<figure><img src="image_1.png" /></figure>

ORGANIC SUPERCONDUCTORS

D.Jérôme

Laboratoire de Physique des Solides (associé au CNRS) Université Paris-Sud, 91405 Orsay, France

The discovery of organic superconductors in 1980 was an happy conclusion of a search for high $T_c$ superconductors (although $T_c$ was modestly in the Kelvin range).

The first generation of organic superconductors belonged to a large family of quasi-one-dimensional conducting cation radical salts. Most of them exhibit a wide range of new phenomena, including the competition between ground states, the influence of a magnetic field on the non ordered phase of a quasi-one-dimensional electron gas and the onset of spin density wave phases at high magnetic field with quantized Hall effect.

The extensive study of the (TM)₂X series has shown that electron interactions are repulsive and of the order of the electron bandwidth. However, the nature of the ground states relies essentially on the balance between charge localisation boosted by the Umklapp scattering and the interchain hopping integral. Second and third generation organic superconductors are two and three dimensional molecular conductors respectively with maximum $T_c$ of 12K (ET)₂X and 33 K (fullerides).

Keywords:A.organic crystals,superconductors

1. Introduction

Crystalline conductors of molecular origin are basically different from the regular metallic crystals. They are solids made of building blocks (the molecules) reflecting on the one hand the properties of the molecular species (ionization energy or electron affinity, characteristic infra- red or Raman active vibration modes, NMR shifts related to individual atomic sites,...) and on the other hand the physical properties of a metal (dp/dT > 0, small Hall coefficient, plasma frequency in the optical reflectance,...).

Contrasting 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 presenting an open shell configuration. Such a situation comes generally from a partial oxidation (reduction) of donor (acceptor) molecule in the formation of a salt with an inorganic anion (cation). In addition, a strong intermolecular overlap of π-orbitals makes the electron delocalization possible over all molecular sites in the crystals (although in some cases the delocalization occurs preferentially along selected cristallographic directions.

Molecular conductors which are the subject of this article should not be confused with extended conjugated polymers and even graphite, where the π-electron system of the extended molecules provides the conducting path-way. The development of molecular conductors as well as the subsequent development of conducting polymers has been stimulated to a large extent by the suggestion made by Little in 1964 that the arrangement of chain conductors in a polarizable medium could promote superconductivity at high temperature [1].

Molecular conductors which are surveyed in this article usually derive from molecules which are small modifications of the flat tetrathiafulvalene (TTF) prototype

fulvalene molecule [2]. These molecules tend to form stacks in the crystalline state. Such a packing optimizes the overlap between molecular orbitals along the stacks. When the Coulomb repulsion (U) does not overcome the energy gained by the band formation, conducting properties can be observed. However, due to the strong directionality of the intermolecular couplings a pronounced one dimensional character is obtained in most organic conductors (they are called one dimensional,(1-D), or quasi one dimensional, (Q-1-D), conductors when the small interchain coupling is taken into account.

Salts of the tetramethyltetrathia(selena)fulvalene, i.e. TMTT(S)F molecule [3] are prototype materials for molecular conductors [4], figure(1). They reveal some of the most typical behavior for 1-D physics which will be reviewed in this article [5,6].

These molecules also give rise to higher dimensionality conductors. In that case, the planar molecules form dimers with a strong overlap between heteroatoms both within each dimer and between neighbouring dimers. The dimers are packed in planes according to a chess-board pattern. This crystal structure gives rise to a large metal-like conductivity within 2-D conducting molecular layers and about $10^4$ less conduction between the sheets [7].

The recently discovered fullerene conductors bear much similarity with organic conducting salts. Neutral C60 molecule possesses a closed shell electronic structure giving rise to an insulating solid. But, very much like organic molecules, when carriers are added to the neutral molecule through its reduction by an alkali cation the salt thus formed may become conducting [8].

Similarly to 1 and 2-D molecular conductors the conductivity of fullerenes is due to the intermolecular overlap of π-molecular orbitals. For all these molecular
<figure><img src="image_2.png" /><figcaption>TMTSF</figcaption></figure>

<figure><img src="image_3.png" /></figure>

1) TMTSF donor molecule and a side view of the prototype (TMTSF)₂X structure.

conductors counter ions do not play any significant role at first sight in the intermolecular conduction process.

This article intends to present a very short review of some physical aspects of organic conductors in which superconductivity has been stabilized [9]. The low dimensionality of these Q-1-D conductors makes the physical properties of these materials extremely diversified and fascinating. Therefore, besides superconductivity, the role of dimensionality, correlations and band filling will also be discussed.

## 2. Materials and Electronic Structure

The salts of the (TM)₂X family where TM is the molecule TMTTF, TMTSF or any mixed (S-Se) molecule represent the archetype of 1-D conductors, figure (1). Superconductivity, when it exists in these materials is restricted to the low temperature domain ($T_c \le 2K$) but a wealth of other instabilities are also observed [6]. All isostructural (TM)₂X compounds exhibit a rather simple band structure with a conduction band built from the highest occupied molecular orbital (HOMO) of the isolated molecule. The conduction band has a width ranging between 0.4 and 1.2 eV [10], figure (2) and its filling is formally 1/4 since one carrier (hole) is provided to each one dimensional unit cell containing two TM entities by the ionization of the monoanion X. However, the three dimensional cristallographic structure of the anions induces a slight dimerization of the bond lengths (and of the overlaps) between molecules along the stacking direction giving rise to a splitting of the conduction band into an half-filled upper band and a filled lower band. The half-filled band character is related to the amount of band dimerization [11]; it is generally more pronounced for sulfur than for selenium compounds and it is known to be reduced under pressure, figure (3).

Some band parameters of these salts are listed in Table I. Typical band widths are in the range of 0.4 and 1.2 eV for sulfur and selenium salts respectively.

<table><caption>**Table I.** Parameters of the calculated dispersion relations for (TMTTF)₂Br and (TMTSF)₂PF₆, figure (2) for the labels using their room temperature and ambient pressure structures (private communication from E. Canadell and to be published in J. Physique I). The calculations were carried out using a basis set of double-ζ Slater type orbitals. Numbers in brackets are the band parameters obtained from the two dimensional multi-ζ calculation in ref [10]. All values are in eV.</caption><thead><tr><th></th><th>(TMTTF)₂Br</th><th>(TMTSF)₂PF₆</th></tr></thead><tbody><tr><td>W1</td><td>0.316 (0.372)</td><td>0.518(0.595)</td></tr><tr><td>WII</td><td>0.417</td><td>0.573</td></tr><tr><td>Wa</td><td>0.797 (0.859)</td><td>1.218(1.366)</td></tr><tr><td>Wb</td><td>0.052 (0.044)</td><td>0.140 (0.114)</td></tr><tr><td>Wb'</td><td>0.257</td><td>0.248</td></tr><tr><td>Wc</td><td>0.002</td><td>0.003</td></tr><tr><td>Wc'</td><td>0.004</td><td>0.002</td></tr></tbody></table>

We shall see that the half filled band character together with the amplitude of the interchain overlap are both important parameters which govern all physical properties in the (TM)₂X family.

## 3. Instabilities in the 1-D Electron Gas, Theoretical Features.

To create a superconducting phase that is stable below a certain critical temperature, one must overcome the strong tendency shown by the low dimensional electron gas (in particular those with a pronounced 1-D character) to undergo a transition toward an insulating ground state exhibiting a density wave periodic potential with wave vector $Q$. The tendency towards instability in a 1-D electron gas is already visible without interactions as shown by the temperature dependence of the non-interacting electron susceptibility [12]:

$$ \chi_o^{1D}(q, \omega) = \sum_k \frac{n(k) - n(k+q)}{\epsilon(k) - \epsilon(k+q)} \quad (1) $$

where $\epsilon(k)$ and $n(k)$ are the excitation energy counted

<figure><img src="image_4.png" /></figure>

2) Energy dispersion of (TMTSF)₂PF₆ and Fermi surface.
<figure><img src="image_5.png" /></figure>

3) Illustration (schematic) of the structural dimerization giving rise to an upper half-filled band in (TM)₂X compounds.

from the Fermi level and the Fermi occupation factor respectively. At low temperature eq.(1) leads to logarithmic divergences,

$$ \chi_o^{1D}(2k_F, T) \approx \log\left(\frac{T}{E_F}\right) \quad \text{and} $$

$$ \chi_o^{1D}(q - 2k_F, T = 0, \omega = 0) \sim \log(v_F \frac{(q - 2k_F)}{E_F}) $$

where $k_F$ is the wave vector which nests the Fermi surface of a 1-D electron gas. In eq. (1) the electron dispersion has been linearized in the vicinity of the Fermi energy, namely

$$ E(k) = E_o + V_F(|k_x| - k_F) \quad (2) $$

Because the FS of TM₂X compounds is open, the major concepts of 1-D physics should apply to these materials, namely, (i) the absence of long range order at finite temperature [13], (ii) the existence of equally divergent correlations between electron states at (+k,↑) and (-k,↓) [14], leading to superconducting pairing (Cooper channel) and electron and hole states at k and k+Q leading to density wave instabilities (Peierls channel) [14]. These two pairing channels coexist in the 1-D electron gas but the divergence of the electron-hole (density wave) channel is, however, limited at low temperature by the existence of a finite coupling $t_⊥$ between near neighbouring chains, which gives rise to a cross over temperature below which 1-D physics no longer applies [15], namely

$$ T_x^o \sim t_\perp / \pi \quad (3) $$

and the 1-D bare susceptibility saturates at low temperature.

$$ \chi_o^{1D}(T < T_{\text{c}}) \sim \log\left(\frac{t_\perp}{\pi E_F}\right) \quad (4) $$

The divergence of the Cooper channel is not affected by the interchain coupling since the equivalence between +k and -k states is preserved even in the presence of a finite transverse coupling.

Going to the situation of the interacting 1-D electron gas, and focussing on low energy excitations, the only pertinent wave vectors to be taken into account for the Coulomb interaction are those which are close to $q = 0$ and $q = 2k_F$ since the electron-hole pair excitation energy is zero for these vectors. Consequently, the Coulomb

interaction is usually approximated by two coupling constants $g_1$ and $g_2$ which are respectively the backward ($q = 2k_F$) and forward ($q = 0$) scattering amplitudes [16].

In addition, there are also $g_3$ 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) [17].

The mean-field treatment of the 1-D electron gas amounts to neglecting the mixture of Peierls and Cooper channels and evaluating correlation functions via the perturbation expansion of ladder diagrams. This summation procedure is not appropriate for the 1-D electron gas as it leads to finite transition temperatures (against the Landau criterion).

However, when superconducting and electron-hole instabilities are treated on equal footing the famous (g₁, g₂) phase diagram for the most stable ground state of a 1D electron gas reveals a common border ($g_1 = 2g_2$) between Peierls and Cooper instabilities ($g_3 = 0$ so far). For $g_1 < 0$, the only stable modulated state in the one exhibiting a charge density wave oscillation (CDW) whereas there is a gap in the spin excitations. On the other hand, the spin modulated ground state is stable only for $g_1 > 0$. There are low-lying spin excitations and the spin susceptibility is finite at $T = 0$.

Response functions of all instabilities contain at all orders in perturbation logarithmic terms such as $\log(\omega/E_F)$, as well as a mixture of Peierls and Cooper divergences.

The only relevant treatment of the 1-D electron gas is the one which retains the mixture of channels as the temperature is lowered. Such a situation is achieved when the band width $E_F$ is renormalized towards a smaller value at low temperature so that the various log (T/E_F) terms of the perturbative expansion become less divergent. The renormalization of the bandwidth is also accompanied by a renormalization of the various coupling constants $g_i$ [16].

Spin excitations (the static susceptibility) rely only on the backward scattering $g_1$ whereas the charge excitations depend on the set of parameters $g_3, g_1-2g_2$. Renormalization procedures lead to hyperbolic trajectories [17], as shown in fig. (4).

$$ g_3^2 - (g_1 - 2g_2)^2 = \text{const} $$

If $|g_3| < g_1 - 2g_2$, then $g_3$ scales to zero and the system evolves towards an attractive pairing instability as $T \to 0$. On the other hand, for $|g_3| > g_1 - 2g_2$ i.e. strong Umklapp situation for an half-filled band, $g_3$ scales towards strong coupling with the concomitant opening of a gap $\Delta\rho$ in the charge excitation spectrum below the temperature $T_\rho$. However, $g_1(T)$ renormalizes to zero when $g_1 > 0$, following the equation,

$$ g_1(T) = \frac{g_1}{1 - \frac{g_1}{\pi V_F} \ln \frac{T}{E_F}} \quad (5) $$

The spin susceptibility is not affected by the charge
localization below $T_ρ$. For the strong or weak $g_3$ situation $χ_S(T)$ then reads:

$$χ_S(T) = χ_S(0)(1 + g_1(T)) \quad (6)$$

where $χ_S(0)$ is the zero temperature susceptibility enhanced above the bare band value by a Stoner factor. This phenomenon is known as the spin-charge separation of the 1-D electron gas. Experimentally, one should not be able to tell from the only knowledge of the temperature dependent susceptibility whether the chain is a metallic conductor or a linear chain of localized spins. It can be shown that the long distance decay of the correlations functions of the interacting 1-D electron gas is determined by a unique exponent which also governs their temperature dependence [18], namely

$$χ_{DW}(2k_F, T)αT^{(K_ρ - 1)} \quad \text{and} \quad χ_{SC}αT^{(1/K_ρ - 1)} \quad (7)$$

Susceptibilities follow power laws dependences at low temperature. Unlike the mean field treatment, no phase transition is observed at finite temperature. Density waves or pairing correlations are divergent at low temperature for $K_ρ < 1$ and $> 1$, respectively. The exponent $K_ρ$ depends on the bare values of the coupling constants, namely, $K_ρ = 1 + \frac{g_1 - 2g_2}{2πV_F}$. Therefore, $K_ρ = 1$ corresponds to the non-interacting situation and $K_ρ = 0$ represents the strong coupling limit with a large gap in the charge degrees of freedom. In addition, the momentum distribution no longer exhibits a step at the Fermi wave vector; this a Luttinger liquid.

$$n_k = n_{k_F} - \beta \mathrm{sign}(k - k_{F}) |k - k_{F}|^{\alpha}$$

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

4) Diagramatic representation of the pair susceptibilities, Cooper $Δ(q, ω)$ and Peierls $π(2k_F + q, ω)$ at lowest order (a) and second order (b) showing the mixture of channels. Renormalization paths for $g_3$ and the charge degree of freedom exponent $K_ρ$.

and the single particle density of states vanishes at $E_F$ according to a power law singularity

$$N(ω) \sim |ω|^{\alpha} \quad \text{where} \quad α = \frac{1}{4}(K_{\rho} + \frac{1}{K_{\rho}} - 2) \quad (8)$$

What is expected from an experimental study of 1-D conductors is the search for the most characteristic features of 1-D physics, namely, the existence of a 1-D regime in temperature, the spin-charge separation, evidences for a Luttinger liquid and the determination of power laws for divergent susceptibilities.

## 4. 1-D Physics : Experimental

A direct consequence of the pronounced 1-D character of the TM2X series is the existence of a wide variety of behaviours that can be observed for the transport properties, depending on parameters such as the chemical composition of the organic molecule or the inorganic anion and the hydrostatic pressure in various members of the family. The incipient instabilities that develop at high temperature in the 1-D regime depend on the intra-chain interactions and on the strength of the Umklapp scattering ($g_3$) whereas the nature of the long range that is stabilized at low temperature depends also on the coupling between the chains.

The generic diagram in figure (5) displays the variety of 1-D regimes and ground states that can be observed in the TM2X series [6]. Special attention has been paid to some key compounds labelled by letters in figure (5). At the left side of the diagram, (TMTTF)2PF6 (a) presents a poor conductivity $σ_{II} \approx 30Ω^{-1}cm^{-1}$ but a metal like conductivity at high temperature and a charge localized (CL) behaviour below $T_ρ \approx 250K$. It is the compound in which the half-filling character is most pronounced within the TM2X series since the overlap alternation amounts to $Δt_{II} / t_{II} \approx 0.38$. The localization which takes place below $T_ρ$ is due to repulsive interactions between 1-D electrons belonging to the half-filled band [19], figure (6). The average distance between electrons is a (the periodicity along the stacking direction). In the same temperature regime the localized electrons of the Heisenberg chain couple to phonons and give rise to 1-D lattice fluctuations at wave vector $2k_F$ (periodicity 2a) detected by X-ray experiments.

<figure><img src="image_9.png" /></figure>

5) Generalized phase diagram for the (TM)2X series. Spin-Peierls (SP), spin density wave (SDW) and superconductivity (SC) are indicated together with the zero pressure location of some prototypical compounds, see text.
Below about 20 K these 1-D lattice fluctuations order three-dimensionally and the system undergoes a phase transition towards a spin-Peierls phase with the loss of the spin degrees of freedom. NMR experiments have shown an activated behaviour for the spin-lattice relaxation time related to the spin-Peierls gap in the spin degrees of freedom [20].

As one moves to the right in the diagram not much is noticed for the temperature dependent susceptibility but the g3 term (proportional to the band dimerization, g3=g1(Δg/EF) becomes less important and the charge localization is shifted to lower temperatures, Tp ≈ 100K for (TMTTF)2Br, (b) [21]. Coupling of weakly localized electrons to phonons looses in strength, although some 2kF 1-D lattice fluctuations are still observed below Tp [22]. The weakly localized electron gas undergoes a transition towards a ground state which is characterized by a modulation of the spin density at wave vector 2kF. NMR experiments have demonstrated both the magnetic nature of the ground state in (TMTTF)2Br and its commensurate character [23,24]. The localized moments order antiferromagnetically along transverse directions and the periodicity is doubled accordingly. The commensurate character of the SDW ground state is also attested by the activation of the relaxation rate since no low lying excitation modes (phasons) are present in this x 2 commensurate ground state to couple to nuclear spins.

The behaviour of the Bechgaard salt (TMTSF)2PF6 (c) [3] is interesting to compare with that of (TMTTF)2Br as its conductivity is metal-like down to a phase transition towards a SDW ground state, figure (7). No 1-D localization is observed before the transition at TSDW (= 12 K). The additional periodicity in the exchange potential in the SDW phase is responsible for the opening of a gap at the Fermi level accompanied by a metal-insulator transition [25]. No doubts the concept of Fermi surface is valid for the compound (TMTSF)2PF6. Furthermore, it is the optimum nesting condition of the Q-1-D Fermi surface which decides for the actual value of the magnetic modulation wave vector Q. 13C and 1H-NMR spectra have unambiguously shown that Q is incommensurate with the underlying lattice [26,27,28]. Furthermore, the existence of low lying phason

<figure><img src="image_10.png" /><figcaption>6) Temperature dependence of the ESR susceptibility of (TMTTF)2Br and ρ(T) for (TMTTF)2PF6 and (TMTTF)2Br.</figcaption></figure>

modes (although pinned by impurities and defects) makes the NMR relaxation rate temperature independent in the SDW state.

The main difference between (TMTSF)2PF6 and (TMTTF)2Br is the weak localization which takes place at high temperature for the latter compound. The interchain interaction which is required for the onset of long range order in (TMTTF)2Br can be ascribed to an antiferromagnetic coupling interaction between weakly localized spins provided by the exchange of a 2kF spin fluctuation between chains [29]. The existence of such an interchain coupling suggests that the correlated jump of an electron-hole pair (a 2kF spin fluctuation) between neighbouring chains is easier than the jump of an individual particle : the electron-hole cross-over temperature TX2 is thus higher than the single particle cross-over at TX1. Actually, it has been suggested that the single particle tunnelling can be impeded by 1-D correlations. The cross-over temperature thus becomes instead of eq.(3) [29]:

$$T_x \sim T_x^\nu \left(\frac{t}{E_F}\right)^{\frac{1-K_\rho}{K_\rho}} \quad (9)$$

The reduction of TX1 in eq.(9) can be quite significant even for relatively small repulsive interactions ($K_ρ ≤ 1$) since t⊥/EF = 0.1 and may lead to a suppression of the single particle cross-over in agreement with the NMR determination for a lower limit value in the vicinity of 10 K [30].

Direct evidences for the existence of a 1-D regime and the role of electron-electron interactions in the (TM)2X series are given by the extensive NMR studies which have been conducted in various compounds.

NMR spin lattice relaxation probes the low energy excitation modes of the 1-D electron gas at q = 0 and q = 2kF.

Therefore, instead of a summation over all q-vectors in the Moriya's formulation [32] for T1 the relaxation rate in a 1-D electron gas simplifies [33],

$$[T_1 T]^{-1} = C_o \chi_s^2(T) + C_1 \quad (10)$$

<figure><img src="image_11.png" /><figcaption>7) Resistivity and susceptibility of (TMTSF)2PF6.</figcaption></figure>
The first term in eq.(9) is linked to the uniform ($q = 0$) spin excitations of the 1-D electron gas ($C_0$ is a constant related to the hyperfine interaction), whereas the second term probes the $q = 2k_F$ spin fluctuations. $C_1(T)$ is related to the $2k_F$ spin susceptibility, namely $C_1(T) \propto \chi_{SDW}$ ($2k_F$, T) $\propto T^{\alpha/2}$. Hence, the contribution to $T_1^{-1}$ coming from $2k_F$ spin fluctuations becomes

$$ \left(\frac{1}{T}\right)_{2k_F} \propto T^{K_\rho} \qquad (11) $$

Figure (8) shows that the temperature independent contribution to $1/T_1$ (extreme situation of strong coupling with $K_\rho = 0$ for a 1-D quantum antiferromagnet) prevails in the whole temperature domain below $T_\rho$ for (TMTTF)₂PF₆ and (TMTTF)₂Br salts.

The temperature dependence of $\chi_{SDW}$ ($2k_F$, T) has been approached with much accuracy in the conductor (TMDTDSF)₂PF₆ (c in fig. 5) built with a mixed sulfurselenium TM molecule [34], figure (9). There, the strong coupling limit, $K_\rho \to 0$, is reached below $T_\rho/2$.

Some temperature dependence of $K_\rho$ is expected at higher temperature according to the renormalization trajectories displayed in fig. (4). There is also evidence from NMR data for significant 1-D $2k_F$ spin fluctuations even in compounds exhibiting a metal-like conduction down to the onset of long range ordering at low temperature. For example $2k_F$ spin fluctuations contribute to $T_1^{-1}$ below 100 K and 30 K in (TMTSF)₂PF₆ and (TMTSF)₂ClO₄ respectively [31]. The data in fig. (8) show that a renormalized "Korringa" behaviour is recovered below 10 K. Therefore, that temperature can be taken as the lower limit value $T_{X1}$ for the 1-D to 3-D single particle cross over regime. The cross-over regime extends up to about 30 K in (TMTSF)₂ClO₄. Thus the enhancement of $T_1^{-1}$ observed between 30 and 10 K is due to spin fluctuations which evolve from 1-D to 2 or 3-D on cooling towards low temperatures.

<figure><img src="image_12.png" /></figure>

8) ¹³C spin lattice relaxation rate versus $T\chi_s^2(T)$ for (TMTTF)₂PF₆ and (TMTTF)₂Br undergoing a charge localization below $T_\rho$. No localization is observed in (TMTSF)₂X compounds (dashed line). $T_1^{-1}$ in (TMTSF)₂ClO₄ reveals a strong deviation from a "Korringa" behaviour below 30 K (insert).

Recent photoemission experiments performed on (TMTSF)₂PF₆ at 50 K have failed to show a Fermi step and a finite single particle density of states at the Fermi level [35]. Instead, N(E) is proportional to (E-E_F)¹.² which leads, according to eq.(8), to $K_\rho \approx 0.2$. Such a small value for $K_\rho$ represents admittedly a rather strong repulsive coupling situation [18] which would be in qualitative agreement with the low temperature $T_1^{-1}$ data in the same compound but yet hard to reconcile with the metal-like character of the conductivity which shows no sign of localization below 100 K [36].

Power law divergences of $\chi_{SDW}$ have also been measured in the 1-D conductor TTF-[Ni(dmit)₂] where the band filling is different from 1/2 [37]. There again $K_\rho = 0.4$ although the TTF chain contributing to the conduction reveals no sign of localization [38].

NMR and susceptibilities have strongly contributed to the understanding of the physical properties of (TM)₂X conductors. Thus at ambient pressure the ratio $g_1/\pi V_F$ is found to be close to unity and it is essentially the same for all compounds in the (TM)₂X family [31]. Furthermore, the amplitude of the AF spin fluctuations and the charge degrees of freedom are governed by the Umklapp coupling term $g_3$ and the combination $g_1-2g_2$. The Umklapp term is dominant for sulfur compounds leading to a localized 1-D antiferromagnet at low temperature but weakens gradually for selenium system (or under pressure). NMR has also provided indications for the dimensionality cross-over regime extending between 30 and 10 K in selenium compounds. The onset of the SDW ground state of (TMTSF)₂PF₆ is linked to an incommensurate vector Q nesting optimally the Q-1-D Fermi surface. Inter-chain interactions between first neighbour chains amounts to adding a term such as $\cos k_\perp b$ in eq.(2). Deviations to Q-1-D perfect nesting (occurring under pressure) can be described including also next near neighbour interactions, namely [39].

$$ E(k) = E_0 + V_F(|k_\perp| - k_F) + 2t_\perp \cos k_y b + 2t_\perp^1 \cos 2k_y b \quad (12) $$

with $t_\perp^1 \approx t_\perp^2 / t_{\parallel}$

The stability of the SDW ground state is suppressed as soon as the deviation to perfect nesting becomes

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

9) Power law temperature dependence of $\chi_{SDW}$($2k_F$,T) derived from NMR in (TMDTDSF)₂PF₆.
<figure><img src="image_14.png" /></figure>

10) Superconducting transition of (TMTSF)₂PF₆ under pressure and (ET)₂Cu(SCN)₂ at ambient pressure. Also shown, the specific heat anomaly and the Meissner expulsion in (TMTSF)₂ClO₄.

significant, i.e. $T_c^1 > T_{SDW}^o$ where $T_{SDW}^o$ is the 1-D mean-field temperature for the onset of the SDW state [40].

Once the SDW ground state is removed increasing the deviation to perfect nesting in eq.(11) either under pressure or chemically, superconductivity becomes stable at low temperature with a maximum critical temperature smaller than 2K [9,41], figure (10). As shown by NMR experiments, 2kF spin fluctuations are still very active when superconductivity sets in. The intra-chain repulsive interactions should prevent s-wave superconducting order.

However, the exchange of 2kF spin fluctuations between chains could promote an attractive pairing for electrons belonging to neighbouring stacks [42]. Such a pairing mechanism would give rise to an anisotropic superconducting gap $\Delta(k) = \Delta_o \cos k_x b$ with lines of zeros on the Fermi surface.

Figure (10) displays the superconducting transition of (TMTSF)₂PF₆ salts observed by resistivity data. What is remarkable in fig.(10) is the strong temperature dependence of $\rho(T)$ above $T_c$, unlike the behaviour of regular metals for which the resistivity is limited at low temperature by temperature independent elastic scattering.

The observation of a Meissner flux expulsion [43] and of a specific heat anomaly [44] support the picture of bulk superconductivity, figure (10). On a C/T vs T² plot, the specific heat of (TMTSF)₂ClO₄ displays a very large anomaly at 1.2K and follows the law: $C/T = \gamma + \beta T$, with $\gamma = 10.5$ mJ·mol⁻¹·K⁻² corresponding to $N(E_F) = 2.1$ states·eV⁻¹·mol⁻¹ [N(E_F) = $3\gamma/2\pi2k_B^2$] and $\beta = 11.4$ mJ·mol⁻¹·K⁻⁴.

The existence of superconductivity in all members of the (TM)₂X series has recently been confirmed, (Orsay

results, to be published in Advanced Materials, 1994).

It is still premature to state that organic superconductivity of 1-D conductors is non-conventional and mediated by a magnetic exchange although several experimental results support this possibility:

i) NMR in the superconducting phase revealed a power law temperature dependence of $T_1^{-1}$ and has failed to show a coherence peak at $T_c$ [45].

ii) Superconductivity is present in all members of the (TM)₂X family but seems to ve very sensitive to small concentration of impurities, cristalline disorder or packing defects [46].

iii) The stability of the superconducting ground state takes advantage of the proximity with the antiferromagnetic ground state as attested by the large negative pressure coefficient $\delta \ln T_c / \delta p = -25\% \text{kbar}^{-1}$ [47].

Attempts to force the stabilization of superconductivity at lower pressures hoping to increase $T_c$ in the compound (TMTSF)₂ReO₄ (where a pressure of 10kbar is necessary to prevent the ReO₄⁻ anion ordering to open a gap at the Fermi level) have failed to raise $T_c$ above 1.2 K [48]. This behaviour illustrates the close interplay that exists between magnetism and superconductivity in the (TM)₂X series.

## 5. High Magnetic Fields Properties

Starting from the superconducting state, increasing magnetic field perpendicular to the stacking axis gives rise to spectacular effects. First, the superconducting state is destroyed above a critical field which is very anisotropic ($H_{c2}^c \sim 0.1T$, $H_{c2}^b \sim 2T$) [49]. At higher fields, an antiferromagnetic ground state is recovered above a threshold field $\sim 3T$ for field $\parallel c^*$ axis [50]. Increasing the field further induces a sequence of phase transitions between SDW ground states. The major peculiarity of these phases is
<figure><img src="image_15.png" /><figcaption>11) Quantized Hall voltage of (TMTSF)₂PF₆ (under 9 kbar) and phase diagram. Integers show the number of filled Landau levels in each subphase.</figcaption></figure>

provided by the field-independent Hall resistance within each subphase [51] which is quantized at the value ρ<sub>XY</sub> = h/2Ne<sup>2</sup>, where N is a small integer, N = 1, 2, ..., 5. (TMTSF)₂PF₆) gives a remarkable illustration for the behaviour of the quantized Hall resistance [52,53], figure (11). Field-induced spin density wave (FISDW) phases correspond to semimetallic phases with a density of carriers which is about 10<sup>2</sup> times smaller that that of the Q-1-D metallic phase at H < 3T. Above 18T, the resistance jumps to a very large value and the Hall resistance becomes large and no longer follows the quantized law. The phase above 18T is thus ascribed to a semiconducting SDW phase with no free carriers at zero temperature i.e. N = 0 (unlike subphases with N > 0 which do contain free carriers).

Similar restorations of SDW phases under magnetic
fields have been achieved with other members of the
(TMTSF)2X series, X = ClO4⁻ [54,55] and ReO4⁻ [56]. The
ClO4⁻ compound has raised much interest since the Hall
plateaus can be indexed by the quantum numbers N = 1, 3,
6 and so forth. The plateau N = 2 is clearly missing and a
sign reversal of the Hall resistance is observed between N =
3 and N = 6 phases. The N = 1 phase of this compound
extends from 8 T up to 27 T [57] and the N = 0 phase has
not yet been clearly identified but is probably shifted above
27 T [58,59], see figure (12). The giant reentrance of
TFISDW in the ClO4⁻ phase diagram seems to be specific to
that compound. No such anomaly for the stability of the
FISDW phases has been detected in the prototype system
(TMTSF)2PF6. The particular ordering of the ClO4 anions
is very likely responsible for this phenomenon. Secondly,
as soon as a moderate magnetic field is applied along the c*
direction the temperature dependence of the longitudinal
resistance no longer remains metal-like. A shallow minimum
is observed instead at low temperature and rises
continuously with magnetic field [60]. Clearly, the magnetic
field induces a weak electron localization which bears some
similarities with the charge localization observed below Tρ
in a sulfur compounds such as (TMTTF)2Br at zero field,
see fig (5). Furthermore, the localization is most efficient
when H//c* and no localization is observed up to 27T as
H//b'.

Figure (12) displays the phase diagram of
(TMTSF)₂ClO₄ showing both the various ground states
corresponding to FISDW phases and the field-induced

localization occuring in the undistorted phase at high
temperature. comparing the development of the localization
under magnetic field and the diagram of FISDW phases it is
clear that a feature such as the reentrance of the FISDW
states above 15 T is not observed for the localization. This

<figure><img src="image_16.png" /><figcaption>12) Phase diagram of (TMTSF)₂ClO₄ showing the FISDW ground states and the restoration of the localization under magnetic field (H//c*). Quantized Hall voltage of (TMTSF)₂ClO₄.</figcaption></figure>
feature enables us to rule out (at least at high fields) 3-D critical fluctuations in the vicinity of the AF phases as the source of localization. Instead, we have proposed that the localization can be explained by the magnetic field limiting the amplitude of the transverse quasi-classical motion of Q-1-D electrons moving along the stacks. The dimensionality cross-over (or the effective interchain overlap) is thus decreased under magnetic field. Hence, the localization at $T_p$ related to a charge gap in a 1-D electron gas is restored by the magnetic field. At zero magnetic field, the cross-over regime (20-30K) is reached at a temperature which is higher than $T_p$ and the charge localization of the 1-D conductor looses its significance.

FISDW phases have been given a very elegant theoretical treatment [61] called the "standard model" which also relies on the field induced one dimensionalization of Q-1-D electron trajectories under a transverse magnetic field [62]. The divergence of $\chi_{SDW}(Q)$ at low temperature which is suppressed by pressure together with the SDW ground state in (TMTSF)₂PF₆, can be reactivated as soon as a magnetic field is applied to Q-1-D electrons along the direction c* of lowest conductivity [39]. The spin susceptibility $\chi_{SDW}(q, H)$ thus displays logarithmically divergent peaks as T approaches zero for the set of different wave vectors defined by $q_x = 2k_F + NeHb/h$, where N is an integer. Crudely speaking, the field restores enough 1-D character in the energy dispersion for the SDW distortion to become stable. A sequence of phases must be crossed before the N = 0 phase (which is believed to correspond to the insulating SDW ground state which is stable at ambient pressure) can be reached above 18T. The integer variable N which is related to the deviation of $q_x$ from $2k_F$ labels a semimetallic subphase containing N fully occupied Landau levels below the Fermi level. As the degeneracy of each Landau level increases linearly with the magnetic field, the density of carriers in the semimetallic pockets increases accordingly and the Hall resistance remains fixed at the quantized value. The situation where the Fermi level falls between N filled levels and upper empty levels minimizes the diamagnetic energy of the 2D carriers. This situation prevails in a finite range of magnetic fields as long as the wave vector of the magnetic modulation of a given

subphase can vary linearly with the field. First order phase transitions, with a finite jump of the modulation vector, are expected between various subphases, these transitions are in agreement with the hysteresis shown in fig. (12).

## 6. Spin Density Wave Collective Conduction

The broken-symmetry ground state characterized by the magnetic modulation (SDW) of wave vector Q is responsible for the activated behaviour of the conductivity in the semiconducting phase of (TMTSF)₂X compounds through the single particle gap $2\Delta$. Moreover, two kinds of collective excitations arise in the SDW condensate [63]. They are (i) magnon excitations that manifest themselves in the antiferromagnetic resonance and (ii) the phase mode of the condensate, which is gapless provided the spin modulation is non-commensurate with the underlying lattice. In (TMTSF)₂PF₆, the wave-vector corresponding to the optimum nesting of the Fermi surface is shown by NMR to be incommensurate although it is very close to the commensurate value (0.5, 0.25, 0). As long as Q is incommensurate with the underlying lattice a rigid motion of the density wave can be achieved without any cost in energy. This translational invariance gives rise to a collective contribution to the conductivity:

$$ \sigma(\omega) = (ne^2/m^*) \delta(\omega - \omega_0) $$

where n is the density of condensed carriers in the SDW state, m* is the effective mass of the condensate and $\omega_0$ is the oscillation frequency of the collective mode, either zero for an ideally pure material, or a finite frequency when the collective mode is pinned by commensurability or impurities [64].

The interaction between the SDW and non magnetic impurities provides a finite pinning energy. A threshold electric field $E_T$ must, therefore, be reached before the condensate can contribute to the conduction collectively. $E_T$ is given by the balance between the energy provided by the electric field when the condensate is moved by the wavelength $2\pi/k_F$ and the pinning energy, namely $E_T \approx \omega_0^2$. In all SDW phases of the (TMTSF)₂X series $E_T$ lies in the millivolt per centimeter range [65].

As shown in figure (13), the conductivity of the

<figure><img src="image_17.png" /><figcaption>13) Typical current-voltage characteristics showing the onset of SDW sliding above the threshold field and the beating between an external AC current and the current oscillation of the sliding SDW state (Shapiro steps).</figcaption></figure>
SDW phase becomes field dependent above $E_T$. Then the total current comprises a regular component excited through the single particle SDW gap and a collective term $I_{SDW}$.

A direct evidence for a spin density wave transport is the detection of a current oscillating at a frequency which is proportional to the DC current carried collectively [66]. The recent observation of such oscillations, the harmonic and sub-harmonic locking of this oscillation to an external ac source and a motional narrowing of the NMR spectrum in the sliding SDW state, figure (14) has provided a firm evidence for the existence of a novel collective transport in an SDW condensate [26].

## 7. Conclusion and other Remarks

The study of new classes of conductors in which organic superconductivity has been discovered has contributed to a better experimental and theoretical understanding of the physics of the 1-D electron gas. Superconductivity in the (TM)₂X series is only one instability among many possible ground states whose stability is governed by the competition between the charge gap (enhanced by Umklapp scattering terms g3) and the amplitude of the transverse coupling $t_⊥$. Intrachain repulsive interactions are important for all compounds ($g1/πv_F=1$) but they give rise to a clear cut spin-charge separation only when the Umklapp contribution is dominant. For strong charge localization at wave vector 4kF the intrachain exchange coupling leads to strong spin-Peierls fluctuations and concomitant 1-D 2kF lattice softening which promote a spin-Peierls lattice instability (prototype example (TMTSF)₂PF₆). When the charge localization is weaker, the interchain AF exchange favors the propagation of 2kF spin correlations and the ground state becomes SDW

<figure><img src="image_18.png" /><figcaption>14) ¹³C-NMR spectrum in the SDW phase of (TMTSF)₂PF₆ showing the motional narrowing arising when a current is passed through the sample.</figcaption></figure>

(commensurate for (TMTTF)₂Br). The ground state becomes an incommensurate SDW state due to a nested Fermi surface if the system becomes 2 or 3-D before the 1-D charge localization can be active (prototype (TMTSF)₂PF₆). Small deviations to perfect nesting at $T < T_c$ suppress the incommensurate SDW ground state but strong AF fluctuations remain active in the conducting phase and the ground state is superconducting with $T_c ≤ 2K$. Pairing of carriers through the exchange between neighbouring chains of AF fluctuations is to be considered for superconducting gap anisotropy and exotic pairing mechanisms that could explain several experimental observations [67].

Charge localization in 1-D conductors remains always behind the curtain since superconductivity turns into a localized state when magnetic fields in the ten Tesla range are applied perpendicular to the conducting direction.

The angular dependent magnetoresistance observed in 1 and 2D conductors were not discussed in that paper. Very briefly, the phenomenon first discovered in 1-D (TMTSF)₂ClO₄ [68] has been studied in more details with (TMTSF)₂PF₆ [69,70]. The sharp dips of magnetoresistance which are observed at angles which fulfill "magic angles" conditions, figure (15) have been explained in terms of particular electron paths which do not sweep the entire Fermi surface for these particular orientations of the magentic field but are restricted to trajectories going through the nodes of the reciprocal lattice [71].

Similarly, we did not discuss the properties of 2-D organic conductors in which superconductivity can be stabilized up to 12K. Giant magnetoresistance oscillations have been detected ($β_{H-(ET)2I_3}$ [73], $β-(ET)2IBr_2$ [74], $κ-(ET)₂Cu(SCN)₂$ [75] and $α-(ET)₂X$, with X = TlHg(SeCN)₄) [76] and peaks of magnetoresistance are also visible at some peculiar orientations of the magnetic field [77]. They have been ascribed to the optimization of the Fermi surface cross section stationarity achieved at these angles [78]. Electron correlations are still remaining rather strong in these materials. They are detected through the blue shift of the $σ(ω)$ oscillator strength [79], a non-Korringa behaviour of the spin-lattice relaxation rate, a large pressure coefficient

<figure><img src="image_19.png" /><figcaption>15) Angular oscillation of the magnetoresistance (Lebed's oscillation) in (TMTSF)₂PF₆, after [69].</figcaption></figure>
of the susceptibility [80] and in some cases the onset of SDW ground states [81].

Superconductivity has been raised up to 33 K in fullerene compounds [82]. The response of the magnetic properties of these 3-D molecular conductors to a pressure-induced modification of the bandwidth has revealed a very minor role of electron correlations which can be related to the remarkably large area available on each organic molecule for electron delocalization [83]. The pairing mechanism in

these 3-D superconductors is probably mediated by high energy intraindual vibrations Thus T<sub>c</sub> is governed by the intermolecular distance which depends on the volume of the alkali atom or on pressure [84].

Acknowledgment - I gratefully acknowledge the
contribution of a large number of colleagues at Orsay and in
several laboratories all over the world. The research activity
has been partly supported by DG XII and DG XIII
European Actions and DRET and CNRS contracts.

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