<figure><img src="image_88.png" /><figcaption>Figure 73. The ⁷⁷Se NMR signal amplitude vs temperature below 30 K showing the onset of a magnetic state below 13 K, in a field of 55.4 kOe. The dashed curve represents the Curie law which should be followed by the amplitude of the NMR signal for a temperature independent linewidth (after reference [11]).</figcaption></figure>

<figure><img src="image_89.png" /><figcaption>Figure 74. Proton NMR line width versus temperature in a field of 10.57 kOe (right scale) and homogeneous linewidth, 1/T₂ (spin echo) (left scale) (after reference [11]).</figcaption></figure>

considerably to the proton spin-lattice relaxation rate $T_1^{-1}$, explaining the observed increase of $T_1^{-1}$ [11]. In addition the antiferromagnetic resonance frequency is shifted far away from the free ESR line [15], making the latter unobservable below the transition [9]. The above results (i) existence of static magnetic moments, (ii) anisotropic magnetic susceptibility, (iii) a spin-flop transition, (iv) the existence of magnetic modes and (v) the absence of a lattice superstructure) are clear evidence for a spin-density wave (i.e. antiferromagnetic) low-temperature state in $(TMTSF)_2PF_6$. Simultaneously with its appearance a gap is opened in the electronic spectrum, demonstrating that the conduction electrons are the driving force of the instability, i.e. $(TMTSF)_2PF_6$ is an itinerant antiferromagnet. The periodicity of the spin structure along the chain direction is then given by the Fermi wave-vector, leading to a doubling of the unit cell, i.e. there are now four TMTSF molecules in the unit cell. As discussed in section 2, the Fermi surface of a quasi-one-dimensional metal is most unstable against a modulation which also doubles the lattice period in the transverse directions, which thus would be expected to take place in $(TMTSF)_2PF_6$.
<figure><img src="image_90.png" /><figcaption>Figure 75. Temperature dependence of the single crystal spin susceptibility of (TMTSF)₂AsF₆ in a magnetic field (H = 3 kOe) lower than the spin-flop field. The axes *b**, *a* and *c** are respectively the easy, intermediate and hard axis of the SDW state stable below 12 K (after reference [14]).</figcaption></figure>

However, a confirmation of a doubling of the transverse lattice constants must await a polarized neutron study. We also remark that, contrary to initial results indicating transitions in conductivity [1] and magnetism [10] at different temperatures, more recent studies show the existence of only one transition [8, 13, 16].

In the low-temperature phase of (TMTSF)₂PF₆ the conductivity has dropped by two orders of magnitude from its maximum value; however it is still fairly high and no activated behaviour could be found [17]. (Tunnelling experiments [18], however, show a gap of about 60 K.) At the same time there is a large magnetoresistance [17] and a very high Hall mobility [8], $\mu_H \gtrsim 10^4 \text{ cm}^2 \text{ V}^{-1} \text{ s}^{-1}$, properties which are not unusual for a small-gap semiconductor.

Like (TMTSF)₂PF₆, (TMTSF)₂AsF₆ also exhibits a metal-semiconductor transition [1, 16] at 12 K (at ambient pressure). Apart from the anisotropic susceptibility [14], already mentioned above (figure 75), both the absence of any indication of superlattice formation [5, 19] and the disappearance of the free-electron ESR line [12] strongly suggest an antiferromagnetic low-temperature state. In this compound resistivity [16] and susceptibility measurements show activation energies of 20–30 K, i.e. a small-gap semiconductor is realized. Both a metal-insulator transition [1] and a vanishing ESR line [12] are also observed in (TMTSF)₂SbF₆ and (TMTSF)₂NO₃, thus implying an antiferromagnetic semiconducting state for these compounds also.
Behaviour quite different from that of TMTSF salts containing the centrosymmetric hexafluorophosphate anions described above is observed in salts with non-centrosymmetric anions like (TMTSF)₂BF₄ and (TMTSF)₂ReO₄. These compounds exhibit metal-insulator transitions at 39 and 180 K, respectively [1, 20], which are much sharper than those occurring in the hexafluorophosphate compounds. For (TMTSF)₂ReO₄ a structural investigation [6, 21] shows that the transition is due to anion ordering: the anion position of the (TMTSF)₂X structure [4] is an inversion centre, a non-centrosymmetric unit at this position will thus have at least two energetically degenerate orientations. (Note that the hexafluoro-anions are centrosymmetric.) In addition there is some effective interaction (for example, electrostatic or mediated by the lattice) which couples anions in different cells favouring an ordered arrangement at sufficiently low temperatures. In the case of (TMTSF)₂ReO₄ this ordering is observed at the metal-insulator transition temperature [6, 21], 180 K. In the ordered phase the orientation of the tetrahedral ReO₄⁻-ions alternates along all three lattice directions, corresponding to a modulation wave-vector (π/a, π/b, π/c). In the metallic state the Fermi wavenumber is 2k_F = π/a, so that the potential created by the anion ordering has the wave-vector which most easily creates a gap on the whole Fermi surface (see section 2). The large electronic gap E_g ≈ 2000 K observed in the insulating phase indicates that this potential is quite strong. For (TMTSF)₂BF₄, having a tetrahedral anion as in (TMTSF)₂ReO₄, no structural analysis has been carried out so far; however, the very sharp transition at relatively high temperature (39 K) compared with hexafluoro-compounds suggests a transition similar to that of (TMTSF)₂ReO₄. Also, the observation of an ESR line below the transition [12] clearly shows that the insulating state is not magnetic. Ordering of the planar anions has also been found in (TMTSF)₂NO₃ around 41 K [6, 21], however with modulation wavenumber (π/a, 0, 0). This wave-vector is not optimal for destroying the Fermi surface (see section 2), and correspondingly only a very weak anomaly of the conductivity is observed [1]. However, the intensity of the superstructure reflections in (TMTSF)₂NO₃ (and (TMTSF)₂ReO₄) is three orders of magnitude larger than superlattice reflections of Peierls-distorted states [6] (see section 4). The tetrahedral anions in the zero-pressure superconductor (TMTSF)₂ClO₄ are not observed to order [6].

The observation of antiferromagnetism in compounds with centrosymmetric anions like (TMTSF)₂PF₆ is in striking contrast with the properties of the same salt in the isostructural sulphur analogue family (TMTTF)₂PF₆ [22]. In this compound below 15 K the typical weak superlattice reflections of a Peierls distortion are found [6], and typical Peierls behaviour is also seen in the magnetic susceptibility [23]. Of course, due to the centrosymmetric PF₆⁻ anion no anion order-disorder transition is possible. A low-temperature metal-insulator transition (T_c ≈ 19 K) is also observed [24] in (TMTTF)₂Br; however its origin (Peierls or antiferromagnetic) is not yet clear. On the other hand, (TMTTF)₂X compounds with non-centrosymmetric anions like [6] (TMTTF)₂ClO₄ or [24] (TMTTF)₂SCN show anion-ordering transitions as in the (TMTSF)₂X family.

Applying pressure to the TMTSF compounds reduces the metal-insulator transition temperatures, both those of antiferromagnetic origin [2, 16] (figure 76) and those arising from anion ordering [20] (figure 77). At pressures around 10 kbar the metal-insulator transition vanishes completely, giving way to the superconducting state to be discussed in the next section (figure 78).
<figure><img src="image_91.png" /><figcaption>Figure 76. (a) Temperature dependence of the resistivity of (TMTSF)₂NO₃ under 6.5 kbar showing the anomaly related to the anion ordering at 40 K and a metal-non metal transition at ≈6 K (left) and under 12 kbar showing the suppression of the low temperature insulating state (right) (after reference [51] and JÉROME, D., and MAZAUD, A. (unpublished work, Orsay)). (b) Temperature dependence of the resistivity of (TMTSF)₂PF₆ showing the suppression of the metal-insulator transition between 8 and 9 kbar (after reference [50]).</figcaption></figure>

The TMTTF compounds are considerably less sensitive to pressure. Though the anion ordering of (TMTTF)₂ClO₄ rapidly vanishes with increasing pressure [26], low-temperature metal-insulator transitions (unrelated to anion ordering) are observed in this compound as well as in (TMTTF)₂PF₆ up to 30 kbar. Only in (TMTTF)₂Br does the metal-insulator transition vanish above 22 kbar [24].

## 5.2. *The superconducting state*

Increasing the hydrostatic pressure on (TMTSF)₂PF₆ the metal-insulator transition temperature becomes very small beyond 8 kbar. At 12 kbar and 0.9 K a phase transition into a state of unmeasurably low resistivity occurs [2] (figure 79), which is suppressed by a magnetic field (applied in the perpendicular direction) of
<figure><img src="image_92.png" /></figure>

<figure><img src="image_93.png" /></figure>

Figure 77. (a) Phase diagram of (TMTSF)₂ReO₄. The metal-insulator transition at 180 K is due to the ordering of the ReO₄⁻ with periodicity 2a along the *a*-axis. Triangles and circles correspond to T<sub>M-1</sub> derived from warming and cooling curves, respectively. Inverted triangles (above 10 kbar) show observation of superconductivity (after reference [20]). (b) X-ray patterns of (TMTSF)₂ReO₄ at 100 K (A) and 200 K (B) respectively below and above the phase transition. Arrows point towards the low temperature superstructure reflections of wave vector (0.5*a*, 0.5*b*, 0.5*c*) in (A) and to the corresponding high temperature *isotropic* precursor scattering in (B). The sample is oriented with *a* and *c* directions in the equatorial plane (after reference [21]).
<figure><img src="image_94.png" /><figcaption>Figure 78. Phase diagram of (TMTSF)₂PF₆ showing Orsay data ○ and ▲, Bell Laboratories data ● (after reference [28]), IBM data ☐ (after reference [35]).</figcaption></figure>

some hundred oersted. These results clearly indicate a superconducting state in (TMTSF)₂PF₆, which is confirmed by the observation of a nearly complete diamagnetism [27] at low frequency (ν = 68 Hz) and especially of the Meissner effect [28]. Superconductivity under pressure around 11 kbar has by now also been observed in (TMTSF)₂X compounds with X = AsF₆ [19], SbF₆ [29], TaF₆ [29] and ReO₄ [20], as well as for X = ClO₄ [3] at ambient pressure.

Superconductivity has also been established in (TMTSF)₂ClO₄ by specific heat measurements using an a.c. calorimetric technique [30]. On a C/T versus T² plot, the molar specific heat (a mole contains N (Avogadro number) molecular units

<figure><img src="image_95.png" /><figcaption>Figure 79. The superconducting transition of (TMTSF)₂PF₆ at 0.9 K under 12 kbar detected by resistance and a.c. susceptibility methods. A niobium marker sample in the same pressure cell (T_c ~ 9.5 K) is used for the purpose of signal calibration.</figcaption></figure>
(TMTSF)₂ClO₄) displays a very large anomaly around 1.2 K (figure 80 (a)). Above T² = 1.5 K² the specific heat obeys the classical relation C/T = γ + βT², where β = 11.4 mJ mol⁻¹ K⁻⁴ and γ = 10.5 mJ mol⁻¹ K⁻² corresponding to N(EF) = 2.1 states eV⁻¹ mole⁻¹ via the relation N(EF) = 3γ/2π²k_B². The C/T versus T plot of the electronic contribution (figure 80 (b)) allows a very accurate determination of both the critical temperature, namely T_c = 1.22 K, and the specific heat jump at the transition, i.e. ΔC/γT_c = 1.67.

Furthermore the data of figure 80 make possible a deeper characterization of the
superconducting state below T_c. At T = 0, the free energy difference between the
normal and superconducting areas gives the condensation energy H_0^2/8π. Therefore
from H_0^2/8 = ∫_0^T_c (C.es - C_n) dT, H_0 = 44 ± 2 Oe is derived. Moreover, a fit of the
data in figure 80, with an exponential temperature dependence of the specific heat at
low temperatures, leads to an energy gap 2δ ≈ 4 K which is in accord with the low
value of the critical temperature. The specific heat also shows a small critical width
for the three-dimensional fluctuations around T_c in agreement with the generalized
Ginzburg-Landau theory of weakly coupled conducting chains [31, 32] (see also
section 2.4.2).

Under a magnetic field applied along the *c*\* axis of the single crystals *T*<sub>c</sub> is depressed at a rate of 1.1 mK Oe<sup>-1</sup>. Therefore, in the low magnetic field regime *H* || *c*\* ≲ 1 kOe, the specific heat follows the law *C/T* = *γ* + *βT*<sup>2</sup> with both par-

<figure><img src="image_96.png" /><figcaption>Figure 80. Temperature dependence of the molar specific heat of (TMTSF)₂ClO₄ lattice and electronic contribution (a) and electronic contribution after subtraction of the lattice contribution (b) (after reference [30]).</figcaption></figure>
ameters $\gamma$ and $\beta$ approximately equal to the values deduced from the zero field data at $T^2 > 2$ K$^2$. These results show that:

(i) The superconducting state of $(TMTSF)_2ClO_4$ is destroyed by a fairly small magnetic field (about 1 kOe) along the $c^*$ direction (see below).

(ii) $N(E_F)$ in the conducting state at $T > T_c$ is hardly affected by magnetic fields of 1 kOe along $c^*$. This latter behaviour is in marked contrast with the high-field properties which will be discussed in section 5.3.3.

The magnetization curves of $(TMTSF)_2PF_6$ [28] are those of a type-II superconductor, and considerable experimental effort has been made to characterize the superconducting state by its upper critical field $H_{c2}$. From the anisotropy of the normal state properties [1] of the $(TMTSF)_2X$ salts (section 3) it was clear that $H_{c2}$ should be anisotropic. Recent investigations [16, 33] of the anisotropy within the $b^*-c^*$ plane show indeed a ratio $H_{b^*}/H_{c^*} \approx 15$ for the upper critical fields along the $b^*$ and $c^*$ directions, respectively, in (TMTSF)AsF$_6$ at 11 kbar (figure 81) and $(TMTSF)_2ClO_4$ at ambient pressure (figure 82), and a somewhat lower value for $(TMTSF)_2PF_6$ at 11 kbar. Using the critical field anisotropy, the known lattice parameters [4], and the Ginzburg-Landau relation (see section 2)

$$ \frac{H_{b^*}}{H_{c^*}} = \frac{bt_b}{ct_c} \qquad (5.1) $$

we can derive $t_b/t_c \approx 30$. This value can be compared with the room temperature anisotropy of the conductivity. In that temperature region the transverse conductivities are always diffusive, so that one expects [34]

$$ \frac{\sigma_{b^*}}{\sigma_{c^*}} = \left( \frac{bt_b}{ct_c} \right)^2 = 225 \qquad (5.2) $$

in good agreement with the experimental value [1, 17]. In spite of the nearly identical anisotropies in the three above compounds, the absolute values of the critical fields are quite different: extrapolating to zero temperature one finds $H_{b^*} = 2$ kG, $H_{c^*} = 200$ G for $(TMTSF)_2PF_6$ [33], but $H_{b^*} = 20$ kG, $H_{c^*} = 1.4$ kG for $(TMTSF)_2AsF_6$ with $(TMTSF)_2ClO_4$ somewhere in between [33]. We also remark that in many investigations [2, 19, 28, 29, 35-37] the orientation of the field in the

<figure><img src="image_97.png" /><figcaption>Figure 81. Anisotropy of the upper critical field $H_c^\perp$ in $(TMTSF)_2AsF$ under 9.5 kbar. The anisotropy ratio is about 15 (after reference [16]).</figcaption></figure>
<figure><img src="image_98.png" /><figcaption>Figure 82. Upper critical field $H_c$ in (TMTSF)₂ClO₄ for different orientations of the magnetic field with respect to the crystal axis (after reference [33]).</figcaption></figure>

$b^*-c^*$ plane is not specified, explaining the wide scatter in reported perpendicular critical fields.

The results of measurements of the upper critical field along the chains, $H_a$, are more difficult to understand. Greene *et al.* [33] found in (TMTSF)₂ClO₄, near the zero-field transition temperature, $H_a/H_{b^*} = 3$, and a linear temperature dependence of $H_a(T)$ with $dH_aT/dT \approx 35 \text{ kG K}^{-1}$. Gubser *et al.* [37] found the same value in one sample, but $dH_a(T)/dT \approx 70 \text{ kG K}^{-1}$ in another.

Unfortunately, the measurement of the transverse critical field by Gubser *et al.* is in an unspecified direction within the $b^*-c^*$ plane, so that a value for the $a-b^*$ anisotropy is not easily deduced. The same order of magnitude of $H_a/H_{b^*}$ as in (TMTSF)₂ClO₄ is also observed in (TMTSF)₂PF₆ [28, 33, 38]. Proceeding as above, from $H_a/H_{b^*}$ a bandstructure anisotropy of $t_a/t_{b^*} = 6$ is derived (for the properties of electrons near the Fermi surface the small dimerization of the structure [4] is unimportant, so that we may for the present purpose consider a three-quarter filled band. Note also that even for $t_a/t_{b^*} = 6$ the Fermi surface is not closed, so that it is consistent to use the analysis of section 2). On the other hand, from plasma edge data [56, 72] $t_a/t_{b^*} \approx 80$ for (TMTSF)₂PF₆ ($t_a = 250$ meV, $t_{b^*} = 3$ meV) and $t_a/t_{b^*} \approx 60$ for (TMTSF)₂ClO₄ have been obtained, whereas from the conductivity anisotropy [1, 17] of (TMTSF)₂PF₆ and the analysis of Soda *et al.* [34] one finds $t_a/t_{b^*} = 30$. In view of the quantitative uncertainties of the results, the agreement between the anisotropies derived from conductivity and plasma edge data is quite reasonable. On the other hand, there is one order of magnitude difference between the plasma edge and critical field values of $t_a/t_{b^*}$, and from the critical field values one would deduce $t_{b^*} = 40$ meV. Such a large value can clearly be ruled out from the plasma edge and conductivity measurements (even the surprisingly low anisotropy observed by Greene *et al.* [33] does not diminish much the discrepancy). We also note that the conductivity anisotropy of (TMTSF)₂PF₆ does not vary with pressure [39, 40], so that the discrepancy cannot be attributed to the fact that the critical field anisotropy is observed at 11 kbar. From the above discussion it is clear that the critical field anisotropy is not related to the bandstructure anisotropy by a simple Ginzburg–
Landau type relation like (5.1), as also remarked by Greene et al. [33]. These authors invoke Pauli paramagnetic limiting [41, 42] of $H_a$. However, the detailed theory of this effect [43, 44] gives the Pauli limit on the critical field near the zero-field transition temperature $T_c$ as

$$ \frac{\mu_B H_p(T)}{2\pi T_c^\circ} = \sqrt{\left(\frac{1 - T/T_c^\circ}{7\zeta(3)}\right)} \qquad (5.3) $$

in the absence of spin-orbit scattering, where $\mu_B$ is the Bohr magneton, and $\zeta(x)$ is Riemann's zeta function. Larger values of $H_p(T)$ are found in the presence of spin-orbit scattering, but the proportionality $H_p(T) \propto (1 - T/T_c^\circ)^{1/2}$ is still valid [44]. On the other hand the usual orbital effect (section 2) leads to $H_{c2} \propto (1 - T/T_c^\circ)$, so that near $T_c^\circ$ Pauli limiting is not the dominating mechanism. This point of view is supported by the fact that measured $H_a(T)$ curves [16, 33] are very linear and never show the downward curvature predicted by (5.3) (apart from temperatures below 0.2 K). In some cases even upward curvature is observed [2, 33]. Also, Pauli limiting should only act to limit the highest ($H_a$) critical field and therefore give a strongly temperature-dependent anisotropy, in contrast to the experimental results. We thus conclude that the unexpectedly low (but temperature-independent) anisotropy cannot be explained by Pauli limiting of $H_a$. A possible alternative explanation could be in terms of strong fluctuation effects (see following section), which may reduce the critical field anisotropy. However, at the present stage, no quantitative results are available, and we can only conclude that the relation between the anisotropies of the upper critical field and the bandstructure is more complicated than the simple Ginzburg-Landau relation (5.1).

The lower critical fields $H_{c1}$ are usually very low. From their magnetization curves of $(TMSF)_2PF_6$ Andres et al. [28] deduce $H_{c1} = 5$ Oe at 0.2 K for a field perpendicular to the chain (a) direction and an order of magnitude lower for parallel field (figure 83) (see also, for example, Mailly et al. [38] and Schwenk et al. [45]).

The superconducting state is extremely sensitive to radiation-induced defects: 0.01% defects completely destroy superconductivity, both in $(TMSF)_2ClO_4$ [33] and in $(TMSF)_2PF_6$ [46] (figure 84). Even though it is probable that a large part of the radiation-induced defects are free radicals and therefore act as magnetic impurities, there is still a very large effect: in rare-earth superconductors 1% of magnetic rare-earth impurities is required to lower the transition temperature by 1 K, by the mechanism described theoretically by Abrikosov and Gorkov [47], i.e. the organic superconductors are 100 times more sensitive to magnetic impurities than these substances, although the effect of magnetic three-dimensional impurities in molybdenum is much stronger. Although this discrepancy seems difficult to explain within the usual theory [47], it is interesting to remark that very large effects of small defect concentrations are also observed in substances with a Peierls type transition, described in section 4. In that case the main origin of the high defect sensitivity is believed to be the easy destruction of interchain coherence, not so much the decrease of the mean-field ordering temperature of a single chain. By analogy it seems likely that the easy destruction of superconductivity is also due to loss of the interchain coherence necessary to establish a long-range ordered state in a quasi-one-dimensional system [32] (note that the effect of non-magnetic impurities on the mean-field Peierls transition temperature is the same as that of magnetic impurities on the superconducting mean-field transition temperature, see section 2).
<figure><img src="image_99.png" /><figcaption>Figure 83. (a) Temperature dependence of the magnetization in (TMTSF)₂ClO₄ measured by a SQUID technique. The shielding curve represents the temperature dependence for the sample first cooled in *zero* field. The upper curve represents the Meissner effect in a field of 0.2 Oe (||*c*|) (after reference [38]). (b) Anisotropy of the magnetization curves (shielding effect) in (TMTSF)₂ClO₄ at T = 50 mK. The dashed lines represent the behaviour of the magnetization for complete flux expulsion taking into account demagnetization factors (1 and 1/2 for parallel and transverse configurations, respectively) (after reference [38]).</figcaption></figure>

The overall pressure-temperature phase diagram of compounds like (TMTSF)₂PF₆ or (TMTSF)₂AsF₆ is shown in figure 78, exhibiting a normal, an antiferromagnetic (insulating) and superconducting phase. The above description of the superconducting state applies at pressures 1–2 kbar above the critical pressure P_c separating antiferromagnetic and superconducting low-temperature states. On the other hand, very near the antiferromagnetic–superconducting crossover the transition into the superconducting state happens in a quite unusual way: Greene and Engler [35] have observed an *increase* of the resistivity by more than a factor 5 between 4 and 1.2 K in (TMTSF)₂PF₆ at P = 6.5 kbar (these authors as well as Andres et al. [28] find P_c ≈ 6.5 kbar, whereas Jérôme et al. [2] find P_c ≈ 9 kbar; the difference may lie in different sample quality or different pressure scales) before the transition into the superconducting state occurs. Similar effects are observed in
<figure><img src="image_100.png" /><figcaption>Figure 84. Effect of electron irradiation on the superconducting transition of (TMTSF)₂PF₆ (after reference [46]).</figcaption></figure>

(TMTSF)₂ClO₄ at ambient pressure [29, 48] and for (TMTSF)₂AsF₆ around 9.5 kbar, where a very detailed investigation has been undertaken [16]. The resulting phase diagram for (TMTSF)₂AsF₆ around 9.5 kbar is shown in figure 85, exhibiting a direct transition between ‘insulating’ and a superconducting state with decreasing temperature. One should note, however, that the dash-dotted line in figure 85 does *not* denote a maximum in ∂log ρ/∂T (the sign of a phase transition, see section 4.1) but rather the onset of increasing resistivity with decreasing temperature. The increase of resistivity is likely to be due to a quite slow onset of antiferromagnetic ordering involving only a part of the conduction electrons so that a gap opens only on a part of the Fermi surface. Another possibility would be short-range anti-ferromagnetic order, which would diminish the density of states at the Fermi surface.

<figure><img src="image_101.png" /><figcaption>Figure 85. The phase diagram of (TMTSF)₂AsF₆. The inset shows the re-entrance of the superconducting state below the SDW state (after reference [16]).</figcaption></figure>
In fact recent EPR data [49] obtained with $(TMTSF)_2ClO_4$ under ambient pressure could support the existence of a 'weakly antiferromagnetic' state in the temperature domain where a resistance upturn is observed. In any case the increase of resistivity by a factor 5 or more indicates a loss of density of states by the same factor. Surprisingly, the superconducting transition temperature is only very slightly affected by this decrease, as shown by the weak pressure dependence in figure 85. The increase of resistivity before the superconducting transition in $(TMTSF)_2ClO_4$ at ambient pressure [29, 44] (figure 86) indicates that for this compound $P_c \approx 1$ kbar, so that, hypothetically, negative pressure would stabilize an antiferromagnetic state in that case. Actually, Parkin et al. [24, 29] showed that there is a correlation between the lattice constant in the *c* direction and $P_c$, a short lattice constant (as in $(TMTSF)_2ClO_4$) favouring superconductivity. In that picture $P_c$ is just the pressure necessary to decrease the *c*-lattice constant to allow a superconducting state. One should also notice that in $(TMTSF)_2ClO_4$ the orientations of the tetrahedral $ClO_4^-$ ions are disordered down to the lowest temperatures [6] (see also recent data in section 5.4), whereas disorder is not possible in compounds with octahedral anions like $PF_6^-$ and $AsF_6^-$.

Upon increasing pressure above 11 kbar there is a decrease of the superconducting transition temperature [28] of $(TMTSF)_2PF_6$, down to 0.2 K at 24 kbar [50] (figure 87), corresponding to the rather high value of $\partial \ln T_c/\partial P \approx -7\% \text{ kbar}^{-1}$ for the pressure dependence of the transition temperature (at 11 kbar). The properties of the superconducting state above 11 kbar are expected to be very similar to those described above.

Apart from the above compounds, superconductivity has now also been observed in $(TMTSF)_2X$ compounds [20, 29, 48] with $X = SbF_6$, $TaF_6$ and $ReO_4$. The properties of the superconducting state are very similar to those discussed above, and some of the main parameters are summarized in table 5. We only mention two peculiarities.

<figure><img src="image_102.png" /><figcaption>Figure 86. Temperature dependence of the resistance of $(TMTSF)_2ClO_4$ under ambient pressure (after CHOI, M. Y., CHAIKIN, P. M., HAEN, P., and GREENE, R. L., 1982, Solid St. Commun., **41**, 225). The upturn of the resistance before the superconducting transition at ≈1.2 K corresponds to the onset of magnetic ordering.</figcaption></figure>
<figure><img src="image_103.png" /><figcaption>Figure 87. The superconducting transition at 0.2 K in (TMTSF)₂PF₆ under 24 kbar (after reference [50]).</figcaption></figure>

(i) In (TMTSF)₂ReO₄ superconductivity does not compete with an antiferromagnetic insulating state, but rather with an anion-ordered insulator [20] (see section 5.1). Still, a superconducting state is stable at pressures above $P_c \approx 10$ kbar.

<table><caption>Table 5. The (TMTSF)₂X family<sup>a</sup>.</caption><thead><tr><th>X</th><th><i>T</i><sub>M-1</sub><br>(K)</th><th>Insulating state</th><th>Anion<br>symmetry</th><th><i>P</i><sub>c</sub><br>(kbar)</th></tr></thead><tbody><tr><td>PF<sub>6</sub></td><td>12</td><td>SDW</td><td>Octahedral</td><td>8</td></tr><tr><td>AsF<sub>6</sub></td><td>12</td><td>SDW</td><td>Octahedral</td><td>9.5</td></tr><tr><td>SbF<sub>6</sub></td><td>17</td><td>SDW</td><td>Octahedral</td><td>11</td></tr><tr><td>TaF<sub>6</sub></td><td>—</td><td>—</td><td>Octahedral</td><td>10</td></tr><tr><td>ClO<sub>4</sub></td><td>3.5<sup>b</sup></td><td>SDW<sup>b</sup></td><td>Tetrahedral</td><td>10<sup>-3</sup></td></tr><tr><td>ReO<sub>4</sub></td><td>180</td><td>PLD<br>(anion ordering)</td><td>Tetrahedral</td><td>9.5</td></tr><tr><td>BF<sub>4</sub></td><td>39</td><td>—</td><td>Tetrahedral</td><td>—</td></tr><tr><td>BrO<sub>4</sub></td><td>220<sup>c</sup></td><td>—</td><td>Tetrahedral</td><td>—</td></tr><tr><td>IO<sub>4</sub></td><td></td><td>Semicond.</td><td>Tetrahedral</td><td>—</td></tr><tr><td>NO<sub>3</sub></td><td>12</td><td>(SDW)</td><td>Planar</td><td>No<br>superconductivity</td></tr></tbody></table>

<footnote><sup>a</sup> BECHGAARD, K., 1982, *Molec. Crystals liq. Crystals*, **79**, 1. (This article provides an overview of the (TMTSF)₂X salts.)</footnote>

<footnote><sup>b</sup> See: TAKAHASHI, T., JÉROME, D., and BECHGAARD, K., 1982, *J. Phys. Lett., Paris*, **43**, L-565, a SDW transition observed at 3.5 K for a fast cooling rate of the sample (superconductivity is observed below 0.9 K). When the sample is cooled very slowly below 50 K, no SDW transition is observed before the system undergoes a transition towards a superconducting state at $T_c \approx 1.2$ K.</footnote>

<footnote><sup>c</sup> TOMIĆ, S., JÉROME, D., MONOD, P., and BECHGAARD, K., 1982, *J. Phys., Paris* (to be published).</footnote>
(ii) Above $P_c \approx 8$ kbar ($\text{TMTSF}$)$_2\text{NO}_3$ remains metallic down to 50 mK, but does not become superconducting [51]. The absence of superconductivity may be due to the fact that under pressure the order-disorder transition of the anions [6] at 40 K considerably reduces the density of states at the Fermi surface, as indicated by a large anomaly in the resistivity [40] (at ambient pressure the anomaly is very weak) [1].

Recent results [25] indicate that organic superconductivity is not limited to the $\text{TMTSF}$ molecule: at 22 kbar ($\text{TMTTF}$)$_2\text{Br}$ has a metallic high-temperature phase and a metal-insulator transition at 10 K, much like ($\text{TMTSF}$)$_2\text{PF}_6$ at low pressure. Increasing the pressure to 25 kbar, the insulating state disappears, and instead around 3 K there is a sharp drop in the resistivity (figure 88). Though the resistivity does not drop to zero (due to sample inhomogeneities?) its magnetic field dependence (figure 88) strongly hints towards a superconducting origin of the effect. Even in the selenium superconducting series a non-zero resistance in the superconducting state [3, 19, 52] is often observed. It is explained by the occurrence of small cracks in the samples, leading to finite resistance in series with the superconducting domains.

Summarizing this section we note the most remarkable properties of the ($\text{TMTSF}$)$_2\text{X}$ family of compounds:

(i) Above a critical pressure $P_c$ ($P_c = 1$ bar in ($\text{TMTSF}$)$_2\text{ClO}_4$) these compounds are type II superconductors with a transition temperature of the order 1 K, showing thus for the first time superconductivity in an organic compound.

(ii) In a small region around $P_c$ both superconducting and antiferromagnetic order occur at the same pressure but it is not clear whether they coexist *at the same temperature* at a given pressure.

(iii) The critical fields are highly anisotropic, though the anisotropy in the $a-b^*$ plane is weaker than expected from the anisotropy of the single-electron properties.

<figure><img src="image_104.png" /><figcaption>Figure 88. Normalized resistance versus temperature for ($\text{TMTTF}$)$_2\text{Br}$ under 25 kbar. The inset shows the result of the application of a magnetic field up to 70 kOe along an undefined direction perpendicular to the *a*-axis (after reference [25]).</figcaption></figure>
(iv) The superconducting state is extremely sensitive to defects. A concentration of 0.01% (probably magnetic) impurities destroys superconductivity.

### 5.3. Superconducting fluctuations

#### 5.3.1. Density of states and conductivity

So far, this chapter has been concerned with the phenomenon of superconductivity occurring at very low temperature ($T < 1.2$ K), when the resistance drops to zero. However, above $T_c$ the electronic properties of $(TMTSF)_2X$ are remarkably different from those of regular metallic conductors. We shall now list some of these remarkable properties.

(i) The longitudinal conductivity exceeds $10^4$ ($\Omega$ cm)$^{-1}$ below 40 K or so in most members of the $(TMTSF)_2X$ family, even at ambient pressure [1] and the conductivity reaches $10^5$ ($\Omega$ cm)$^{-1}$ around 4.2 K for $(TMTSF)_2PF_6$, $(TMTSF)_2AsF_6$, etc. under pressure [2], and for $(TMTSF)_2ClO_4$ at ambient pressure [3]. These values of $\sigma_\|$ are about 10 times larger than the maximum conductivity observable in other conducting charge transfer compounds at low temperature. Given the electron density of $10^{22}$ charges cm$^{-3}$, a mobility of $\approx 2 \times 10^3$ cm$^2$ V$^{-1}$ s$^{-1}$ is derived at 1.3 K (for $\sigma_\| = 5 \times 10^5$ ($\Omega$ cm)$^{-1}$) leading to an approximate mean-free path of 700 intermolecular spacings along the *a* axis.

(ii) The transverse conductivity displays about the same temperature dependence as the longitudinal component down to 50 K and increases somewhat faster at low temperatures [39, 53] (figure 89).

(iii) The longitudinal low temperature ($1.2 < T < 30$ K) conductivity is extraordinarily affected by the application of a magnetic field along the transverse $c^*$ direction [53] but the magnetoresistance is reduced for $H\|$ $b^*$ [16] and is even practically non-existent up to 75 kOe as $H\|$ $a$ (figure 90).

<figure><img src="image_105.png" /><figcaption>Figure 89. Temperature dependence of the conductivity anisotropy (for the two components of the conductivity measured on the same sample) for $(TMTSF)_2PF_6$ at 12 kbar. The transverse direction is undefined in this experiment).</figcaption></figure>
(iv) The temperature dependence of the electrical resistivity is still large below 4.2 K, in a temperature domain where the resistance of all three-dimensional metallic conductors is limited by residual impurities [2] (figure 79).

Similarly, high conductivity has been observed in the pressure-stabilized conducting state of TMTSF-DMTCNQ [54, 55] with a similar response to the application of a magnetic field (figure 69).

Admittedly the simplest picture which can account for the previously presented transport phenomena is that of light carriers on anisotropic Fermi surfaces moving along stacks of great perfection. However, we find such an interpretation rather unlikely considering

(i) the optical effective mass along the chain axis close to the free electron mass [56],

(ii) the extraordinary sensitivity of the resistance at low temperature in the conducting regime to the presence of defects since $\Delta\rho/\rho_0 \approx 30\%$ per percentage of defects [46],

(iii) the impossibility for the band structure of single-chain conductors such as $(TMTSF)_2X$ to show semimetallic Fermi surfaces at low temperature, in contrast to two-chain conductors with large interchain tunnelling like HMTSF-TCNQ [57].

When TMTSF-DMTCNQ had been stabilized in a highly conducting state at low temperature under pressure the existence of a gradual growth of fluctuating superconducting pairing had been claimed to arise below 30 K [54].

<figure><img src="image_106.png" /><figcaption>Figure 90. Temperature dependence of the resistance of $(TMTSF)_2PF_6$ at 12 kbar under a magnetic field of 55 kOe applied (probably along the *c*\* direction) (after reference [53]).</figcaption></figure>
When the tunnelling integral $t_⊥$ between chains in a quasi-one-dimensional superconductor is small, the transition into the long-range ordered superconducting state occurs at a temperature well below the mean-field transition temperature $T_c^°$ of an individual chain. In that case the transition temperature is [32] (see also section 2.4.2).

$$T_c \approx E_c(T_\perp/E_c) \frac{1}{1-\eta/2} \qquad (5.4)$$

Here $E_c$ is a cut-off of the order of the Fermi energy, and $\eta$ determines the importance of quantum fluctuations of the superconducting order parameter ($\eta = 0$ is the classical limit). For a superconductor, $\eta$ should lie between 1 and 2, indicating the importance of quantum fluctuations [32]. More precisely, current theories of the (TMTSF)₂X superconductors (see section 5.4) imply $\eta \lesssim 2$. In view of the uncertainties of the parameters $\eta$ and $E_c$, a quantitative evaluation of $T_c$, based on (5.4) is not reliable, however, the large conductivity [1, 17] and optical [56] anisotropies make a $T_c$ well below $T_c^°$ reasonable for these compounds. Above $T_c$, up to a temperature $T_1$ of the order of (but somewhat larger than) $T_c^°$, strong superconducting fluctuations on individual chains are expected, leading to two principal features [58] (section 2.4): a decreased quasi particle density of states near the Fermi level (the pseudo-gap) [59], and a collective contribution to the conductivity [39, 60].

The pseudo-gap can be crudely viewed [59] as electrons leaving the normal Fermi fluid, combining to form virtually-bound pair states in a superfluid at an energy above the Fermi level with time constant $\tau_N$ and decaying back to the Fermi sea with the Ginzburg-Landau decay time $\tau_s$. The change of the density of states at the Fermi level is given by a detailed balance argument, namely $[N(E_F) - N_0]/N_0 = \tau_s/\tau_N$. There is evidence from several experiments from the existence of well-marked pseudo-gap effects in the (TMTSF)₂X series which we shall review briefly.

Electron quantum tunnelling is by far the most sensitive probe of the quasi-particle density of states [61]. The transition probability of single electrons, which obey Fermi statistics, to tunnel between two metallic electrodes separated by a thin insulating barrier, is proportional to the density of final states. Thus, provided one of the electrodes is in a superconducting state, the transition probability remains small as long as the bias voltage of the junction does not exceed $\pm\Delta/e$; where $2\Delta$ is a measure of the pairing energy in the superconducting state. Since the density of tunnelling states is given by $N_T(E) = N(E_F)[E^2/(E^2 - \Delta^2)]^{1/2}$ in the BCS model, the junction resistance exhibits sharp minima at bias voltages $\pm\Delta/e$ corresponding to the gap edge singularities.

Measurements of the tunnelling resistance can thus be used to determine changes in the density of states of quasiparticles in a one-dimensional conductor above $T_c$, which indicate the development of the pseudo-gap below $T_1$. Such measurements have successfully been made with (TMTSF)₂PF₆ [62] using the Schottky barrier geometry, namely studying the tunnel resistance of quasiparticles between a (TMTSF)₂PF₆ single crystal and an evaporated N-doped degenerate semiconductor through the intrinsic potential barrier existing at the interface (figure 91). As the junction is put under a pressure of 11 kbar and cooled down to very low temperatures the dynamic resistance of the junction $dV/dI$ versus applied bias displays the characteristic shape usually observed when one of the electrodes is a superconductor (figure 92). The response to a magnetic field lends additional support
<figure><img src="image_107.png" /><figcaption>Figure 91. Band structure in the region of a contact between a metal and a *n*-type semiconductor (Schottky barrier). The upper diagram represents the situation of a normal metallic electrode and the lower diagrams display the density of states close to the Fermi level for the superconducting electrode of a Schottky junction with zero bias of the junction (middle) and –*e*Δ or +*e*Δ bias, left and right respectively (after reference [62]).</figcaption></figure>

to the superconducting origin of the tunnelling characteristics. A magnetic field significantly decreases the amplitude of the resistance peak. A decrease of the resistance peak is also noticed on warming, but the tunnelling characteristics persist up to 5 and even up to 15 K or so in some cases (figure 93).

The tunnelling characteristic in figure 93 has been attributed to the presence of a pseudo-gap in the density of quasiparticle below 15 K which reflects the establishment of one-dimensional, short-range ordered superconducting pairing in contrast with three-dimensional, long range ordered superconductivity which is observed in (TMTSF)₂PF₆ at 11 kbar only below T₃ = 1 K. The width of the resistance anomaly (figure 92) gives an estimate of the intrachain pairing energy 2Δ = 3.6 meV. This energy is indeed much larger than T_c, but according to (5.4) the large difference between Δ ≈ 40 K and T_c can be attributed to the weak interchain coupling t⊥/t∥. If the value of T₁ is derived somewhat arbitrarily from a BCS relation such as 2Δ/kT₁ = 3.5, one obtains T₁ ≈ 10 K. Therefore, the phase transition is suppressed by fluctuations of the phase of the superconducting order parameter by a factor of nearly 10, and following the Landau–Ginzburg picture of one-dimensional conductors deviations from mean-field behaviour are already important below 20 K. Furthermore, the onset of three-dimensional order can also be detected by electron tunnelling, since at T < T₃, namely T = 50 mK, decreasing the amplitude of the modulating current by a factor 20 results in an increase of the zero bias resistance peak by a factor more than 2 [62]. The narrow gap 2δ developing below T₃ appears
<figure><img src="image_108.png" /></figure>

Figure 92. Dynamic resistance $dV/dI$ versus bias voltage for a $(TMTSF)_2PF_6-n/GaSb$ tunnel contact at $T = 50$ mK and $P = 11$ kbar. The dashed curve displays the same dynamic resistance with an increased (×5) voltage axis sensitivity. The distance between the two dynamic resistance minima is 3.6 meV (after references [62]).

therefore as a fine structure in the centre of the broad pseudo-gap $2\Delta$. This results in good agreement with the value $2\delta \approx 0.35$ meV derived in $(TMTSF)_2ClO_4$ from specific heat data [30], see section 5.2.

A similar zero-bias anomaly has been observed under ambient pressure with $(TMTSF)_2ClO_4$-GaSb contacts. There, a pseudo-gap of amplitude $2\Delta \sim 3-4$ meV is also visible at a temperature larger than the three-dimensional ordering temperature ($T_c \approx 1.2$ K). Besides the tunnelling characteristics presented in figure 92, namely a maximum of differential resistance at zero bias, other behaviour has also been observed. Figure 94 displays two different behaviours observed with $(TMTSF)_2PF_6$-GaSb junctions [63]. State 1 corresponds to the tunnelling characteristics of electrons through a normal-insulator-superconductor (N-I-S) junction as discussed previously. In some cases a zero bias resistance minimum develops at low temperature (state 2 in figure 94). State 2, exhibiting resistance anomalies at bias voltages of $\pm\Delta$ and $\pm 2\Delta$, can be understood as superposition of single electron tunnelling (N-I-S) and of tunnelling between two superconducting domains separated by a thin insulating region (S-I-S junction). Behaviours such as those displayed in figure 94 (state 2) can be explained either if both superconducting
<figure><img src="image_109.png" /><figcaption>Figure 93. (a) Temperature dependence of the zero bias tunnel resistance with a modulation current amplitude $\Delta I = 10^{-7}$ A. (b) Magnetic field dependence of the zero bias tunnel resistance at $T = 50$ mK with a modulation current amplitude $\Delta I = 5 \times 10^{-9}$ A (after reference [62]).</figcaption></figure>

domains belong to the organic superconductor [63] or if the evaporated film of tellurium doped GaSb possibly becomes superconducting below 8 K [33]. It has been checked [64] that junctions of similarly evaporated films of n-doped GaSb on gold or copper substrates do not show any zero-bias resistance anomalies down to 1.2 K like those displayed in figures 92 and 94. Therefore we feel confident that the anomalies of $dV/dI$ in the vicinity of the zero bias are indeed related to the development of superconductivity in the organic electrode.

Once in a junction of relatively low resistance (10 Ω) at high temperature the zero bias contact resistance dropped sharply to zero below 12 K (figure 95). This result may be tentatively explained noting that in the temperature region of strong superconducting fluctuations (i.e. below $T_c$) any increase of interchain coupling will lead to an increase of $T_c$, due to a partial suppression of fluctuations. In the present case, the increased interchain coupling may be thought to be due to foreign atoms which may diffuse from the evaporated GaSb-layer into the organic solid and form local interchain bridges [63] (figure 96). The calculation of this effect by one of the authors [65] shows that a concentration of about 4% per molecule of these crosslinks with $t_⊥$ equal locally to $t_‖$ is sufficient to explain an observed stabilization of
<figure><img src="image_110.png" /></figure>

Figure 94. The two different behaviours of a $(TMTSF)_2PF_6$-n/GaSb junction. State 1 corresponds to the conventional normal-insulating-superconducting (N-I-S) sequence. The distance between $dV/dI$ minima corresponds approximately to the pseudo-gap $2\Delta$ of the one-dimensional fluctuating superconductor. State 2, often obtained after the application of an electric spike to the junction in state 1, shows resistance anomalies at $\pm\Delta$ and approximately $\pm 2\Delta$ together with a sharp resistance minimum at zero bias. State 2 could correspond to a set of N-I-S and S-I-S junctions in parallel at the interface between $(TMTSF)_2PF_6$ and the evaporated semiconductor. The tunnelling of the electrons in the N-I-S and S-I-S junctions would give rise to the $\pm\Delta$ and $\pm 2\Delta$ resistance anomalies respectively. The zero bias resistance minimum could be explained by the electron pair tunnelling in the S-I-S junctions (Josephson current) (after reference [63]).

<figure><img src="image_111.png" /></figure>

Figure 95. Dynamic resistance of a low resistance $(TMTSF)_2PF_6$-n/GaSb junction at zero bias versus temperature. It looks as if the contact becomes superconducting around 12 K. The residual resistance of 0.2 Ω at lower temperatures corresponds to the series resistance of the silver dag contacts and non-superconducting bulk $(TMTSF)_2PF_6$ (after reference [63]).
<figure><img src="image_112.png" /><figcaption>Figure 96. Schematic picture of interchain chemical bridging.</figcaption></figure>

superconductivity around the characteristic temperature $T_1$ for the development of strong one-dimensional fluctuations.

Further point contact junctions of $(TMTSF)_2ClO_4$ with chrysocal needles displaying differential resistance characteristics with a resistance minimum at zero bias are extremely sensitive to a microwave ($\nu = 9.6$ GHz) irradiation (figure 97) [66]. Together with the above results, this may be understood as a Josephson effect occurring in the S–I–S regions of the junction. However no Josephson current can be observed at zero bias, as a consequence of strong thermal fluctuations suppressing the d.c. Josephson current in high resistance ($\approx 20$ KΩ) junctions [67].

Other studies of the electronic properties of $(TMTSF)_2PF_6$ between 4.2 and 77 K have led to an interpretation which corroborates the tunnelling data. Such is the case for thermal conductivity experiments performed on $(TMTSF)_2PF_6$ under 12 kbar [68]. The thermal conductance displays a very large drop at low temperature, beginning around 40 K. Since the Wiedemann–Franz law is fairly well obeyed at high temperature, namely, $\sigma$ and $\kappa$ increase both with the same rate at 300 K between 1 atm and 12 kbar, we feel fairly confident that the electronic contribution is dominant in the total thermal conductivity at 40 K. The drop of the thermal conductivity below 40 K can therefore be related to the opening of a pseudo-gap at the Fermi level. A drop of thermal conductivity has also been detected in $(TMTSF)_2ClO_4$ under ambient pressure below 50 K [69] (figure 98(a)). Furthermore the thermal conduction at low temperature is strongly dependent on the application of a magnetic field. When a field of 5T is applied (along a direction 10° off longitudinal alignment) the thermal conduction stays constant below 30 K (figure 98(b)). Data obtained with a magnetic field aligned with a transverse (unspecified) direction of the crystal lead to a qualitatively similar increase of the thermal conduction at much smaller values of the magnetic field. The sensitivity of the thermal conductivity to magnetic field is an essential piece of information which allows us to rule out the lattice as the origin for the drop of $\kappa$ below 50 K. Consequently the low-temperature behaviour of $\kappa$ must be attributed to a decrease of the electronic contribution. Furthermore, the increase of $\kappa$ in a magnetic field *cannot be explained by the Wiedemann-Franz law* since the electrical conductivity is enhanced (and not decreased) by a magnetic field.

Further evidence for the existence of a large pseudo-gap in the density of quasiparticle states at low temperature has been provided by the far-infrared reflectance spectrum of $(TMTSF)_2ClO_4$ [70]. The absorption shows a marked increase at the energy of $\approx 3.8$ meV (figure 99(a)). The FIR threshold can be seen
<figure><img src="image_113.png" /><figcaption>Figure 97. Dynamic resistance of a (TMTSF)₂ClO₄-chrysocal point junction put in a microwave cavity with different level of microwave power (ν = 9.6 GHz) at T = 3.9 K (a) and T = 25 K (b). The zero-bias dynamic resistance is about 20 Ω without irradiation (after reference [63]).</figcaption></figure>

up to 15 K or so. In addition, magneto-transmission experiments [71] show a marked decrease in transmission upon application of a magnetic field of 0.2 T below the energy of 3.8 meV (figure 99 (b)). A similar technique when applied to a niobium marker sample leads to the well-known result, namely a superconducting gap 2Δ = 2.9 meV. Summarizing, thermal conductivity and far-infrared data strongly suggest the existence of a well-developed pseudo-gap of width 2Δ ≈ 3.8 meV (42 K) in (TMTSF)₂ClO₄.

Finally, far-infrared reflectance properties of (TMTSF)₂PF₆ at 25 K under ambient pressure [72] support the existence of a pseudo-gap around 50 cm⁻¹ (figure 100 (a)) and a very large contribution to the frequency-dependent conductivity which is sharply peaked around zero frequency (figure 100 (b)). We wish to point out that not all physical measurements are as sensitive to the opening of a pseudo-gap as those described above. This is the case, in particular, for the Seeback coefficient which, for an ordinary conductor, behaves like T/T<sub>F</sub> at low temperature, whereas it behaves like (T - T<sub>c</sub>°)/T<sub>F</sub> in the presence of one-dimensional superconducting fluctuations. As the Seeback coefficient is rather small at low temperature its quasi-insensitivity to X-ray irradiation in (TMTSF)₂ClO₄ [33] (supposed to suppress the superconducting precursor effects, see section 5.3.2) cannot be taken as a very significant proof for the absence of a pseudo-gap in pure systems.
<figure><img src="image_114.png" /><figcaption>Figure 98. Thermal conductivity of (TMTSF)₂ClO₄ (a) and below 50 K (b) with the effect of a magnetic field of 50 kOe aligned along the longitudinal axis (after reference [69]).</figcaption></figure>

In the presence of a gap (or pseudo-gap) at the Fermi level the density of quasi-particles able to contribute to the conduction is likely to decrease with decreasing temperatures, due to their virtual condensation into fluctuating superconducting domains. The condensed pairs can be accelerated by an electric field and contribute to the d.c. conductivity. Since the excess conduction (or paraconduction) is proportional to $\langle|\Delta|^2\rangle$, it becomes negligible above $T_c$ in most three-dimensional superconductors. But the problem is quite different in a quasi-one-dimensional superconductor with a small transverse coherence length (i.e. $\xi_\perp \ll \xi_\|$). The
<figure><img src="image_115.png" /><figcaption>Figure 99. (a) Far infrared absorption spectrum of (TMTSF)₂ClO₄ at T = 2 K. The light is polarized along the longitudinal crystal axis (T. Timusk, private communication). (b) Far infrared frequency dependence of the magneto-absorption of Nb and Pb grids (left) and of a (TMTSF)₂ClO₄ grid (right). In both cases there is an onset of magneto-absorption below the energy gap as superconductivity is destroyed by the magnetic field. The arrow marked 1/g denotes the position of the grid spacing frequency (after reference [71]).</figcaption></figure>
<figure><img src="image_116.png" /></figure>

Figure 100. Chain axis reflectance of (TMTSF)₂PF₆ at various temperature (a) and frequency dependent conductivity (b). These data emphasize the low far infrared conductivity at low temperature. [σ(25 K, 10 cm⁻¹) ≈ 300 (Ω cm)⁻¹] compared to the large d.c. conductivity [σ(25 K, d.c.) > 10⁴ (Ω cm)⁻¹] (after reference [72]).
paraconductivity of a quasi-one-dimensional superconductor worked out by Schulz et al. [39] leads to the following contribution:

$$ \sigma_{\parallel}' = \left( \frac{v_F}{2t_{\perp}a} \right)^2 \frac{e^2 m \alpha'}{16} \xi_{\parallel}(T) \left[ 1 - \frac{a}{(a^2 + 4\pi\xi_{\perp}^2)^{1/2}} \right], \quad (5.5) $$

$$ \sigma_{\perp}' = \frac{e^2 m \alpha'}{16} \xi_{\parallel}(T) \left[ 1 - \frac{3a}{2(a^2 + 4\pi\xi_{\perp}^2)^{1/2}} + \frac{a^3}{2(a^2 + 4\pi\xi_{\perp}^2)^{3/2}} \right] \quad (5.6) $$

for the longitudinal and transverse paraconductivity, respectively. Here $v_F$ is the Fermi velocity, $a$ the interchain distance, $m$ the carrier mass and $\alpha'$ a constant depending on the microscopic properties of the system, especially on the single-particle scattering time.

We now consider two limiting cases.

(i) Near the transition temperature, $\xi_{\perp}$ is much larger than $a$, i.e. the fluctuations are three-dimensional. Then, one finds the typical square root divergence of a three-dimensional superconductor, however $\sigma_{\parallel}'$ is enhanced over its value in an isotropic superconductor by a factor $(v_F/2t_{\perp}a)^2$, which may be very large in organic conductors due to small transverse coherence length compared to $\xi_{\parallel}$ which leads to a large mean square amplitude of the superconducting order parameter. The anisotropy of the paraconductivity is $(v_F/2t_{\perp}a)^2$.

(ii) Sufficiently far above $T_c$ one has $\xi_{\perp} < a$, i.e. one-dimensional fluctuations. Then (5.5) reduces to the well-known form of a one-dimensional superconductor. However, a transverse supercurrent between adjacent chains requires a well-developed order parameter on the two chains. As the probability for this to happen decreases with increasing temperature, for $\xi_{\perp} < a$, $\sigma_{\parallel}^{\text{ID}}$ decreases much faster than $\sigma_{\parallel}^{\text{ID}}$. Correspondingly, the anisotropy of the fluctuation conductivity increases with increasing temperature. An approximate expression for (5.5) in the one-dimensional limit is

$$ \sigma_{\parallel}' = \frac{\pi e^2}{16 d_{\perp}^2} \xi_{\parallel}(0) t^{-3/2}, \quad (5.7) $$

where $t = (T - T_1)/T_1$ and $\xi_{\parallel}(0)$ is the zero temperature Landau-Ginzburg coherence length. A behaviour such as that of (5.5) and (5.6) is likely to explain the behaviour of the experimental data below 50 K for $(TMTSF)_2PF_6$ under 12 kbar (figure 101). A numerical estimate of the paraconduction (equation (5.7)) leads to $\sigma_{\parallel}' > 10^4 \text{ cm}^{-1}$ at $T = 25$ K, given $\xi_{\parallel}(0)/d_{\perp} \gtrsim 50$ and $T_1 \approx 12$ K according to the tunnelling data in $(TMTSF)_2PF_6$ under pressure [62]. The drop of the conductivity anisotropy at low temperature (see figure 89) is also in agreement with the paraconductivity picture, since $\sigma_{\parallel}/\sigma_{\perp} \approx (T - T_1)/T_1$ in the one-dimensional regime [53]. In this picture $\sigma_{\parallel}/\sigma_{\perp}$ begins to decrease when the paraconduction becomes the dominant conduction channel. Accordingly (figure 89) the conduction of superconducting fluctuating origin is dominant below about 30 K. The fluctuating regime and the three-dimensional ordered ground state need not be of the same nature. The transition temperature to a long-range ordered state (and the type of order) depend both on intra- and inter-chain interactions whereas the dominant fluctuations are determined only by the intrachain interactions.
<figure><img src="image_117.png" /><figcaption>Figure 101. Temperature dependence of the (TMTSF)₂PF₆ longitudinal resistivity under 12 kbar.</figcaption></figure>

Therefore, a situation of coexistence between a superconductive one-dimensional precursor regime and an SDW three-dimensional ordered ground state is perfectly conceivable. We believe it is the situation which prevails in (TMTSF)₂PF₆ and related compounds with hexafluorate anions at low pressure.

The possibility of an SDW origin for the pseudo-gap observed up to 30 K or so in (TMTSF)₂ClO₄ and (TMTSF)₂PF₆ seems rather unlikely for the following reasons:

(i) A commensurate fluctuating SDW regime should not give rise to the large d.c. conduction [73] which is observed below 50 K in the (TMTSF)₂X series (X = PF₆, AsF₆, ..., ClO₄, ...).

(ii) The magnetic field enhances the stability of the SDW state whenever it is observed in the (TMTSF)₂X family [13, 74], whereas the tunnelling [62] thermal conductivity [69] and FIR [71] data show a suppression of the pseudo-gap.

Finally we wish to re-emphasize at the end of a section dealing mainly with the question of the precursor regime the existence of two energy scales, revealed by specific heat and tunnelling experiments.

(i) A small energy scale of the order of a few Kelvin observed from specific heat data which is related to the establishment of three-dimensional order requiring a small condensation energy $H_0^2/8\pi \approx 2 \times 10^{16}$ eV mol$^{-1}$. The nearly mean-field behaviour of the superconducting three-dimensional ordering is not unexpected: the critical temperature region associated with the onset of three-dimensional ordering in the immediate vicinity of $T_c$ is expected to be quite narrow [31, 32], and due to quantum fluctuations the relation between $T_c$ and the condensation energy is BCS-like for $\eta \lesssim 2$ (cf. equation (5.4)) [32]. Indeed, the analysis of the paraconductivity in (TMTSF)₂PF₆ [39] above $T_c$ has located the cross-over temperature between the one-dimensional regime and a higher dimensional regime as being around 2 K. It is this small energy scale which determines, together with the overlap integrals, the values
of critical fields $H_{c2}$ for the suppression of three-dimensional ordering. They are in the 1 and 15 kOe range for $H\|c^*$ and $H\|b^*$, respectively.

(ii) A larger energy scale which is related to the formation of electron pairs within isolated chains with the formation of a gap $2\Delta \approx 40$ K in the density of states as shown by tunnelling data. Crudely speaking the one-dimensional condenstation energy is about 100 times the condensation energy related to three-dimensional ordering. We wish to re-emphasize that for quasi-one-dimensional superconductors the comparison between the one-dimensional condenstation energy, $N(0)\Delta^2/2$, and the orbital critical fields $H_{c2}$ is meaningless.

### 5.3.2. *Effects of irradiation*

The extreme sensitivity of the low-temperature conduction properties of $(TMSF)_2PF_6$ or TMSF-DMTCNQ to irradiation is another outstanding property of these one-dimensional conductors [46, 75]. As already shown in section 5.2 the superconducting transition temperature is unusually strongly depressed by impurities (presumably magnetic) created by irradiation (electron X-ray, protons, etc.). Moreover, the residual conductivity, i.e. the finite conductivity at low temperature just above the superconducting transition (when it still exists) is sensitive to the defects (figure 102(a)). According to Bouffard et al. [46] the superconducting transition temperature of $(TMSF)_2PF_6$ falls abruptly to $T=0$ K when the ratio $\sigma(T_c)/\sigma(200$ K) becomes less than, say 5 (figure 102(a)) (for comparison in pristine crystals this ratio is larger than 50). Even more striking sensitivity of $T_c$ to damage is observed by other authors [33, 75], since there the three-dimensional ordered superconducting state is suppressed by irradiation with much less decrease of the conductivity at low temperature (see figure 102(b)). As shown on figure 103 the unusually high sensitivity of the low-temperature resistivity to defects vanishes smoothly above about 50 K. The dependence of the conduction on irradiation has been related to the suppression of the one-dimensional superconducting fluctuations by impurities [46]. It must be pointed out however, that the extremely large sensitivity of the superconducting transition temperature to irradiation is due to the breaking of coupling occurring between order parameters of neighbouring chains. A higher concentration of impurities is required to destroy the pairing mechanism between $+k_F$ and $-k_F$ states of electrons on a given chain which gives rise to paraconductivity.

A very similar response of the low temperature resistivity to irradiation has been found in TMSF-DMTCNQ under a pressure higher than 10 kbar (figure 104) [46]. This effect has also been related to the presence of one-dimensional superconducting fluctuations up to about 40 K, in this compound, although no superconducting state (i.e. zero resistance) has yet been observed at low temperature. The absence of a three-dimensional superconducting state in TMSF-DMTCNQ may be understood in terms of a large concentration of paramagnetic centres localized on the DMTCNQ chain [78]. It can also be ascribed to the one-dimensional character which is more pronounced in TMSF-DMTCNQ than in the $(TMSF)_2X$ series [76] and to the interchain hybridization decreasing $N(E_F)$ at low temperature (section 3.3) for two-chain conductors. Figure 105 shows the effect of irradiation on the resistance of TMSF-DMTCNQ ($P=15$ kbar) at various temperatures [46]. At $T < 80$ K, two regimes are clearly observed.
<figure><img src="image_118.png" /><figcaption>Figure 102. (a) Superconducting transition temperature of (TMTSF)₂PF₆ versus the normalized value of the conductivity just above the transition (after reference [46]). (b) Temperature dependence of the resistivity of (TMTSF)₂PF₆ under 11 kbar for various amounts of radiation induced damage (after reference [33]).</figcaption></figure>

(i) At a low concentration of defects a strong effect of impurities, possibly due to the destruction of the paraconductive channel.

(ii) The second regime, at $c \ge 0.03\%$ characterized by a smaller sensitivity to irradiation has been related to breaks occurring along the conducting chains. The rate of damage is actually similar to the rate observed in the conducting regime of TMTSF-DMTCNQ stabilized by a large concentration of defects
<figure><img src="image_119.png" /></figure>

Figure 103. Temperature dependence of normalized resistance for pristine (TMTSF)₂PF₆ (full line) and irradiated (TMTSF)₂PF₆ (concentration of molecular defects, ○ = 0.02%, △ = 0.02%, ■ = 0.03%, ∇ = 0.03%, ▲ = 0.05%, □ = > 0.05%). The broken line shows the behaviour of pure TMTSD-DMTCNQ (after reference [46]).

<figure><img src="image_120.png" /></figure>

Figure 104. Defect concentration dependence of the longitudinal resistance of TMTSF-DMTCNQ at various temperatures, normalized to the pristine crystal resistance (P = 15 kbar) for X-ray irradiation (after [46]). The dotted line indicates the slope of the damage curve for an irradiation at 21 K with fast neutrons (after reference [79]).

[79] at low pressure with no superconducting fluctuations present. Very similar data have been obtained with (TMTSF)₂PF₆. From figure 104 the paraconductive contribution to the measured conductivity can be derived and the ratio σ'_‖/σ_{sp} is plotted versus temperature on figure 105. According to this estimate, at about 35 K the paraconductivity is already equal to the single-particle conductivity and is 10 times larger at 10 K. Furthermore, the temperature dependence of σ'_‖ is very satisfactorily explained by the one-dimensional limit of the paraconductivity theory (see section 5.3.1) namely
<figure><img src="image_121.png" /></figure>

Figure 105. Temperature dependence of the ratio between the superconducting fluctuation conductivity and the single particle conductivity in pure crystals derived from irradiation experiments in TMTSF-DMTCNQ under 15 kbar (black dots), and an irradiated sample with $5 \times 10^{-4}$ defect concentration (open dots) (after reference [46]). The continuous line is a fit of the data with the Ginzburg-Landau theory of one-dimensional superconducting fluctuations [39].

$\sigma_{\parallel}^{\prime}/\sigma_{\text{sp}} \approx T^{-1/2}(\ln T/T_1)$ with $T_1 = 7$ K. In conclusion, the exceptional behaviour under irradiation of the resistance of the compounds exhibiting a very high conductivity at low temperature ($\sigma_{\parallel} > 10^5 (\Omega \text{cm})^{-1}$) appears to be well explained by the development of one-dimensional superconducting fluctuations in a very wide range of temperatures above helium temperature. Other authors using 2 MeV proton irradiation on $(\text{TMTSF})_2\text{PF}_6$ under pressure reported an equally large sensitivity to radiated damage [75].

### 5.3.3. *The fluctuating regime under high magnetic field*

Magnetotransport studies of the $(\text{TMTSF})_2\text{X}$ superconducting series reveal several interesting features:

(i) An appreciable magnetoresistance is observed at fields larger than $H_{c2}$ whenever they are applied along directions perpendicular to the high conductivity axis [80, 16]. Moreover the transverse magnetoresistance of $(\text{TMTSF})_2\text{PF}_6$ is highly anisotropic for $H$ in the $b^*-c^*$ crystal plane, tending to saturation for $H \perp c^*$ and rising steeply with field for $H||c^*$. Quantitative measurements performed on $(\text{TMTSF})_2\text{AsF}_6$ under 11 kbar at 2.5 K [16] give $\Delta\rho/\rho_0 = 3.9$ and 0.15 for fields applied, respectively, along $c^*$ and $b^*$. Moreover, the temperature dependence of the resistivity exhibits a negative temperature coefficient below about 15 K for high fields along $c^*$ (figure 90) [53].

(ii) Quantum oscillations of the resistance (de Haas-Shubnikov effect) have been reported in $(\text{TMTSF})_2\text{PF}_5$ [81] under pressure at 1.1 K, but only at fields larger than an onset field $H_{\text{on}}$, of $\approx 65$ kOe (with $H||c^*$) (figure 106). The angular dependence of the quantum oscillations implies the existence of very small closed orbits in the $a-b$ plane, corresponding to only 1% of a Brillouin zone cross-section, perpendicular to the $c^*$ direction. However it must be
<figure><img src="image_122.png" /></figure>

Figure 106. (a) The third-harmonic ($\nu = 150$ Hz) of the resistive signal of $(TMTSF)_2ClO_4$ recorded with a 50 Hz field modulation versus magnetic field ($\|c\|$) at $P = 6.9$ kbar and $T = 1.1$ K. The oscillations of magnetoresistance vanish abruptly below the threshold field of ≈65 kOe (after reference [81]). (b) Successive integers for the peaks in (a) versus inverse magnetic field. The Shubnikov-de Haas frequency is given by $dn/d(1/H)$ (after reference [81]).

emphasized that such a semimetallic two-dimensional Fermi surface, namely elongated cigars or tubes along the $c^*$ axis, is present *provided* the field is larger than the onset field, or if $H \cos\theta > H_{on}$ where $\theta$ is the angle between the magnetic field and the $c^*$ direction. The data also imply that (when quantum oscillations are detected) the material is a compensated conductor with equal numbers of electrons and holes. Furthermore, a similar behaviour of the conducting state under high magnetic field has been observed in $(TMTSF)_2ClO_4$ at ambient pressure [82]. At $T = 1.0$ K, the onset field for the observation of quantum oscillations is 60 kOe when the magnetic field is aligned with the $c^*$ axis. None of the magnetotransport properties are characteristic of the behaviour of a regular conducting state above a superconducting transition.

The high field investigation of the $(TMTSF)_2ClO_4$ specific heat reveals very interesting features in the temperature dependence of the electronic specific heat [30]. Figure 107 shows the behaviour of the specific heat for a field of 63 kOe aligned with $c^*$ direction. In figure 107 (a) the total sample specific heat (including the lattice contribution) is displayed on a $C/T$ versus $T^2$ plot. After subtraction of a lattice contribution derived from the zero field data the electronic contribution is plotted on figure 107 (b). This figure exhibits several striking features.

(i) The electronic specific heat vanishes as $T$ goes to zero implying therefore that the ground state of $(TMTSF)_2ClO_4$ under high magnetic field contains only a very low density of states at the Fermi level. Obviously this low density of states cannot be attributed to a single particle gap resulting from superconductivity since $H_{c2}(H\|c^*)$ is about 0.5 kOe for $(TMTSF)_2ClO_4$ around 0.4 K as discussed in section 5.2. Therefore, we are forced to conclude that the data of figure 107 (b) imply the stabilization of a low density of states phase (a semiconducting or more likely a semimetallic phase) in $(TMTSF)_2ClO_4$ at very low temperatures under field.

(ii) Going up in temperature, figure 107 (b) reveals a specific heat anomaly at $T \approx 1.4$ K which marks the existence of a phase transition between the low-
<figure><img src="image_123.png" /><figcaption>Figure 107. Temperature dependence of the molar specific heat of (TMTSF)₂ClO₄ under a magnetic field of 63 kOe along the *c*-direction. (a) lattice plus electronic contributions, (b) electronic contribution (after reference [30]).</figcaption></figure>

temperature low density of carriers and the high-temperature conducting state.

(iii) The electronic specific heat of the conducting state above 1.4 K is about 70% larger than the corresponding value in zero field. Consequently the density of states of (TMTSF)₂ClO₄ measured at *T* ≈ 2 K is enhanced by the application of a magnetic field. This latter feature is in marked contrast with the density of states above *T*_c found to be approximately field independent in the limit of the low applied fields (see section 5.2). We shall now proceed to a discussion in the high field of specific heat data of (TMTSF)₂ClO₄ within the framework of the electronic properties of the other members of the (TMTSF)₂X series.

We first note that the low temperature behaviour of (TMTSF)₂ClO₄ at ambient pressure closely resembles that of the pressure stabilized conducting state of (TMTSF)₂PF₆ (at *P* ≳ 9 kbar or so) discussed above. Consequently, we believe that the phase transition observed by NMR in (TMTSF)₂PF₆ under pressure and high magnetic field at about 1–2 K [83] is similar to the transition observed in the specific heat study at 1.4 K and 63 kOe for (TMTSF)₂ClO₄. Recent ⁷⁷Se NMR experiments [74] performed on (TMTSF)₂ClO₄ single crystals have confirmed the magnetic character of the state with a low density of carriers which is stabilized by a large magnetic field (figure 108). Going one step further, the high field specific heat results indicate that the onset of Shubnikov–de Haas oscillations in (TMTSF)₂PF₆ under
<figure><img src="image_124.png" /><figcaption>Figure 108. Temperature dependence of the NMR ⁷⁷Se signal in (TMTSF)₂ClO₄ under ambient pressure, (a) line-width and (b) NMR signal intensity; the dashed horizontal line shows the Curie law behaviour which could be expected if there were no magnetic ordering occurring around 3.5 K (after reference [74]).</figcaption></figure>

pressure [81] coincides with a phase transition between a low field ($H < 60$ kOe) metallic state and a low carrier-density high field ($H > 60$ kOe) state. Oscillations appear only in the low carrier-density state and suggest that this state is of semimetallic nature. Moreover from the behaviour of the proton spin-lattice relaxation [83] rate which shows an extra source of relaxation in the vicinity of the metallic-semimetallic transition one can suggest that the semimetallic state is probably magnetic. The relaxation peak can be accounted by the presence of low-lying magnetic modes. It is therefore tempting to suppose that the semimetallic states restored under high field ($H \gtrsim 60$ kOe) either in (TMTSF)₂ClO₄ under ambient pressure or in (TMTSF)₂PF₆ under $P \gtrsim 9$ kbar bear close similarities. A semimetallic band picture for a modulated state is quite possible in a quasi-one-dimensional single-chain conductor in the presence of a warped Fermi surface, especially if the wave-vector of the distortion is $(2k_F, 0, 0)$ instead of the vector $(2k_F, \pi/b, \pi/c)$ which is expected to lead to a better nesting of the Fermi surface [84]. Such a situation has already been encountered for the anion ordered phase of (TMTSF)₂NO₃ below 40 K [51]. The comparison between magnetoresistance and specific heat data under high
magnetic field suggests that the restoration of a semimetallic phase occurring at a field where $\Delta\rho/\rho_0 \gtrsim 10$ ($T = 2.5$ K, $H\|c^*$), is a direct consequence of the suppression of the one-dimensional superconducting fluctuating regime. These phenomena are in direct correspondence with the character of quasi-one-dimensional matter in which an electron gas undergoes at low temperature a transition against the formation of an ordered state, either dielectric with a spin or lattice modulation, or superconducting. The high field investigation of the $(TMTSF)_2X$ family thus reemphasizes the existence of two energy scales governing the onset of the superconductivity instability.

Summarizing this section, we claim that the development of the superconducting instability at low temperature in the $(TMTSF)_2X$ series is largely influenced by the low dimensional character of the electronic motion. In this respect several experimental investigations of either $(TMTSF)_2ClO_4$, $(TMTSF)_2PF_6$ or $(TMTSF)_2AsF_6$ under pressure have answered some key questions.

(i) Schottky tunnelling, far-infrared reflectance or magnetotransmission and thermal conductivity studies lead to a very consistent picture: short-range superconducting pairing takes place along the chains (without interchain coherence) at temperatures as high as 15 K. The impact of this short-range pairing manifests itself in the quasiparticle density of states by the observation of a pseudo-gap at the Fermi level below 15 K or so in $(TMTSF)_2PF_6$ under pressure. Moreover the ratio $2\Delta/kT_c$ of about 40 for $(TMTSF)_2PF_6$ under 11 kbar and for $(TMTSF)_2ClO_4$ under ambient pressure indicates that the actual critical temperature $T_c$ below which both a zero resistance state and magnetic flux expulsion are observed is not determined solely by the strength of the superconducting pairing within each chain. The superconducting phase transition occurs when coherence is established between the phases of the order parameters on adjacent chains. As far as superconductivity is concerned, the transfer coupling which matters most is the Josephson coupling $t_{\perp}^2/t_\parallel$ which can transfer an electron pair between the chains. There exists a broad temperature domain extending up to 30 K or so in which the electronic properties of conductors like $(TMTSF)_2ClO_4$ or $(TMTSF)_2PF_6$ are, to a large extent, dominated by the onset of intrachain superconducting order.

(ii) The collective-mode related to the onset of one-dimensional short-range ordering accounts for the unusually large conductivity at low temperature above $T_c$, which is extraordinarily dependent on magnetic field or irradiation damage.

(iii) The high field electron specific heat establishes that the quantum oscillations observed above 65 kOe in $(TMTSF)_2PF_6$ under $P > P_c$, implying a two-dimensional Fermi surface, are related to the high field restoration of a magnetically ordered semimetallic ground state.

(iv) Finally one may hope to take advantage of the one-dimensional fluctuation regime and stabilize the three-dimensional ordered superconducting state at a temperature related to the amplitude of the intrachain pairing energy (i.e. 12 K, for $(TMTSF)_2PF_6$ under 11 kbar) provided the interchain coupling is enhanced via a chemical cross-linking establishing electron bridges between the stacks of molecules. Preliminary results on $(TMTSF)_2PF_6$ tend to support the view that $T_c$ enhancement is possible following this approach.
### 5.4. Origin of phase transitions

One of the most interesting features of the $(TMTSF)_2X$ class of compounds is the competition between superconductivity and an antiferromagnetic (SDW) phase (figures 78 and 85). A somewhat similar behaviour occurs in some metallic compounds like $(RE)_xMo_6S_8$ [85–87], $(RE)_xMo_6Se_8$ [88] (RE is one of the rare earth elements like Gd, Tb, Er or Dy, and $x \approx 1$), or $SmRH_4B_4$ [89], and is usually due to magnetic ordering of localized ($f$-) electrons, whereas the superconductivity is due to $s$-electrons [90]. The situation is quite different in the $(TMTSF)_2X$ compounds: there is only one conduction band, with a Fermi surface roughly represented by figure 30 (b). This is evident from the fact that the antiferromagnetic phase is insulating, due to a $2k_F$-periodicity of the magnetic ordering which completely destroys the Fermi surface. Moreover, at pressures very close to the magnetism–superconductivity boundary a rise of resistivity indicates the onset of partial magnetic ordering at temperatures above the superconducting transition [16, 35, 42, 48]. Such a situation is quite hard to understand in the framework of the conventional mean-field (Hartree–Fock) type theories of instabilities of an electron gas: in that picture magnetic ordering is due to repulsive electron–electron interactions [91], whereas singlet superconductivity arises from an attractive effective electron–electron interaction [92]. Further evidence for repulsive short-range forces comes from the temperature dependence of the magnetic susceptibility [93]: the spin susceptibility of the $(TMTSF)_2X$ compounds in the metallic regime is enhanced over its free electron value, and such a behaviour is only expected for repulsive short-range interactions [94, 95], i.e. $g_1 > 0$ in the terminology of section 2.2. On the other hand, for $g_1 > 0$ there is a gap in the spin excitation spectrum, and consequently the spin susceptibility is activated [95], in disagreement with experiment.

How can one reconcile the evidence for repulsive electron–electron interactions with the existence of superconductivity? One possibility is to assume that the superconductivity in the organic compounds is of triplet rather than of singlet type. In the mean-field approximation triplet superconductivity is stable even for purely repulsive electron–electron interactions (section 2.2.1). However, the notorious high sensitivity of triplet superconductivity to non-magnetic impurities [96] makes this possibility highly unlikely. (In this context, the high sensitivity of the superconducting state to irradiation-induced defects (section 5.3.2) might be taken as an indication of triplet superconductivity, however, so far convincing evidence for or against triplet superconductivity is lacking.) On the other hand, a fully consistent treatment of the one-dimensional electron gas [97] shows that even for repulsive bare interactions there may be strong superconducting fluctuations of the singlet type, which are only logarithmically weaker than the triplet fluctuations (figure 5). The transition from the antiferromagnetic to the superconducting low-temperature state of the $(TMTSF)_2X$ compounds then would correspond to a crossing of the $g_1 = 2g_2$ line in the $g_1 - g_2$ phase diagram, figure 5. On both sides of the line some three-dimensional coupling is necessary to suppress order parameter fluctuations and to stabilize a long-range-ordered state. Further, retardation effects due to phonon-mediated interactions (see below) favour singlet over triplet superconductivity and thus lead to a transition from antiferromagnetism to singlet superconductivity as the $g_1 - 2g_2 = 0$ line is crossed [98]. This picture has been further elaborated by Emery et al. [99], who include the umklapp scattering $g_3$. The presence of a weak umklapp scattering, due to the dimerization of the organic stacks, was first pointed out by
Barisic and Brazovskii [93]. In the presence of umklapp scattering the superconductivity-antiferromagnetism boundary is shifted to $g_1 - 2g_2 = |g_3|$. Emery et al. then argue that increasing pressure decreases the dimerization and therefore $g_3$, leading to a crossover from an antiferromagnetic to a superconductivity ground state. This picture qualitatively explains the rapid decrease of the antiferromagnetic transition temperature near the critical pressure (figure 78). Horovitz et al. [98] argue that due to crystal symmetry direct electron-electron umklapp scattering is forbidden and only electron-phonon umklapp scattering occurs. This symmetry argument seems however to be incorrect [99].

The above picture assumes that the bare interaction constant fulfils $g_1 \approx 2g_2$. On the other hand, for Coulomb interactions one has $g_2 \propto q^{-2}$, and even in the presence of screening one expects $g_2 > g_1$ [93]. To obtain $g_1 \approx 2g_2$ a phonon-mediated electron-electron interaction may be invoked, which acts to reduce the effective $g_1$ and $g_2$ at energy transfers below the corresponding phonon frequencies, as discussed in section 4.3. Thus for the reduction of the low-momentum-transfer interaction $g_2$ only optical or intramolecular phonon modes are expected to be effective. Strong interactions between intramolecular phonons and conduction electrons are indeed not unusual in organic conductors (see section 4.3), and it seems therefore reasonable to assume that a combination of Coulomb repulsion and intramolecular-phonon-mediated attraction places the $(TMTSF)_2X$ in the region $g_1 > 0$, $g_1 \approx 2g_2$ of the $g_1 - g_2$ phase diagram (figure 5). However, a fully microscopic picture of the origin of the interactions $g_1$, $g_2$ and $g_3$, which should also be able to explain the strong decrease of the superconducting transition temperature with pressure [35, 50], and which would greatly help our understanding of the properties of the $(TMTSF)_2X$ series, is lacking so far. We note that especially high-frequency intramolecular phonons will generally not be mere nuclear motions, but also involve a considerable polarization of the molecular charge distribution. Thus the possible role of these modes in the $(TMTSF)_2X$ compounds partly justifies Little's original idea [100] about attractive electron-electron interaction mediated by polarizable molecules.

In the immediate vicinity of the antiferromagnetism-superconductivity critical pressure one observes generally a considerable increase in resistivity followed by the superconducting transition at lower temperature [16, 35, 48] as well as antiferromagnetic resonance in the special case of $(TMTSF)_2ClO_4$ [49] (which is thought to have a critical pressure of 1 bar). These results may be understood assuming that the interaction constants $g_1$ and $g_2$ vary slightly over the Fermi surface, due to interchain interactions. Then some parts of the Fermi surface may fulfil $g_1 - 2g_2 < |g_3|$, leading to antiferromagnetism, whereas on other parts one has $g_1 - 2g_2 > |g_3|$, leading to the superconducting transition. A mean-field theory containing two distinct types of Fermi surface has been investigated by Machida and Matsubara [101]. Although the applicability of mean-field theory seems doubtful in the presence of experimental evidence for strong fluctuation effects, and two distinct parts of the Fermi surface are surely not a correct description away from the critical pressure where the whole Fermi surface is destroyed by the antiferromagnetism, leading to an insulating state, the qualitative features of that theory may well apply to $(TMTSF)_2X$ compounds near the critical pressure. Especially, the interesting question whether the antiferromagnetic order persists in the superconducting phase is not yet experimentally answered.
<figure><img src="image_125.png" /><figcaption>Figure 109. Proposed phase diagram of (TMTSF)₂ClO₄, (a) for the ‘quenched’ state, (b) for the ‘relaxed’ state, after [74]. The results of specific heat measurements [30] and the onset of Shubnikov-de Haas oscillations [82] are also included. The definition of the terminology ‘quenched (relaxed)’ state is given in reference [74] and is also briefly discussed in section 5.4.</figcaption></figure>

We finally comment on the anion order-disorder transitions of the type observed in (TMTSF)₂ReO₄ (figure 77 (a)). Contrary to the Peierls-transitions observed in the TTF-TCNQ series (section 4.2.1), the precursor effects of this transition, observed in X-ray scattering [21], always show three-dimensional correlations, i.e. the correlation length is (nearly) isotropic. This indicates that the strongly anisotropic electronic properties only play a minor role and that the transition is not driven by the Peierls mechanism, but rather by an isotropic (electrostatic or lattice mediated) interaction [20] between the anions. In that picture a gap at the Fermi level only opens because the longitudinal component of the wave-vector of the anion ordering happens to be $2k_F$.

However, anion ordering does not necessarily lead to a doubling of the lattice periodicity along the chain direction. For instance, recent EPR and resistive data [102, 103] have demonstrated the existence of a phase transition at 24 K in (TMTSF)₂ClO₄. Furthermore X-ray diffuse scattering studies [104] have shown that ClO₄ ions order at the transition in a ‘uniform’ way which *does not modify* the lattice periodicity (the periodicity remains at below 24 K).

A ‘slow’-cooling of (TMTSF)₂ClO₄ down to low temperature through the 24 K transition [74, 103] leads to a well anion-ordered state [104] in which the superconducting transition temperature is maximized at $T_c \approx 1.2$ K (see figure
109 (b)). However, fast cooling freezes-in some residual anion disorder at low temperature (Q-state in figure 109 (a)) which may weaken the one-dimensional superconducting divergence [105], and enables the onset of SDW ordering at ≈3.5 K with the subsequent lowering of the superconducting transition below 0.9 or so.

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## 6. Concluding remarks

The intense investigation of the physical properties of organic conductors which started in the early seventies has shown how strongly the compounds synthesized so far, namely those belonging to the TTF-TCNQ or (TMTSF)₂X series, are influenced by the quasi-one-dimensional nature of the electronic motion. More than any other inorganic conductors exhibiting some features related to linear chains like the A15 compounds or transition metal trichalcogenides (with the exception of the mixed-valence platinum chain compounds), these organic conductors have been shown experimentally to be influenced by the laws of one-dimensional physics.

(i) Upon cooling to low temperatures spontaneous symmetry breaking occurs (with very few exceptions). The observed low-temperature states are of very different nature: either a charge (or spin) density wave of superconductivity is found.

(ii) There exists an extended range of temperatures above the critical temperature for the onset of three-dimensional long-range order where short-range order on individual chains is well developed. In this precursor regime finite lifetime collective modes contribute to the electric conductivity: fluctuating Fröhlich conductivity, provided the periodicity of the charge density wave is incommensurate with the lattice, or the paraconductivity of superconductors.

Charge density wave fluctuations may be visualized directly by diffraction techniques and as such their existence has been easily accepted by the scientific community, although their contribution to the conduction has been firmly established only by far-infrared reflectance data and conductivity studies under high pressure.

Concerning superconducting fluctuations extending up to 30 K or so in the (TMTSF)₂X series, their existence has been highly controversial since unlike Fröhlich fluctuations direct visualization is impossible. However, there is a large amount of data obtained with the help of different techniques supporting the existence of one-dimensional short range superconducting order accompanied by a strong depression of the density of states at the Fermi level up to about 10 to 15 times the phase transition temperature.

Theoretically, the theory of one-dimensional electron systems (the so-called 'g-ology'), modified to include interchain couplings, is able to explain qualitatively the different occurring low-temperature phases. In the low-temperature superconducting phase quantum fluctuations are still effective in suppressing the order-parameter well below its mean-field value. The often complicated phase-diagram of the two-chain TTF-TCNQ-like compounds has been quite successfully explained using the Landau theory of phase transitions.
In spite of the spectacular progress made in the experimental investigation and theoretical understanding of these rather special conductors, several open problems remain. The phase diagram of two-chain conductors is not yet fully understood, especially in the neighbourhood of the commensurability domain. The respective role of electron-electron, electron-lattice, electron-intramolecular-phonon and interchain interactions in determining the kind of low temperature ordered state as well as their relation to various model parameters (as for example employed in the 'g-ology' picture) is not yet understood.

The role of the crystal structure, band filling (or 2:1 stoichiometry) and high pressure in controlling the balance between magnetic (insulating) and superconducting ground states in the (TMTSF)₂X series is still unclear. A particularly interesting problem is the possible coexistence of magnetism and superconductivity in the vicinity of the border line separating these two instabilities.

The possibility of a triplet rather than singlet superconducting ground state in the (TMTSF)₂X compounds cannot yet be excluded. A more thorough experimental investigation of the fluctuating regime is for example the study of the collective mode lifetime. Several experimental techniques are difficult to apply on organic superconductors. In particular tunnelling techniques and far-infrared studies which have shown to be so powerful but also deserve a stronger experimental effort in the near future.

Finally, the chemical modification of (TMTSF)₂X constitutes a big challenge, either to increase the ability of the electrons to jump from chain to chain and thus to stabilize superconducting fluctuations at higher temperatures, or to find better organic superconductors by the use of new organic molecules instead of TMTSF.

## Acknowledgements

The development of research in organic conductors and superconductors at Orsay has been due to the close collaboration of K. Bechgaard, J. Friedel and M. Ribault. We wish to emphasize the fruitful cooperation of all our colleagues in the high-pressure and in the X-ray scattering groups, as well as that of numerous visiting scientists at Orsay.