\sigma_{\perp}/\sigma_{\parallel}$ is no longer a constant, might possibly be due to this effect in some cases.
<figure><img src="image_7.png" /><figcaption>Fig. 5. $(T_1 T)^{-1}$ versus $H_0^{-1/2}$ in TTF-TCNQ ($D_4$). $T_1$ is the spin-lattice relaxation time and $T$ the absolute temperature. (1 kgauss = $10^{-1}$ tesla).</figcaption></figure>

But, in the TTF-TCNQ and $(TMTSF)_2X$ families discussed below, there is direct evidence of strong one-dimensional fluctuations in that range, and they are sufficient to explain the deviations of conductivities from the high temperatures regime.

Another possible argument in favour of a cross-over from incoherent to coherent regimes at really low temperatures is that the mechanism of delocalization should not be effective in that range. However, when there is metallic conduction in that range of temperature, as in the $(TMTSF)_2X$ family, we shall see that there are strong current-carrying fluctuations which not only provide a metallic current of their own but help delocalize the independent electrons by Kaveh's mechanism.

Finally a direct proof of the disappearance of the one-dimensional regime would be the observation of de Haas-van Alphen or cyclotron resonances. Pending more systematic studies of this kind, we shall assume a one-dimensional regime to hold in all the metallic range above $T_3$. In the compounds of interest here, this seems to be a diffuse incoherent one-dimensional coupling above $T_1$; whether it becomes a coherent one-dimensional coupling near $T_3$ does not seem settled in all cases and relies on observations such as transverse plasma edges (Jacobsen et al. 1981).

## 2.4. Electron-lattice interactions. The TTF-TCNQ family

### 2.4.1. Peierls instability and Kohn anomaly

All the first chain compounds studied showed at low temperature an insulating phase (Zeller 1973), and not a superconducting one (fig. 3).

It was tempting to think this insulating state to be due to localization by disorder. But n.m.r. measurements by Mrs. Launois on the Pt sites of KCP, the first and most disordered of the structures studied, clearly showed a Knight-shift in the metallic state which merely disappeared in the insulating one, with no sign of the magnetic disorder expected to go with localization (Niedoba et al. 1973).

These KCP samples, subsequently studied by X-rays (Comès et al. 1973 a), then showed clearly a phonon mode along the chains, of large amplitude between $T_1$ and $T_3$ and incoherent in phase from chain to chain, that condensed below $T_3$ into a three-dimensional static lattice modulation; the modulation along each chain, with the same
wavelength as above $T_3$, was of opposite phase to those of the nearest neighbouring chains. The dynamic and static nature of these lattice modulations, for $T \ge T_3$ respectively, was later checked and studied by neutron inelastic scattering.

As realized from the start, these observations and similar ones made later on the TTF-TCNQ family (Denoyer et al. 1975, Kagoshima et al. 1975, Comès et al. 1975) pointed to the existence of an active electron-lattice coupling, responsible for a zero degree 'Peierls instability' to a non-conducting state (Peierls 1955) and to a high temperature giant 'Kohn anomaly' in the metallic state (Kohn 1959).

Peierls first pointed out that any small lattice modulation of an isolated chain, with a wave length $\lambda$ would, by modulating the transfer integrals $t_\parallel$, couple the electronic states of energies $E(k)$ and $E(k - 2\pi/\lambda)$ and of parallel spins, and thus produce gaps at $k = \pm \pi/\lambda$ (fig. 6). If $k = k_M$ (hence $\lambda = \frac{1}{2}\lambda_M$) the gap occurs at the Fermi level; all occupied electronic states have their energy lowered by the modulation. This is the most stable configuration and it leads to an insulating state. One might think that the lowering of energy being of the order of half the energy gap $\Delta_\parallel$ over a region in $k$ proportional to $\Delta_\parallel$, the total lowering of energy would vary as $\Delta_\parallel^2$, and thus as the square of the amplitude of modulation $u_0$. For, as is easily shown:

$$ \Delta_\parallel = 2 \left| \frac{dt}{da} \sin k_M a \right| \frac{u_0}{a} \quad (12) $$

This reasoning however neglects the tail ends of the gap, for $k$ far from $\pm k_M$, where a development in $\Delta_\parallel/\xi_k$ is valid; $\xi_k$ is the unperturbed electron energy measured from the Fermi level. This gives an energy:

$$ 4 \sum_{k=0}^{k_M} \left\{ -\sqrt{(\xi_k^2 + \Delta_\parallel^2) - \xi_k^2} \right\} \approx 2\Delta_\parallel^2 \sum_{\xi_k}^{k_M} \frac{1}{\xi_k} $$

Using the fact that at 0 K this is valid up to a cut-off energy $|\xi_c| \approx 1/2\Delta_\parallel$, and that $\xi_k$ varies nearly linearly with $k$ over a region of energy up to $w$, the modulation induces a lowering of energy (fig. 6):

$$ -K\Delta_\parallel^2 \ln \frac{2w}{\Delta_\parallel} \quad (13) $$

where $K = 2\pi/\lambda$. Owing to the logarithmic term, this always overweighs any elastic

<figure><img src="image_8.png" /><figcaption>Fig. 6. Peierls insulator.</figcaption></figure>
reaction $L\Delta_{\parallel}^2$ of the medium. Minimizing with respect to $\Delta_{\parallel}$ gives:

$$ \Delta_{\parallel} = w \exp\left(-\left(\frac{1}{2} + \frac{L}{K}\right)\right) \approx w \exp\frac{1}{n(E_M)g_1} \quad (14) $$

where $n(E_M)$ is again the density of states at the Fermi level, and:

$$ g_1 = -\frac{V_{ep}^2}{\hbar\omega_e} \quad (15) $$

is the (negative) coupling constant, $V_{ep} = 2dt_{\parallel}/da \sin k_M a$ is the electron-lattice coupling and

$$ \hbar\omega_e = E_{abb'} \quad (16) $$

is an elastic energy per atom or molecule (E elastic constant).

In the high temperature range ($T > T_1$, fig. 3), $\Delta_{\parallel}$ must be replaced by $k_B T$ as the cutoff energy $|\xi_c|$ and thus in the logarithm, equation (13). This leads to a soft phonon mode $2k_M$ of frequency:

$$ \begin{aligned} \omega'(2k_M) &= \omega(2k_M) \left(1 - \frac{K}{2L} \ln \frac{w}{k_B T}\right) \\ &= \omega(2k_M) \left(1 - \frac{n(E_M)|g_1|}{2} \ln \frac{w}{k_B T}\right) \end{aligned} $$

### 2.4.2. One-dimensional fluctuations and three-dimensional couplings

At the intermediate temperatures ($T_3 < T < T_1$, fig. 3) one expects large one-dimensional fluctuations of the $2k_M$ Peierls instability. In the general case where the wave length of the modulation is incommensurate with the lattice period $a$, the modulation has a phase that can vary continuously. The situation is similar to that of superconductivity and equation (2) applies.

If now neighbouring chains are weakly coupled, one expects a three-dimensional static ordering of the $2k_M$ modulation up to a critical temperature given by equation (5). Two cases can occur for this transverse coupling:

(1) If the Fermi surface is warped by a marked electron transfer $t_{\perp}$, such that $\hbar^{-1}t_{\perp}\tau_{\parallel} > 1$, this warping will change with the relative phases of the modulations of neighbouring chains and stabilize one arrangement with respect to the others (Weger 1978).

(2) If the Fermi surface is planar and diffuse ($\hbar^{-1}t_{\perp}\tau_{\parallel} < 1$), the interchain coupling will only be due to the long-range electrostatic interactions between the chains (Barisic 1972, Saub et al. 1976, Bak and Emery 1976, Bjelis and Barisic 1977), for a lattice modulation of a chain, when not a simple dimerization, produces an excess charge-density wave, such that the negative ions are associated with the compressed regions, or locally larger band width, if the band is less than half-full and vice versa for a more than half-full band, see (fig. 7). Then, as first pointed out by Barisic (1972), the Coulomb coupling between these charges will tend to stabilize modulations of opposite phases on neighbouring chains in cases such as KCP with one type of chain; and the coupling energy $\alpha_{\perp}$ will be directly proportional to the square of the amplitude $u_0$ of modulation. If the modulations were in phase, the Coulomb interactions between the charges would, on the contrary, compensate for the logarithmic term due to lattice modulation and
<figure><img src="image_9.png" /><figcaption>Fig. 7. Charge-density wave associated to a lattice modulation.</figcaption></figure>

thus in general destabilize the modulation. In the high temperature disordered state, these interactions average to zero. But at temperatures not too much above $T_3$, they can produce a weak short-range order of the phases on neighbouring chains, which has been observed (Kagoshima et al. 1978).

### 2.4.3. Incommensurability. Fröhlich conductivity

The wavelength $\lambda$ of the Peierls modulation should vary continuously with the filling of the electron band. And this can be varied by doping (as in KCP) or, in charge-transfer compounds with two types of chains such as TTF-TCNQ, by applying pressure. It is now well established that this is so, except near to some simple 'commensurate' values of $\lambda$, where the compound keeps to such values over a range of dopings or pressure.

In the continuously varying (incommensurate) state, the modulation has a phase that can, in principle, be varied freely without changing the energy of the modulation. The experimental observation by inelastic neutron scattering of such zero-energy 'phasons' in the low temperature ordered state is still a matter of controversy. As first pointed out by Fröhlich (1954), another way of observing this motion could be to apply an electric field. For the motion of a modulation at a velocity $v=\omega/k$ should carry a current of charge proportional to $v$, and this could couple with the electric field to absorb energy (Bardeen 1973).

Indeed such a motion produces an effective perturbing potential of the form $V=2V_k \cos(Kz - \omega t)$ which couples the time-dependent Bloch functions $|k(z,t)\rangle$ into functions:

$$|\psi\rangle = \sum_k a_k(t) |k(z,t)\rangle \text{ such that, with } H_0 |k\rangle = i\hbar \frac{\partial}{\partial t} |k\rangle \text{ and } (H_0 + V) |\psi\rangle = i\hbar \frac{\partial}{\partial t} |\psi\rangle$$

$$\frac{da_k}{dt} = -\frac{i}{\hbar} \sum_{k'} \langle k(z,t) | V | k'(z,t) \rangle a_{k'}(t)$$

If $K=2k_M$, $a_k$ and $a_{k-k}$ are coupled strongly if $E_k \approx E_{k-2k_M}$, that is, if $k \approx k_M$. Their coupled equations then give, to first order in $V$ and if Umklapp processes can be neglected:

$$a_k = \alpha_k \exp \left( -\frac{i}{\hbar} \left( \hbar\Omega - \frac{\hbar\omega}{2} - E_k \right) t \right)$$

$$a_{k-K} = \alpha_{k-K} \exp \left( -\frac{i}{\hbar} \left( \hbar\Omega + \frac{\hbar\omega}{2} - E_{k-K} \right) t \right)$$
with

$$ \hbar\Omega = \frac{1}{2} [E_k + E_{k-\mathbf{k}} \pm \{(E_k - E_{k-\mathbf{k}} + \hbar\omega)^2 + 4|v_K|^2\}^{1/2}] $$

where $v_K$ is proportional to $V_k$.

$\hbar\Omega$ plays the role of an effective energy and is pictured in fig. 8. The motion of this Fröhlich mode shifts the energy curve and the energy gaps by $\delta k = \frac{1}{2}\hbar\omega dE_k/dk_M$ along the $k$ axis. This shift can be shown to produce an average current proportional to $\delta k$, thus to $\omega$ and therefore to the velocity $\omega/K$ of the Fröhlich mode.

This effect arises from the same electron-phonon terms as the incoherent phonon drag (Weger and Gutfreund 1978, Kaveh et al. 1978a). But it is different at low temperatures and could be termed a coherent phonon drag. No such conductivity is observed in the three-dimensional lattice-modulated state of any of the linear chain compounds. In the TTF-TCNQ family, the reason might be simply because they contain pairs of chains with equal numbers of electrons and holes; by a coherent motion of their modulations, these should give a total current equal to zero. But more generally such motions are expected to couple to lattice defects, especially dislocations; these should hamper strongly their long-range coherent motion (Lee et al. 1974, Friedel 1977). A motion should still be possible, in single-chain compounds, under a large enough electric field.

The situation should be quite different in the one-dimensional fluctuation range ($T_3 < T < T_1$, fig. 3) (Bardeen 1973, Patton and Sham 1974, Schulz 1980). Such fluctuations should also carry a current, at least at temperatures low enough for them to have a physical significance in time and space. Furthermore, because they are localized on a finite region of a single chain, these fluctuations should be less sensitive to pinning by defects, especially those widely spaced such as dislocations; and in two-chain compounds such as TTF-TCNQ, they should be free to move independently on the two types of chains. The current produced follows much the same law as in the static regime; it affects only $\sigma_{\parallel}$, giving a contribution which should increase with the amplitude of the fluctuations but vary also with their relaxation rate.

A detailed analysis must consider two further terms.

The fluctuations induce a 'pseudo-gap' effect, which reduces the number of independent electrons, and thus reduces $\sigma_{\parallel}$ and $\sigma_{\perp}$ (Weger and Gutfreund 1978, Kaveh et al. 1978a).

Also, the collisions of the remaining independent electrons on the fluctuations produce an incoherent phonon drag effect which increases the current $\sigma_{\parallel}$ of the independent electrons (Weger 1979, Gutfreund and Weger 1977).

<figure><img src="image_10.png" /><figcaption>Fig. 8. Fröhlich mode.</figcaption></figure>
As $\sigma_\parallel$ increases strongly on lowering $T$ in the fluctuating ranges, it is clear that the reduction of $\sigma_\parallel$ due to the pseudo-gap is a small correction with respect to the contributions due to the fluctuations. Furthermore, the decrease of $\sigma_\perp/\sigma_\parallel$ with lowering $T$ in this range is an indication of the role of fluctuations.

A direct proof comes from the fact that the fluctuations are expected to be blocked by the lattice, and the contributions to $\sigma_\parallel$ due to their motion disappear, if the static modulations are commensurate with the lattice. Such a case occurs for instance for TTF-TCNQ under pressure (fig. 9): neutron measurements have shown a $\times 3$ commensuration in the static (three-dimensional) modulation (Megtert et al. 1979), at a pressure range where the critical temperature $T_3$ is larger (Friend et al. 1978); for the same pressure range, there is a strong drop of $\sigma_\parallel$ (but not of $\sigma_\perp$) above $T_3$, which fades out with increasing $T$ towards $T_1$ (Andrieux et al. 1979 c). This drop can be taken as a direct measurement of the part of $\sigma_\parallel$ related to the motion of the fluctuations in the

<figure><img src="image_11.png" /><figcaption>Fig. 9. Pinning or lattice modulations in TTF-TCNQ with increasing pressure. (a) Parallel and (b) transverse conductivities; (c) $T_3(P)$; (d) $2k_M(P)$. (1 bar $= 10^5$ Nm$^{-2}$.)</figcaption></figure>
incommensurate state. The ratio of the incoherent to coherent phonon drag effects should increase with the number of effective independent electrons and thus with temperature; but no simple experiment can measure the contribution of each effect. Finally, the larger $T_3$ for the commensurate phase, and the fact that it is preferred to the incommensurate ones over a finite range of pressures, can be related to the blocking of the phase of the modulation in the commensurate phase. This makes a commensurate modulation analogous to an Ising model: equation (6) applies, leading to a larger $T_3$ than equation (5) if $\alpha_{\perp} \ll \alpha_{\parallel}$.

The total longitudinal conductivity can be written approximately as follows:

$$\sigma_{\parallel} = \sigma^{\mathrm{F}} + \sigma_{\parallel}^{\mathrm{p}}$$

where $\sigma^{\mathrm{F}}$ and $\sigma_{\parallel}^{\mathrm{p}}$ are respectively the fluctuating Fröhlich and single particle contributions (Jérôme 1980, Jérôme and Schulz 1981).

By decomposing the conductivity of TTF-TCNQ at ambient pressure according to this equation and assuming a similar temperature dependence for $\sigma_{\parallel}^{\mathrm{p}}$ and $\sigma_{\perp}$ (see equation 11), it has been shown that $\sigma^{\mathrm{F}}$ is the dominant conduction process below about 150 K, the temperature at which 1-D short range ordering becomes clearly visible in the X-ray diffraction patterns. At room temperature, the single particle channel dominates $\sigma_{\parallel}$ and consequently the normal and parallel components of the conductivity are affected in the same way by pressure, as described by equations (11). Furthermore, the conductivity anisotropy, which remains constant down to about 150 K shows a marked increase on further cooling since then $\sigma^{\mathrm{F}} \gg \sigma_{\parallel}^{\mathrm{p}}$. Calculation of the fluctuating sliding conduction (or fluctuating Fröhlich conduction) has been performed at high temperature including the effect of commensurability pinning forces (Schulz 1980). The results show that provided the fluctuating domains of length $\xi_{\parallel}$ along the chains are sufficiently long-lived to permit the commensurability pinning forces to have an appreciable effect, even a weak commensurability potential may considerably reduce the sliding conduction. The sliding conduction can also be suppressed by pinning to defects such as those induced by irradiation (Bouffard et al. 1981). A concentration of about $2 \times 10^{-3}$ molecular defects is sufficient to destroy completely the collective contribution to the total conductivity, whereas the conduction due to single particles is practically unaffected.

Detailed far infrared measurements on TTF-TCNQ at low temperatures (Tanner et al. 1981) provide additional support for a fluctuating sliding-mode mechanism for the conductivity above $T_3$ and up to 150 K. Derivations of $\sigma^{\mathrm{F}}$ either from the Kramers-Kronig analysis of the far infrared reflectance data or from the comparative temperature dependence of $\sigma_{\parallel}$ and $\sigma_{\perp}$ lead to a reasonable agreement: the conductivity of about 5000 to 7000 $\Omega^{-1}$ cm$^{-1}$ at 60 K in ordinary crystals is due (up to about 80%) to a collective mode at zero frequency (of life-time $\tau_c \simeq 1.6 \times 10^{-12}$ s). Notice that the fluctuation life-time at 60 K is still shorter than the electron life-time $\tau_{\perp}$ on a given chain, as derived from n.m.r. This guarantees the one-dimensional nature of the Fröhlich fluctuations.

As seen in Section 2.2, the temperature $T_1$ sets the temperature scale for the development of one-dimensional fluctuations, towards a low temperature ordered state. In TTF-TCNQ, precursor effects become visible by X-rays below 150 K (Comès 1977) thus setting an upper limit to $T_1$. The three-dimensional ordering temperature $T_3 = 54$ K is obviously lower than $T_1$. The depression is however not as large as that observed for the superconducting instability, probably on account of the large
interchain electrostatic coupling between neighbouring charge-density waves associated with the lattice modulation along the chains (Saub *et al.* 1976).

## 2.5. Electron-electron coupling

### 2.5.1 TTF-TCNQ

There is evidence in the TTF-TCNQ family of corrective effects due to sizeable electron-electron coupling.

If Coulomb interactions between chains are strong enough to produce the three-dimensional coupling responsible for $T_3$, it is likely that Coulomb interactions within a chain are not negligible even above $T_3$.

The strength of the *B* term in the n.m.r. studies, indicates large forward scattering. Part at least of this could be due to electron-electron scattering along the chains or along neighbouring chains, although Kaveh (1981) pointed out that part might also be due to phonon scattering. Further, the strength of the magnetic susceptibility, suggests a sizeable Stoner enhancement (Torrance 1977, Jerome and Caron 1979).

Finally, there are the ‘$4k_M$’ static or fluctuating lattice modulations observed in some compounds (Pouget *et al.* 1976). They vary with *T* and *P* in a way different from the $2k_M$ ones, which indicates that they cannot be mere harmonics of the $2k_M$ modulations (thus, they are observed at temperatures higher than the $2k_M$ ones). A simple way of explaining them is to assume that there are $2k_M$ spin modulations which could produce a small $4k_M$ lattice modulation via an intermolecular exchange interaction; the $2k_M$ spin modulation could also produce a $4k_M$ charge modulation either by a classical spin-orbit coupling or via electron-electron correlations; this in turn can couple to a lattice $4k_M$ modulation via an electron-phonon coupling. A similar effect is well known for the antiferromagnetic phase of chromium (Lomer 1962). These explanations are due to Weger and Gutfreund (1979). The spin modulation is not necessarily saturated; if small it can therefore coexist with a $2k_M$ lattice modulation.

However, another interpretation which was initially proposed by Blandin (Emery 1976, Kondo and Yamaji 1977) suggests that electron-electron repulsions should be strong enough to split the band into two halves, thus essentially preventing two carriers from sitting in the same delocalized state, and multiplying by two the Fermi wave number in the corresponding ‘Hubbard’ sub-band. The simultaneous observation of $2k_M$ lattice modulation would be difficult to reconcile with this picture. In any case, a careful search for magnetic phases or fluctuations should be made in these compounds.

A more thorough experimental and theoretical study of coexistence of spin and lattice modulations is anyway in order.

### 2.5.2. The $(TMTSF)_2X$ family

The $(TMTSF)_2X$ family described below shows definitely at low temperatures a static spin modulated state (with small molecular moments). Under pressure, this low temperature state is replaced by a superconductive state. No evidence of static or fluctuating lattice modulation is observed in these compounds (Pouget *et al.* 1982). As explained below, these new observations were in a way expected in the theoretical analysis which was developed to compare these various types of instabilities. They are coherent with repulsive effective electron-electron interactions, as expected from direct Coulomb interaction. The intramolecular Coulomb interactions can be estimated from the difference of the two successive redox potentials. As expected for fairly large and polarizable molecules, this is not large in TMTSF molecules, being of the order of
0.3 eV. But neither is the band width, which seems less than 1 eV. A more thorough study of these microscopic parameters is certainly due.

## 2.6. Comparing the instabilities

### 2.6.1. Isolated chains. The $g_1g_2$ model

Starting with Gorkov (Byschkov et al. 1966, Menyhard and Solyom 1973, Solyom 1973, Gutfreund et al. 1974, Lee 1975), a number of studies were made of isolated conducting chains, taking into account that the (unperturbed) Fermi electrons could interact by either forward or backward scattering. This is the so-called $g_1g_2$ model.

Consider first more generally a metal with Bloch functions $|k, \sigma\rangle$ which interact through $V$. Following BCS, we concentrate on the collisions through $V$ between electrons of opposite momenta and opposite spins. They involve the following pair integrals:

$$ \langle k\sigma, -k-\sigma | V | k'\sigma, -k'-\sigma \rangle = g(k', k) $$

The summation over all occupied $k'$ pairs leads to a $k$ pair creation (or annihilation) operator. With reference to an unperturbed ground state of energy taken as zero, the corresponding reduced Hamiltonian can be written

$$ H_{\text{eff}} = \sum_{k\sigma} \xi_k |k\sigma\rangle\langle k\sigma| + \sum_{k\sigma} \xi_{\bar{k}} |\overline{k\sigma}\rangle\langle \overline{k\sigma}| + \sum_{k\sigma} \Delta_k (|k\sigma\rangle\langle -\overline{k-\sigma}| + |\overline{k\sigma}\rangle\langle -k-\sigma|) $$

where $\langle -\overline{k-\sigma} |$ is the destruction operator of a positive hole of energy:

$$ \xi_{-\overline{k}} = \xi_{\overline{k}} = -\xi_k $$

Advantage has been taken of the fact that creating an electron in the ground state is the same as destroying a positive hole.

In a way very similar to the Peierls instability of fig. 5, each state $|k\sigma\rangle$ of energy $\xi_k$ is replaced by a state:

$u_k|k\sigma\rangle - v_k|\overline{k-\sigma}\rangle$ of energy:

$$ \varepsilon_k = \pm (\xi_k^2 + \Delta_k^2)^{1/2} $$

with:

$$ u_k = \frac{1}{\sqrt{2}} \left( 1 + \frac{\xi_k}{\varepsilon_k} \right)^{1/2} $$

$$ v_k = \frac{1}{\sqrt{2}} \left( 1 - \frac{\xi_k}{\varepsilon_k} \right)^{1/2} \frac{\Delta_k}{|\Delta_k|} $$

In this Bogoliubov transformation, the $-$ sign in $\varepsilon_k$ refers to the lower branch, occupied at 0 K, of fig. 10. At 0 K, the self-consistency equation for $\Delta_k$ is ($k'$ unoccupied pairs):

$$ \Delta_k = \sum_{k'} g(k', k) u_{k'} v_{k'} = -\frac{1}{2} n(E_M) \int_{-\omega}^{\omega} g(k', k) \Delta_k \frac{d\xi_{k'}}{|\xi_{k'}|} \approx -\frac{1}{2} n(E_M) \Delta_k \bar{g} \ln \frac{2\omega}{|\Delta_k|} $$

$g$ is assumed effective up to a cut-off frequency of excitation $\pm \omega$, and $\bar{g}$ is an average
<figure><img src="image_12.png" /><figcaption>Fig. 10. BCS superconductivity.</figcaption></figure>

value of the $g_s$ involved. Hence, with $\Delta_k = \text{const} = \Delta_\parallel$,

$$ |\Delta_\parallel| \approx 2\omega \exp \frac{2}{n(E_M)\bar{g}} \quad (17) $$

This classical demonstration leads to the following remarks.

In the purely linear conductors considered here, there are two parts in $g:g_1$, referring to backward scatterings ($k' \approx k \pm 2k_M$); $g_2$ referring to forward scatterings ($k' \approx k$). Thus:

$$ \bar{g} = g_1 + g_2 \quad (18) $$

As pointed out earlier, the lattice modulation of a one-dimensional conductor gives rise to a gap opening and a change of energy which can be formally written in the same way if we assimilate the cutoff energies $\omega$ and $w$. The gap term $\Delta_k |k\sigma\rangle\langle k'\sigma|$ now couples electron states; and the $g$ can be considered as a matrix element of an effective electron-electron interaction:

$$ g_i = \langle k\sigma, k'\sigma | V | k'\sigma, k\sigma \rangle $$

Adding to the $g_1$ term (due at least partly to the electron-lattice coupling) a $g_2$ term due to electron-electron repulsion, one finds that equation (17) holds with:

$$ \bar{g} = 2g_1 - g_2 \quad (19) $$

In a similar way, a spin modulation can be related to a gap term $\Delta_k (|k\uparrow\rangle\langle k'\downarrow| + |k\downarrow\rangle\langle k'\uparrow|)$ due to an effective $g_i = \langle k\sigma, k' - \sigma | V | k' - \sigma, k\sigma \rangle$ and leading to the gap equation (17) with:

$$ \bar{g} = -g_2 \quad (20) $$

Also, to the singlet superconductivity considered so far one can add a triplet superconductivity with an effective

$$ g_i = \langle k\sigma, -k\sigma | V | k'\sigma, -k\sigma \rangle (\sigma = \uparrow \text{ or } \downarrow \text{ for } S_z = \pm 1) $$

or

$$ g_i = \langle k\uparrow, -k\downarrow | 1 | -k\uparrow, k\downarrow \rangle \quad (\text{for } S_z = 0). $$

Then equation (17) is valid with:

$$ \bar{g} = -g_1 + g_2 \quad (21) $$
The standard two beam perturbation technique used here is only valid for weak couplings $\bar{g}$. It also assumes a gap that goes to zero with $\bar{g}$. According to (17), this requires negative couplings $\bar{g}$. Thus for positive $\bar{g}$s, the restriction to one type of coupling (e.g. Cooper's pairs for superconductivity) is no longer physically reasonable. The condition $\bar{g}<0$ then indicates in what region of the $(g_1g_2)$ plane modulation can occur at 0 K. We can also compare their relative stability at 0 K, the most stable being the one with the most negative $\bar{g}$ if we assume all couplings to have the same cut-off energy ($\omega$ or $w$). This leads to fig. 11.

This diagram had a powerful impact on the development of the field. It first suggested that a direct transition from spin modulation (SM) to superconductivity could be produced by a small variation of the parameters, for example by applying pressure. This in turn suggested that the superconductivity observed in the $(TMSF)_2X$ family was due, as was the spin modulation, to repulsive interactions (positive $g_s$), that is, to direct electron-electron repulsion, and not, to attractive interactions as assumed in the usual BCS theory or as suggested by Little. It finally also suggested that the superconductivity considered was of the triplet kind.

There are however a number of points which make fig. 11 at best suggestive, but not
demonstrative.

(1) The mean field approximation used here neglects the quantum (phase) fluctuations which are actually important in the one-dimensional case at low temperatures (Schulz and Bourbonnais 1983). However, extrapolations from high temperatures including these fluctuations seem to lead to essentially the same type of $g_1g_2$ diagram. Also the inclusion of weak interchain couplings should suppress these phase fluctuations and probably make the mean field approximation valid even at low temperatures.

(2) The assumption of equal values of $g_1$, $g_2$ for the four types of coupling in a given compound originates from the idea that, in the four cases, the couplings are due to the same interaction, in fact the direct electron-electron interaction. This assumption is prevalent in the literature since Overhauser's work (1960). Even in that case, the assumption of constant $g_1$ and $g_2$ can be questioned; only a description of the dynamical screening of long range Coulomb interactions can explain why the condition $2g_2 < g_1$ necessary for superconductivity can be obtained. Moreover, as stressed above, the direct Coulomb interactions are not the main coupling terms for the

<figure><img src="image_13.png" /><figcaption>Fig. 11. $g_1g_2$ diagram. SM, spin modulation; LM, lattice modulation; SS, singlet superconductivity; TS, triplet superconductivity.</figcaption></figure>
lattice modulation (wrongly named 'charge-density wave'); and the leading term in $g_1$, in that case, given by equation (15) and due to electron-lattice couplings, differs from the corresponding term for BCS phonon-mediated singlet superconductivity by the fact that $\omega_e \gg \omega_p$, the average phonon energy. It is clear also that an explicit spin dependence of the $g_{is}$ would act differently on the various couplings.

(3) Even if the $g_1$ values were equal in the four modulations, it is clear that the cut-off energies ($\omega$ and $w$) are very different depending on the nature of the couplings: they are of the order of phonon frequencies for phonon-mediated interactions responsible for BCS superconductivity, while they are of the order of the Fermi energy for all other cases.

(4) The presence of the negative ions doubles the period along the chains. This weak dimerization produces a folding of the $E(k)$ curves, with a small gap $\Delta$ for half-filled bands.

One can associate with the corresponding Umklapp processes a new coupling constant $g_3 \simeq g_1\Delta/E_F$ ($E_F$ Fermi energy) (Barisic and Brazovskii 1981). As pointed out recently (Emery et al. 1982), $g_3$ shifts the borderline $g_1=2g_2$ parallel to itself; but the effect is only appreciable at temperatures small compared to $\Delta$; thus $g_3$ might contribute to the low temperature stabilization of the spin modulated phase versus the superconductive one. According to Horovitz et al. (1981), a non-vanishing $g_3$ also allows a direct borderline between the spin-modulated and singlet superconductive phases: this point might be of importance, as there is so far no conclusive evidence of a triplet nature of the observed superconductivity.

### 2.6.2. Coupled chains

An analysis of the three-dimensional order observed at low temperatures requires a finite coupling between chains. Suzumura and Fukuyama (1978) for instance have extended the ($g_1g_2$) model by assuming a coherent coupling due to a finite transfer energy $t_\perp$. Figure 11 is then still approximately followed by systems showing this three-dimensional order. But this further step raises new questions.

Fukuyama's assumption is quite reasonable for the singlet superconductive phase. As stressed above, it is not likely for the lattice modulation in the diffusive range where $\hbar^{-1}t_\perp\tau_\parallel \ll 1$. For spin modulations, besides the Anderson coupling term $t_\perp^2/U$ (U, effective Coulomb repulsion), one expects the classical exchange $J'=J-2t_\perp S$ to dominate at long range, where $J=\langle\psi_a(1)\psi_b(2)\rangle e^2/r_{12}|\psi_b(1)\psi_a(2)\rangle$ and $S=\langle\psi_a|\psi_b\rangle$; these three expressions have the same exponential decay with interchain distance; but this is multiplied by different polynomial functions of distance, so that some cross-over might occur with pressure. Even in the superconducting phases, it is not clear that the interchain coupling has the same nature as the intrachain one: the interchain coupling could be singlet while the intrachain one is triplet; it might be phonon-mediated; and in that case a superconducting state with no interchain current could subsist at $t_\perp=0$, through coupling between electrons on neighbouring chains. It is however likely that this last mechanism would not lead to a Meissner effect.

The strengths of the interchain and intrachain couplings $\alpha_\parallel\alpha_\perp$ have therefore no necessary direct relation in any of the four instabilities discussed here. One can quite possibly have:

$$ \alpha_\parallel^A \gg \alpha_\parallel^B \text{ but } \alpha_\perp^A \ll \alpha_\perp^B $$

In such a case, instability B might appear below $T_3$. But at temperatures well above $T_3$
the one-dimensional A-fluctuations will dominate. This seems to be the case in the $(TMTSF)_2PF_6$ family at normal pressure, where A is the superconductivity and B the spin modulation.

### 3. Precursor regime

In this Section we will primarily discuss the development of very large precursor effects in organic superconductors, observed between $T_1$ and $T_3$. The precursor effects related to the Peierls instability have been summarized in the previous Section. However before discussing the properties of the $(TMTSF)_2X$ series we shall recall some features of a very interesting charge-transfer complex which has marked an important step towards the achievement of organic superconductivity.

#### 3.1. *The case of TMTSF-DMTCNQ*

The charge-transfer compound TMTSF-DMTCNQ (Jacobsen et al. 1978) was the first organic compound studied in which a truly conducting state is stabilized under pressure down to helium temperatures. Application of high pressures quickly suppresses the sharp metal-insulator transition seen at 42 K at ambient pressure, and above 10 kbar (1 GPa)$\sigma_\|\|$ exceeds $10^5 \Omega^{-1} cm^{-1}$ at $T=1.2$ K (Andrieux et al. 1979 a). Moreover, the overall temperature dependences of $\sigma_\|\|$ and $\sigma_\perp$ are qualitatively similar under pressure (Andrieux et al. 1979 b) in contrast with the behaviour of TTF-TCNQ (Jerome and Schulz 1981); an enormous magnetoresistance in both components of the resistance is found for fields perpendicular to the molecular stacking axis and essentially no magnetoresistance for the parallel geometry up to 75 kOe (7.5 tesla), fig. 12.

To explain the conductivity of TMTSF-DMTCNQ at low temperatures in terms of a Fröhlich conduction process similar to that of TTF-TCNQ, coherence lengths of the order of 1000 Å would be required. Unless an accidental cancellation of the Coulomb interchain coupling exists for the TMTSF-DMTCNQ structure, very large coherence

<figure><img src="image_14.png" /><figcaption>Fig. 12. Temperature-dependence of the resistivity of TMTSF-DMTCNQ under 13 kbar in zero applied field and under a field of 75 kOe (equivalent to 7.5 tesla) applied perpendicular to the high conductivity axis.</figcaption></figure>
lengths would inevitably lead to the three-dimensional ordering of the one-dimensional lattice modulation at high temperature. Therefore it is unlikely that motion of charge-density waves can explain the high conduction of TMTSF-DMTCNQ under pressure. No more satisfying is the interpretation of $\sigma_\parallel$ within a single-particle scattering model, considering (i) the extraordinary sensitivity of the low temperature resistance ($P > 10$ kbar) to defects, $\Delta\rho \sim 300~\mu\Omega$ cm per defect percent and (ii) the violation of the Matthiessen's rule (Bouffard *et al.* 1981 b).

For these reasons two further possibilities have been proposed to account for the properties of TMTSF-DMTCNQ.

According to Hardebusch *et al.* (1979) the Fermi surface of TMTSF-TMTCNQ under pressure and at low temperature may be semimetallic with small pockets of electrons and holes resulting from the interchain hybridization in a two-chain conductor in the limit $t_\perp\tau_\parallel > h$. Such a picture followed a similar explanation by Weger (1976) of the quasi-two-dimensional compound HMTSF-TCNQ.

The second point of view is a more speculative and altogether more exciting prospect (Andrieux *et al.* 1979 a, b); it postulates the gradual growth at low temperatures of a superconducting pairing between the $+k_M$ and $-k_M$ electron states. If not yet fully settled, this lively discussion was a powerful factor for looking at somewhat similar one-chain compounds, namely the $(TMTSF)_2X$ series. The determination by Pouget (1981) of a quarter-band filling, the smallest ever found so far in organic conductors and the existence of one-dimensional scattering at wave vectors $2k_F$ and also $4k_F$ (the only case of $4k_F$ scattering among selenium organic conductors) stimulated the synthesis of the series of TMTSF pseudo-halogen salts $(TMTSF)_2X$ (Bechgaard *et al.* 1980) in which a similar commensurate band filling can be achieved by stoichiometry. The search for such compounds was inspired by the recent study of the corresponding sulphur series (Galigne *et al.* 1978).

### 3.2. The $(TMTSF)_2X$ series

$(TMTSF)_2X$ are cation radical salts of TMTSF (tetramethyltetraselenafulvalene) with inorganic anions $X=NO_3$, $BF_4$, $PF_6$, $AsF_6$, $SbF_6$, $ClO_4$, etc.... The anion is negatively charged in these salts and the corresponding positive charge is shared between two organic molecules. The structure consists of a triclinic columnar arrangement of TMTSF molecules in the form of sheets of donor stacks separated by anion sheets. The unit cell along the stacking direction contains two donor molecules related by a pseudo-translation $a/2$ followed by a rotation of $180^\circ$ around the $a$-axis (fig. 13).

However, the consideration of the non-planarity of the molecules and of the full three-dimensional triclinic structure indicates that the periodicity of the potential seen by the charges moving along the TMTSF stacks is fixed by the distance $a$ between the anions. The average intermolecular distance are slightly dimerized, for instance, 0.08% and 0.027% in $(TMTSF)_2PF_6$ and $(TMTSF)_2ClO_4$ respectively. The dimerization is however more pronounced in the sulphur series; it amounts to 0.6% in $(TMTSF)_2Br$ (Galigne *et al.* 1978).

The positive charges are delocalized along the $a$-axis as a result of overlap of the $\pi$ wave functions along the stacking direction and give rise to a half-filled band of commensurability $\times 2$. When the $(TMTSF)_2X$ family is considered from the point of view of commensurability, the conductivity behaviour is quite remarkable since in most salts of that family it is enhanced at low temperature ($T \sim 20$ K) with respect to the room temperature value by a factor greater than one hundred.
<figure><img src="image_15.png" /><figcaption>Fig. 13. View of the $(TMTSF)_2ClO_4$ crystal structure along the stacking direction (a) and $(TMTSF)_2PF_6$ perpendicular to the stacking direction (b).</figcaption></figure>

Polarized reflectance studies on $(TMTSF)_2PF_6$ (Jacobsen *et al.* 1981) show a metallic-like optical spectrum with a plasma frequency at $11200 \text{ cm}^{-1}$ for $E \parallel a$ and an optical relaxation rate of $1920 \text{ cm}^{-1}$ at $T = 300$ K (fig. 14). No significant change of the plasma frequency is seen at low temperature ($T = 25$ K), but the relaxation rate becomes $580 \text{ cm}^{-1}$ or the optical conductivity increases by a factor 3-5, which is about 30 times smaller than the enhancement of the d.c. conduction but in fair agreement with the value of the frequency-dependent conductivity in the far infrared regime ($\omega \approx 10 \text{ cm}^{-1}$) (Jacobsen *et al.* 1982).
The structure of the (TMTSF)₂X (Thorup *et al.* 1981) confers a significant anisotropy on the band structure. According to the d.c. and microwave conductivity (Bechgaard *et al.* 1980), the *a* to *b* transfer integral anisotropy amounts to ≈ 30, in (TMTSF)₂PF₆ at ambient temperature following the relation (11) which applies whenever the transverse conduction is diffusive.

In the context of the optical studies, (TMTSF)₂PF₆ has been considered as a pseudo-two-dimensional conductor at low temperatures and indeed a well defined plasmon resonance is observed at 25 K with the *E*-field polarized along the transverse *b*-direction at the frequency of 2360 cm⁻¹ (fig. 14). However the longitudinal bandwidth derived from optical studies is about 1 eV and the effective mass anisotropy *m*<sub>b</sub>*\**/*m*<sub>a</sub>*\**/≈20 (Jacobsen *et al.* 1982). Although strong two-dimensional coupling is supported by the existence of short interchain Se–Se contacts (3·879 Å in the TMTSF

<figure><img src="image_16.png" /><figcaption>Fig. 14. Polarized reflectance of (TMTSF)₂PF₆ at (a) 300 K; (b) 100 K and (c) 25 K. (After Jacobsen *et al.* 1981.)</figcaption></figure>
sheets), the directionality of the orbitals must be taken into account in calculating the transverse overall overlap (Wudl 1981).

The observation of the mid-infrared plasma-like resonance with $E \parallel b$ does not mean that the Fermi surface of the conducting state has closed orbits (this would require $m_b^* \approx m_a^*$). As mentioned in Section 2, short-range order develops at 25 K with coherence extending over a small number of neighbouring chains. ($TMTSF)_2PF_6$ lies in the coherent transverse conduction regime at low temperature.

We tend to believe that the high-field quantum oscillations of the resistance observed in the conducting state of $(TMTSF)_2PF_6$ under high pressure are actually due to the recovery of a semimetallic state under high field (Jerome 1981, Garoche et al. 1982). The anisotropy of the band structure manifests itself also in the behaviour of the critical fields of the superconducting state (see Section 4.3).

Consequently, in order of magnitude, it is reasonable to assume that the band parameters of $(TMTSF)_2X$ are within the ratios 1,1/20, 1/300 for $t_{\parallel} t_{\perp}^{b*}$ and $t_{\perp}^{c*}$ respectively.

<figure><img src="image_17.png" /><figcaption>Fig. 15. Temperature-dependence of the resistivity of four members of the $(TMTSF)_2X$ series under ambient atmosphere.</figcaption></figure>
The (TMTSF)₂X family shows a wide spread of electronic properties since the X=ReO₄, BF₄, PF₆, NO₃ salts undergo metal-insulator transition at 180, 40, 12 and 10 K respectively (fig. 15) while the perchlorate X=ClO₄ stays highly conducting with σ|| > 10⁵ Ω⁻¹ cm⁻¹ down to helium temperature (Bechgaard et al. 1981). Some of the differences between the various salts can be attributed to the nature of the anions. These anions do not contribute to the conduction process because they are too widely separated for any significant overlap along the *a*-axis, but they do create a potential seen by the electrons of the conducting chains. Since the anions lie at an inversion centre of the three-dimensional structure, centrosymmetric ions such as PF₆, AsF₆ etc... cannot lead to any doubling of the potential periodicity along the *a*-direction, but non-centrosymmetric ions with for example a planar structure (X=NO₃), or tetrahedral structure (X=ClO₄ or BF₄) can give rise to a doubling of the periodicity along the stacking axis. If an alternate ordering of the ions occurs, the periodicity of the electron potential becomes 2a and consequently a gap is opened at the Fermi level of the one-dimensional band structure. Such ion ordering has been observed experimentally by X-ray diffuse scattering in several salts (Pouget et al. 1982).

### 3.3. The one-dimensional pseudo-gap in (TMTSF)₂PF₆

The possibility of large one-dimensional superconducting fluctuations at low temperatures has been proposed to explain the high conductivity observed in TMTSF-DMTCNQ (Andrieux et al. 1979 b) and (TMTSF)₂PF₆ (Jerome et al. 1980). As already mentioned in Section 2.4.3, one-dimensional short-range ordering can induce a marked depression of the density of states N(E_M) at the Fermi surface in a temperature domain centred around T₁ and extending down to T₃, giving rise to a so-called 'pseudo-gap'. As far as superconductivity is concerned the pseudo-gap can be very crudely viewed as electrons leaving a normal Fermi fluid combining to form virtually-bound pair states in a superfluid at an energy above the Fermi level with time constant τ_N and decaying back to the Fermi sea with the Ginzburg-Landau decay time τ_s = πT₁/8T(T-T₁). The change of the density of states at the Fermi level is given by a detailed balance argument, namely {N(E_M)-N₀}/N₀ = τ_s/τ_N.

We shall now summarize the experimental data that show that there is a very pronounced pseudo-gap effect in the (TMTSF)₂X series.

Amongst the various techniques that can be used to determine the superconducting pairing energy, namely infrared spectroscopy, specific heat, nuclear spin-lattice relaxation and acoustic attenuation, electron quantum tunnelling is by far the most sensitive probe of the quasi-particle density of states. The transition probability of single electrons, which obey Fermi statistics, to tunnel between two metallic electrodes separated by a thin insulating barrier is proportional to the density of final states. Thus provided one electrode is in a superconducting state, the transition probability remains small as long as the bias voltage of the junction does not exceed ±Δ/e, where 2Δ is a measure of the pairing energy in the superconducting state. Since the density of tunnelling states is given by N_T(E)=N(E_M)[E²/(E²-Δ²)]¹/² in the BCS model, the junction resistance exhibits sharp minima at bias voltages ±Δ/e corresponding to the gap edge singularities

Measurements of the tunnelling resistance can also be used to determine changes in the density of states of quasi-particles in a one-dimensional conductor above T₃, which signal the development of the pseudo-gap below T₁. Such measurements have been made successfully with (TMTSF)₂PF₆ (More et al. 1981). The experiments have been performed using the Schottky barrier geometry, namely studying the tunnel resistance
of quasi-particles between a $(TMTSF)_2PF_6$ single crystal and an evaporated n-doped degenerate semiconductor through the intrinsic potential barrier existing at the interface. As the junction is put under a pressure of 11 kbar and cooled to very low temperatures the plot of the dynamic resistance of the junction $dV/dI$ versus applied bias displays the characteristic shape usually observed when one of the electrodes is a superconductor (fig. 16). The application of a magnetic field lends additional support to the superconducting origin of the tunnelling characteristics. A magnetic field decreases significantly the amplitude of the resistance peak. A decrease of the resistance peak is also noticed on warming, but the tunnelling characteristics persist up to 5 K and even up to 15 K or so in some cases.

The tunnelling characteristics in fig. 16 have been attributed to the presence of a pseudo-gap in the density of quasi-particle states below 15 K which reflects the establishment of one-dimensional short-range ordered superconducting pairing, in contrast with three-dimensional superconductivity which is observed in $(TMTSF)_2PF_6$ at 11 kbar only below $T_3=1$ K, (see Section 4). The width of the zero bias resistance anomaly (fig. 16) gives an estimate of the intrachain pairing energy $2\Delta=3.6$ meV. This energy is indeed much larger than $T_3$, but according to equation (5)

<figure><img src="image_18.png" /><figcaption>Fig. 16. The dynamic characteristic $dV/dI$ of a $(TMTSF)_2PF_6$—n/GaSb tunnel contact at $T=0.05$ K and $P=11$ kbar (1.1 GPa). The tunnel junction resistance is about 5 to 10 kΩ under ambient conditions.</figcaption></figure>
the large difference between $\sigma_{\parallel} \approx 40$ K and $T_3$ can be attributed to the weak interchain coupling $t_{\perp}^2/t_{\parallel}$. If the value for $T_1$ is derived somewhat arbitrarily from a BCS relation such as $2\Delta/kT_1 = 3.5$, one obtains $T_1 \approx 10$ K. Therefore, the phase transition is depressed by a factor of nearly 10, by one-dimensional fluctuations of the phase of the superconducting order parameter; and following the Landau-Ginzburg picture of one-dimensional conductors, deviations from mean-field behaviour are already important below 20 K.

Similar behaviour of the Schottky tunnelling has been obtained under ambient pressure with $(TMTSF)_2ClO_4$-GaSb junctions. There, a pseudo-gap of amplitude $2\Delta \sim 3.4$ meV is visible at a temperature larger than the three-dimensional ordering temperature ($T_3 \sim 1.2$ K).

Moreover, the existence of the pseudo-gap at low temperature for $(TMTSF)_2PF_6$ is not restricted to the pressure range in which a three-dimensional ordered superconducting state is stable, since far infrared reflectance experiments under ambient pressure (and at $T=25$ K) have also clearly revealed a gap around $50 \text{ cm}^{-1}$ (Jacobsen et al. 1982).

### 3.4. Paraconductivity in $(TMTSF)_2PF_6$

Apart from the transition into a superconducting state which will be discussed in Section 4, there are several unusual features in the temperature dependence of the conductivity of $(TMTSF)_2PF_6$ above 9 kbar (Schulz et al. 1981 a): (i) At helium temperatures, $\sigma_{\parallel}$ exceeds $10^5 \Omega^{-1} \text{ cm}^{-1}$ (fig. 17) which is an order of magnitude larger than the maximum conductivity observed in other organic conductors, for instance in TSF-TCNQ at $T \approx 30$ K and in some HMTSF-TCNQ crystals at helium temperature under pressure, where $\sigma_{\parallel}$ levels off at $\approx 10^4 \Omega^{-1} \text{ cm}^{-1}$. Moreover, $\sigma_{\parallel}$ increases strongly with temperature in the whole temperature range down to $T_3$, in striking contrast with usual metals, where below about 10 K the conductivity reaches its temperature-independent, impurity-limited value. (ii) The conduction anisotropy remains practically constant between 300 and 100 K but displays a significant decrease below 40 K (fig. 18). (iii) The conductivity at low temperature ($T \sim 30$ K) is extraordinarily affected

<figure><img src="image_19.png" /><figcaption>Fig. 17. Temperature-dependence of the $(TMTSF)_2PF_6$ longitudinal resistivity under 12 kbar (1.2 GPa).</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Fig. 18. Temperature-dependence of the conductivity anisotropy of (TMTSF)₂PF₆ under 12 kbar (1.2 GPa).</figcaption></figure>

by the application of a magnetic field along the transverse c*-direction. The magnetoresistance amplitude is however much smaller for $H \parallel b^*$ and *a* (fig. 19). (iv) The Kramers-Kronig analysis of the far-infrared reflectance data of (TMTSF)₂PF₆ (under ambient pressure) has shown the existence at $T=25$ K of a zero-frequency collective mode (with a life-time $\tau_c > 5 \times 10^{-13}$ s, which is significantly larger than the single particle life-time $\tau_{sp} \approx 3 \times 10^{-14}$ s) (Jacobsen *et al.* 1982) (fig. 20).

In view of this unusual conductivity behaviour and of the large pseudo-gap (short-range superconducting order) effect displayed by tunnelling experiments, it is tempting to correlate the high conductivity to an excess conductivity attributed to the acceleration by the electric field of long-life-time superconducting pairs created by fluctuations above $T_3$. This excess conductivity, also called paraconductivity, is

<figure><img src="image_21.png" /><figcaption>Fig. 19. Effect of a transverse magnetic field on the resistivity of (TMTSF)₂PF₆ under 12 kbar (1.2 GPa). The applied field in the upper curve is 5.5 tesla.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Fig. 20. Chain axis reflectance of (TMTSF)₂PF₆ at various temperatures and frequency dependent conductivity as obtained from chain axis reflectance. Notice the logarithmic frequency scale (after Jacobsen et al. 1982).</figcaption></figure>

proportional to $\langle |\Delta^2| \rangle$ i.e. it is practically negligible above $T_c$ in ordinary three-dimensional superconductors. The situation can be drastically different in one-dimensional conductors since there the paraconductivity is enhanced by a geometrical factor $(\xi_{\parallel}(0)/d_{\perp})^2$, where $\xi_{\parallel}(0)$ and $d_{\perp}$ are respectively the zero-temperature Landau-Ginzburg longitudinal coherence length and the interchain distance. This may be as large as $10^5$ in superconductors such as (TMTSF)₂PF₆.

The paraconductivity of quasi-one-dimensional conductors has been calculated by Schulz using the time-dependent Landau-Ginzburg theory (Schulz et al. 1981 a). Though this approach suffers from serious limitations in one-dimensional conductors, it provides a convenient starting point for the understanding of fluctuation effects. The summation of the Azlamazov-Larkin diagram leads to the following paraconductivity:

$$ \sigma'_{\parallel} = \left( \frac{v_M}{2t_{\perp}d_{\perp}} \right)^2 \frac{e^2m\alpha'}{16} \xi_{\parallel}(T) \left[ 1 - \frac{d_{\perp}}{(d_{\perp}^2 + 4\pi\xi_{\perp}^2)^{1/2}} \right] \quad (22 \text{ a}) $$

$$ \sigma'_{\perp} = \frac{e^2 m \alpha'}{16} \xi_{\parallel}(T) \left[ 1 - \frac{3 d_{\perp}}{2(d_{\perp}^2 + 4\pi\xi_{\perp}^2)^{1/2}} + \frac{d_{\perp}^3}{2(d_{\perp}^2 + 4\pi\xi_{\perp}^2)^{3/2}} \right] \quad (22 \text{ b}) $$

corresponding to the longitudinal and transverse components respectively where $\alpha'$ is a factor depending on the single-particle scattering time. Near $T_3$, as $\xi_{\perp} > d_{\perp}$, $\sigma'_{\parallel}$ and $\sigma'_{\perp}$ in (22) both become proportional to $(T - T_3)^{-1/2}$ (the typical square root divergence of three-dimensional superconductors). However in this temperature domain $\sigma'_{\parallel}$ is enhanced over its value in a three-dimensional superconductor by a factor $(v_M/2t_{\perp}d_{\perp})^2 \approx (t_{\parallel}/t_{\perp})^2$. In this three-dimensional domain of fluctuations, the paraconductivity anisotropy becomes $(\sigma'_{\parallel}/\sigma'_{\perp}) \approx (t_{\parallel}/t_{\perp})^2$.
Sufficiently above $T_3$, $\xi_{\perp} < d_{\perp}$ (when fluctuations of adjacent chains are one-dimensional) equation (22 a) reduces to the well-known form for a one-dimensional superconductor, namely:

$$ (\sigma'_{\parallel})_{1d} = \frac{\pi e^2}{16 d_{\perp}^2} \xi_{\parallel}(0) \epsilon^{-3/2} \quad \text{where} \quad \epsilon = \frac{T - T_1}{T_1} \qquad (23) $$

A transverse supercurrent between adjacent chains requires a well developed order parameter on the two chains. As the probability for this to happen decreases with increasing temperature, $(\sigma'_{\perp})_{1d}$ decrease much faster than $(\sigma'_{\parallel})_{1d}$. The anisotropy of paraconductivity in the one-dimensional regime increases linearly with the temperature (Schulz et al. 1981a, Jerome 1981):

$$ (\sigma'_{\parallel}/\sigma'_{\perp})_{1d} \approx \frac{T - T_1}{T_1} \qquad (24) $$

according to equations (22 a, b).

A numerical estimate of equation 22 leads to

$$ (\sigma'_{\parallel})_{1d} \approx 800 \frac{\xi_{\parallel}(0)}{d} \Omega^{-1} \text{ cm}^{-1} \text{ at } T=2T_1, \text{ taking } d_{\perp}=4 \text{ Å.} $$

Thus superconducting fluctuations could very easily give a conductivity in excess of $10^5 \Omega^{-1} \text{ cm}^{-1}$, since $\xi_{\parallel}(0)/d_{\perp}$ may well be over 100, according to critical field determinations (see Section 4). The behaviour of $\sigma_{\parallel}/\sigma_{\perp}$ upon cooling is fairly well understood in terms of equation (24) and provides additional evidence for the onset of fluctuations below 40 K. Furthermore the conductivity of superconducting origin exceeds the single-particle contribution $\sigma_0 \approx 10^4 \Omega^{-1} \text{ cm}^{-1}$ below 40 K or so. At helium temperature the single-particle conductivity is negligible with respect to the paraconduction which becomes the dominant conduction channel.

# 4. Three-dimensionally ordered states

## 4.1. The spin-modulated insulating state

Most of the conduction salts of the (TMTSF)₂X family exhibit two possible ground states. The phase diagram of (TMTSF)₂PF₆, which is displayed in fig. (21), shows that, under pressure, superconductivity becomes the ground state above 9 kbar or so. Superconducting and insulating ground states share a common phase separation which is a quasi-vertical line in the phase diagram of fig. (21). The nature of the insulating state is of particular interest with respect to the discussion of the diagram on fig. (21). X-ray diffuse scattering experiments on (TMTSF)₂AsF₆ have failed to detect any significant softening of 2k_M phonon modes in the vicinity of the metal-insulator transition (M-I transition) which are expected for a Peierls-like transition. Moreover, several experiments have firmly established that the insulating state of (TMTSF)₂PF₆ and (TMTSF)₂AsF₆ is magnetic. Contrary to the ground states of TTF-TCNQ, TSF-TCNQ, etc... where the spin susceptibility vanishes, there remains a finite but anisotropic spin susceptibility below $T_{M-I}$ in (TMTSF)₂PF₆ and (TMTSF)₂AsF₆. The insulating state of these two salts is not a Peierls state but is instead a spin-modulated state.

Two sets of experiments have characterized the nature of the insulating state. First the existence of internal magnetic fields has been substantiated by n.m.r. experiments performed on selenium nuclei (Andrieux et al. 1981). The ⁷⁷Se n.m.r. line-width
<figure><img src="image_23.png" /><figcaption>Fig. 21. Phase diagram of (TMTSF)₂PF₆ featuring the insulating (or semimetallic spin modulated state, the three-dimensional ordered superconducting state and the conducting state (probably dominated by large superconducting fluctuations). (After Jerome 1982.)</figcaption></figure>

broadens abruptly around 12 K and below 11 K the n.m.r. signal can no longer be detected (fig. 22). The proton line-width also shows a significant increase at $T_{M-1}$ but remains visible in the insulating state (Andrieux et al. 1981). Since both nuclei have $I = \frac{1}{2}$ spins, the only possible interpretation of the line broadening is the existence of local magnetic fields. Secondly, single-crystal static susceptibility measurements performed on (TMTSF)₂AsF₆ and (TMTSF)₂PF₆ have shown an anisotropic spin susceptibility at low fields ($H<4$ kOe; 0.4 tesla), (fig. 23) below $T_{M-1}$ (Mortensen et al. 1982). The easy axis of the magnetization is close to the *b*-axis of the crystal structure whereas *a* and *c* are respectively the medium and hard axis. Field dependences show that the magnetization undergoes a spin-flop transition when the magnetic field along *b*\* (the easy axis) exceeds 4.5 kOe (0.45 tesla). All these data are typical of an antiferromagnetic ground state.

Slater (1951) considering the particular case of a half-filled band conductor has shown that the self-consistent Hartree-Fock potential of the electrons cannot have the full translational symmetry of the lattice in the case of antiferromagnetic ordering in a

<figure><img src="image_24.png" /><figcaption>Fig. 22. Intensity of the <sup>77</sup>Se n.m.r. signal in (TMTSF)₂PF₆ versus temperature, showing the vanishing (through line broadening) of the signal at the metal-insulator transition temperature.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Fig. 23. Spin susceptibility of $(TMTSF)_2AsF_6$ as deduced from static measurements with field parallel to $a$, $b^*$ and $c^*$. The magnetic field of 3 kgauss ($\equiv 0.3$ tesla) is less than the spin-flop field (after Mortensen *et al.* 1982). A susceptibility of $10^{-4}$ e.m.u. $mol^{-1} \equiv 10^{-5} J T^{-2} mol^{-1}$.</figcaption></figure>

metal, but that the exchange term of the potential exhibits a periodicity twice the lattice periodicity. In particular for the half-filled band situation of $(TMTSF)_2X$ the new periodicity of the antiferromagnetic state becomes $2a$ and gaps open on the Fermi surface at the points connected by the spanning vector. In quasi-one-dimensional conductors such as $(TMTSF)_2X$ the spanning vector is likely to be $Q=(2k_M, \pi/b, \pi/c)$ according to the 'warped' Fermi surface model discussed by Horovitz *et al.* (1975) for the Peierls transition. The insulating character of the antiferromagnetic state suggests that the energy gap covers most of the $Q$-one-dimensional Fermi surface, which implies that the transverse tunnelling couplings cannot be very large.

For an antiferromagnet, the eigen-modes for electron spin resonance are no longer the Zeeman spin resonance modes but instead modes which involve exchange and anisotropy fields. Torrance *et al.* (1982) have found that below $T_{M-1}=12$ K, $(TMTSF)_2PF_6$ behaves like an antiferromagnet with an easy plane and a hard axis ($c^*$-direction). Accordingly, antiferromagnetic resonance is observed under zero applied magnetic field at the frequencies $\omega_+$ and $\omega_- \approx 35$ and 12 GHz respectively at $T=4.2$ K). The shift of the e.p.r. line away from the Zeeman value occurring below $T_{M-1}$ can therefore be taken as a clue to the magnetic nature of the dielectric state.

An enhancement of the spin-lattice nuclear relaxation is also observed in the insulating state (Andrieux *et al.* 1981) in sharp contrast with the drop of relaxation in a Peierls insulator. The enhanced relaxation is due to the coupling between the nuclear spins and the fluctuations of the spin modulations.
## 4.2. *The anion-ordered insulating state*

Not all conducting salts of the $(TMTSF)_2X$ show a spin-modulated insulating state at low temperature under ambient pressure. For instance in the perchlorate salt, the resistance continues to fall at helium temperatures and finally drops to a zero value at a superconducting transition. For the salts of non-centrosymmetric anions like $NO_3$, $BF_4$ and $ReO_4$, the crystal structure imposes at 0 K either a statistical disorder of the anions with unit cell length *a*, or an ordering which leads to a doubling of the cell length along the *a*-axis. This ion ordering has been clearly observed in $(TMTSF)_2ReO_4$ at 180 K (Pouget *et al.* 1982) where it coincides with the occurrence of an insulating state. From the point of view of transport properties, ion ordering through a doubling of the *a*-unit cell or one-dimensional lattice softening of the $2k_M$ mode cannot be distinguished, since they both trigger the occurrence of a gap at $\pm k_M$. But X-ray diffraction experiments can clearly discriminate these different states since the intensity of the satellite spots in the three-dimensional ordered state which is related to the square of the displacement amplitude is much bigger for an ion ordering than for a lattice modulation.

As far as $(TMTSF)_2NO_3$ is concerned, the $NO_3$ ordering below 40 K is characterized by the wave vector $(2k_M, 0, 0)$ (Pouget *et al.* 1982) which does not represent a vector nesting the entire surface of the warped three-dimensional Fermi surface. Therefore, the ordering is only accompanied by a small decrease of the density of states at the Fermi level, which does not prevent the occurrence of a semimetal-insulator transition at lower temperature (Jerome 1982).

## 4.3. *The superconducting state*

Under a sufficiently large pressure (fig. 24) superconductivity has been observed in most members of the $(TMTSF)_2X$ series (Jerome *et al.* 1980, Parkin *et al.* 1981), namely $X=PF_6$, $AsF_6$, $SbF_6$, $TaF_6$, $ReO_4$ with $(TMTSF)_2ClO_4$ showing superconductivity at ambient pressure (Bechgaard *et al.* 1981). Superconductivity has been firmly established by a zero resistance state (Jerome *et al.* 1980), quasi-complete diamagnetic shielding (Ribault *et al.* 1980), the flux expulsion in small applied magnetic fields (Andres *et al.* 1980) (figs. 25 and 26), and the observation of a specific heat anomaly (Garoche *et al.* 1982) (fig. 27). These experiments guarantee that superconductivity below $T_3$ is a bulk phenomenon. Throughout the series of superconducting compounds, the superconducting state exhibits similar behaviour: there is a critical pressure $P_c$ above which the three-dimensional superconducting state is stabilized at a maximum $T_3$ of $\approx 1.2-1.3$ K.

The comparison between the critical pressure for the stabilization of the superconducting ground state and the average intermolecular dimerization throughout the $(TMTSF)_2X$ series suggests that the stability of the dielectric ground state is related to the amplitude of the dimerization and in turn to the amplitude of the Umklapp scattering term $g_3 = g_1\Delta/E_F$ (where $\Delta$ is the amplitude of the dimerization gap) (Barisic and Brazovskii 1980). However another possible correlation between $P_c$ and the lattice parameters has been proposed by Parkin *et al.* (1981); *c* characterizes the separation of the TMTSF sheets and it is determined by the anion size. Thus, whereas the lattice parameters *a* and *b* vary little ($<0.1\%$) and in no systematic way as the anion is varied from the smallest $ClO_4$ to the largest $TaF_6$, *c* monotonically increases by about 5%. $P_c$ can be correlated with *c* and depends therefore on the interchain coupling strength via the inorganic anions. One expects that the conductivity anisotropy with respect to the *a*-axis will be largest along the *c*\* direction. The largest critical pressure of about 9 kbar
<figure><img src="image_26.png" /><figcaption>Fig. 24. Sketch of the $(TMTSF)_2AsF_6$ resistivity showing a conductor to insulator transition at 12 K under zero pressure and the stabilization of a highly conducting state at low temperature under 11 kbar (1·1 GPa).</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Fig. 25. Evidence for superconductivity in $(TMTSF)_2PF_6$ under 12 kbar (1·2 GPa) from (a) a zero-resistance state and (b) a.c. susceptibility by comparison with a niobium marker sample ((b), upper panel).</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>Fig. 26. D.C. magnetization of (TMTSF)₂ClO₄ with a magnetic field applied along the *c*\* direction. The dashed line refers to the diamagnetic shielding. (After Mailly and Ribault 1982).</figcaption></figure>

in (TMTSF)₂TaF₆ would require a change of *c* of some 0.5% kbar⁻¹ which compares well with values ranging from 0.5 to 0.27% kbar⁻¹ measured in TTF-TCNQ.

The band-structure anisotropy is not to be neglected in the behaviour of the superconducting state below *T*₃. The weak interchain tunnelling coupling makes the magnetic field penetration relatively easy, so that superconductivity is of type II. The lower critical field *H*⊥c₁ for fields perpendicular to the chain axis is so low that only rough estimates can be made from DC magnetization data. Careful SQUID-assisted measurements on (TMTSF)₂ClO₄, with screening of the Earth's magnetic field, have led to *H*⊥c₂ ~ 0.5 Oe (5 × 10⁻⁵ tesla) (at *T*=0.05 K with *H*∥*b*\*) (Mailly and Ribault 1982). The upper critical field *H*⊥c₂ which is easily derived from resistive methods reveals a very pronounced anisotropy related to the band-structure anisotropy. As far as the *b*-*c* plane is concerned, there exists an overall consensus regarding the various experimental results, for *H*<sup>b\*</sup><sub>c₂</sub>/*H*<sup>c\*</sup><sub>c₂</sub> ≈ 15–20 with *H*<sup>b\*</sup><sub>c₂</sub>=18 kOe (1.8 tesla) at *T*=0.2 K for (TMTSF)₂AsF₆ under 11 kbar (*T*₃ = 1.15 K) (Brusetti *et al.* 1982, Greene *et al.* 1982) (fig. 28).

The experimental situation is still very controversial for the *a* to *b* anisotropy, since the slope d*H*<sup>a</sup><sub>c₂</sub>/d *T* in (TMTSF)₂ClO₄ (under ambient pressure) varies from 7.5 kOe/K up to 70 kOe/K (0.75 tesla/K up to 7.0 tesla/K) according to the authors (Gubser *et al.* 1981). The scatter of the experimental values may be due to the severe experimental
<figure><img src="image_29.png" /><figcaption>Fig. 27. Electronic molar heat capacity of (TMTSF)₂ClO₄ single crystals in (a) zero applied field and (b) under 63 kOe) (6·3 tesla), along *c** (after Garoche et al. 1982).</figcaption></figure>

requirements in aligning the sample with the magnetic field for the determination of the parallel critical fields.

A convenient way to treat anisotropy effects in superconductivity is the Ginzburg-Landau formalism developed for low-dimensional conductors.

For open Fermi surfaces, corresponding to quasi-one-dimensional (coherent) conductors provided $t_⊥ \ll t_{||}$, the anisotropy of the coherence length (order parameter) is $\xi_{||}(T)/\xi_{⊥}(T) = t_{||}/t_⊥$ (Schulz et al. 1981 a). Therefore, this model when applied to the temperature dependences of $H_{c2}^⊥$ and $H_{c2}^‖$ near $T_3$ in (TMTSF)₂ClO₄ leads to $\xi_{||}(0)/\xi_{⊥}(0) \approx 25$, since $dH_{c2}^⊥/dT = \Phi_0/2\pi\xi_{||}(0)\xi_{⊥}(0)$ at $T_c$ and $dH_{c2}^‖/dT = \Phi_02\pi\xi_{⊥}^2(0)T_c$ (where the $⊥$ direction is not specified). The resulting value for the band-structure anisotropy agrees fairly well with the typical high temperature conductivity anisotropy in the (TMTSF)₂X family, namely $\sigma_{||}/\sigma_⊥ = (t_{||}/t_⊥)^2 \approx 300-1000$ (Bechgaard et al. 1980).

The assumption of closed Fermi surfaces which would require a $t_⊥$ not very much smaller than $t_{||}$ (a semimetallic picture) would lead to $\xi_{||}/\xi_⊥ = (t_{||}/t_⊥)^{1/2}$ (Schulz et al.
<figure><img src="image_30.png" /><figcaption>Fig. 28. Transverse critical fields $H_{c2}$ in $(TMTSF)_2AsF_6$ (after Brusetti et al. 1982). The longitudinal critical field of $(TMTSF)_2ClO_4$ (after Gubser et al. 1981) has also been drawn for comparison. An applied field of 1 kOe is equivalent to $10^{-1}$ tesla.</figcaption></figure>

1981 a). The lower limit to the a to b anisotropy of $\xi$ in $(TMTSF)_2ClO_4$ is about 6. This would lead to a band-structure anisotropy $t_{\parallel}^a/t_{\perp}^b \approx 36$ which is inconsistent with the starting assumption ($t_{\perp}^b \approx t_{\parallel}^a$) for closed Fermi surfaces.

### 4.4. Specific heat data and three-dimensional ordering

The specific heat of $(TMTSF)_2ClO_4$ which has been measured between 0.4 and 2 K, exhibits several striking features (Garoche et al. 1982):

(i) A large anomaly of the electronic contribution at 1.22 K which is related to the onset of three-dimensional superconducting order (fig. 27), and which follows a nearly mean field behaviour.

(ii) A density of states at the Fermi level derived from the electronic contribution to the total specific heat at $T > T_3$ which is significantly enhanced by the application of a transverse magnetic field.

(iii) The restoration under large magnetic field of a ground state exhibiting a very low density of states at the Fermi level (either a narrow gap semiconducting or a semimetallic state).

The specific heat anomaly suggests a small critical width for the three-dimensional fluctuations around the three-dimensional transition in agreement with the generalized Ginzburg-Landau theory of weakly coupled metallic chains by Menyhard (1978). The field-dependent electronic density of states at the Fermi level corresponds to the gradual suppression of the pseudo-gap by the magnetic field (already observed by electron-tunnelling experiments in $(TMTSF)_2PF_6$ (More et al. 1981)).

Furthermore, the restoration of a semimetallic ground state by the magnetic field seems a direct consequence of the suppression of one-dimensional fluctuations.
Therefore, the specific heat data reinforce the description of the superconductivity in the (TMTSF)₂X family in terms of one-dimensional physics.

### 4.5. Pressure and irradiation effects on organic superconductivity

Figure 21 shows the pressure dependence of $T_3$ for superconductivity, namely $\partial \ln T_3 / \partial P \approx -7\% \text{ kbar}^{-1}$ after the data of Schulz et al. (1981 b), Andres et al. (1980) and Greene et al. (1980). A similar pressure dependence of $T_3$ has been observed in (TMTSF)₂ClO₄ (Parkin et al. 1981).

Since bandwidths derived by optical and thermopower data lead to $W \approx 0.5-1 \text{ eV}$, the $T_3$ pressure coefficient should be compared with that of d-band superconductors. In those metals, where the electron-electron pairing can be ascribed to a phonon-mediated attraction, $T_c$ can either decrease by hardening or acoustic phonons or increase by the enhancement of the electron-phonon matrix element. The combination of these effects may lead to an increase or a decrease in $T_c$ with pressure depending on the metal studied, as observed experimentally.

However, as far as organic superconductors are concerned one can investigate some possible reasons for the large pressure coefficients of $T_3$. Band broadening under pressure seems to be small, following the results of plasma-edge measurements under pressure in TTF-TCNQ or TSF-TCNQ (Welber et al. 1978) and the Seebeck coefficient of (TMTSF)₂PF₆ ($\approx 1/E_F$), which does not exhibit any significant pressure dependence between 1 atm and 12 kbar (C. Weyl, D. Jerome and K. Bechgaard 1981, unpublished).

Moreover in molecular conductors, intramolecular phonons are known to be strongly coupled to the electrons and such modes have been shown to be favourable to superconductivity (Horovitz 1977). However, both the frequency and strength of coupling of these modes to the electrons are molecular properties and are therefore little dependent on pressure.

Other possibilities for the pressure dependence of $T_3$ are directly related to the spirit of quasi-one-dimensional theories.

Current theories of quasi-one-dimensional predict an increase of $T_3$ with increasing $t_\perp$. As $t_\perp$ is increased under pressure one would expect $\partial T_3 / \partial P > 0$, in contrast with the experimental results.

However, the apparent discrepancy between the observed pressure dependence of $T_3$ and the most naive quasi-one-dimensional picture (assuming for instance $\alpha_\parallel$ unaffected by pressure) may not rule out the interpretation of organic superconductivity in terms of one-dimensional physics (Greene et al. 1980) since: (i) as $t_\perp$ is weakly increased under pressure, $t_\parallel$ increases at the same rate, the conduction anisotropy at room temperature remaining practically unchanged up to 12 kbar in (TMTSF)₂PF₆. Therefore the Josephson coupling is not dramatically enhanced; and (ii) under 24 kbar, whereas $T_3$ is strongly depressed by pressure ($T_3 \approx 0.2 \text{ K}$), the behaviour of the resistance of (TMTSF)₂PF₆ reveals a paraconduction (fig. 29) as strong (at least) as the contribution observed under 12 kbar (or even ambient pressure) in (TMTSF)₂ClO₄ (Schulz et al. 1981 b).

This latter fact suggests that $\xi_\parallel$ (to which the paraconduction is proportional (equation (23)) could be increased under pressure and consequently $\alpha_\parallel$ or $T_1$ decreased since $\xi_\parallel$ is inversely proportional to the longitudinal pairing energy.

Another appropriate theoretical framework within which to consider the pressure dependence of $T_3$ is the calculation of phase transition temperature of quasi-one-dimensional conductors made by Suzumura and Fukuyama (1978). They considered finite tunnelling coupling between individual chains as represented by the Byschkov,
<figure><img src="image_31.png" /><figcaption>Fig. 29. Resistivity of $(TMSF)_2PF_6$ under 24 kbar (2.4 GPa) (after Schulz *et al.* 1981 b).</figcaption></figure>

Gorkov and Dzyaloshinskii model (1966) (with $g_1 > 0$, which seems to be the situation in the $(TMSF)_2X$ family following the existence of a magnetic state). $g_1$ can be expected to decrease somewhat under pressure if an analogy is drawn with the magnetic properties of TTF-TCNQ. This statement is supported by the weakening of the nuclear spin-lattice relaxation under pressure (Soda *et al.* 1977). Consequently, according to Suzumura and Fukuyama's calculation a small decrease in $g_1$ could easily result in a large drop in the phase transition temperature even though a concomitant decrease in $g_2$ favours a small enhancement.

The previous discussion of the pressure-dependence of $T_3$ reveals clearly that more experiments are needed to establish or invalidate the use of quasi-one-dimensional theories for the $(TMSF)_2X$ series.

The last (but not least) striking effect which we shall discuss in this article is the extreme sensitivity of organic superconductivity to the defects created by X-ray irradiation. A defect concentration of 0.1% per molecule is large enough to suppress the zero-resistance state and to smear out the precursor superconducting regime (Bouffard *et al.* 1982). Some of the defects created by X-ray irradiation carry an unpaired spin as shown by e.p.r. investigations. Thus, they are pair-breaking. A concentration of 0.1% unpaired spins will suppress $T_3$ if the exchange coupling between the electrons and the loca spins is of the order of 0.1 eV with $E_F \sim 0.1$ eV.

## 5. Conclusion

The recent work on organic chain conductors stresses a number of interesting points.

(1) In the polarizable organic molecules used as building blocks, the intermolecular Coulomb interactions are modest. But the conductivity along the chains is achieved by transfer between (non-bonding) double bonds, leading to narrow bands. The ratio between these two quantities, namely Coulomb interactions and band widths, might be favourable to a type of superconductivity directly related to the Coulomb repulsions, and not to the phonon-mediated attractions, as in normal BCS superconductors. This might open the possibility of stronger superconductive interactions along the chains, with high
mean field transition temperatures $T_1$. However, there is so far no conclusive evidence for such a triplet superconductivity, and the observed $T_{1s}$, although high, are not exceptional.

(2) In chain superconductors, thermal superconducting fluctuations can reach high temperatures of order $T_1$, and this might be of practical use.

(3) A coherent tunnelling of Cooper pairs between neighbouring chains produces a sizeable three-dimensional superconductive temperature $T_3$, which could raise by increasing the strength of this transverse coupling, either uniformly or at local points between the chains.

It is also clear that more efforts in the field of organic superconductivity can be worthwhile in two directions:

(1) arrangements of other organic molecules with varying degrees of intra-atomic Coulomb interactions and interatomic transfer integrals;

(2) other topologies, such as three-dimensional networks of conducting chains or piling up of conducting planes.

From a wider point of view, it seems that purely organic materials can have on the one hand all the variety of conducting and magnetic behaviour of more classical metals or semiconductors, but with a great flexibility associated with the use of organic molecules, and leading to a variety of topologies, couplings and vibrations, and on the other hand the unique possibility of exhibiting properties related to one-dimensional physics, namely a ground-state showing always a broken symmetry (either insulating or superconducting).

## Acknowledgments

The opening of this promising field has been due to the collaboration of many people, chemists as well as physicists, theoreticians and experimentalists.

The close cooperation of Orsay with several groups in Organic Chemistry, especially K. Bechgaard from the Oersted Institute and L. Giral and J. M. Fabre from the University of Montpellier, was essential.

As always in a complex field, the point of view reported here can but be sketchy and one-sided. We wish to stress that Orsay's contribution would not have developed so positively without the initial impetus and the inspired collaboration of M. Weger, the late G. Soda, J. R. Cooper and R. H. Friend.

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