# Organic superconductors: from (TMTSF)2PF6 to Fullerenes

Denis Jerome

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## To cite this version:

Denis Jerome. Organic superconductors: from (TMTSF)2PF6 to Fullerenes. Jean-Pierre Farges. Organic conductors, fundamentals and applications, M. Dekker, 1994. hal-05000226

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Distributed under a Creative Commons Attribution 4.0 International License
# ORGANIC CONDUCTORS
## FUNDAMENTALS AND APPLICATIONS

EDITED BY

JEAN-PIERRE FARGES

Université de Nice - Sophia Antipolis
Nice, France

Marcel Dekker, Inc.

New York • Basel • Hong Kong

Copyright © 1994 by Marcel Dekker, Inc.
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# Organic Superconductors: From $(TMTSF)_2PF_6$ to Fullerenes

**Denis Jérôme**

Université Paris-Sud, Orsay, France

## I. INTRODUCTION

Molecular conductors are basically different from the common metallic conductors, in that they are solids made of building blocks (the molecules) presenting their own specific properties: that is, molecular orbitals with their ionization energy or electron affinity, characteristic infrared or Raman active modes, and nuclear magnetic resonance (NMR) chemical shifts of different atomic sites. At variance with conventional molecular crystals made of neutral organic molecules held together by weak van der Waals forces, organic conductors contain molecules with unpaired carriers in π molecular orbitals (open shells) which allow the delocalization over all molecular sites in the crystal via a strong intermolecular π overlap. The open-shell character comes from a partial oxidation (reduction) of donor (acceptor) molecules in the formation of a salt with an inorganic anion (cation). These molecular conductors are therefore different from the extended conjugated polymers or graphite, where the π system of the extended molecules provides the conducting pathway.

The development of molecular conductors (superconductors) has been stimulated to a large extent by the suggestion of Little in 1964 [1] that the arrangement of chain conductors in a polarizable medium could give rise to superconductivity at high temperature. The prerequisite for the formation of a conductor is first, the requirement for having charged molecules
in the solids, and second, to allow these charges to delocalize between molecular entities. Charging the molecules is achieved either by a charge-transfer reaction (see our subsequent discussion of dmit superconductors) or by the formation of an organic salt. For the charges to be delocalized between the molecules, the on-site Coulomb repulsion must not overcome the energy, which is gained by the formation of energy bands in the solid. Thus a condition such as $U \le 4t_\parallel$ should prevail where $U$ and $4t_\parallel$ are the on-site repulsion and the bandwidth, respectively, for the molecular crystal to be a metal-like conductor.

In a rough approximation the on-site Coulomb interaction can be estimated from the Coulomb repulsion between two additional electrons located on the same molecule. If it is possible for the two electrons to be far from each other in a large molecule of size 10 to 20 Å, the Coulomb repulsion is on the order of 1 to 5 eV and the dielectric screening by the environment may reduce it by an order of magnitude, making it about 0.3 to 0.5 eV. However, the most stringent criterion is established by the strong tendency toward the formation of insulating phases at low temperature, as they may arise from possibilities of distortions opening energy gaps over the entire Fermi surface in conductors of low dimensionality.

Although the superconducting instability requires three-dimensional long-range order, it is customary to make the difference between the physical properties at low temperature, which are three-dimensional (and possibly very anisotropic) and those at high temperature, where one- or two-dimensional features may prevail [2]. Both the geometry of the building bricks and their packing in a three-dimensional lattice govern the dimensionality of the transport properties of these particular conductors, which ranges from one to three (Fig. 1). The intermolecular overlap between molecular orbitals is optimized by the stacking of the planar molecules on top of each other like pancakes or by the packing of similar molecules in two-dimensional conducting sheets.

As far as one-dimensional conductors are concerned, the planar shape of the molecules is responsible for their particular crystal structure (Fig. 1). All these molecules are modifications of the flat fulvalene skeleton [3]. Noticeable intermolecular overlap is allowed when they form stacks and the conduction band derives from $\pi$ orbitals directed along the stacks. The overlap of the molecular orbitals is greatest along the stacks of molecules and much weaker between them, making the conductivity greater along the stacking axis than along any other directions. Typical bandwidths in segregated stacks of molecules are in the range 0.4 to 1 eV [4].

The salts of the tetramethylselenafulvalene (TMTSF) molecule [5] are considered as prototype materials for one-dimensional molecular conductors. The overlap between organic stacks along the transverse b direction
<figure><img src="image_4.png" /><figcaption>**Figure 1** Prototype molecules TTF (tetrathiafulvalene and TMTSF (tetramethyl-tetraselenafulvalene) which are used for the elaboration of organic conductors TTF-TCNQ or superconductors (TMTSF)₂X. View of the crystal structure of some molecular superconductors: (a) (TMTSF)₂PF₆; (b) (alkali)₃C₆₀; (c) (BEDT-TTF)₂Cu(SCN)₂ side view; (d) view along the axis perpendicular to the conducting plane.</figcaption></figure>
amounts to $t_{\perp b} \approx 20$ to 30 meV, which is about one-tenth the value for the intrastack overlap ($t_{\parallel} \approx 200-300$ meV), and makes the conduction band about 1 eV wide. The overlap is negligible along the c direction. Such a marked anisotropy causes the Fermi surface to be nearly planar, intersecting the $k_{\parallel}$ axis at wave vectors $\pm k_{\text{F}}$, where the Fermi wave vector for a one-dimensional conductor is related to the density of carriers $\rho$ per unit cell by the relation $2k_{\text{F}} = \rho\pi/a$ [6].

These one-dimensional salts are members of a much broader isostructural family: the (TM)₂X series, which includes salts of selenium containing molecules with a variety of inorganic anions as well as sulfur-containing molecules [7]. The stabilization of superconductivity below a certain critical temperature $T_c$ requires overcoming the strong divergence developing at low temperature in a one-dimensional electron gas toward an insulating behavior, which is discussed later in the chapter [8]. (TMTSF)₂PF₆ is actually the compound in which superconductivity has been observed for the first time albeit under a pressure of 9 kbar [9]. However, (TMTSF)₂ClO₄ is the only compound pertaining to the one-dimensional family undergoing a transition toward superconductivity at $T_c = 1.2$ K under ambient pressure [10]. These are rather low values for a transition toward superconductivity, but this feature is closely related to the competition between superconductivity and other instabilities arising in one-dimensional conductors, as we shall see below.

In another interesting series of one-dimensional conductors where the cation is an arene such as perylene, the anion is a transition-metal-coordinated complex $[M(mnt)₂]⁻$ which provides an additional degree of freedom, depending on whether the metal atom carries a spin $M = Ni, Pd, Pt, \text{etc.}$ or not for $M = \text{Au}$ [11]. Superconductivity has also been found in another series of materials based on transition metal complexes $M(dmit)₂$, where $M = \text{Ni}$ or $\text{Pd}$ with $T_c = 1.6$ K under 6 kbar [12].

Higher-dimensionality conductors also derive from similar planar molecules, which form dimers with strong overlap between heteroatoms both within each dimer and between neighboring dimers (Fig. 1). This crystal structure gives rise to considerable conductivity within molecular layers and about $10^4$ less conduction between the sheets. They are considered as two-dimensional conductors (see Ref. 13 for an overview). Their Fermi surface is tubular with $t_{(a,b)} = 100$ to 150 meV, $t_{\perp c} \approx 0.5$ meV and consequently, $T_c$ rises above 8 K in $(BEDT-TTF)₂I₃$ [14,15] and even up to 10 to 12 K for anions such as $Cu(SCN)₂⁻$ [16] and $Cu[N(CN)₂]Cl⁻$ [17] or $Br⁻$ [18] since superconductivity is no longer competing in these two-dimensional conductors with typical one-dimensional instabilities.

The recent discovery of spherical molecules containing 60 carbon atoms arranged over the surface of a sphere, each atom being linked to three
neighbors forming the C₆₀ molecule, has opened the gate to isotropic mo-
lecular conductors (the so-called fullerene materials) [19]. Three among
the four valence electrons of each carbon atom are engaged in sp² bonding
with the three neighbors. There remains one electron per carbon atom in
a p-state which is delocalized over the entire surface of the sphere in π
molecular orbitals. The neutral C₆₀ molecule possesses a closed-shell elec-
tronic structure giving rise to an insulating molecular solid. However, pop-
ulating the lowest unoccupied molecular levels (the LUMO) with carriers
given by inorganic cations such as alkali or alkaline earth metals gives rise
to a partly filled conduction band. This band is half-filled in the prototype
material A₃C₆₀ (A₃ = K, Rb, Cs₂ Rb, etc.) [20]. The C₆₀ spheres are
packed in a face-centered-cubic lattice, leaving cavities with octahedral and
tetrahedral local symmetry which are all occupied in the A₃C₆₀ compound (Fig.
1). Very much like one- or two-dimensional molecular conductors, the
conductivity in fullerenes is due to the overlap of π molecular orbitals
between near-neighbor molecules, counterions playing essentially no role
in the conduction process. The shape of the molecules together with the
crystal structure make A₃C₆₀ a three-dimensional conductor with a con-
duction band that is very narrow (≤0.5 eV) due to the weakness of inter-
molecular interactions. However, it is the elaboration of a preparation
route based on graphite sublimation and allowing a mass production of C₆₀
molecules that boosted the discovery of superconductivity in this new family
of molecular conductors [21]. Superconductivity ranges from 19 K in K₃C₆₀
[22] up to 28 K in Rb₃C₆₀ [23]. The highest T_c value stabilized at the moment
for a molecular conductor is 33 K in Cs₂RbC₆₀ [24].

## II. ONE-DIMENSIONAL ELECTRON GAS

A. Mean-Field Solution of the One-Dimensional Problem

Although a mean-field approximation will not provide the correct answer
to the problem of interacting one-dimensional electrons, a brief survey
using such an approach is still useful to show that the competition between
superconducting and dielectric order is the basic problem of one-dimen-
sional conductors [25a,b].

We are concerned with low-energy processes (near the Fermi energy).
Consequently, the only pertinent wave vectors of electron–hole pair ex-
citations will be those close to q = 0 and 2k_F since the electron–hole pair
excitation energy is zero for these vectors in the one-dimensional nonin-
teracting electron gas. The Fourier transform of the Coulomb interaction
will contain terms near q = 0 and q = 2k_F which are approximated by
various constants $g_i$ in the interaction term of the Hamiltonian (g-ology model) [26] (Fig. 2a).

The one-dimensional Hamiltonian reads.

$$H = H_0 + H_{\text{int}} \qquad (1)$$

<figure><img src="image_5.png" /><figcaption>**Figure 2** (a) Linearized electron dispersion in the Luttinger approximation (A) and diagrammatic representation of elementary interactions $g_1 \cdots g_4$ (B). Solid and dashed lines represent electrons near $k_F$ and $-k_F$, respectively. The $g_3$ interaction exists only in case of a half-filled band; (b) Diagrammatic representation of the Cooper pair susceptibility $\Delta(q, \omega)$ and the density wave susceptibility $\Pi(2k_F + q, \omega)$ at lowest order (C) and second order (D) which shows the mixture between Cooper and Peierls channels.</figcaption></figure>
where the noninteracting contribution is linearized in $q$ near the Fermi energy (Luttinger approximation), namely,

$$H_0 = \sum_{k,\sigma} V_F \left[ (k - k_F)a_{k,\sigma}^{\dagger}a_{k,\sigma} + (-k - k_F)b_{k,\sigma}^{\dagger}b_{k,\sigma} \right] \quad (2)$$

and

$$H_{\text{int}} = \sum_{k_1 k_2} g_1 a_{k_1 \sigma}^{\dagger} b_{k_2 \sigma'} a_{k_2 + 2k_F + Q, q, \sigma'} b_{k_1 - 2k_F - Q, \sigma} + g_2 a_{k_1 \sigma}^{\dagger} b_{k_2 \sigma'}^{\dagger} b_{k_2 + Q, \sigma' a_{k_1 - Q, \sigma}} \quad (3)$$

where $g_1$ and $g_2$ are the amplitudes of the elementary scattering amplitude [with change of momentum $q = 0$, forward ($g_2$) and $q = 2k_F$, backward ($g_3$)].

In addition to the $g_1$ and $g_2$ terms in Eq. (3) there are $g_3$ terms involving the transfer of two particles from one side of the Fermi surface to the other. Since the total momentum transfer is $4k_F$, this process is allowed only if $4k_F$ is a reciprocal lattice vector (Umklapp scattering).

We now define new operators $O_\alpha(q)$, which are the Fourier components of operators characterizing the various ordered states of the interacting one-dimensional gas, namely $\alpha = \text{CDW, SDW, or singlet, triplet super-conductivity SS and TS, for example:}$

$$O_{\text{CDW}}(q) = \sum_{k,\sigma} b_{k-q,\sigma}^{\dagger} a_{k,\sigma} \quad (4a)$$

$$O_{\text{SDW},i}(q) = \sum_{k,\sigma,\sigma'} b_{k-q,\sigma}^{\dagger} \sigma_{i}^{\sigma\sigma'} a_{k,\sigma'} \quad (4b)$$

where $\sigma_i^{\sigma\sigma'}$ are the elements of the Pauli matrices for $i = x, y, z$.

Using the definitions, Eqs. (4), $H_{\text{int}}$ can be rewritten

$$H_{\text{int}} = \sum_{q} (g_1 - \frac{1}{2}g_2) O_{\text{CDW}}^{+}(q) O_{\text{CDW}}(q) + g_2 \sum_{q,i} O_{\text{SDW},i}^{+}(q) O_{\text{SDW},i}(q) \quad (5)$$

One can also rewrite operators such as $O_{\text{CDW}}(q)$ like

$$O_{\text{CDW}}^{+}(q) O_{\text{CDW}}(q) = (O_{\text{CDW}}^{+}\langle O_{\text{CDW}} \rangle + O_{\text{CDW}}\langle O_{\text{CDW}}^{+} \rangle) - |\langle O_{\text{CDW}} \rangle|^2 \\ + [O_{\text{CDW}}^{+}(q) - \langle O_{\text{CDW}}^{+}(q) \rangle] [O_{\text{CDW}}(q) - \langle O_{\text{CDW}}(q) \rangle] \quad (6)$$

In the mean-field approximation, the possibility for fluctuations of the order parameter is forbidden. Therefore, the last term in Eq. (6) is zero and $H_{\text{int}}$ reduces to the mean-field Hamiltonian.
$$ \bar{H}_{\text{int}} = \sum_{k} \langle O_{\text{CDW}}(q) \rangle (a_{k_F+k\uparrow}^{\dagger} b_{k_F+k-q\uparrow} + a_{k_F+k\downarrow}^{\dagger} b_{k_F+k-q\downarrow}) \\ + \langle O_{\text{CDW}}^{\dagger}(q) \rangle (b_{k_F+k-q\uparrow}^{\dagger} a_{k_F+k\uparrow} + b_{k_F+k-q\downarrow}^{\dagger} a_{k_F+k\downarrow}) \quad (7) $$

The Hamiltonian (7) can be diagonalized by a Bogoliubov transformation. A gap $\Delta_{\text{CDW}}$ is opened at the Fermi surface for $q = 2k_F$.

$$ \Delta_{\text{CDW}} = 2E_F \exp\left(-\frac{1}{\lambda_{\text{CDW}}}\right) \quad \text{where} \quad \lambda_{\text{CDW}} = -\frac{2g_1 - g_2}{2\pi V_F} $$

The gap $\Delta_{\text{CDW}}$ is finite if $\Delta_{\text{CDW}} > 0$, which sets the condition,

$$ 2g_1 - g_2 < 0 $$

One could have looked similarly for a finite value of the SDW order parameter which is obtained when the condition $g_2 > 0$ is fulfilled. Consequently, no electron-hole instability exists in the domain of the $g_1, g_2$ plane limited by

$$ g_2 < 0 \quad \text{and} \quad g_1 > \frac{g_2}{2} $$

In addition, the stability of singlet and triplet superconductivity requires $g_1 + g_2 < 0$ and $g_1 - g_2 > 0$, respectively. If all four instabilities are now treated on an equal footing, the diagram leading to the most stable ground state is as given in Fig. 3. Singlet (SS, CDW) and triplet (TS, SDW) phases occur for $g_1 < 0$ and $g_1 > 0$, respectively. Superconducting and density wave orders are separated by the line $g_1 = 2g_2$.

The diagram for the stability of different ground states in quasi-one-dimensional interacting electron gases has had a powerful impact on the development of the field. This was the first suggestion for a common border between electron-electron pairing instability and the spin-modulated ground state. It has also been suggested that superconductivity could be stabilized in one-dimensional conductors by repulsive interactions.

## B. Beyond Mean-Field Approximation

However, the mean-field treatment of the one-dimensional system suffers from serious drawbacks. The first and most important of these is the prediction of long-range order at a finite temperature, whereas in a one-dimensional system fluctuations are known to destroy long-range order [25a], as can easily be illustrated by the stability of a ferromagnetic Ising
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 3 (a) Ground states of the one-dimensional mean-field model. SM, LM, TS, and SS indicate spin modulation, lattice modulation, and triplet and singlet superconductivity, respectively. (b) The most divergent correlations of the one-dimensional Hamiltonian in the presence of Umklapp scattering SDW correlations are more divergent than CDW correlations in the $g_1 > 0$ region.</figcaption></figure>

chain. Starting with a configuration corresponding to ferromagnetic order, the reversal of a spin leads to a change in internal energy $J (>0)$. If a concentration $n$ of such reversed spins exists in the chain, the change of free energy with respect to the ferromagnetic configuration reads [2]

$$ \Delta F = nJ + kT[n \log n + (1 - n)\log(1 - n)] \quad (8) $$

Its minimization $\delta \Delta F / \delta n = 0$ when $n$ is small leads to $n = \exp[-(1 + J/kT)]$ or equivalently, to the temperature-dependent correlation length:

$$ \frac{\xi(T)}{a} = \exp\left(1 + \frac{J}{kT}\right) \quad (9) $$

$\xi(T)$ reaches infinity only at zero temperature. Thermal fluctuations destroy the onset of long-range order at any finite temperature.

The previous treatment deals with a one-component order parameter (such as for a commensurate Peierls distortion) but does not apply to situations where the order parameter is complex with an amplitude and a phase (superconductivity, incommensurate Peierls, or spin density wave transitions). The latter situation is analogous to classical moments which can rotate freely in an *XY* plane. The coherence length of the *XY* model is less strongly divergent at low temperature than for the Ising model,
namely:

$$ \frac{\xi(T)}{a} = \frac{J}{kT} \qquad (10) $$

The second drawback of the mean-field approximation is neglect of electron-phonon interaction, which should contribute attractively to the $g_1$ contribution and lead to a lattice modulation (Peierls instability). However, a ground state of such a nature is not stabilized in the TM$_2$X family. The absence of long-range order is deeply rooted in the one-dimensional problem and arises when the effect of interactions on the different response functions is calculated perturbatively [26].

Response functions $R(\omega)$ for all instabilities contain at all orders in perturbation logarithmic terms, such as $\ln(\omega/E_F)$, which give rise to infrared divergencies. Similar arguments apply to the bare couplings $g_1$, which must be replaced by effective couplings (vertex) representing perturbative expansions of the bare coupling. The perturbation expansion reveals a mixture of divergent contributions coming from electron-electron pairs and electron-hole pair divergences. An example of mixture of two types of instabilities in Cooper and Peierls susceptibilities is shown for typical diagrams in Fig. 2b. The mixture of diverging channels is specific to a one-dimensional conductor and is responsible for the absence of long-range order. The one-dimensional problem (i.e., the determination of the temperature dependence of response functions) can be solved using various approximations [26].

In the diagrammatic expansion, the most trivial approximation neglects the mixture of channels. For example, calculating the Cooper pair susceptibility, we neglect contributions coming from density wave fluctuations; this approximation amounts to the mean-field treatment. Some improvements have been provided by the summation of a specific class of diagrams (the parquet diagrams) [25b], which takes into account the channel mixture but still gives rise to long-range order at $T > 0$.

An improved treatment is obtained by renormalization group methods, which lead to the absence of long-range order at any finite temperature [26]. However, interactions now become temperature dependent. A very simple illustration of the one-dimensional behavior is provided by the first-order renormalization calculation of $g_1$ and $g_2$, which become

$$ g_1(T) = \frac{g_1}{1 - (g_1/\pi v_F) \ln(T/E_F)} \qquad (11a) $$

$$ g_2(T) = g_2 - \frac{1}{2}g_1 - g_1(T)] \qquad (11b) $$

where $g_1$ and $g_2$ are the amplitudes taken at high temperature ($T > E_F$).
The response functions $R_i(q, \omega)$ for instabilities $i$ are found to behave like

$$R_i(q, \omega) \approx |\omega^2 - V_F^2 q^2|^{-\alpha_i/2} \quad \text{at } T = 0 \quad (12a)$$

and

$$R_i(0, 0) \approx T^{-\alpha_i} \quad \text{at } \omega = q = 0 \quad (12b)$$

where $\alpha_i$ is an exponent for the divergence of correlations in the channel $i$ (CDW, SDW, TS, SS) which depends on the bare couplings $g_1$ and $g_2$. The diagram for the most diverging fluctuations is analogous to the mean-field diagram in Fig. 3. In the one-dimensional regime, the temperature dependence of response functions are characterized by power law dependences, Eq. (12b).

### C. Spin-Charge Separation

Together with the power law dependence and the absence of phase transitions, the separation of the Hamiltonian into spin and charge parts is a peculiar feature of one-dimensional conductors. The concept of spin-charge separation can be visualized very easily with a one-dimensional half-filled band model in the limit of large $U$ preventing double-site occupancy. Strong short-range antiferromagnetic order due to $J \approx t^2/U$ is expected in such a chain.

Removing a carrier at $t = 0$ creates both additional charge and spin locally. After some time the charge will have evolved due to the kinetic term of the Hamiltonian without flipping the spins. Therefore, the original situation of a coupled charge-spin excitation will decay in time into a charge excitation carrying no spin (holon) and at a different place a spin excitation carrying no charge (spinon).

<table><tr><td>↑↓↑↓↑↓↑↓↑</td><td>Ground state</td></tr><tr><td>↑↓0↓↑↓↑↓↑</td><td>Excited state at t = 0</td></tr><tr><td>↑↓↓↑↓↑0↓↑</td><td>Excited state at time t<br/>[spinon (↓↓) + holon (0)]</td></tr></table>

We may also consider a one-dimensional chain of electrons with one electron per site in the large $U$ limit of the Hubbard model ($U > t$). The ground state of the chain is insulating since it costs an energy $U$ to move an electron from one unit cell to its neighbor. The spins adopt an antiferromagnetic configuration which provides the antisymmetrization of the ground-state wavefunction. Double occupancy of $k$ states in the conduction band is thus forbidden by coulombic repulsions.

If we now consider the band picture, each $k$ state is doubly occupied when $U$ is negligible. But when $U$ is switched on, one may consider that
double occupancy is forbidden, and therefore for a given concentration of carriers the number of (singly) occupied $k$ states below the Fermi energy should double. These carriers can thus be defined as a noninteracting gas of spinless fermions [27]. The Fermi vector for these new particles becomes $2k_F$ instead of $k_F$ in the initial noninteracting gas. This effect gives rise to the well-known $4k_F$ diffusion in diffuse x-ray scattering [28].

The band splits into a lower singly degenerate Hubbard band and an other singly degenerate Hubbard band at the energy $U$ above the former band. When the site occupancy is $\rho = 1$, the lower Hubbard band is fully occupied and conducting configurations correspond to the excitation of carriers into the upper band at the energy $U$. The ground state of such a system is thus a Mott-Hubbard insulator.

There exists a major difference between a band insulator and a Mott-Hubbard insulator as far as spin excitations are concerned. Flipping a spin in a band insulator is forbidden by the Pauli principle and requires an excitation of the carrier through the energy gap. Therefore, spin and charge degrees of freedom are coupled. However, flipping a spin in a Mott insulator requires only the exchange energy $J \approx t^2/U$, which is much smaller than the energy $U$ that is necessary for conduction when $U > t$.

Spin and charge excitations are thus decoupled by coulombic interactions in the one-dimensional electron gas. However, the one-dimensional Fermion system is not a Fermi liquid, as indicated by the behavior of the momentum distribution function, which does not exhibit a Fermi step at $k_F$ and presents a single-particle density of states vanishing according to a power law singularity at $E_F$. This is a Luttinger liquid [29] with

$$n_k \approx n_{k_F} - \beta \operatorname{sign}(k - k_F)|k - k_F|^\alpha, \quad \beta > 0$$

$$N(\omega) \approx |\omega|^\alpha \quad \text{where} \quad \alpha = \frac{1}{4} \left( K_\rho + \frac{1}{K_\rho} - 2 \right)$$

$K_\rho$ determines the long-distance decay of all correlation functions of the system. The temperature dependence of the response functions becomes

$$\chi_{\text{SDW,CDW}}(2k_F) \approx T^{K_\rho - 1} \quad \text{and} \quad \chi_{\text{SS,TS}} \approx T^{1/K_\rho - 1} \qquad (13)$$

The noninteracting gas corresponds to $K_\rho \approx 1$ and density waves or pairing correlations are divergent at low temperature for $K_\rho < 1$ and $> 1$, respectively. The observation of power law dependences for the response functions in an extended domain of temperatures should thus be taken as the signature of one-dimensionality. One-dimensionality is indeed limited at low temperature by the crossover regime toward a physics at higher dimensionality.
The response functions that diverge logarithmically at low temperature in the one-dimensional noninteracting electron gas level off at a value $log(t_⊥/πE_F)$ corresponding to $T ≈ t_⊥$ [30] when the interchain coupling $t_⊥$ is taken into account. Band structure calculations would thus suggest a bare crossover temperature for the single-particle motion of $T_x^p ≈ 80$ K or so. However, it has been suggested [31] that the single-particle interchain tunneling can be impeded by one-dimensional correlations. The crossover regime is further reduced and reads

$$T_x \sim T_x^o \left( \frac{t_⊥}{E_F} \right)^{(1-K_p)/K_p} \quad (14)$$

This effect can be quite significant even for relatively small coupling $K_p$ $≤ 1$ and may lead to a suppression of the dimensionality crossover down to, say, $T_x ≈ 10$ K given $t_⊥/E_F ≈ 10$.

Below the crossover, both channels are decoupled. A Fermi-liquid behavior is recovered using renormalized values of the interactions at the crossover temperature. We may expect mean-field treatments for dimensionality 2 or 3 to become valid at $T < T_x$ (see, e.g., the critical exponents for the SDW instability in $(TMTSF)_2PF_6$ (see Section III.B).

The existence of a finite interchain interaction in a quasi-one-dimensional system is a prerequisite for a phase transition. However, there exists no necessary relation between the strengths of intra- and interchain couplings. It is thus conceivable to observe for SDW and SC instabilities the existence (if not the predominance) of magnetic fluctuations above the superconducting transition or the reverse as well. Such situations are encountered in the study of the $(TMTSF)_2X$ series.

### III. THE TM₂X FAMILY

#### A. Electronic Structure and Phase Diagram

As mentioned above, all members of the TM₂X series are isostructural compounds and no major difference could be expected at first glance between the four sulfur $(TMTTF)_2PF_6$ and the four selenium $(TMTSF)_2ClO_4$ compounds. They both exhibit the typical zigzag molecular packing (tri clinic symmetry) with two molecular units per unit cell (i.e., a formally half-filled band).

A model band structure has been calculated for TM₂X materials by Grant [32] using a tight-binding scheme with a few simplifications. An important one is the use of the highest occupied molecular orbital (HOMO) of the isolated molecules as the only state that is relevant for the band structure close to the Fermi level. The band structure calculated with six
near-neighbor transfer integrals obtained from quantum chemistry, which are summarized in Table 1 [33,34], is displayed in Fig. 4.

The energy dispersion along XV and ΓY in Fig. 4a is the signature of the nonzero interstack coupling, and the gap along XV reflects the existence of the structural dimerization. An overall decreasing tendency for the band dimerization going from $(TMTTF)_2PF_6$ to $(TMTSF)_2PF_6$ is indicated in Table 1. The conduction band can accommodate four electrons per $TM_2$ units. Therefore, removing one carrier per unit cell leads to a half-filled situation for the upper band in Fig. 4b.

$(TMTSF)_2ClO_4$ is a fairly good metal-like conductor at room temperature ($\sigma_{||} \approx 500 \ \Omega^{-1} \cdot \text{cm}^{-1}$) and undergoes a transition toward a superconducting ground state at 1.2 K [10]. On the other hand, $(TMTTF)_2PF_6$ is a poor conductor ($\sigma_{||} (300 \text{ K}) \approx 30 \ \Omega^{-1} \cdot \text{cm}^{-1}$) which shows a semi-conducting behavior on cooling [35]. The ground state of the latter system is somewhat reminiscent of a Peierls state, as it displays (below 20 K) a lattice modulation with periodicity $2a$ along the stacking axis ($2k_F$ wave vector) [36]. In addition, there is zero-spin susceptibility in this ground state since the isolated spins are dimerized in a spin Peierls state [37].

For both systems the temperature dependence of the spin susceptibility is about similar, with a 40% monotonous drop from 300 K to 50 K [37,38]. Clearly, there is no close relation between transport and magnetic properties in the high-$T$ regime of these salts. This is an experimental illustration of the spin-charge separation concept of one-dimensional conductors.

The compounds mentioned above are only two members of the $TM_2X$ series, whose generic phase diagram is shown in Fig. 5a. The names of compounds along the $x$ axis indicate their location at atmospheric pressure in the generalized diagram. In addition, the pressure scale shows that it is also possible for a given compound to be shifted throughout the entire

<table><caption>**Table 1** Transfer Integrals Intrastacks ($t_s$) and Interstacks ($t_i$) of Selected One-Dimensional Conductors. All Energies are in Units of $10^{-3}$ eV</caption><thead><tr><th></th><th>t<sub>s</sub></th><th>t<sub>i</sub></th><th>Δt<sub>s</sub>/t<sub>s</sub><sup>a</sup></th></tr></thead><tbody><tr><td>(TMTSF)<sub>2</sub>PF<sub>6</sub></td><td>365</td><td>26.2</td><td>0.16</td></tr><tr><td>(TMTSF)<sub>2</sub>ClO<sub>4</sub></td><td>366</td><td>21.6</td><td>0.14</td></tr><tr><td>(TMTTF)<sub>2</sub>Br</td><td>240</td><td>10</td><td>0.13</td></tr><tr><td>(TMTTF)<sub>2</sub>PF<sub>6</sub></td><td>115</td><td>12.3</td><td>0.38</td></tr></tbody></table>

<footnote><sup>a</sup>The value of $\Delta t_s/t_s$ reflects the amplitude of the structural dimerization.</footnote>

<footnote>Source: Ref. 33.</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 4** (a) Fermi surface (schematic) and band structure for (TMTSF)₂X (after Ref. 32) and for β(ET)₂X (after Ref. 32b); (b) schematic illustration of the structural dimerization giving rise to a half-filled upper band in (TMTSF)₂X compounds.</figcaption></figure>

diagram by applying hydrostatic pressure with a concomitant evolution of their physical properties. Materials shown in Fig. 5a all have centrosymmetric anions (with the noticeable exception of $ClO_4^-$), while noncentrosymmetric anions give rise to further problems (see Section III.E).

The long-range order which is stabilized at low temperature depends on the nature of the incipient instabilities that develop at high temperature along each stack on a short-range scale without any transverse correlation (i.e., one-dimensional fluctuations). It is therefore more illustrative to present the physical properties versus temperature for a few representative
<figure><img src="image_9.png" /><figcaption>**Figure 5** (a) Generalized phase diagram for the TM₂X series. Notations SP, SDW, and SC refer to spin-Peierls, spin density wave, and superconducting ground states, respectively. The dashed line marks the limit between metal-like and localized (CL) behavior. The zero-pressure locations of several prototype compounds are indicated [after Ref. 8]. (b) Electronic structure of the one-dimensional electron gas showing the 2kF wave vector for noninteracting particles and the 4kF wave vector in the lower Hubbard band of strongly interacting particles.</figcaption></figure>

systems as shown in Fig. 5. We shall pay attention to spin and charge degrees of freedom making use of the experimental data obtained from transport, susceptibility, and NMR experiments.

## B. Nuclear Magnetic Resonance and Transport Properties

NMR spin lattice relaxation measurements provide very direct information about the Fourier transform of the spin susceptibility χ(q,ω) in a one-dimensional conductor [39]. The spin degrees of freedom constitute a relaxation channel for nuclear spin due to the modulation of the hyperfine interaction by the electron spin time dependence, which is given generally
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

in *D* dimensions by the Moriya relation [40],

$$ (TT)^{-1} = 2\gamma_n^2 |A|^2 \int dqD \frac{\chi_{\perp}^{\parallel}(q, \omega_n)}{\omega_n} \qquad (15) $$

where $\chi_{\perp}^{\parallel}(q, \omega_n)$ is the imaginary part of the transverse spin susceptibility taken at the nuclear Larmor frequency $\omega_n$ and integrated over all $q$ vectors. The summation over $q$ simplifies for a one-dimensional conductor, only $q = 0$ and $2k_F$ wave vectors giving rise to low-lying excited states. Hence $T_1^{-1}$ for one-dimensional conductors contains only $q = 0$ and $q = 2k_F$ contributions probing uniform and antiferromagnetic correlations of the electron gas, respectively [41,42]. $T_1^{-1}$ can be written formally as

$$ [T_1 T]^{-1} = C_0 \chi_s^2(T) + C_1(T) \qquad (16) $$

The first contribution in Eq. (16) probes the uniform ($q = 0$) spin correlations, $\chi_s(T)$ is the Faraday spin susceptibility, and $C_0$ is a constant depending on the hyperfine parameters. The second term probes the $q = 2k_F$ contribution. The $2k_F$ contribution can be temperature dependent or
independent, depending on the strength of the antiferromagnetic fluctuations since $C_1(T)$ is proportional to $\chi(2k_F, T)$, which follows a one-dimensional power law:

$$ \chi(2k_F, T) \sim \left( \frac{T}{E_F^*} \right)^{-\gamma_{1D}} \qquad (17) $$

where $\gamma_{1D}$ is an exponent ($> 0$) describing the divergence of the antiferromagnetic fluctuations in the one-dimensional regime ($T > T_x$) and $E_F^*$ is a renormalized cutoff ($E_F^* < E_F$) [43]. The relation between $\gamma_{1D}$ and the correlation exponent $K_\rho$ defined earlier reads

$$ 1 - \gamma_{1D} = K_{\rho} \qquad (18) $$

The extreme situation of strongly repulsive interactions in a one-dimensional quantum antiferromagnet corresponds to $\gamma_{1D}$ (or $K_\rho = 0$) and according to Eq. (16), $T_1^{-1}(2k_F)$ becomes temperature independent. The small $q$ spin excitations of the one-dimensional electron gas are not singular at low temperature, and as the temperature is increased, uniform fluctuations grow at the expense of diminishing antiferromagnetic correlations.

At the left side of the diagram, the compound $(TMTTF)_2PF_6$ exhibits metal-like properties above room temperature but below a temperature $T_\rho$, where the resistivity passes through a shallow minimum [$T_\rho \approx 250$ K for $(TMTTF)_2PF_6$], a loss of the charge degrees of freedom is observed, although the spin susceptibility remains unaffected (one-dimensional quantum antiferromagnet). The loss of the charge degrees of freedom has been explained in terms of coulombic repulsions in a one-dimensional conductor [44]. For the situation of one particle per site, there exists a correlation gap $\Delta_\rho$ in the charge excitation spectrum (the Mott-Hubbard gap). Therefore, when $T$ becomes smaller than $\Delta_\rho/k_B$ charge excitations become frozen out. Hence the chain of localized spins with periodicity $a$ ($4k_F$ wave vector); Fig. 5b is described by the Heisenberg linear model, where the antiferromagnetic interaction given by $2t_\parallel^2/U$ in the strong-coupling limit depends on the lattice spacing through the overlap integral $t_\parallel$. Thus the system of localized spins may gain energy by dimerizing, giving rise at low temperature to one-dimensional lattice fluctuations with a doubled periodicity ($2a$ or wave vector $2k_F$) which are detected by x-ray experiments [36]. Simultaneously, the spin susceptibility follows fairly well the temperature dependence of the Bonner-Fisher model [45] down to a temperature $T_{sp} \approx 60$ K for $(TMTTF)_2PF_6$ [37], below which one-dimensional lattice fluctua-
tions are strongly developed and $\chi_s$ is decreased with respect to the one-dimensional quantum antiferromagnet by the opening of the spin-Peierls pseudogap at the Fermi level. Finally, below 20 K, these one-dimensional $2k_F$ fluctuations order three-dimensionally and the system undergoes a phase transition at $T_{sp} = 20$ K toward a nonmagnetic singlet ground state (total spin = 0) in which the lattice is tetramerized [36].

The NMR data of the ¹³C relaxation rate of (TMTTF)₂PF₆ are shown in a $T_1^{-1}$ versus $T\chi_s^2(T)$ diagram [41b,46], where $\chi_s^2$ is the spin susceptibility obtained from EPR or Faraday spin susceptibility experiments (Fig. 6). $T_1^{-1}(T)$ data follow very closely the law given by Eq. (17). The very good agreement with a law $C_1 \sim T^{-1}$, which is observed between 40 and 200 K, indicates that the $4k_F$ localized spins adopt the dynamics of a one-dimensional quantum antiferromagnet, with a weight of $2k_F$ spin correlations, which is dominant even up to room temperature ($K_p = 0$). The spin-Peierls ground state is accompanied by an activation of the relaxation, reflecting the freezing of the spin degrees of freedom in a spin-singlet ground state.

Moving to the right in the generic diagram, the metallic character becomes predominant as both the room-temperature conductivity increases and the positive temperature dependence of the resistivity extends to lower temperature [$T_p \approx 100$ K for (TMTTF)₂Br] [47] and also for the mixed S-Se(TMDTDSF)₂PF₆ salt [48]. The role of one-dimensional coulombic repulsions is attenuated in the latter salts, as shown by the Mott localization taking place only below 100 K or so. Once more the temperature-dependent spin susceptibility above 100 K [47,48] does not depart significantly from that of (TMTTF)₂PF₆ [37].

However, the nuclear relaxation rate behaves differently since the $T^{-1} \sim C_0\chi_s^2(T)$ law is followed accurately at all temperatures $T > T_p$, whereas the $2k_F$ contribution to $T_1^{-1}$ grows only below $T_p$ and gives rise to a temperature-independent term below $T \approx T_p/2$. The temperature dependence of the $2k_F$ contribution can be used to extract the exponent $\gamma_{1D}$ (or $K_p$) for one-dimensional correlation functions. According to Fig. 7, $\gamma_{1D}$ reaches the strong-coupling limit $\gamma_{1D} = 1$ below $T_p/2$, and the vanishing $2k_F$ part above $T_p$ is also consistent with the power law dependence of Eq. (17) with $0 < \gamma_{1D} < 1$ in this high-temperature domain.

The data for (TMTTF)₂Br are somewhat less detailed, but they are qualitatively similar to those of the mixed S-Se compound. The observation of one-dimensional $2k_F$ lattice fluctuations is very marginal in these two compounds, with only a weak intensity arising below 60 K, which vanishes below 20 K in (TMDTDSF)₂PF₆ [49]. However, the diffuse scattering does not condense into Bragg peaks as for the spin-Peierls transition
<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

**Figure 6** (a) Resistivity [after Ref. 47] and (b) EPR spin susceptibility of (TMTTF)₂X compounds [after Ref. 46] undergoing a one-dimensional Mott-Hubbard localization below Tp; (c) electron spin susceptibility of (TMTTF)₂PF₆ [after Ref. 38]; (d) plot of the ¹³C spin lattice relaxation rate versus Tχs²(T) for three compounds in the TM₂X series. The temperature below which the charges are localized is indicated by Tp. No localization is observed for Se compounds (dashed line) above the SDW or SC ordering. (From Ref. 41b.)
<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

**Figure 7** Properties of the mixed compounds (TMDTDSF)₂PF₆: (a) Faraday susceptibility and resistivity (insert); (b)⁷⁷Se $T_1^{-1}$ versus temperature profile showing the onset of a SDW ground state at 7 K; (c)⁷⁷$T_1^{-1}$ versus $T\chi_s^2$, revealing the 2kF SDW contribution below $T_p$ which is plotted versus temperature in part (d). (e) 2kF one-dimensional scattering amplitude and linewidth, which is the signature of spin-Peierls fluctuations vanishing below 2 K when SDW fluctuations are taking over (after Ref. 49). (From Refs. 48 and 51.)
<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /><figcaption>Figure 7 Continued</figcaption></figure>
<figure><img src="image_20.png" /></figure>

in $(TMTTF)_2PF_6$. Instead, the divergences of the relaxation rate which are observed at $T_N$ = 19 and 7 K for $(TMTTF)_2Br$ [50] and $(TMDTDSF)_2PF_6$ [51], respectively, show the signature of a ground state with low-lying magnetic excitation modes. This ground state is characterized by a finite modulation of the spin density with a wave vector **Q** that is called a spin density wave phase.

The ¹³C-NMR line-shape analysis at $T < T_N$ in $(TMTTF)_2Br$ has proved that the periodicity of the magnetic modulation is commensurate with the underlying lattice [52]. The spin lattice relaxation is activated in the SDW ground state since there exists a finite energy gap in the phason modes of this commensurate SDW phase (Fig. 8).

One may notice that the three-dimensional magnetic fluctuations are competing in $(TMDTDSF)_2PF_6$ against the $2k_F$ lattice fluctuations and finally, impose a magnetic long-range order at 7 K [51] instead of the spin-Peierls ground state, which could have been expected as a natural consequence of the one-dimensional $2k_F$ fluctuations visible at high temperature.

As far as all selenium molecular compounds are concerned, the shallow minimum of the resistivity is no longer observed, but different behaviors are observed for the representative compounds $(TMTSF)_2PF_6$ and $(TMTSF)_2ClO_4$. The conductivity of the former compound reaches $10^6$ $(\Omega \cdot cm)^{-1}$ at 12 K and then vanishes abruptly as the system undergoes a metal-insulator transition [15] toward an antiferromagnetic (SDW) ground state, as indicated by NMR [53] and susceptibility measurements [54]. Furthermore, ¹³C NMR has proved the incommensurate nature of this SDW ground state [52]. It is the presence of this additional periodicity in
<figure><img src="image_21.png" /><figcaption>**Figure 8** $^{13}$C-NMR spectrum of the 100% $^{13}$C-enriched central double bond of TMTTF in (TMTTF)₂Br at T = 300 K (a) and of the NMR spectrum (b) showing the existence of a commensurate spin modulation with two distinct magnetic sites and an activated relaxation rate (c). (From Ref. 52.)</figcaption></figure>

the exchange potential that is the cause of the gap opening at the Fermi surface (M–I transition). The various components of the wave vector Q are determined as far as the PF₆ salt is concerned by the best nesting conditions of the Fermi surface of the quasi-one-dimensional conductor. These conditions do not necessarily coincide with commensurate values [55]. Finally, slowly cooled (TMTSF)₂ClO₄ does not reveal any M–I and magnetic transition before the superconducting ground state is stabilized at 1.2 K [10].

Although the plot $T_1^{-1}$ versus $T\chi_s^2$ in Fig. 6 does not reveal any finite intercept as $T\chi^2 \to 0$, the presentation of the NMR data as $\log T_1^{-1}$ versus $\log \chi_s^2$ in Fig. 9 [41b] shows a significant deviation to the law $T_1^{-1} \approx T\chi_s^2$ below 150 and 30 K in PF₆ and ClO₄ salts, respectively. A quantitative analysis of the $T_1^{-1}$ temperature profile leads to the power law exponent $\gamma_{1D} \sim 0.90$ below 100 K in (TMTSF)₂PF₆ and a similar value for ClO₄ below 30 K.
<figure><img src="image_22.png" /><figcaption>Figure 10 (a) Low-temperature $^{77}$Se $T_1^{-1}$ versus T data of (TMTSF)$_2$ClO$_4$ in the relaxed state. (After Ref. 41b.) (b) Temperature dependence of $^{77}$Se $T_1^{-1}$ in (TMTSF)$_2$PF$_6$ at P = 1 bar, 5.5 kbar, 9 kbar, and 11 kbar. (After Ref. 57.)</figcaption></figure>

type of relaxation:

$$T_1^{-1} \sim T \left( \frac{T_x}{E_F^*} \right)^{-\gamma_{1D}} \quad (19)$$

where $0 \le \gamma_{1D} \le 1$.

The data of ClO$_4$ and PF$_6$ under pressure [41b] (Fig. 10) reveal that $T_x \approx 8$ K, which can be reconciled with the observed Korringa enhancement of 10 and a renormalized Fermi energy $E_F^* \approx 100$ K. Below $T_x$ the two-dimensional electron gas undergoes a transition toward either a superconducting ground state (ClO$_4$ or PF$_6$ under pressure) or an antiferromagnetic ground state (TMTTF)$_2$Br or (TMTSF)$_2$PF$_6$. Under such circumstances the nuclear relaxation is enhanced by three-dimensional critical fluctuations in
<figure><img src="image_23.png" /></figure>

the vicinity of $T_N$ with the critical exponent $\frac{1}{2}$ [56]:

$$T_1^{-1} \approx (T - T_N)^{-1/2} \qquad (20)$$

The critical divergence of Eq. (20) is followed accurately for the onset of SDW ground states arising either from a quasi-one-dimensional metal as for $(TMTSF)_2PF_6$ or from a one-dimensional Mott-Hubbard paramagnetic insulator [i.e., $(TMTTF)_2Br$ at ambient pressure or $(TMTTF)_2PF_6$ under 13 kbar] [57] (Fig. 11).

Relaxation and susceptibility data have been used to extract the values of the interactions governing the spin degrees of freedom, with $\chi_s(T)$ given
<figure><img src="image_24.png" /><figcaption>**Figure 11** (a) $^{13}$C-$T_1^{-1}$ of (TMTTF)₂PF₆ at ambient pressure showing a spin-Peierls transition below 20 K and under 13 kbar, where a SDW ground state is stabilized with a $(T - T_N)^{-1/2}$ divergence of $T_1^{-1}$ as shown by the insert. (b) Log-log plot of $T_1^{-1}$ versus $T - T_N$ in the critical domain of (TMTTF)₂PF₆ at 1 bar (right scale, dots) and 5.5 kbar (left scale, squares). (After Ref. 41b.)</figcaption></figure>
[26] by

$$ \chi_s(T) = \chi_0 \left[ 1 - \frac{g_1(T)}{2} \chi_0 \right]^{-1} \quad (21) $$

where $\chi_0$ is the bare susceptibility of the one-dimensional electron gas and $g_1(T)$ is the renormalized backward-scattering amplitude given by Eq. (11a). Values of $g_1/\pi\nu_F$ for several TM$_2$X salts and for TTF Ni[(dmit)$_2$]$_2$ are reported on Table 2 using $E_F$ data derived from NMR experiments or band structure calculations.

We can see from Table 2 that the amplitude of $g_1/\pi\nu_F$ is nearly the same throughout the S and Se series, as expected from the similar amplitudes for the temperature variation of $\chi_s(T)$ up to 300 K in both cases. There is no scaling under pressure between the uniform susceptibility of S and Se compounds. However, such a scaling exists for $2k_F$-spin correlations, which depend primarily on the amplitude of the Umklapp scattering $g_3$. When $g_3$ is larger or smaller than $2g_2 - g_1$, the electron gas evolves toward a strong-coupling situation at low temperature with strongly developed AF correlations or toward the paramagnetic weak-coupling limit, respectively [58]. A crude approximation for $g_3$ is $g_3 = g_1(\Delta/E_F)$ [44], where $\Delta$ is given by the bandgap of the dimerized structure. Therefore, band parameters in Table 1 give $g_3/\pi\nu_F$ equal to 0.38 and 0.16 $g_1/\pi\nu_F$ for S and Se compounds,

<table><caption>**Table 2** Parameters Describing the Behavior of Transport and Magnetic Properties of Some TM$_2$X Compounds in the Conducting (or Localized) Phase at High Temperature$^a$</caption><thead><tr><th>Compound</th><th>T<sub>p</sub><br>(K)</th><th>K<sub>p</sub></th><th>g<sub>1</sub>/πν<sub>F</sub><br>(at 0 K, 1 bar)</th><th>E<sub>F</sub><br>(K)</th></tr></thead><tbody><tr><td>TMTTF<sub>2</sub>PF<sub>6</sub></td><td>250</td><td>0<br>T &lt; 250 K</td><td>1.05</td><td>1600</td></tr><tr><td>TMDTDSF<sub>2</sub>PF<sub>6</sub></td><td>100</td><td>0<br>T &lt; 50 K</td><td></td><td></td></tr><tr><td>TMTTF<sub>2</sub>Br</td><td>100</td><td>0<br>T &lt; 50 K</td><td>0.94</td><td>1900</td></tr><tr><td>TMTSF<sub>2</sub>PF<sub>6</sub></td><td>No</td><td>0.1<br>T &lt; 100 K</td><td>1.16</td><td>3100</td></tr><tr><td>TMTSF<sub>2</sub>ClO<sub>4</sub></td><td>No</td><td>0.2<br>T &lt; 30 K</td><td></td><td></td></tr><tr><td>TTF-Ni[(dmit)]<sub>2</sub>]<sub>2</sub></td><td>No</td><td>0.4</td><td>1.1</td><td>1500</td></tr></tbody></table>

<footnote>$^a$K$_p$, $g_1$, and $E_F$ have been derived from NMR experiments.</footnote>

<footnote>Source: Ref. 41b.</footnote>
respectively. The dimerization gap of S compounds is expected to be sup-pressed under sufficiently high pressure [33,59]; then T<sub>p</sub> goes to zero and the amplitude of AF fluctuations joins smoothly the one found in the selenide series at low pressure, which are less dimerized.

C. Transport Properties in the Metallic Regime

The room-temperature conductivity of different organic superconductors
can vary in a large proportion, depending on the specific compound [6].
The conductivity is highest in the four Se-based systems, (TMTSF)<sub>2</sub>X(σ<sub>||</sub>
≈ 600 (Ω · cm)<sup>-1</sup>). The room-temperature conductivity decreases moving
toward the left in the generic diagram of Fig. 5. The conductivity is gov-
erned by the character of longitudinal and transverse charge motions.

The transverse charge motion is incoherent for a quasi-one-dimensional
conductor as long as the condition t<sub>⊥</sub> << ħ/τ<sub>||</sub> is fulfilled (where the intra-
chain scattering time provides a broadening to the quasi-one-dimensional
Fermi surface). This situation is encountered at high temperature. The
interchain hopping rate is thus determined by the one-dimensional Fermi
golden rule [60]:

$$
\frac{1}{\tau_{\perp}} = \frac{|t_{\perp}|^2}{\hbar^2 \tau_{\parallel}} \qquad (22)
$$

leading to a temperature-independent value for the conductivity anisotropy:

$$
\frac{\sigma_{\parallel}}{\sigma_{\perp}} \sim \left( \frac{t_{\parallel}a}{t_{\perp}b} \right)^2 \qquad (23)
$$

The experimental data of (TMTSF)₂PF₆ show that the prediction of a
temperature-independent law is followed over an extended temperature
domain down to ≈ 25 K (Fig. 12).

For ħ/τₗ < t⊥ the warping of the Fermi surface is pertinent and the
transverse motion becomes coherent. Thus, assuming a Fermi surface
broadening of E<sub>F</sub>/3 ≈ 1000 K at room temperature and decreasing down
to 10 K at 25 K, according to the temperature dependence of the resistivity
[5], a value of 10 K is obtained for the interchain coupling. This is a much
smaller value than what can be derived from band structure calculations
(i.e., 20 meV). The difference can be explained by the role of intrachain
correlations improving the one-dimensional confinement, decreasing the
effective interchain coupling and the crossover temperature [42].

The resistivity anisotropy of the TM<sub>2</sub>X series is typically 1:200:30,000
along the *a*, *b*, and *c* axes, respectively, leading to the overlap anisotropy
*t<sub>a</sub>:*t<sub>b</sub>:*t<sub>c</sub>* = 10:1:0.1 with the assumption of incoherent transverse conduction
<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /><figcaption>**Figure 12** (a) Temperature dependence of the longitudinal resistivity of (TMTSF)₂PF₆ obtained with the clamped contact technique. (From W. Kang, private communication, and O. Traetteberg, Thesis, Univ. Orsay, 1993.) (b) The conductivity anisotropy is temperature independent as long as the transverse motion remains incoherent. Data for (TMTSF)₂PF₆ under 12 kbar. (After Ref. 6.)</figcaption></figure>
at room temperature according to Eq. (23). Until very recently the temperature dependence of the longitudinal resistivity could not be measured with much accuracy because of voltage jumps occurring during the cooling procedure. However, a new contacting technique has significantly improved the quality of the data [61].

The temperature dependence of the longitudinal resistivity of $(TMSF)_2PF_6$ (or $AsF_6$) follows a power law $T^{1.4}$ from 300 K down to $\approx 100$ K. Below 35 K, the resistivity follows the law $\rho_o + AT^2$, which is valid down to the metal-insulator (SDW) transition [62] (Fig. 12). The term $\rho_o$ is close to zero for samples displaying large resistance ratios (RR) between room temperature and the temperature corresponding to the minimum of the resistance. Very different RR values can be found in a given batch of samples ranging from 800 to 30. For a sample with RR = 260, values of $\rho_o = 0$ and $A = 0.04 \ \mu\Omega \cdot cm \cdot K^{-2}$ are found [62] (Fig. 13). This result suggests that the inelastic electron scattering is dominant in the conducting state at low temperature.

The very large pressure coefficient of the susceptibility (Fig. 14a) and conductivity in the metallic regime ($\partial \ln \sigma / \partial P \approx 25\% \ kbar^{-1}$ for the $(TMSF)_2X$ series around room temperature [6]) raises a serious problem for the comparison with theory, which usually computes constant-volume temperature dependences. Hence the temperature dependence at constant pressure that is observed in actual experiments must be transformed into constant-volume data since the change of volume (due to the thermal expansion) cannot be ignored between 300 and 50 K. No detailed determinations of the constant-volume resistivity have been performed so far. However, a crude estimate of the intrinsic temperature dependence can be performed using the thermal expansion and the pressure dependence of the *a* axis at various temperatures [59] (Fig. 14b).

A dependence close to a linear law is observed down to 100 K. At low temperature, both the thermal expansion and the pressure coefficient are small. Therefore, the constant-volume temperature dependence of the resistivity does not deviate from the quadratic law observed under constant pressure. At this stage it is interesting to stress that the theory of the resistivity in a half-filled band conductor [63], including the strength of the coulombic repulsions as derived from NMR data (Section III.B), should lead to a more localized behavior than that observed experimentally in Fig. 14.

The Hall effect has been measured in some organic conductors. As far as $(TMSF)_2ClO_4$ is concerned, the low-field Hall constant measured at low temperature is small and positive (holes) in fair agreement with the value derived from the kinetic theory of the Hall constant for a one-
<figure><img src="image_27.png" /><figcaption>Figure 13 Temperature dependence of the resistivity of (TMTSF)₂AsF₆, log-log plot (a) and versus T² (b) following the law ρ = ρ₀ + AT₂, where ρ₀ = 0 for the nonalloyed sample.</figcaption></figure>

dimensional conductor [64]:

$$R_H = - \frac{\pi \rho}{2n|e|c \tan(\pi \rho / 2)} \quad (24)$$

where n is the carrier density. Taking n = 1.4 × 10²¹ cm⁻³, Eq. (24) leads to R_H+ = + 3.4 × 10⁻⁹ m³/A · s with ρ = ½ in Eq. (24), as the weak
<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /><figcaption>**Figure 14** (a) Pressure dependence of the spin susceptibility $\chi \sim (T_1T)^{-1/2}$ from NMR data. (From Ref. 41b.) (b) Constant-pressure and constant-volume temperature dependences of the resistivity of $(TMSF)_2AsF_6$ derived point by point from the constant-pressure data of Fig. 12. The lattice parameters are from Ref. 33 and the pressure coefficient of the conductivity from Ref. 57.</figcaption></figure>
dimerization of the TMTSF stacks does not modify significantly the band structure close to $k_F$. This value can be compared with the experimental determination at low-magnetic fields of $4 \times 10^{-9} \text{ m}^3/\text{A} \cdot \text{s}$ [65] (Fig. 15a).

The thermopower of single-chain conductors pertaining to the (TMTSF)₂X series ranges around 20 µV/K at room temperature [66]; it follows a quasi-linear temperature dependence and extrapolates to a value near zero at 0 K (Fig. 15b). The use of a one-dimensional tight-binding formulation [67a],

$$S = - \frac{\pi^2 k_B T}{3|e|} \frac{\cos(\pi\rho/2)}{2|t|\sin^2(\pi\rho/2)} \quad (25)$$

for a ⅜-filled band leads to a tight-binding band width of ≈ 1 eV. Both Hall effect and thermopower data suggest a rather simple picture for the electronic Fermi gas of (TMTSF)₂X superconductors very close to the straightforward one-dimensional model of two planar Fermi surfaces.

### D. Suppression of the SDW Phase Under Pressure

Although the symmetry of TM₂X salts is triclinic, most physical features can be explained by taking an orthorhombic symmetry with the tight-binding energy dispersion written as

$$E(\mathbf{k}) = -2t_a \cos k_x a - 2t_b \cos k_y b - 2t_c \cos k_z c \quad (26)$$

with $t_a: t_b: t_c \approx 100:10:0.3$ and $t_a \approx 0.2$ eV for (TMTSF)₂X compounds [68].

The dispersion along the conducting axis can be linearized without significant modifications in the physical properties of the model provided that $kT < t_a$. Thus Eq. (26) becomes

$$E(\mathbf{k}) = E_0 + v_F(|k_x| - k_F) - 2t_{\perp} \cos k_y b - 2t'_{\perp} \cos 2k_y b \quad (27)$$

The linearized dispersion equation (27) leads to the same definition of the Fermi surface $E(\mathbf{k}) = E_F$ as that of Eq. (26) provided that $t'_{\perp} \approx t_{\perp}^2/t_a$, with $v_F \approx 2at_a$. We have neglected the dispersion along the c direction of very weak coupling.

Equation (27) leads to a Fermi surface that is only slightly distorted from the sinusoidal shape. If the last contribution in Eq. (27) is forgotten, both sheets of Fermi surface nest perfectly with the nesting vector $Q_t = (2k_F, \pi/b)$. Then the spin susceptibility $\chi_0(\mathbf{q})$ becomes large near $\mathbf{q} = Q_t$ and diverges logarithmically as $T \to 0$. A spin-modulated (SDW) ground state should thus be established below an ordering temperature $T_N \le T_0$, where $T_0$ is the one-dimensional mean-field temperature for the SDW instability:

$$T_0 \sim E_F \exp \left[ - \frac{1}{N(E_F)U} \right] \quad (28)$$
<figure><img src="image_30.png" /><figcaption>**Figure 15** (a) Temperature dependence of the Hall voltage in the “relaxed” state of $(TMTSF)_2ClO_4$ at $T = 0.35$ K. The outset of a FISDW phase is visible at 4 T. (From W. Kang, private communication.) (b) Thermopower data of $(TMTSF)_2ClO_4$. (After Ref. 67b.)</figcaption></figure>

The Fermi surface departs from perfect nesting when the second harmonic contribution in Eq. (27) becomes a relevant contribution, namely if $t'_⊥ ≈ T_0$. Thus the nesting of the Fermi surface is frustrated and the susceptibility $χ_0(q)$ no longer presents any logarithmic divergence at $q = Q_r$ as $T → 0$ but only a relative (nondivergent as $T → 0$) maximum at a
vector $Q_0$ (close to $Q_r$), which optimizes the nesting of the nonsinusoidal Fermi surface (Fig. 16). This intermediate situation for $t'_⊥$ can still lead to a SDW instability which is suppressed when $t'_⊥$ is increased (by applying pressure or changing the anion $PF_6 \rightarrow ClO_4$). There thus exists a critical value for $t'_⊥$ above which a nondistorted quasi-one-dimensional conductor is stabilized at low temperature (Fig. 16). This critical value amounts to about $t'_{⊥cr} \approx T_0 (\approx 1 - 2$ meV). Given $t'_⊥ \approx 20$ meV, the band structure anisotropy makes $t'_⊥$ very close to the critical value already at ambient pressure. This situation explains why only a modest pressure of 9 kbar is enough to suppress the SDW ground state.

<figure><img src="image_31.png" /><figcaption>**Figure 16** (a) The SDW ground state of (TMTSF)₂X compounds is stable as long as $t'_b < t'_{cr}$. (b) At $t'_b = 0$, there is a logarithmic divergence of $\chi_0(\mathbf{q})$ at $Q = (2k_F, \pi/b)$. (c) $\chi_0(\mathbf{q})$ is no longer divergent at $t'_b = 0$; the maximum of $\chi_0(\mathbf{q})$ occurs at $Q_0 = (2k_F + Q_‖, q_⊥)$. (After Ref. 68a.)</figcaption></figure>
### E. Anion-Ordering Phase Transition

Another kind of semiconducting ground state is observed in the salts of the TM₂X series when the anion is noncentrosymmetric. That the sites occupied by anions are inversion centers of the structure thus implies two possible orientations for noncentrosymmetric anions. The anions are disordered at room temperature and both orientations have the same statistical occupancy. However, entropy is gained at low temperature if they order at a temperature T AO. The new periodicity of the lattice is important for the one-dimensional electron gas, since a doubling of the anion periodicity along the stacking axis opens a gap at the Fermi level that stabilizes an anion-ordered semiconducting ground state [69].

Various ground states of salts with noncentrosymmetric anions can be classified according to their anion periodicity. In (TMTSF)₂ClO₄, the anion order at T AO = 24 K is characterized by the wave vector (0, ½, 0) [70]. The periodicity is doubled along the *b* direction but is left unchanged along the stacking axis (Fig. 17a,b). This kind of ordering should thus have a minor influence on the conducting properties, although it may significantly disturb the stability of the field-induced SDW phases (see Section VII.A).

The perfect long-range anion ordering of ClO₄ anions cannot be reached if the compound is quickly cooled through the anion-ordering temperature (<1 K/min). The compound is thus stabilized in a “quenched” state namely Q state at low temperature at variance with the “relaxed” state that is obtained after a slow cooling procedure (>1 K/min) [71]. The Fermi surface of the Q state does not present gaps at *k*<sub>y</sub> = ±π/2*b*. It thus resembles the quasi-one-dimensional Fermi surface of (TMTSF)₂PF₆ [72], and a SDW ground state establishes below 6 K (Fig. 17c) [71]. The low value of the resistivity in the Q-SDW ground state of (TMTSF)₂ClO₄ suggests that it may be more appropriately described by a semimetal.

The anion ordering in (TMTSF)₂ReO₄ (½, ½, 0) [73] opens a large gap at 180 K which stabilizes an insulating ground state [6]. Moreover, the type of anion ordering can be changed by a pressure that modifies the balance between anion-anion electrostatic coupling and anion-organic stack coupling. For example, the anion order of (TMTSF)₂ReO₄ becomes (0, ½, ½) at low temperature under a pressure of 12 kbar, and a superconducting state is stable below 1.3 K [74] (Fig. 17d).

The situation for (TMTSF)₂NO₃ is somewhat different since the triangular NO₃ anion adopts a (½, 0, 0) periodicity below T AO = 45 K [74]. This new lattice symmetry implies a folding of the band structure about planes *k*<sub>x</sub> = ±π/2*a*. Consequently, the anion ordering triggers a transition between a quasi-one-dimensional conductor at high temperature and a two-
<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /><figcaption>**Figure 17** (a) Anion ordering (0, ½, 0) in (TMTSF)₂ClO₄ at 24 K (from Ref. 75). (b) Different ground states are obtained in slow-cooled and quenched samples. (After S. Tomic, Thesis, Univ. Orsay, 1987, and Ref. 71.) (c, d) Phase diagram for the anion ordering [from Ref. 74] and transport properties of (TMTSF)₂ReO₄. (From S. S. P. Parkin, D. Jérome, and K. Bechgaard, *Mol. Cryst. Liq. Cryst.* 79:213 (1982).)</figcaption></figure>
<figure><img src="image_34.png" /><figcaption>Figure 17 Continued</figcaption></figure>

dimensional semimetal below 45 K. Hence the antiferromagnetic ground state that establishes below 9 K in this compound should be considered as the result of an excitonic instability of nested two-dimensional Fermi surfaces [76] instead of the quasi-one-dimensional instability observed in $(TMTSF)_2PF_6$ (Fig. 18). The difference in anion-ordering structure may be responsible for the absence of field-induced SDW phases in $(TMTSF)_2NO_3$ (see Section VII.A).

# IV. SUPERCONDUCTING PROPERTIES OF MOLECULAR CONDUCTORS

## A. Superconducting Parameters of Molecular Superconductors

Differences between the two major series of superconducting materials, $TM_2X$ and $ET_2X$, are clear cut. A salient feature of superconductivity in the quasi-one-dimensional $TM_2X$ series is the common border existing between superconducting and magnetic phases, which provides an upper limited value smaller than 2 K. This limitation is apparently independent of the anion symmetry since attempts to force the stabilization of superconductivity at lower pressures in the compound $(TMTSF)_2ReO_4$, where a pressure of 12 kbar is necessary to prevent the $ReO_4$ anion ordering at low temperature from opening a gap at the Fermi level have failed to raise
<figure><img src="image_35.png" /><figcaption>**Figure 18** Fermi surface of (TMTSF)₂NO₃ (schematic), in the quasi-one-dimensional regime (a) and below the anion ordering transition (b).</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>**Figure 19** Hidden SDW ground state of (TMTSF)₂ReO₄ obtained after depres-
surization of the sample at low temperature. (After Ref. 77.)</figcaption></figure>
$T_c$ above 1.2 K [77] (Fig. 19). The behavior of $(TMTSF)_2ReO_4$ emphasizes the deeply rooted competition that exists between superconductivity and magnetism in the $TM_2X$ series. It is clear that the low values of $T_c$ are a direct consequence of the one-dimensional character of the Fermi surface. It is a situation where fluctuations can be very efficient in suppressing the superconducting long-range order down to low temperature. Consequently, $T_c$ values higher than 1 to 2 K should be searched for in materials where the Fermi surface is less favorable for the stabilization of antiferromagnetism at low temperature.

Figure 20 displays the superconducting transition of $(TMTSF)_2PF_6$ salts observed by resistivity data [6,9]. What is remarkable in Fig. 20 is the strong temperature dependence of $\rho(T)$ above $T_c$, unlike the behavior of regular metals, for which the resistivity is limited at low temperature by temperature-independent elastic scattering. The behavior of the temperature dependent resistivity at low temperature in $TM_2X$ compounds has been ascribed to a precursor effect above $T_c$ [6].

The observation of a Meissner flux expulsion [78] and of specific heat anomaly [79] supports the picture of bulk superconductivity (Fig. 21). On a $C/T$ versus $T^2$ plot, the specific heat of $(TMTSF)_2ClO_4$ displays a very large anomaly at 1.2 K and follows the law

$$ \frac{C}{T} = \gamma + \beta T^2 \qquad (29) $$

with $\gamma = 10.5 \text{ mJ} \cdot \text{mol}^{-1} \cdot \text{K}^{-2}$ corresponding to $N(E_F) = 2.1$ states $\cdot \text{eV}^{-1} \cdot \text{mol}^{-1}$ [$N(E_F) = 3\gamma/2\pi k_B^2$] and $\beta = 11.4 \text{ mJ} \cdot \text{mol}^{-1} \cdot \text{K}^{-4}$. Superconductivity in the $TM_2X$ series is not restricted to be containing molecules as shown by the stabilization of a superconducting state below 0.8 K in $(TMTTF)_2Br$ under a pressure of 25 kbar, (see Note added in proof).

The $(ET)_2X$ series with two-dimensional Fermi surfaces has indeed provided superconducting compounds with higher $T_c$: up to 8.1 K for $X = I_3^-$ [14,15] and even up to 10 to 12 K for $X = Cu(SCN)_2^-$ [16], $Cu[N(CN)_2]Br^-$ [18] (Fig. 22). Some parameters describing the superconducting state of molecular superconductors are summarized in Table 3.

It is not yet clear whether these layered superconductors should be considered as three-dimensional anisotropic superconductors or as two-dimensional superconductors with a three-dimensional ordering resulting from a Josephson coupling between the layers [80]. The nature of the pairing mechanism for one- and two-dimensional superconductors has obviously been the matter of several conjectures, but this question is still open.

The most salient feature of one-dimensional organic superconductivity is the strong sensitivity to the presence of impurities and even nonmagnetic lattice defects as observed in both families of superconductors [81] (the
<figure><img src="image_37.png" /><figcaption>**Figure 20** Superconducting transition of $(TMSF)_2PF_6$ under 8 kbar and $\kappa$-(ET)$_2Cu(SCN)_2$ at ambient pressure.</figcaption></figure>

latter being unusual in conventional superconductors), which could point toward the existence of an anisotropic gap in the superconducting state. Pairing of carriers through the exchange between neighboring chains of an antiferromagnetic fluctuation is a possibility for gap anisotropy in the $TM_2X$ series [82]. The absence of spin-lattice relaxation time anomaly at $T_c$ in the $TM_2X$ [83] series (Fig. 22) can also support the possibility of gap anisotropy [84].
<figure><img src="image_38.png" /><figcaption>**Figure 21** Superconductivity in (TMTSF)₂ClO₄ by transport [10], magnetic [78], and specific heat data [79].</figcaption></figure>

It is also remarkable to point out that superconductivity has also been discovered in a family of charge-transfer compounds based on the molecule M(dmit)₂, with M = Ni, Pd with T_c = 1.6 K under 7 kbar [12] (Fig. 23). In this interesting series, structural and NMR studies support the picture of a one-dimensional Fermi surface, and the ground state competing with superconductivity is a charge density wave modulation driven by the Ni(dmit)₂ stacks [85].

## B. Isotope Effect

Isotopic shift experiments are also important in organic superconductors, but their interpretation in all three classes must be treated with great caution, as many extrinsic effects may affect T_c in these materials. The isotopic deuterium labeling of methyl or ethylene groups located at the outskirt of the molecule may result in a significant T_c shift, which can be related to the known very large pressure coefficient of T_c in organic su-
<figure><img src="image_39.png" /></figure>

**Figure 22** (a) The superconducting transition temperature is very sensitive to defects created by proton irradiation (left) (after Ref. 81a) and neutron irradiation (right) (after Ref. 81b). (b) Schematic representation of the interchain exchange of $2k_F$ spin fluctuations proposed by Bourbonnais and Caron [82] and temperature dependence of $^1H-T_1^{-1}$ in $(TMTSF)_2ClO_4$, showing the absence of any Hebel-Slichter anomaly at $T_c$ and suggesting a power law temperature dependence in the superconducting state [83].

perconductors. A variety of data have been published in deuterated TM and ET salts [86,87].

$T_c$ is also known to be extremely sensitive to disorder and sample purity [81,88]. Therefore, the most reliable experiments are those in which the isotopic substitution is performed on an active site of the molecule and
<table><caption>**Table 3** Parameters Describing the Superconducting Phase of Molecular Superconductors</caption><thead><tr><th></th><th>One-<br>dimensional</th><th>Two-<br>dimensional</th><th>Three-<br>dimensional</th></tr></thead><tbody><tr><td>Compounds</td><td>TM<sub>2</sub>X</td><td>κ-ET<sub>2</sub>X</td><td>A<sub>3</sub>C<sub>60</sub></td></tr><tr><td>T<sub>c</sub> (K)</td><td>X = ClO<sub>4</sub><br>1.2</td><td>X = Cu(SCN)<sub>2</sub><br>9<br>X = Cu[N(CN)<sub>2</sub>]Br<br>12</td><td>A = K<br>19<br>A = Rb<sub>2</sub>Cs<br>33</td></tr><tr><td>H<sub>c1</sub>(0) (10<sup>-4</sup> T)</td><td>a 0.2<br>b 1<br>c 10</td><td>(a, b) 1<br>c 45</td><td>1.3<br>(K<sub>3</sub>C<sub>60</sub>)</td></tr><tr><td>H<sub>c2</sub>(0) (T)</td><td>a 2.8<br>b 2.1<br>c 0.16</td><td>(a, b) 30<br>c 2</td><td>49<br>(K<sub>3</sub>C<sub>60</sub>)</td></tr><tr><td>ξ(0) (Å)</td><td>a 700<br>b 335<br>c 20</td><td>(a, b) 174<br>c 7</td><td>26<br>(K<sub>3</sub>C<sub>60</sub>)</td></tr><tr><td>λ(0) (Å)</td><td></td><td>5000-10,000</td><td>2400-4800<br>(K<sub>3</sub>C<sub>60</sub>)</td></tr></tbody></table>

<figure><img src="image_40.png" /><figcaption>**Figure 23** Superconductivity in TTF [Ni(dmit)<sub>2</sub>] under 7 kbar. (After Ref. 12.)</figcaption></figure>
does not induce any lattice parameter modification. Two such studies have been reported, with ¹³C substitution of the central double bond of the ET molecule in κ phases of Cu[N(CN)₂]Br and Cu(SCN)₂ salts [89] and on the β<sub>H</sub> phase of I₃ [90].

As far as κ phases are concerned, the isotope shift was found to be zero within a 0.1% accuracy [89] and confirmed in Ref. 90, whereas β<sub>H</sub>(ET)<sub>2</sub>I<sub>3</sub> has revealed an isotopic downshift ΔT<sub>c</sub>/T<sub>c</sub> in between −1.25 and −3.75% [90]. In both cases the isotope shift of the Raman-active a<sub>g</sub> C≡C stretching modes of the ET molecule were found to be in agreement with molecular dynamics predictions Δω/ω = −1.8%.

Within the canonical weak-coupling BCS theory assuming the C≡C modes to be the most strongly coupled to electrons should lead to a temperature shift of ΔT<sub>c</sub>/T<sub>c</sub> = Δω/ω = −1.8%, which lies within the experimental window found for β<sub>H</sub>(ET)<sub>2</sub>I<sub>3</sub> but is in clear disagreement with the κ phase reports. The present experimental status could support a pairing mechanism through intramolecular vibrations involving C≡C bonds in β<sub>H</sub>(ET)<sub>2</sub>I<sub>3</sub> [91], but such a mechanism seems to be ruled out for the κ phases, which have a 30% higher T<sub>c</sub> but do not show marked isotopic shifts for T<sub>c</sub>.

A recent study of the isotope shift in ³⁴S-substituted κ phases [92] has ruled out either a dominant role for the C—S stretching motions or a conventional BCS isotope effect involving the ET molecules as the relevant mass entity. There is still the possibility for the pairing in κ phases to be related to a non-phonon-mediated mechanism such as the interplane exchange of spin fluctuations as proposed by Bourbonnais and Caron [82] in the context of one-dimensional superconductors.

Understanding the pairing mechanism is apparently less of a problem for superconducting fullerenes. Although experiments on isotopic enrichment of C₆₀ with ¹³C atoms have led to a standard sign of the shift (T_c is smaller in ¹³C-enriched samples), there remains much uncertainty about its amplitude, with α ranging from ~0.3 up to ~1.4 [93a] (where α is related to the C₆₀ molecular mass by T_c ~ M⁻⁶ and amounts to 0.5 in a BCS formulation), high-pressure experiments have played an important role in clarifying the origin of the attractive pairing interaction.

### C. Role of Pressure in Fullerenes

The critical temperatures of doped fullerenes pertaining to the series (MₓM'₁₋ₓ)₃C₆₀ correlates very well with the lattice parameter of the face-centered cubic lattice [20]. As shown in Fig. 24, the transition temperature decreases as the C₆₀ separation is decreased either by changing the composition or by the application of high pressure. This empirical law suggests
<figure><img src="image_41.png" /><figcaption>Figure 24 Pressure dependence of $T_c$ in alkali-doped fullerenes. The $K_3C_{60}$ data are shifted in the pressure leak to take into account the smaller lattice parameter. (After Ref. 93b.)</figcaption></figure>

that $T_c$ is essentially governed by the interball overlaps (or the density of states at Fermi level). In the weak-coupling BCS model for phonon-mediated electron pairing, $T_c$ is given by

$$T_c = \Omega_{ph} e^{-1/\text{VN}(E_F)} \quad (30)$$

where $\Omega_{ph}$ is the energy of the phonons and $V$ the electron-phonon coupling strength.

The pressure dependence of $N(E_F)$ has been measured recently through the pressure dependence of the ¹³C-NMR Knight shift in $K_3C_{60}$ [94]. In Fig. 25, a plot of $\ln T_c(P)$ versus $K(P)$ is presented. As shown by this plot, linear behavior is effectively observed, which intersects the y axis at $\Omega_{ph} = 600$ K and $\lambda = N(E_F)V = 0.3$ at ambient pressure [94]. Thus the value of $T_c$ appears to be governed by $N(E_F)$ and the pressure data suggest that high-frequency intraball phonons are likely to be involved in the superconductivity of fullerenes [20,94].

# V. OPTICAL PROPERTIES

## A. Visible and Near-Infrared Properties

Optical studies of organic superconductors have provided a wealth of information related to the bandwidth, the dimensionality of the electron
<figure><img src="image_42.png" /><figcaption>Figure 25 (a) Linear relation between $(1/T_1)^{1/2}$ and the $^{13}$C-NMR frequency in $K_3C_{60}$ under pressure leading to the determination of the $C_{60}^3$ chemical shift and (b) ln $T_c$ versus the reciprocal Knight shift under pressure. The continuous line is the BCS prediction. (After Ref. 94.)</figcaption></figure>

gas, and the role of electron correlations [95]. Most optical experiments measure the reflection coefficient, which is given in terms of the complex index $N = n + ik$ by the relation

$$R = \left| \frac{\tilde{N} - 1}{\tilde{N} + 1} \right|^2 = \left| \frac{\sqrt{\tilde{\epsilon}} - 1}{\sqrt{\tilde{\epsilon}} + 1} \right|^2 \quad (31)$$

where the complex dielectric constant reads

$$\tilde{\epsilon}(\omega) = \epsilon(\omega) + i \frac{\sigma(\omega)}{\epsilon_0 \omega} \quad (32)$$

Assuming that we have a metal in which the dominant contribution to the dielectric constant comes from intraband optical transitions $\epsilon(\omega)$ becomes

$$\tilde{\epsilon}(\omega) = \epsilon_{\infty} - \frac{\omega_{p0}^2}{\omega(\omega + i\gamma)} \quad (33)$$

In Eq. (33), $\epsilon_\infty$ is the core polarizability arising from high-frequency transitions and $\gamma$ is the damping rate. The plasma frequency $\omega_{p0}$ is given by

$$\omega_{p0}^2 = \frac{Ne^2}{\epsilon_0} \frac{1}{m^*} \quad (34)$$
where *N* is the density of carriers in the band and *m*\* is the effective mass in the field direction.

A one-dimensional band with band filling defined by $\rho$ (0.5 for the $TM_2$ salts forgetting about the structural dimerization) and a molecular volume $V$ would lead to

$$ \omega_{po}^2 = \frac{4t_{\parallel}a^2e^2}{\pi\epsilon_0\hbar^2V} \sin\frac{\pi\rho}{2} \quad (35) $$

The main goal of optical studies is to determine to what extent the law given by Eq. (33), namely the Drude behavior, is followed for one-dimensional conductors in the presence of strong correlations. Figure 26 gives a typical example of Drude behavior taken from the work of Jacobsen [95].

The conductivity sum rule is an important consequence of the Drude model, namely,

$$ \int_{0}^{\infty} \sigma(\omega) d\omega = \omega_{po}^2 \quad (36) $$

for intraband transitions in a noninteracting electron gas. However, the observation of a well-defined plasma edge cannot be taken as proof of good metallic character. This is shown experimentally by the reflectance spectrum of a variety of ($TM_2$)$_X$ and ($ET$)$_X$ salts exhibiting quite different conductivity properties at room temperature (see Fig. 26 and Table 4).

The Drude parameters in Table 4 have been derived by fitting the reflectance spectra with the Drude-like dielectric constant, Eq. (33). It has been stressed by Jacobsen [98] and Yamaji [68b] that in the limit of highly anisotropic one-dimensional conductors the ratio of transfer integrals is not proportional to the ratio of plasma frequencies but to its square instead.

The near-infrared reflectance provides the response to plasmon oscillations of the electron gas (which are uniform excitations). This region of the spectrum is, however, not sensitive to the strength of the short-range coulombic interactions, which prevent conductivity in a Mott-Hubbard insulating state. This is illustrated by the frequency-dependent conductivity $\sigma(\omega)$ measured in various salts exhibiting very different values of the conductivity at room temperature (Fig. 27). The peak of the conductivity at the frequency $\omega_o$ correlates with the metallic character: namely, a low frequency of the peak position corresponds to a high dc conductivity and vice versa. The structures below $\omega_o$ are attributed to the coupling with intramolecular modes.
<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /><figcaption>**Figure 26** (a) Single-crystal optical reflectance data for organic conduction with decreasing conductivity from top to bottom (from Ref. 95); (b) optical reflectance of (TMTSF)₂PF₆ at different temperatures in the conducting regime (from Refs. 95 and 98); (c) anisotropy of the optical reflectance (from Ref. 98); (d) optical reflectance of β(ET)₂I₃ (from Ref. 97).</figcaption></figure>
<figure><img src="image_45.png" /><figcaption>Figure 26 Continued</figcaption></figure>

<table><caption>**Table 4** Interpretation of the Reflectance Data of Organic Conductors in Terms of a Drude Model</caption><thead><tr><th>Compound</th><th>σ (300 K)<br>(Ω·cm)<sup>-1</sup></th><th>ω<sub>p</sub> (cm<sup>-1</sup>)</th><th>γ (cm<sup>-1</sup>)</th><th>t(eV)</th><th>ε<sub>∞</sub></th></tr></thead><tbody><tr><td>(TMTTF)<sub>2</sub>PF<sub>6</sub></td><td>20</td><td>8860</td><td>1380</td><td>0.2</td><td>2.50</td></tr><tr><td>(TMTTF)<sub>2</sub>Br</td><td>250</td><td></td><td></td><td></td><td></td></tr><tr><td>(TMTSF)<sub>2</sub>AsF<sub>6</sub></td><td>500</td><td>9900</td><td>1230</td><td>0.25</td><td>2.56</td></tr><tr><td>(TMTSF)<sub>2</sub>ClO<sub>4</sub></td><td>700</td><td>9900</td><td></td><td></td><td></td></tr><tr><td>β(ET)<sub>2</sub>I<sub>3</sub></td><td>30</td><td>9100</td><td></td><td>0.17</td><td></td></tr></tbody></table>
<figure><img src="image_46.png" /></figure>

<figure><img src="image_47.png" /><figcaption>Figure 27 Frequency-dependent conductivity of various TM₂X compounds and β(ET)₂I₃. DBTSF-TCNQ F₄ is a Mott insulator at 300 K (after Ref. 96).</figcaption></figure>

An estimate of the role of Coulomb correlations can be reached by computing the experimental intraband oscillator strength (see Ref. 96),

$$I\rho = \int \sigma(\omega) d\omega \qquad (37)$$

and comparing it either to the expected value for a noninteracting electron gas $I_o = \frac{2}{p_o}$ or to the sum rule for a gas of noninteracting spinless fermions in the limit of $U \to \infty$. For the latter situation, the plasma frequency is given by a formula such as Eq. (35), although the states are now occupied
up to ± $2k_F$ and the density of states is halved as compared to the $U = 0$ limit. Therefore, the oscillator strength of a one-dimensional Mott-Hubbard insulator should read

$$I_{\infty} = \omega_{\text{po}}^2 \cos \frac{\pi \rho}{2} \qquad (38)$$

where $\rho = 0.5$ for the TM$_2$X series.

As shown in Table 5, a comparison between the Mott-Hubbard insulating compound (TMTTF)$_2$PF$_6$ and the metallic conductor (TMTSF)$_2$ClO$_4$ shows that the experimental oscillator strength $I_\rho$ is close to the Mott-Hubbard insulator in (TMTTF)$_2$PF$_6$, but for the (TMTSF)$_2$ClO$_4$ conductor it is closer to the $U = 0$ limit. Table 5 shows that the suppression of oscillator strength correlates with the enhancement of the magnetic susceptibility over the Pauli value calculated with band parameters obtained from the plasma edge analysis.

Correlation effects are likely to be quite important in the compound $\beta$(ET)$_2$I$_3$ since they also are narrowband conductors. However, the reason why these strong interactions do not materialize in a Mott-Hubbard insulator could be attributed to the absence of one-dimensional character for this system, which precludes establishment of a Mott-Hubbard localized state.

The temperature dependence of reflectance and conductivity spectra of $\beta$(ET)$_2$ I$_3$ are very illuminating indeed (Fig. 27). At room temperature $\sigma(\omega)$ reveals a semiconducting behavior with a peak of conductivity at $2000 \text{ cm}^{-1}$ while an approximate metallic behavior is obtained at low temperature; the plasma frequency remains almost temperature independent.

The shift of oscillator strength toward higher frequencies due to large on-site coulombic repulsion corresponds physically in the limit $U \gg 4t$ to charge-transfer transitions into states with doubly occupied sites. Using

<table><caption>**Table 5** Comparison Between the Experimental Oscillator Strength $I_\rho$ and the Sum Rule Computed in the Limit of Noninteracting Electrons $I_0$ and Spinless Fermions $I_\infty$</caption><thead><tr><th rowspan="2">Compound</th><th>I<sub>0</sub> (10<sup>7</sup> cm<sup>-2</sup>)</th><th rowspan="2">I<sub>∞</sub> (10<sup>7</sup> cm<sup>-2</sup>)</th><th rowspan="2">I<sub>ρ</sub> (10<sup>7</sup> cm<sup>-2</sup>)</th><th rowspan="2">χ<sub>s</sub>/χ<sub>ρ</sub></th></tr><tr><th>from reflectance edge</th></tr></thead><tbody><tr><td>(TMTTF)<sub>2</sub>PF<sub>6</sub></td><td>7.9</td><td>5.6</td><td>5.8</td><td>3.3</td></tr><tr><td>(TMTSF)<sub>2</sub>ClO<sub>4</sub></td><td>9.8</td><td>6.9</td><td>8.3</td><td>2.0</td></tr><tr><td>β(ET)<sub>2</sub>I<sub>3</sub>(300 K)</td><td>8.3</td><td>5.9</td><td>6.2</td><td>~3</td></tr><tr><td>β(ET)<sub>2</sub>I<sub>3</sub>(40 K)</td><td>9.2</td><td>6.5</td><td>6.2</td><td>~3</td></tr></tbody></table>
a Hubbard model Schulz [29] has computed the correlation exponent $K_ρ$ and the weight of the dc conductivity in the total conductivity integrated from zero to infinity as a function of band filling for various $U/t$ ratios. Figure 28 reveals a blue shift of the oscillator strength due to coulombic interactions. The role of interactions is dominant near $ρ = 1$ and at exact half filling all the oscillator strength is shifted above a frequency corresponding to the correlation gap $Δ (≈ kTρ)$. Close to quarter-band filling the sensitivity of the optical conductivity to correlations actually becomes much weaker.

## B. Far-Infrared Properties

Using data obtained by far-infrared spectroscopy (FIR) we shall see that an interpretation of the conductivity of conducting $TM_2X$ compounds at low temperature in the framework of a single-particle picture triggers some interesting problems. According to Drude's description of the conductivity, $σ(ω)$ should be constant at low frequencies [$≈ σ(0)$] and should fall off as $1/ω^2$ at a frequency of the order $γ = 1/τ$, namely,

$$σ(ω) = \frac{ω_p^2 τ}{1 + ω^2 τ^2} \qquad (39)$$

The dc conductivity of all these systems becomes very large near 12 K [$σ(0) ≥ 3 × 10^5 (Ω·cm)^{-1}$] while the experimental reflectance edge is

<figure><img src="image_48.png" /><figcaption>**Figure 28** Relative weight of the dc peak in the total oscillator strength of the conductivity versus the band filling at different ratios of $U/t$, increasing from 1 to 256. The insert gives the weight of the dc peak versus bandfilling for different values of $U/t$, increasing from 1 to 16, top to bottom. (After Ref. 29.)</figcaption></figure>
not affected by temperature [96]. Thus the use of Eq. (39) leads to a value $1/\tau(12\,\text{K}) \sim 4\,\text{cm}^{-1}$, which agrees reasonably well with the direct estimate of the electron lifetime at 12 K, based on the observed temperature dependence of $\sigma_{dc}$ [10]. However, such a small value for $1/\tau$ predicts [Eq. (39)] the existence of a large conductivity in the FIR regime, say $\sigma \approx 4$ to $5.10^3 (\Omega \cdot \text{cm})^{-1}$ at $40\,\text{cm}^{-1}$. This is about 10 times larger than the measured FIR conductivity (obtained via a Kramers-Krönig analysis of the FIR reflectance data) [99] (see Fig. 29). To reconcile the observed values of dc and FIR conductivities, the existence at low temperature of a collective mode centered around zero frequency has been proposed by several authors [6,100].

The oscillator strength of this mode can be derived using the frequency dependence of the dielectric constant of a damped oscillator at zero frequency, which reads

$$ \epsilon_1(\omega) = \epsilon_H - \left(\frac{\Omega_p}{\omega}\right)^2 \quad (40) $$

<figure><img src="image_49.png" /><figcaption>Figure 29 Far-infrared conductivity of $(TMTSF)_2ClO_4$, relaxed phase. The dashed line is the Drude law with $1/\tau \sim 3.5\,\text{cm}^{-1}$ and $\omega_p = 10^4\,\text{cm}^{-1}$. (After Ref. 99.)</figcaption></figure>
where $\epsilon_H \sim 500$ takes into account the high-frequency contribution to $\epsilon_1(\omega)$. The dielectric constant crosses zero at $\approx 10 \text{ cm}^{-1}$ (data for $(TMTSF)_2\text{AsF}_6$ [101] (Fig. 30). Hence a value of $500 \text{ cm}^{-1}$ is obtained for the plasma frequency of the collective mode. The Drude analysis of the conductivity of this mode gives rise to

$$ \sigma(0) = \Omega_p^2 \tau_c \qquad (41) $$

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /><figcaption>**Figure 30** Far-infrared data of $(TMTSF)_2\text{AsF}_6$, showing the FIR gap from conductivity and the zero crossing of the dielectric function at $\approx 6 \text{ cm}^{-1}$ which is relevant for the low-frequency collective mode. (After Ref. 100.)</figcaption></figure>
where $1/τ_c$ is the width of the collective mode. We thus obtain from the experimental values of $Ω_p$ and $σ(0) ∼ 3 × 10^5 (Ω · cm)^{-1}$ a collective mode width of $1/τ_c ≲ 0.5 cm^{-1}$ at $T ≈ 12$ K.

Therefore, in the frequency range ($ω ∼ 30$ to $40$ cm$^{-1}$) the collective mode contribution to the conductivity becomes negligible compared to the single-particle contribution (see Fig. 30) and the single-particle contribution in the oscillator strength is shifted down to the narrow zero-frequency mode.

The assumption of such a collective contribution to the low-temperature conductivity can be reconciled with direct measurement of the microwave conductivity at 20 K between 4.5 and 35 GHz [102]. As shown in Fig. 31, the development of the collective mode toward low temperatures can be predicted assuming a temperature-independent oscillator strength.

The physical origin of this zero-frequency collective mode is not yet clarified. Two possible mechanisms could be considered: (1) the onset of superconducting fluctuations below 30 K, and (2) the contribution of sliding

<figure><img src="image_52.png" /></figure>

**Figure 31** Schematic picture showing the development of the collective mode at low temperature. $T = 20$ K data are from Refs. 102 and 5. At $T = 10$ and 2 K only dc data are known and $σ(ω)$ is drawn assuming a temperature-independent oscillator strength of the collective mode.
spin density wave fluctuations to the conductivity. The latter assumption is tempting since some examples of the contribution of fluctuating density waves to the conductivity of a conducting regime are already known (Fröhlich fluctuating conduction in TTF-TCNQ [6]) and also a three-dimensional ordered depinned SDW contribution to the conduction in the SDW state of $(TMTSF)_2PF_6$, $AsF_6$ (see the following section). However, a magnetic density wave origin for the collective mode is hard to reconcile with a similar behavior of the FIR conductivity observed in $(TMTSF)_2ClO_4$, where no SDW ground state is stable for the relaxed phase, which undergoes the superconducting transition below 1.2 K and also with the large and positive magnetoresistance coefficient which is measured between 12 and 30 K in $(TMTSF)_2PF_6$ materials. It is thus tempting to attribute the large value of the conductivity to the development of superconducting correlations [6].

Using different microwave measurements of the conductivity from 3 up to 150 GHz (0.1 up to 5 cm⁻¹), Donovan et al. [103] reported nearly the same temperature dependence at all frequencies between 300 and 20 K. Consequently, they found no evidence for a narrow dc collective mode implied by the FIR studies. The question of the existence of a strongly conductive mode is still open, but an accurate determination of the temperature dependence of the resistivity which may not have been achieved in all cases is a prerequisite before comparing dc and high-frequency data.

# VI. SPIN DENSITY WAVE COLLECTIVE CONDUCTION

## A. Nonlinear Conduction and Narrowband Noise Generation

The antiferromagnetic state (Fig. 5) also reveals a lot of new physical properties. It is the nesting of the Fermi surface with the wave vector $Q$ that drives the transition from a metal at $T > 12$ K for $(TMTSF)_2PF_6$ into an insulator exhibiting a magnetic modulation or spin density wave at the same wave vector $Q$ [6]. The low-field susceptibility of the SDW phase behaves very much alike that of a regular antiferromagnet. For $(TMTSF)_2AsF_6$ easy, intermediate, and hard axes are located approximately along the *b*, *a*, and *c* axes, respectively (Fig. 32) and a spin-flip transition arises around 5 kOe [104] (Fig. 32). Antiferromagnetic resonance studies have also been carried out in the SDW ground states of $(TMTSF)_2X$ [105a] and of some S compounds [105b].

The broken symmetry of the SDW ground state characterized by a magnetic modulation at wave vector $Q$ in $(TMTSF)_2X$, $X = PF_6$, $NO_3$, and so on, is responsible for the activated behavior of the conductivity through the single-particle gap $2Δ$. However, two kinds of collective excitations arise in the SDW condensate. They are, first, the magnon exci-
<figure><img src="image_53.png" /><figcaption>**Figure 32** (a) Spin susceptibility of (TMTSF)₂AsF₆ measured at a field lower than the spin-flop field and (b) measured versus field. The spin-flop field is around 4 kOe. After (Ref. 104).</figcaption></figure>

<figure><img src="image_54.png" /></figure>
tations, which manifest themselves in the antiferromagnetic resonance [105], and second, the phase mode of the condensate, which is gapless provided that the spin modulation is noncommensurate, with the underlying lattice [106]. In $(TMTSF)_2PF_6$, the wave vector corresponding to the optimum nesting of the Fermi surface is incommensurate, as shown by various measurements sensitive to local magnetic fields. Therefore, as long as $Q$ is incommensurate with the underlying lattice, a phason mode begins at $\omega = 0$ for $q = 0$; in other words, the electrons can slide freely without any cost in energy, almost as in a superconductor [106] (Fig. 33a). This translational invariance gives rise to a collective contribution to the conductivity:

$$ \sigma(\omega) = \frac{ne^2}{m^*} \delta(\omega - \omega_0) \qquad (42) $$

where $n$ is the density of condensed carriers in the SDW state, $m^*$ the effective mass of the condensate, and $\omega_0$ is either zero, in an ideally pure material, or a finite frequency, when the collective mode is pinned by commensurability or impurities. (For a review, see Ref. 107.) This finite frequency is determined by the relation

$$ \omega_0^2 \sim V^2 n_i N(E_F) \Delta \qquad (43) $$

where $n_i$ and $V$ are the concentration of impurity centers and the SDW-impurity coupling potential, respectively.

However, in real materials the interaction between a SDW (or the associated CDW) and impurities or crystal defects provides a finite pinning energy, and a threshold electric field $E_T$ must be reached before the condensate can contribute to the conduction at zero frequency. It is given by the balance between the energy provided by the electric field when the condensate is moved by the length $2\pi/k_F$ and the pinning energy:

$$ eE_T \propto \omega_0^2 \qquad (44a) $$

and

$$ E_T = \frac{m^* \omega_0^2}{m 4eV_F} \qquad (44b) $$

Describing the motion of the pinned SDW state by a harmonic oscillator and assuming that the displacement of the collective mode by ISDW leads to depinning, the following relation between the static dielectric constant and the threshold field is obtained:

$$ \epsilon E_T = \text{const} \qquad (44c) $$
<figure><img src="image_55.png" /></figure>

<figure><img src="image_56.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_57.png" /></figure>

**Figure 33** (a) Frequency-dependent conductivity in the absence of pinning (A) and with pinning and damping (B) (after Ref. 107); (b) nonlinear conductivity in the SDW phase of $(TMTSF)_2AsF_6$; (c) current-voltage characteristics. (d) The $\epsilon E_\perp = \text{const.}$ relation is verified in the series of alloys $(TMTSF)_2AsF_{6(1-x)}$ SbF$_{6x}$ when the threshold field $E_T$ is determined by the amount of lattice defects $x(1-x)$, see also Fig. 34b. (After O. Traetteberg, thesis Univ Paris Sud (1993) unpublished.)
<figure><img src="image_58.png" /></figure>

<figure><img src="image_59.png" /></figure>

In all SDW phases of the (TMTSF)₂X series E_T is restricted to the range of a few mV/cm [108] (Fig. 33b), and for an electric field of twice the threshold field the collective current becomes a significant fraction of the total current (Fig. 33c). As shown in Fig. 33d, the relation εE_T = const is followed very accurately over several decades in the series of alloys to be discussed below. Furthermore, the linear increase of E_T versus the density of lattice defects suggests that the phase of the spin modulation is locally adjusted at each defect site to provide the largest pinning energy (strong pinning limit) [109a,b] (Fig. 34).
<figure><img src="image_60.png" /><figcaption>Figure 34 (a) Weak (top) and strong pinning (bottom) of a density wave. The phase of the modulation is locally adjusted at each defect site in the strong-pinning limit. (b) Threshold field in the $(TMTSF)_2AsF_{6(1-x)}$ SbF$_{6x}$ series (after Ref. 120); (c) threshold field of $(TMTSF)_2PF_6$ versus the concentration of defects created by x-ray irradiation (after Ref. 109c).</figcaption></figure>

Direct evidence for a spin density wave transport is the detection of a current oscillating at a frequency that is proportional to the dc current carried collectively. The recent observation of such oscillations; the harmonic and subharmonic locking of this oscillation to an external ac source and a motional narrowing of the NMR spectrum in the sliding SDW state have established firm evidence for the existence of a novel collective transport in a SDW condensate.

When a dc current flowing through the SDW state overcomes the threshold current, which is necessary for sliding the current density $j_{SDW}$ carried by each chain is proportional to the phase winding rate $v_{\phi} = \frac{1}{2}\pi d\phi/dt$: that
is,

$$j_{\text{SDW}} = 2ef_c v_{\phi} \quad (45)$$

where $f_c$ is the condensate density, reaching unity at 0 K. Furthermore, the sliding current generates a current oscillation at a frequency $v_n$. Direct observation of the ac current is relatively easy in a sliding CDW. However, the amplitude of this coherent noise generation being much weaker in a SDW state has only been detected in $(TMTSF)_2PF_6$ either by an interference between an external frequency $v_e$ whenever the ratio $v_n/v_e = p/q$ where $p$ and $q$ are two small integers (Fig. 35) or by a Fourier analysis [110,111a,b] (Fig. 36a). In addition, the proportionality law between the collective current and the noise frequency expected from Eq. (45) is very well followed (Fig. 36b). Similar results have been obtained in the quenched SDW state of $(TMTSF)_2ClO_4$ via direct observation of the coherent noise oscillation [112]. However in both cases, the ratio $j_{\text{SDW}}/v_n$ has been found significantly smaller than expected (about 8 to 20 times) from Eq. (45) using dc current data. This discrepancy has been attributed to an underestimation of $j_{\text{SDW}}$ due to a strongly inhomogeneous current distribution [110]. As far as CDWs are concerned [113], measured $j_{\text{CDW}}/v_n$ ratios give $2e$ at low temperature, which means that the winding rate frequency and the coherent noise frequency are equal. For such a situation direct measurement of $v_\phi$ has been achieved by study of the motional narrowing of the NMR line in the DW state [114]. It has been recognized from motional narrowing experiments in $Rb_{0.3}MoO_3$ that $v_\phi = v_n$ [115]. However, the origin of these oscillations is still controversial, as it could be related to the pinning of the CDW state by local impurities (the washboard frequency) or to the conversion of normal into collective electrons at the sample's electrodes with the creation of a new SDW wavefront.

As far as a SDW state is concerned, the two interpretations of the noise generation lead to different values for $v_n/v_\phi = \alpha$: the washboard model [116] and the current contact conversion mechanism [117], leading to $\alpha = 2$ and 1, respectively (since the wave vector of the CDW associated with the SDW is now $4k_F$ instead of $2k_F$ [118]).

The period of the pinning SDW potential has been derived from two sets of NMR experiments which have been performed recently in a sliding SDW state. First, measurement of the SDW velocity from the proton spin echo amplitude of $(TMTSF)_2PF_6$ [119] (Fig. 37) as a function of the dc current (above threshold field) and comparison with the SDW noise spectrum lead to a pinning potential period that is half the SDW wavelength (namely, $\alpha = 2$). Second, the result $\alpha = 1$ is inferred from a study of the amplitude of the motionally narrowed $^{13}$C spectrum in $(TMTSF)_2PF_6$ and of a SDW current and noise spectrum [111] (Fig. 38).
<figure><img src="image_61.png" /></figure>

<figure><img src="image_62.png" /></figure>

**Figure 35** (a) Detection of a beating between the ac current provided by an external source and the internal current oscillation in the sliding SDW state; (b) linear dependence of the SDW current against the frequency of the internal current oscillation, harmonics, and subharmonics. (After Ref. 110.)
<figure><img src="image_63.png" /><figcaption>**Figure 36** (a) Conduction noise amplitude versus frequency for various dc currents; (b) nonlinear conductivity and SDW current in the same $(TMTSF)_2PF_6$ sample, $T = 4.2$ K as in (a). (After Ref. 111a.)</figcaption></figure>

The weak pinning limit is encountered in the solid solution of two kinds of $TM_2X$ salts with anions of similar shape (albeit different in their anionic volume): $(TMTSF)_2AsF_{6(1-x)}$ Sb$F_{6x}$ [120]. The pinning induced by such defects is illustrated by the dependence of $E_T$ on the defect concentration (Fig. 39). It is clear, considering the slope $dE_T/dx$ for irradiation [109c] and solid solution-induced defects [120] that the latter kind of defect represents a weak pinning situation where the phase of the distortion is not fully adjusted to every defect at variance with the strong-coupling limit [109a,b].
<figure><img src="image_64.png" /><figcaption>Figure 37 The proton line width in (TMTSF)₂PF₆ at 4.2 K narrows as the current flowing through the sample increases while the constant magnetization shows that heating by the current is negligible. (After Ref. 119.)</figcaption></figure>

The collective current in a sliding SDW state is not a superconducting current, as there exists a finite damping parameter $\gamma$ which is usually defined by the nonlinear dc conductivity measured at $E = 2E_T$ by the relation

$$ \gamma^{-1} = \sigma_{\text{SDW}}^{\text{dc}} (2E_{\text{T}}) - \sigma_{n} \quad (46) $$

where $\sigma_n$ is the ohmic conductivity. The temperature dependences of $\sigma_{\text{SDW}}^{\text{dc}}$ and $\sigma_n$ for the system (TMTSF)₂AsF₆(1-x)SbF₆x have revealed the existence of several damping mechanisms [120] (Fig. 39):

1. The good scaling between $\sigma_{\text{SDW}}$ and $\sigma_n$ that is observed between 2 and 4 K can be understood in terms of a collective-mode damping induced by thermally activated quasiparticles (as already known for CDW systems).

2. Above a crossover temperature (which is 4 K for the sample $x = 0.07$), $\sigma_{\text{SDW}}$ becomes temperature independent and, furthermore, scales with $E_T^{-1}$ when the concentration of pinning centers is varied. This behavior has suggested that the quasiparticle damping (small at high temperature) is overcome by a temperature-independent damping which is governed by the SDW-defect interaction.

## B. Electodynamic Response

The electrodynamic response of the SDW phase is dominated by a strong resonance of $\sigma(\omega)$ centered around $0.1 \text{ cm}^{-1}$ (3 GHz) in (TMTSF)₂PF₆
<figure><img src="image_65.png" /><figcaption>**Figure 38** $^{13}$C-NMR spectra in the SDW phase of (TMTSF)$_2$PF$_6$ as a function of the total current at T = 4.2 K. (After Ref. 111b.)</figcaption></figure>

[112] which has been attributed to the oscillation of the pinned phase mode. A weaker resonance at $\omega \approx 30 \text{ cm}^{-1}$ comes from the single-particle excitation across the gap $\Delta$. In the clean limit situation $(1/\tau \ll \Delta)$ which is fulfilled, nearly all the oscillator strength is associated with the collective mode resonance at $\omega_0$.

Two sum rules corresponding to the two resonances can be established theoretically: the collective contribution,

$$ \int_{0}^{\infty} \sigma_{\text{SDW}} d\omega = \frac{n^2 e}{m^*} \qquad (47) $$

and second, the total contribution,

$$ \int_{0}^{\infty} (\sigma_{\text{SDW}} + \sigma_{\text{SP}}) d\omega = \frac{n^2 e}{m} \qquad (48) $$

where m is the bare electron mass.
<figure><img src="image_66.png" /><figcaption>**Figure 39** Temperature dependence of the normal conductivity (dashed line) and SDW conductivity (continuous line). [After O. Traetteberg et al., J. Phys. IV C-2:15 (1993).]</figcaption></figure>

Experimentally (Fig. 40), the oscillator strength is about $10^{-2}$ the value expected from Eq. (48). It has been suggested that this reduction could be attributed to the strong effect of coulombic interactions on the electro-dynamics of a SDW state rather than a large enhancement of the effective mass condensate with respect to the free-electron value, as it is the rule in a CDW ground state [122].

# VII. LOW-DIMENSIONAL ELECTRONS IN HIGH MAGNETIC FIELDS

## A. Field-Induced SDW Phases

The response of the quasi-one-dimensional electron gas to the application of a strong magnetic field has provided new and unexpected results: the transition from a quasi-one-dimensional conductor to a quasi-two-dimensional antiferromagnetic semimetal induced by a magnetic field at low temperature. The instability of the quasi-one-dimensional electron gas at high field can be inferred from observation of magnetoresistance oscillations above a certain threshold field in $(TMSF)_2PF_6$ under pressure [123] (Fig. 41). Moreover, NMR data have shown that the ground state which is stabilized by the magnetic field is characterized by the existence of a spin density wave [123] (Fig. 42). These phenomena are fairly well understood with a model in which, crudely speaking, the field restores enough
<figure><img src="image_67.png" /><figcaption>**Figure 40** Pinned mode resonance of the SDW state in $(TMTSF)_2PF_6$. (After Ref. 121.)</figcaption></figure>

one-dimensional character in the energy dispersion for the SDW distortion to become stable at a finite temperature [125].

Again using the model sketched in Section III.D to explain the suppression of the SDW phase under pressure, the divergence of $\chi(\mathbf{q})$ at low temperature can, however, be reactivated as soon as a magnetic field is applied to this quasi-one-dimensional conducting electron gas along the direction $c^*$ of lowest conductivity. $\chi(\mathbf{q}, H)$ thus displays logarithmically divergent peaks at $T \to 0$ for the set of different wave vectors defined by the relation

$$Q_x = 2k_F + \frac{NeHb}{h} \qquad (49)$$

where N is an integer [126]. Crudely speaking, the field restores enough one-dimensional character in the energy dispersion for the SDW distortion to become stable at nonzero temperature (Fig. 43a).

However, the experimental data of $(TMTSF)_2PF_6$ show that the problem is slightly more sophisticated [127,128]. A sequence of phases must be
<figure><img src="image_68.png" /><figcaption>**Figure 41** Detection of FISDW phases in (TMTSF)₂ClO₄ via transport measurements. (After Ref. 123.)</figcaption></figure>

crossed before the so-called “N = 0” phase (which is believed to correspond to the insulating SDW ground state stable at ambient pressure) can be reached above 18 T (Fig. 44). The integer N, which is related to the deviation of Qₓ from 2k_F, labels the phase containing N fully occupied Landau levels. N = 0 means that there is no density of states at the Fermi level, namely, a gap opened over the whole Fermi surface. Phases with N = 1, 2, 3, ... which are observed below 18 T correspond to semimetallic phases with a very low density of carriers (≈ 10⁻² per unit cell) and a tubular Fermi surface along the c* axis. In these subphases 1, 2, or 3, ... Landau levels are completely filled, respectively.

The situation where the Fermi level falls between completely filled and completely empty levels minimizes the diamagnetic energy of the two-dimensional carriers. This situation prevails in a finite range of magnetic fields as long as the wave vector of the magnetic modulation within a given subphase can vary linearly with the field according to Eq. (49). First-order phase transitions are expected between various subphases, in agreement with the hysteresis shown in Fig. 44.

A direct consequence of the field-dependent Q vector is the quantization of the Hall resistance at $ρ_{xy} = h/2Ne^2$ (where the factor 2 comes from spin
<figure><img src="image_69.png" /><figcaption>**Figure 42** Proof for the magnetic nature of the new phases stabilized under magnetic field in (TMTSF)₂ClO₄. The ⁷⁷Se-NMR line broadens and vanishes (a), whereas the relaxation rate diverges (b). (After Ref. 124.)</figcaption></figure>

degeneracy). The (TMTSF)₂PF₆ Hall data (Fig. 44) agree quantitatively with the quantized nesting theory (Fig. 43b). The almost perfect agreement between these data and the standard model is, however, not observed in other members of the (TMTSF)₂X series with noncentrosymmetric anions (X = ClO₄, ReO₄) [129,131].

The ClO₄ compound is particularly interesting for several reasons. First, the Hall resistance exhibits well-defined plateaus versus magnetic field and these plateaus can be indexed by the quantum numbers N = 1, 3, 6, ... [130]. The plateau N = 2 is apparently missing in the data of Fig. 45 and a sign reversal of the Hall resistance is observed between N = 3 and 6. Other authors have attributed a fractional number N = ⅓ to the phase which is remarkably stable in magnetic field above 8 T [132]. Second, the standard quantized nesting model does not provide an interpretation for the destruction of the SDW observed at very high field with reentrance of the undistorted phase. A non-mean-field approach based on the enhancement of the fluctuations at high fields has been proposed to explain this reentrance [133]. However, a tentative interpretation for the reentrance of the FISDW phase of (TMTSF)₂ClO₄ which is based on the modification of the Fermi surface inferred by the ordering of the anion at 24 K has been suggested [134]. The wave vector that characterizes the ClO₄ ordering is known from x-ray studies to be (0,π/b,0) [69]. This means that the opening of gaps at the points (2k_F, ±π/b) in a two-dimensional picture of the Fermi
<figure><img src="image_70.png" /><figcaption>**Figure 43** (a) The noninteracting spin susceptibility ($T = 0$) shows a nondivergent maximum at the optimum nesting vector ($2k_F + Q_\|, q_\perp$)) (top). Logarithmic divergences are tested under a finite magnetic field (bottom). (b) In each FISDW subphase, the theory predicts a density of carriers, which varies linearly against magnetic field, and the Hall voltage, which is constant. (After Ref. 68a.)</figcaption></figure>

surface is likely to perturb the possibilities of nesting close to the transverse vector $Q_t = (2k_F, \pi/b)$.

In the standard model of FISDW states the quasi-one-dimensional electron gas is one-dimensionalized by the magnetic field. It is only when the quasi-one-dimensional Fermi surface is made sufficiently flat by a large magnetic field that the $N = 0$ state becomes stable. In the $N = 0$ insulating state a gap opens over the entire Fermi surface and the modulation wave vector of this phase should be closer to the transverse vector ($2k_F, \pi/b$) than are the characteristic vectors of any semimetallic phases with $N = 1, 2, \dots$. We may reasonably assume that the nesting of the Fermi surface is rather poor in the vicinity of $Q_t$ (a consequence of the anion ordering). Therefore, the stabilization of the $N = 0$ state is *not* energetically favorable.
<figure><img src="image_71.png" /><figcaption>**Figure 44** Quantized Hall voltage of $(TMTSF)_2PF_6$ and phase diagram. The integers indicate the number of filled Landau levels in each subphase. (After Ref. 127.)</figcaption></figure>
<figure><img src="image_72.png" /></figure>

**Figure 45** Quantized Hall voltage in $(TMTSF)_2ClO_4$ (two samples) (after Ref. 130) and phase diagram (from Ref. 132). A new phase line has been discovered around 1 K and 30 T (after R. C. Yu et al, *Phys. Rev. Lett*, 65:2458 (1990)). The phase that extends from 8 to 27 T is presumably $N = 1$ [130].
in $(\text{TMTSF})_2\text{ClO}_4$, unlike $(\text{TMTSF})_2\text{PF}_6$, where the Fermi surface is not affected by any anion-ordering potential. Consequently, the $N=1$ phase of $(\text{TMTSF})_2\text{ClO}_4$ should remain stable in magnetic field since the semi-metallic electron gas ($N=1$) cannot undergo any phase transition toward a state of lower energy. However, for the diamagnetic energy to be minimized, the only occupied Landau level has to be kept entirely full. As the degeneracy of the Landau level increases linearly with the magnetic field, the density of carriers must adapt to the change of the magnetic field. Such a situation can be achieved only if the magnetic distortion is weakened when the magnetic field is increased with a concomitant decrease in $T_{\text{SDW}}$.

The effect of anion ordering on the stability of FISDW phases has been confirmed quantitatively by calculation of the spin susceptibility following the standard approach and including an additional periodic potential with wave vector $Q = (0, \frac{1}{2}, 0)$ [135]. After this numerical computation even phases ($N=0, 2$) are suppressed, whereas odd phases $N=1, 3$ are not. The same model also explains the normal-phase reentrance above 17 T. However, the predicted oscillation for $T_c$ versus the magnetic field in the $N=0$ phase is still lacking in the experimental data.

The quantization of the Hall resistance in the FISDW phases is indeed very reminiscent of the quantum Hall effect in the two-dimensional electron gas [136]. There is, however, an important difference between these two phenomena. In both cases the quantization requires a reservoir of non-conducting electronic states. This reservoir is provided either by localized states in the gap between conducting Landau levels or by the electron-hole (spin modulation) condensate for the two-dimensional electron gas and the FISDW of organics, respectively.

## B. Angular-Dependent Magnetoresistance

Another new phenomenon has been discovered in the study of the angle dependence of magnetotransport properties of quasi-one-dimensional conductors such as $(\text{TMTSF})_2\text{ClO}_4$ and $(\text{TMTSF})_2\text{PF}_6$. The first report of this effect was made by Boebinger et al. [137] and Naughton et al. [138], who observed fine dips in the magnetoresistance of $(\text{TMTSF})_2\text{ClO}_4$ at characteristic angles when the magnetic field is rotated in the $b'$c* plane (which is perpendicular to the conducting a axis; Fig. 46). The presence of magnetoresistance dips in the metallic phase of $(\text{TMTSF})_2\text{ClO}_4$ [139] is clearly not ruled by the orientation dependence of the FISDW phase diagram since the threshold fields are governed only by the component of the external field along the c* direction, as expected for two-dimensional systems. The unusual angular dependence of the magnetoresistance is related to “magic” angles [139] which satisfy the commensurability relation, given
<figure><img src="image_73.png" /><figcaption>**Figure 46** Angular oscillations of the magnetoresistance in (TMTSF)₂ClO₄ [139] showing features at magic angles given by Eq. (50).</figcaption></figure>

in an approximate form by

$$ \tan \theta = \frac{m b}{n c} \qquad (50) $$

where *b* and *c* refer to the lattice periodicities and *m*, *n* are integers. The existence of fine-structure behavior in the magnetoresistance was first predicted theoretically by Lebed and Bak [140].

In a titled magnetic field $H = (0, H \sin \theta, H \cos \theta)$ the electron can move along constant-energy trajectories. This motion is controlled by two periodic potentials with wave vectors $G_y = (e/\hbar) bH \cos \theta$ and $G_z = (e/\hbar) cH \sin \theta$. When the field angle satisfies the commensurability condition, $G_y/G_z = n/m$. The two periods become commensurate, and for “magic” angles given by Eq. (50), the periodic electron trajectories do not sweep out the entire Fermi surface as they do for arbitrary angles (Fig. 47).

This effect has been evidenced most clearly in (TMTSF)₂PF₆, which is also a prototype system for the stabilization of FISDW [127, 128] (Fig. 48). Very sharp dips of magnetoresistance are observed in the metallic phase at angles that satisfy the relation $m/n = 0, \pm 1, \pm 2$ [141, 142]. A commensurability effect at the same angles is also seen in the FISDW state, although peaks instead of dips are observed there [141] and a drop in the FISDW threshold field is observed at magic angles, in agreement with a suggestion of Lebed [143] that at angles given by Eq. (50), the coupling
<figure><img src="image_74.png" /><figcaption>**Figure 47** Electron trajectory on the Fermi surface of a quasi one-dimensional conductor under magnetic field satisfying the commensurability condition. (After Ref. 144.)</figcaption></figure>

<figure><img src="image_75.png" /><figcaption>**Figure 48** Lebed oscillations in (TMTSF)₂PF₆. (Data from Refs. 141 (left) and 142 (right).)</figcaption></figure>

along the third direction is eliminated and the electron gas becomes actually two-dimensional, making the FISDW state more stable at magic angles.

Several suggestions have been proposed to explain the fine structure of the angular dependence of the magnetoresistance. According to Lebed and Bak [140], a resonance of the electron scattering rate is expected at commensurate angles, but it should give rise to peaks in magnetoresistance instead of the minima that are observed. Kagoshima et al. [144] have shown the existence of a finite average velocity leading to a conductivity enhancement
along the magnetic field direction at magic angles. Other models predicting resistance dips have been proposed by Maki [145] and Chaikin [146].

### C. Two-Dimensional Superconductors Under High Fields

Other spectacular effects of low dimensionality are observed in the series of $(ET)_2X$ conductors with a two-dimensional tubelike Fermi surface. Figure 49 presents magnetoresistance data in the phase of $\beta_H$-(ET)$_2I_3$ which can be stabilized at low temperature following a special procedure [147]. Unlike the phase $\beta_L(T_c = 1.2$ K) obtained after a regular cooling down, the $\beta_H$ phase, which can be stabilized at atmospheric pressure after the special cooling procedure under pressure [148] (Fig. 49a), is free of any incommensurate lattice modulation and $T_c$ rises up to 8.1 K (Fig. 49a) [15]. The oscillation of extremely large amplitude which is shown above 9 T (Fig. 49b) in the $\beta_H$ phase is periodic in $1/H$ with a fundamental field $H_0 = 3730$ T, which corresponds to the cross section of the tubular Fermi surface by a plane perpendicular to the field (H is thus perpendicular to the conducting sheets) agrees with a density of carriers of $\approx 1$ per formula unit. In addition, the low-frequency beating is related to the deviation from perfect two-dimensionality. The low-frequency $H_1 = 36.8$ T provides an estimate for the interlayer coupling, namely $t_c \approx 0.5$ meV (if $t_\parallel = 70$ meV). With the value of $T_c$ derived from the experiment, the discrete Landau levels are actually broadened into minibands of width $4t_c = 2$ meV due to the energy dispersion along the $k_Z$ direction. It has been recognized [149] that the discretization can be restored when the magnetic field is no longer perpendicular to the conducting planes (i.e., $H\parallel c^*$) but tilted by an angle $\phi$ from the $c^*$ axis, such as (Fig. 50a)

$$ck_F \tan \phi = \pi(n - \frac{1}{4}), \quad \text{with } n = 1, 2, 3, \dots \qquad (51)$$

At magic angles given by the previous relation, the cross section of the Fermi surface normal to the magnetic field becomes independent of the $k_Z$ coordinate; the minibands are nearly discretized (the width of each miniband becomes smaller than the splitting between Landau levels). This model has explained the angle-dependent magnetoresistance oscillations which are observed in $\beta(ET)_2IBr_2$ [150] Fig. 50b) and in other two-dimensional $(ET)_2X$ salts [144,151] (Fig. 51).

## VIII. CONCLUSIONS

The study of the new class of synthetic conductors in which superconductivity has been discovered has contributed to a better experimental and theoretical understanding of the physics in low-dimensional electron gases
<figure><img src="image_76.png" /><figcaption>**Figure 49** (a) Superconducting transition of β<sub>H</sub>(ET)<sub>2</sub>I<sub>3</sub> at 8.1 K after a cooling procedure indicated in the inset. (From Ref. 15.) (b) Giant oscillations of the transverse magnetoresistance of β<sub>H</sub>(ET)<sub>2</sub>I<sub>3</sub>. (From Ref. 147.) The insert shows that fields corresponding to extrema are periodic versus 1/H. The low-frequency beating is due to the finite but small interlayer coupling.</figcaption></figure>
<figure><img src="image_77.png" /><figcaption>**Figure 50** (a) The cross section of the Fermi surface normal to the magnetic field is independent of $k_z$ at field orientations satisfying relation (51). (After Ref. 149.) (b) Angular oscillations of the magnetoresistance of the two-dimensional conductor $\beta(\text{ET})_2\text{IBr}_2$. (After Ref. 150.) (c) Magic angles determined by the peaks of the magnetoresistance in (b) follow very accurately the relation (51).</figcaption></figure>

(one- or two-dimensional), although there is still no firm explanation for the driving mechanism leading to the attractive pairing in organic superconductors. The interplay between antiferromagnetic and superconducting ground states in the TM$_2$X series, the role of coulombic interactions in the magnetic properties of the low-dimensional electron gas above $T_c$, and the
<figure><img src="image_78.png" /><figcaption>**Figure 51** Angular oscillations of the κ-phase (ET)₂NH₄Hg (SCN)₄. (From Ref. 144.)</figcaption></figure>

strong sensitivity of T_c to nonmagnetic impurities in both one- and two-dimensional organic superconductors are all factors playing in favor of a nonconventional superconducting mechanism. However, it is still premature to establish a close connection between the mechanism of superconductivity in existing organic superconductors and the model proposed 25 years ago by Little. Moreover, the stabilization of spin density wave phases at high magnetic field and the coherent motion of a SDW state under high electric field are remarkable products of the developments of these new compounds.

The recent discovery of superconductivity in compounds of fullerenes doped with alkali and alkaline-earth metals is very encouraging for the future of organic superconductors in which conductivity is governed by the formation of π orbitals. The fullerenes, in which the superconductivity is presumably mediated by high-energy intraball vibrations, represent a good compromise between a weak interball interaction and the on-site coulombic repulsion, which is sufficiently small to prevent the stabilization of an insulating Wigner crystal.

Several important and recent aspects of the physical properties of these molecular superconductors have been left by the roadside. It is the case, for example, for the physics of quasi-one-dimensional conductors at very high magnetic field when the undistorted phase exhibiting a metal-like behavior at zero field should become localized [152]. Reentrance of a superconducting instability is also expected at very low temperature under
high magnetic field [153]. The problem of the mixed superconducting phase of the two-dimensional organic compounds has not been discussed, although a magnetization irreversibility line, a vortex melting transition [154, 155], a slow dynamics of the vortex lines [156], and an intrinsic vortex pinning [157] have been observed experimentally in $(ET)_2X$ superconductors.

The entire series of perylene salts comprising parallel segregated stacks of conducting chains and insulating spins are presented in Chapter 7. Transport and magnetic properties of organic superconductors as well as earlier compounds such as TTF-TCNQ and related systems are discussed in Chapter 9. A detailed presentation of the theory of one-dimensional conductors can be found in Chapter 2. The prospects for applications of organic superconductors are still in their infancy (see also Chapter 16). The future will depend very much on new openings for materials preparation.

However, when large areas of organic superconductors become available, it may be possible to take advantage of the Josephson effect and to use arrays of Josephson circuits prepared by lithographic techniques for the fabrication of low-current devices such as high-density memories and microwave detectors [158]. The very narrow electronic paramagnetic resonance of some organic conductors can be used in sensitive magnetic sensors [159].

## ACKNOWLEDGMENTS

I gratefully acknowledge the constructive and fruitful cooperation which has been very active between Orsay and several laboratories in physics and chemistry all over Europe, Québec, and South Korea. The list of colleagues and co-workers who have been involved in the research on molecular superconductors is so long that I would rather refer the reader to the reference list, where the names of colleagues who should be thanked for their cooperation can easily be found. This research has been supported partially by EEC Esprit Basic Research Action 3121, Stimulation Program CI-CT 90-0863 at DG XII, DRET contract 88/198, and CNRS-GDR C60.

## NOTE ADDED IN PROOF

The compound $(TMTTF)_2Br$ occupies a key position in the generalized $TM_2X$ phase diagram of Fig. 5. The small amplitude of the lattice dimerization, see Table 1, goes along with the observed weak one-dimensional $4k_F$ localization occurring below 80 K as displayed in Fig. 6a. Furthermore, the commensurate character of the spin modulation of the SDW ground state is very likely related to the localized character of the spins and to the concomitant absence of Fermi surface in the temperature regime 80–15 K.
K. However, the one-dimensional localization at a temperature higher than the onset of the SDW state is known to be suppressed by a pressure exceeding 5 kbar or so [160]. ¹H-NMR data [161] have revealed a temperature independent spin-lattice relaxation rate in the SDW phase under 13 kbar (besides the strong temperature dependence related to the three-dimensional critical divergence) at variance with the activated behavior, Fig. 8, observed at 1 bar in the same temperature regime. As already reported in the study of (TMTSF)₂PF₆ at 1 bar a temperature independent relaxation rate is a manifestation of the existence of low lying spin excitations such as phason modes, [52]. We may understand the behavior of NMR data under pressure as the signature of a pressure induced commensurate-incommensurate cross-over between 1 bar and 13 kbar. The commensurability cross over may be closely related to the suppression of the 4k<sub>F</sub> localization as suggested by Fig. 5.

The investigation of transport properties under pressure has shown that the SDW ground state can be suppressed above a critical pressure of 25 kbar [160]. (TMTTF)₂Br was thus found to retain a strong metallic character down to 1.2 K. Furthermore, some samples showed a slight drop of the resistance down to a nonzero value near 3.5 K which was ascribed to a possible signature of superconductivity [160].

A recent reinvestigation of this compound down to 0.4 K has now proved unambiguously the existence of a transition toward a zero resistance state at $T_c$ = 0.8 K under 25 kbar with a 10–90% width of 0.4 K [162]. It is possible that the 3.5 K anomaly observed previously was only related to precursor effects of a three-dimensional superconducting transition occurring below 1 K. This new finding of superconductivity in the (TMTTF)₂X family gives strong indications for TM₂X compounds to be considered as a whole as far as the theoretical interpretation is concerned.

However there remains an important problem with this material since FISDW phases are missing in (TMTTF)₂Br under 25 kbar at 0.4 K up to 20 Tesla. The absence is puzzling in comparison with other quasi one-dimensional superconductors in which the threshold fields for the stabilization of FISDW phases have always been located below 10T at $T = 0.5$ K, Figs. 44, 45 and reference 131 for (TMTSF)₂ReO₄. Within the standard theory of FISDW phases the logarithmic divergence in temperature of the susceptibility under magnetic field at a wave vector governed by the magnetic field is preserved provided the amount of non-nesting along the weak coupling direction is smaller than the cyclotron energy, namely $t_c^2/t_a < \hbar\omega_c$.

The latter condition is easily fulfilled for all compounds undergoing FISDW transitions below 10T since $t_c \sim 1$ meV according to a calculation [163].

As far as (TMTTF)₂Br is concerned the c-axis parameter is the smallest within the TM₂X series [164,165]. The dominant role of interchain coupling
in (TMTTF)₂Br is also supported by the broadest EPR line width among compounds of the (TMTTF)₂X series [47]. This situation could lead to an increased three-dimensional coupling with a possible suppression of FISDW phases below 20T.

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