factor 2.5 between room temperature and the 150 K region. If instead of the nearly free electron model with moderate electron correlations, a strong correlation picture is used, namely $U \gg 4t_\parallel$ in the Hubbard model, the spin diffusion constant becomes [67]

$$D = \frac{2.3J}{\hbar}, \qquad (3.18)$$

where $J$ is an exchange interaction describing the low lying excitations of an effective Heisenberg exchange hamiltonian given by relation (3.10).

In this model a temperature dependence of $J$ and consequently of $D$ is expected only if $k_B T < J$. As with $U/4t_\parallel \approx 0.8-1$, which is derived with the RPA approximation [70], the exchange interaction $J$ amounts to $\approx 300$ K, both moderate and strong coupling models can qualitatively explain the temperature dependence of the spin diffusion constant in TTF-TCNQ. A better understanding of the role of correlations in TTF-TCNQ via NMR relaxation requires a more refined treatment in which the electron-electron interaction is no longer approximated by a $\delta$-function potential [73].

### 3.3. Electronic properties including the interchain coupling

#### 3.3.1. The three-dimensional band structure

In actual conductors like TTF-TCNQ electronic conduction does not proceed solely via a purely one-dimensional motion: the existence of a small but not zero overlap between wavefunctions of neighbouring chains enables the electrons to hop from chain to chain. This situation has already been discussed in section 2.3. The salient effect of interchain tunnelling coupling on the properties of single-chain conductors is to make the distortion with the wave vector $(2k_F^\circ, \pi/a, \pi/a)$ more stable than the one with the wave vector $(2k_F^\circ, 0, 0)$ [74] (figure 28). The upper limit of the overlap integral ratio $\eta = t_\perp/t_\parallel$ for the existence of a Peierls transition at non-

<figure><img src="image_35.png" /><figcaption>Figure 28. The Fermi surface of a quasi-one-dimensional conductor showing the warping of the initially planar surface. The wave vector nesting the Fermi surface is $(2k_F^\circ, \pi/a, \pi/a)$.</figcaption></figure>
zero temperature is $\eta_c = 3(T_p^\circ/T_F)^{1/2}$, where $T_p^\circ$ is the mean-field transition temperature of an isolated chain.

According to the derivation of $\eta_c$, the suppression of the Peierls state via increased three-dimensional coupling requires a transverse coupling larger than, say, one-fifth of the longitudinal coupling. This condition does not seem to be fulfilled in any of the known charge-transfer compounds. The non-existence of a Peierls ground state in $(TMSF)_2ClO_4$ (see section 5) is presumably not due to a hypothetically large interchain coupling in this particular conductor but instead to the divergence of the isolated chain superconducting channel (see sections 2 and 5). There is however another consequence of finite interchain tunnelling which is a characteristic of two-chain conductors only.

In the absence of transverse coupling between TTF(F) chains and TCNQ(Q) chains, the bands derived from the TTF and TCNQ molecular orbitals must cross the Fermi level where states $\psi_Q(k_F)$ and $\psi_F(k_F)$ are degenerate (figure 29). However the existence of any finite interchain coupling between F and Q chains mixes the states of same $k_F$ vector which become non-degenerate [75]. They form a bonding state $(\psi_Q + \psi_F)/\sqrt{2}$ and an antibonding state $(\psi_Q - \psi_F)/\sqrt{2}$ which are $2t_\perp$ apart in energy (the covalency effect) [76, 77] (figure 29).

The magnitude of the gap created by this covalency effect is dependent on the transverse momentum $k_a$ being maximum at $k_a = 0$ and zero at the zone boundary $k_a = \pm\pi/a$, by symmetry, with a tight-binding band model. However, if coupling

<figure><img src="image_36.png" /><figcaption>Figure 29. Energy level diagram in conducting charge-transfer compounds such as TTF-TCNQ; non-interacting chains (top left), including coupling between chains of the same nature, including a warping of the Fermi surface as shown in figure 30 (top right), including coupling between chains of different nature (covalency gap), (bottom left) and a schematic picture of the semimetallic Fermi surface in the low temperature three-dimensional regime (after reference [76] of section 3).</figcaption></figure>
<figure><img src="image_37.png" /><figcaption>Figure 30. Effect of a finite transverse coupling $t_⊥$ on a ‘single-chain’ conductor. (a) strong coupling (coherent three-dimensional coupling); (b) moderate coupling (coherent one-dimensional coupling); (c) weak coupling (incoherent one-dimensional coupling). (After FRIEDEL, J., and JÉROME, D., 1982, *Contemp. Phys.*, **23** (in press).)</figcaption></figure>

between like chains along the *c* direction is allowed, the number of states corresponding to the same *k*-vector is doubled. The conjugated effects of coupling between like and unlike chains are able to transform the original planar Fermi surface of non-interacting chains into a semimetallic three-dimensional Fermi surface. The semimetallic Fermi surface appears as small pockets of electrons and holes comprising a volume which is much smaller than the original Brillouin zone [76, 78, 79] (figure 29). As the Fermi energy of these electron or hole pockets is small, of the order of $t_γ$ or so, there is no hope of observing a three-dimensional Fermi surface in a quasi-one-dimensional conductor unless the temperature is low enough ($k_B T < t_⊥$). Among charge transfer compounds undergoing a Peierls transition, the condition $k_B T_p < t_⊥$ seems to be fulfilled only for HMTSF-TCNQ, where covalency effects due to strong Se···N bonds are especially large compared to the interchain coupling of the TTF-TCNQ series [80]. Besides specific properties such as the diamagnetic susceptibility (see section 3.3.4), the covalency gap contributes to a decrease of density of states at the Fermi level, which may account for the lowering of the Peierls transition at 57, 29 and 24 K in TMTSF, TSF and HMTSF-TCNQ, respectively, as the interchain coupling increases.

There are two families of molecular conductors in which covalency effects do not alter the density of states at the Fermi level:
(i) Radical-ion salts such as NMP-TCNQ, where probably both NMP and TCNQ stacks contribute to the conduction, but where the single electron available per formula unit sets the Fermi level far from the energy where both bands cross each other [29] (see figure 19).

(ii) The radical-ion salts of the $(TMTSF)_2X$ [39], $(TMTTF)_2X$ [81] or $Qn(TCNQ)_2$ series [4], which are single-chain conductors.

### 3.3.2. The three-dimensional escape time $\tau_\perp$

A consequence of the finite interchain overlap integral shows up by a finite transverse conductivity $\sigma_\perp$ which varies with $T$ in a way similar to $\sigma_\parallel(T)$, although it usually exhibits (in the TTF-TCNQ series especially) a somewhat weaker $T$-dependence [82].

The transfer of electron (or holes) between neighbouring chains can be defined by a hopping frequency $h/\tau_\perp$, where $\tau_\perp$ is the average time spent by an electron on a given chain [68, 70]. Obviously a one-dimensional motion requires $\tau_\perp \gg \tau_\parallel$, where $\tau_\parallel$ is the collision time for the motion of the electrons along the chains. $\tau_\parallel$ may be derived from the longitudinal d.c. conductivity, as long as no collective mode of the one-dimensional electron gas (Fröhlich sliding mode or superconducting fluctuations) contributes to the conduction. Weger and coworkers [70, 78] have given a very simple derivation for the hopping frequency $\hbar/\tau_\perp$ (the one-dimensional Golden Rule), which we shall briefly sketch here. We consider a situation where an electron sits on chain 1 at $t=0$ in the state $|\psi_1\rangle$. $t_\perp$ produces a progressive transfer to a neighbouring chain such as 2. For small $t$'s, the electron wavefunction reads:

$$|\psi(t)\rangle = \cos\frac{t_\perp t}{\hbar}|\psi_1\rangle + \sin\frac{t_\perp t}{\hbar}|\psi_2\rangle \quad (3.19)$$

At time $\tau_\parallel \ll \hbar/t_\perp$ the electron wavefunction loses its coherency. The electron has acquired a small probability to be on chain 2, equal to $|\langle\psi_2|\psi(\tau_\parallel)\rangle|^2$, or according to (3.19),

$$\sin^2 \frac{\tau_\parallel t_\perp}{\hbar} \approx \frac{t_\perp^2 \tau_\parallel^2}{\hbar^2}.$$

Its rate of transfer per unit time is thus,

$$\frac{1}{\tau_\perp} \approx \frac{2\pi}{\tau_\parallel} \left( \frac{\tau_\parallel^2 t_\perp^2}{\hbar^2} \right) = \frac{2\pi t_\perp^2 \tau_\parallel}{\hbar}. \quad (3.20)$$

Obviously the derivation leading to (3.20) is justified only when the limit $t_\perp\tau_\parallel/\hbar \ll 1$. If $t_\perp\tau_\parallel/\hbar \gtrsim 1$ the electron oscillates back and forth several times before being scattered by the diffusion process giving rise to the scattering time $\tau_\parallel$. In such a case the electron wavefunction involves the superposition of electron wavefunctions on first, second, etc. ... neighbouring chains: the motion between chains is coherent. If $t_\perp\tau_\parallel/\hbar < 1$, the transverse motion proceeds via successive incoherent interchain hoppings, this is obviously a diffusive process. Very crudely speaking the condition

$$t_\perp \tau_\parallel / \hbar \approx 1 \quad (3.21)$$

defines the transition between the coherent and the diffusive regime for the transverse conductivity.
In this diffusive regime, the transverse conductivity then reads, according to the Einstein relation [70]:

$$ \sigma_{\perp} = N(E_F)k_B T e^2 \frac{D_{\perp}}{k_B T}. \qquad (3.22) $$

$N(E_F)k_B T$ is the number of thermally excited carriers of mobility $eD_{\perp}/k_B T$ available for a transverse hopping. The diffusion coefficient is $D_{\perp} = b^2 \tau_{\perp}^{-1}$ for a tetragonal array of chains with interchain distance $b$. Therefore the transverse conductivity becomes:

$$ \sigma_{\perp} = N(E_F) e^2 b^2 \tau_{\perp}^{-1}. \qquad (3.23) $$

Since the longitudinal (coherent-like) conduction ($\hbar/\tau_{\parallel} < E_F$) is given by:

$$ \sigma_{\parallel} = \frac{4e^2 v_F \tau_{\parallel}}{\hbar b^2} \qquad (3.24) $$

the anisotropy of conductivity becomes, with (3.23) and (3.24) and the definitions $N(E_F) = 4/\hbar v_F b^2$ and $\hbar v_F \approx at_{\parallel}$,

$$ \frac{\sigma_{\parallel}}{\sigma_{\perp}} = \left(\frac{t_{\parallel}}{t_{\perp}}\right)^2 \left(\frac{a}{b}\right)^2. \qquad (3.25) $$

Equation (3.25) establishes that whenever the collective modes do not contribute to $\sigma_{\parallel}$, the conductivity anisotropy is approximately given by the square of the band structure anisotropy. It should thus not be strongly temperature dependent [70].

Quasi-one-dimensional conductors can be classified in four different regimes according to the relative values of $\hbar\tau_{\parallel}^{-1}$, $t_{\perp}$ and $t_{\parallel}$.

(i) Coherent three-dimensional couplings: $\hbar\tau_{\parallel}^{-1} < \tau_{\perp} \approx t_{\parallel}$. This is an extreme case which occurs for very strong interchain couplings. This produces a bi- or tri-dimensional conductor with closed Fermi surface. The chains lose most of their physical identity, which only produces some (small) anisotropy of conductivity (figure 30).

(ii) Transverse coherent and longitudinal coherent couplings: $\hbar\tau_{\parallel}^{-1} < t_{\perp} < t_{\parallel}$. This case occurs for moderate interchain couplings, $t_{\perp}$ warps the planar Fermi surfaces of independent chains over regions of $k_Z$ of order $t_{\perp}/(\delta E/\delta k_Z)_{E_F} \approx \tau_{\perp}/2t_{\parallel}a$ (figure 30). These warped planes correspond to two separate and infinite sheets of the Fermi surface for 'single-chain' conductors, for example the $(TMTS_F)_2X$ series. In 'two-chain' conductors a semimetallic band structure may occur as a result of couplings between like and unlike chains (see section 3.3.1). Quantum oscillations can be observed with a magnetic field applied perpendicular to the chain axis.

(iii) Transverse incoherent and longitudinal coherent couplings: $\tau_{\perp} < \hbar\tau_{\parallel}^{-1} < t_{\parallel}$. This case occurs for weak interchain couplings. The Fermi surface is then made of two diffuse planes as for independent chains (figure 30). This seems to be the case of TTF-TCNQ and the related compounds in the conducting regime and of the $(TMTS_F)_2X$ series, at least at high temperature.

(iv) Longitudinal incoherent couplings: $t_{\parallel} \lesssim \hbar\tau_{\parallel}^{-1}$. This case is probably encountered above room temperature in TTF-TCNQ when the mean free path becomes so short that $\Lambda k_F < 1$.
### 3.3.3. NMR measurement of $\tau_\perp$

A direct measurement of the hopping frequency $\hbar/\tau_\perp$ is offered by NMR measurements [68, 70], since in the presence of a finite escape probability for the electron on a given chain, the relaxation rates saturate in low fields [68]. If the autocorrelation function of the spin density which gives the power spectrum of (3.13) is truncated at large times by a single exponential decay, (3.14) should be replaced by:

$$ \phi_q(t) = \frac{\chi_s}{2} \exp(-Dq^2t) \exp(-t/\tau_\perp). \quad (3.26) $$

Taking the Fourier transform of (3.26) and summing over $q$, we get the contribution to $(T_1 T)^{-1}$ coming from the $q \approx 0$ components

$$ (T_1 T)^{-1} \sim A^2 \chi_s \sqrt{\frac{\tau_\perp}{D}} g(\omega_e), \quad (3.27) $$

where

$$ g(\omega_e) = \sqrt{\frac{1 + \sqrt{(1 + \omega_e^2 \tau_\perp^2)}}{2(1 + \omega_e^2 \tau_\perp^2)}}. \quad (3.28) $$

The interchain coupling does not lead to any change in $(T_1 T)^{-1}$ for the coherent propagation of the spin excitations because the hopping rate $1/\tau_\perp$ is much smaller than the Fermi energy. Therefore, according to the field dependence of $g(\omega_e)$ (equation (3.27)) there exists a cross-over value of $\omega_e$ at

$$ \omega_e \tau_\perp \approx 1 \quad (3.29) $$

between a high-field one-dimensional regime $(1/T_1 \propto H^{-1/2})$ and a low-field three-dimensional regime $(1/T_1 = \text{const})$. Consequently, provided the escape time falls within the limits of the feasibility of NMR experiments, $10^{-12}$ s $\lesssim \tau \lesssim 5 \times 10^{-10}$ s, the field dependence of the nuclear relaxation rates of a quasi-one-dimensional conductor allows a remarkably direct derivation of $\tau_\perp$. Finally the relaxation of a quasi-one-dimensional conductor becomes:

$$ (T_1 T)^{-1} = C_1 g(\omega_e) + C_2, \quad (3.30) $$

where the unique parameter entering $g(\omega_e)$ is the escape time $\tau_\perp$ (equation (3.28)). $C_1$ and $C_2$ depend on the model taken for the description of the quasi-one-dimensional electron gas but are field independent. Data of proton relaxation rates of selectively deuterated TTF-TCNQ (figure 27) show that the field dependence of (3.30) is fairly well obeyed between 300 and 110 K. A fit of the data on figure 27 allows the determination of the three parameters $\tau_\perp$, $C_1$ and $C_2$.

$C_1$ and $C_2$ are related to the electron scattering time $\tau_\parallel$ and to the spin susceptibility possibly enhanced by Coulombic correlations. The derivation of $\tau_\parallel$ from the NMR data depends on the model chosen for the description of the one-dimensional electron-gas. A random phase approximation has been used by Soda et al. [68]. On the other hand the derivation of $\tau_\perp$ does not depend on the actual spin dynamics in the one-dimensional regime. For example, at room temperature $\tau_\perp \approx 8 \times 10^{-12}$ s is found for electrons of both chains of TTF-TCNQ. This escape time is admittedly much smaller than the electron scattering time derived from d.c. conductivity or optical data $\tau_\parallel \approx 3 \times 10^{-15}$ s. These values are consistent
with the condition $t_{\perp}\tau_{\parallel}/\hbar = (\tau_{\parallel}/\tau_{\perp})^{1/2} \ll 1$. TTF-TCNQ belongs to case (iii) previously described.

The definition of the transverse conduction (equation (3.23)) shows that similar temperature dependences are expected for $\tau_{\perp}^{-1}$ and $\sigma_{\perp}$ neglecting the comparatively weak temperature dependence introduced by the overlap integrals and the lattice constants. Such behaviour is actually observed for TTF-TCNQ (figure 31). As the temperature is lowered, the hopping frequency increases and follows the single-particle longitudinal conduction (but not the total d.c. conductivity if collective mode contributions are significant at low temperature, see sections 4 and 5). For TTF-TCNQ, the NMR derivation of $\tau_{\perp}^{-1}$ and the optical determination of $\sigma_{\parallel}$ lead to an effective transverse coupling of $t_{\perp} \sim 5$ meV, and subsequently to interchain matrix elements which agree fairly well with the results of molecular orbital calculations [28]. Extensive investigations of the $T_1$ field dependences have been performed in other charge transfer compounds, for example in deuterated Qn(TCNQ)$_2$ where $T_1^{-1} \propto H^{-1/2}$ is observed with no sign of saturation down to the nuclear Larmor frequency of 3 HHz [83], and in deuterated NMP-TCNQ which exhibits some kind of saturation at low fields [84]. Qn(TCNQ)$_2$ seems to be the most one-dimensional compound studied by nuclear relaxation since the escape time of the electrons in the TCNQ stacks is larger than $3 \times 10^{-11}$ s. The effective transverse coupling of Qn(TCNQ)$_2$ is presumably smaller than 1 meV. The frequency dependence of $T_1$ in HMTSF-TCNQ up to 276 MHz [62] at room temperature indicates a much shorter hopping time (the upper limit of $\tau_{\perp}$ being $4 \times 10^{-13}$ s or $t_{\perp} \gtrsim 20$ meV) which is due to the short interchain Se-N bonds along the a-direction of the HMTSF-TCNQ structure.

As the temperature is lowered $\tau_{\parallel}$ may increase so much that it reaches a cross-over regime $\hbar^{-1}t_{\perp}\tau_{\parallel} \approx 1$ below which the Fermi surface loses its one-dimensional character and becomes two- or three-dimensional. This situation is encountered in

<figure><img src="image_38.png" /><figcaption>Figure 31. Temperature dependence of the interchain hopping rate in TTF-TCNQ(D$_4$) [●] and TTF(D$_4$)-TCNQ [▲]. Normalized conductivity along a- and c-axis is also displayed (after reference [70] of section 3).</figcaption></figure>
HMTSF–TCNQ, due to the already large value of the ratio $\tau_\parallel/\tau_\perp > 1.5 \times 10^{-2}$ instead of $\approx 10^{-3}$ in TTF–TCNQ at room temperature, it is also encountered in $(TMTSF)_2X$ at low temperature (section 5).

### 3.3.4. HMTSF–TCNQ: a special case

Below, say 100 K the static susceptibility of HMTSF–TCNQ [76] quickly drops to a diamagnetic value which is much lower ($\chi(T=0) = -5.25 \times 10^{-4}$ e.m.u. mol$^{-1}$) than that expected either from the calculation of the HMTSF core diamagnetism ($\chi_{calc}(T=0) = -2.85 \times 10^{-4}$ e.m.u. mol$^{-1}$) or from the measurement of the susceptibility of neutral HMTSF molecules ($\chi_{exp}(T=0) = -2.98 \times 10^{-4}$ e.m.u. mol$^{-1}$). An NMR investigation of HMTSF–TCNQ [62] shows that the Pauli contribution is particularly small at 300 K, $\chi_s(300K) \approx 1.5 \times 10^{-4}$ e.m.u. mol$^{-1}$ and decreases steadily down to $\approx 0.4 \times 10^{-4}$ e.m.u. mol$^{-1}$ at zero temperature. The comparison between static susceptibility data and NMR results show therefore that a strong diamagnetic contribution to susceptibility arises below 100 K, and reaches $-1.29 \times 10^{-4}$ e.m.u. mol$^{-1}$ below 30 K (figure 32). This strong low-temperature diamagnetism has been attributed to a Landau–Peierls orbital diamagnetism arising from a semimetallic-like Fermi surface comprising small cylindrical (or ellipsoidal) pockets of electrons and holes, with Fermi energies in the range 10–50 K [79]. The

<figure><img src="image_39.png" /></figure>

Figure 32. Temperature dependence of the residual static susceptibility after subtraction of the Curie–Weiss part and of a core diamagnetic susceptibility $\chi_d^0 = -2.83 \times 10^{-4}$ e.m.u. mol$^{-1}$ estimated from Pascal constants. The static susceptibility includes two components the Pauli contribution $\chi_P$ derived from nuclear spin–lattice relaxation and a Landau–Peierls orbital diamagnetism arising below 100 K or so (after reference [62] of section 3).
strong interchain coupling of HMTSF-TCNQ favours a three- (or two-) dimensional character at low temperature. But at high temperature, namely $k_B T$ larger than $t_⊥$ or so, wave packets of energy $k_B T$ can be formed with states of all possible $k_⊥$ values. These states can thus be localized on individual chains in real space and the band structure becomes one-dimensional. Several properties of HMTSF-TCNQ can be explained in a very natural way by the one-dimensional to three-dimensional cross-over of the Fermi surface below 100 K accompanied by the occurrence of closed electron orbits.

(i) The conductivity of HMTSF-TCNQ increases from $≈ 1800 (Ω cm)^{-1}$ at room temperature up to a very smooth maximum at $≈ 800 (Ω cm)^{-1}$ between 100 and 30 K [37] (figure 33). Around 24 K the conductivity drops sharply, becoming two orders of magnitude less at 4.2 K in good quality crystals [85] (figure 33). The smaller drop of the conductivity which has been reported for one batch in preceding studies [37, 86] has been attributed to the presence of crystalline defects in those crystals [87]. The one-dimensional to three-dimensional cross-over which is located around 100 K, according to the susceptibility data, does not severely affect the b-axis conduction (longitudinal conductivity) $σ_∥$, given by $ne^2A/k_F$ [86]. Both the density of carriers and $k_F$ are roughly similarly affected by the one-dimensional to three-dimensional transition, decreasing approximately by 2 or 3 orders of magnitude, whereas the mean-free path is given essentially by the distance between scattering centres, supposedly only weakly temperature dependent below 100 K.

(ii) The Hall constant measurement [45] has provided a very direct illustration of the sequence of transformation occurring in HMTSF-TCNQ on cooling. The Hall constant indicates a metallic behaviour near ambient temperature (figure 23) ($R_H ∼ 2 × 10^{-11}$ V cm A$^{-1}$ Oe$^{-1}$), the more mobile carriers being holes on the HMTSF chain, with a density of approximately 0.7 per formula unit. The increase of the Hall constant clearly observed below 150 K is related to the drop in the density of carriers accompanying the one- to three-dimensional transition. Around 40 K the density of carriers (holes) is about 1/50 per formula unit. The transition at 24 K is seen by the Hall effect changing sign, becoming electron-like and large at helium temperature ($R_H$ (4.2 K) $≈ -2 × 10^{-8}$ V cm A$^{-1}$ Oe$^{-1}$).

(iii) The magnetoresistance which has also been measured as a function of the temperature [45, 85] is large and anisotropic below the transition at 24 K, being about twice as large for the transverse field parallel to the low conductivity direction (the direction of weakest interchain coupling) than for the field aligned with the intermediate conductivity direction (the direction for the strong Se···N bonds) (figure 33). The mobility of the carriers at low temperature is large, ranging from $2 × 10^3$ to $10^4$ cm$^2$ V$^{-1}$ s$^{-1}$. Susceptibility, transport and magnetotransport data are all consistent with the picture of a semimetallic Fermi surface in HMTSF-TCNQ at low temperatures [75].

A model of semimetallic Fermi surface for the undistorted three-dimensional Fermi surface of HMTSF-TCNQ, i.e. at $T > 24$ K has been proposed by Weger [79]. The semimetallic Fermi surface consists of small section cylinders or elongated ellipsoids in the c-direction. However some problems remain as the formation of a three-dimensional short range Peierls ordering at 24 K suggested by X-ray diffuse scattering experiments [88] is likely to modify the proposed semimetallic Fermi surface. It is clear that the question of the phase transition at 24 K in HMTSF-TCNQ is not yet settled but perhaps the most convincing piece of evidence for a definitely semimetallic nature of the Fermi surface at helium temperature comes
<figure><img src="image_40.png" /><figcaption>Figure 33. (a) Normalized resistance curve $R(T, P)/R(300, 0)$ for HMTSF-TCNQ under pressure up to 14 kbar, according to the initial pressure study [86]. The dashed line shows the data of [37]. (b) ‘Good quality’ HMTSF-TCNQ crystal showing no stabilization of a semimetallic state above 4 kbar. The dotted line corresponds to the stabilization of a semimetallic state observed in poor quality crystals as shown in (a) (after unpublished data of M. Miljak at Orsay). (c) Temperature dependence of the resistivity in a ‘good quality’ HMTSF-TCNQ crystal indicating the existence of a phase transition (maximum slope of $d\ln R/dT$) at 24 K and effect of a magnetic field of 78 kOe parallel to the low conductivity axis (after reference [85] of section 3).</figcaption></figure>

from the observation of quantum oscillations in the transverse magnetoresistance at fields higher than 30 kOe [89] (figure 34). The high-field limit ($\omega_c\tau \approx 1$), $\omega_c$ being the cyclotron frequency and $\tau$ the three-dimensional electron scattering time, is attained in a field of about 30 kOe at low temperature. The single frequency of the oscillations which seems to be somewhat sample and pressure dependent leads to an average cross-section of cylindrical or ellipsoidal Fermi surfaces about a thousand times smaller than the area of the Brillouin zone in the a-b plane, namely $3 \times 10^{12} \text{ cm}^{-2}$.
<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

Figure 34. (a) $(1/R)$ $(dR/dH)$ for HMTSF-TCNQ at 10 kbar with $H \parallel c$ and the current flowing along the longitudinal axis, after subtraction of a quadratic background magnetoresistance. The initial resistance $R$ was taken as the resistance in zero field at 4.2 K. Data at 1.2, 2.4 and 4.2 K are offset by 0.5, 1.0 and $1.5 \times 10^{-4}$ respectively (after reference [89] of section 3). (b) $1/H$ for maxima (integer) and minima (half-integer) in $dR/dH$ plotted against quantum number for HMTSF-TCNQ (after reference [89] of section 3).

The density of carriers $\approx 1/1000$ per formula unit in the small electrons and holes pocket is therefore in good agreement with the magnetotransport data [75].

The observation of magnetoresistance quantum oscillations in $(TMTSF)_2PF_6$ [90] and $(TMTSF)_2ClO_4$ [91, 92] only above a certain threshold magnetic field (of the order of 60 kOe at $T = 1$ K) is due to the restoration of a semimetallic SDW state in these single-chain conductors under very high field. The problem of the $(TMTSF)_2X$ series will be treated extensively in section 5.

Hall effect and resistance data show that the low temperature semimetallic state of HMTSF-TCNQ is hardly modified under a pressure of 5–7 kbar. Therefore, the
early experimental evidence [86] showing the stabilization of a metallic state in HMTSF-TCNQ (under a pressure larger than 4 kbar or so) with $dρ/dT > 0$ down to 50 mK ($σ_{max} ≈ 10^4 (Ω cm)^{-1}$) must be taken with a grain of salt. This stabilization is probably not intrinsic, but due to the coexistence of a decrease of the phase transition under pressure observed between 1 atm and 4 kbar and the possible presence of defects in the crystals used in the work of reference [86]. Defects have been recognized to severely affect the transport properties of a Peierls state [93]. Finally, in contrast to HMTSF-TCNQ, investigation of the magnetoresistance in TTF-TCNQ [94] has revealed practically no effect upon application of a transverse magnetic field. Above 54 K, the magnetoresistance of the longitudinal resistance can hardly be detected; $Δρ/ρ < 0.1%$ in a field of 50 kOe. This result is expected for the one-dimensional band structure of the conductor in the limit $ω_cτ ≪ 1$.

### 3.3.5. Electron spin resonance in quasi-one-dimensional conductors

Electron spin resonance has been used extensively for the study of one-dimensional conductors [95]. This may appear somewhat surprising at first glance since it is well known that conduction electron spin resonance (CESR) usually gives rise to very broad resonance lines in most metallic conductors besides a few exceptions like the light alkali metals [96] or very high purity metals at low temperature [97]. For isotropic metals the dominant process leading to electron spin-lattice relaxation, $1/T_1$, comes from the modulation of the spin-orbit interaction by lattice vibrations [98]. There exists a proportionality relation between the non-spin/flip electron scattering time $τ$ (describing the conductivity for instance) and the spin/flip time $T_1$ (giving rise to the CESR width $1/T_1$), namely

$$τ ≈ c^2 T_1, \qquad (3.31)$$

where c is the mixing coefficient between up and down spin Bloch states ($|+⟩+|c⟩−⟩u_k \exp(ikr)$) due to the perturbation of the spin-orbit interaction. Furthermore, Elliot [99] has shown that the c coefficient in (3.31) is related to the g-shift of the CESR line. The CESR width reads [100]

$$\Delta H = \frac{1}{T_1} \approx \frac{(\Delta g)^2}{\tau}. \qquad (3.32)$$

The use of (3.32) for organic conductors would admittedly lead to results in contradiction with the actual experimental situation [95]:

(i) The measured g-shifts ($Δg ≈ 2 × 10^{-3}$) and electron scattering times $τ ∼ 3 × 10^{-15}$ s provide, through (3.32), linewidths about 2 orders of magnitude larger than those observed in TTF-TCNQ or related compounds ($ΔH ≈ 6, 4$ and 500 Oe in TTF-TCNQ, TMTTF-TCNQ and TSF-TCNQ, respectively) (figure 35).

(ii) The linewidth, according to (3.32) should follow the temperature dependence of the resistivity, and therefore decrease at low temperature in the conducting regime. For TTF-TCNQ instead, $ΔH$ increases by a factor 3 between 300 and 60 K (figure 35).

The remarkably narrow CESR line of most one-dimensional conductors has been attributed to a considerable weakening of the efficiency of the spin-orbit induced relaxation in one-dimensional conductors. On the other hand, there is a strong
<figure><img src="image_43.png" /><figcaption>Figure 35. Temperature dependence of the ESR linewidth for (a) TTF-TCNQ (after TOMKIEWICZ, Y., 1979, *Phys. Rev. B*, **19**, 4038, (b) (TMTTF)₂Br and (TMTTF)₂PF₆ (after COULON, C., DELHAES, P., FLANDROIS, S., LAGNIER, R., BONJOUR, E., and FABRE, J. M., *J. Phys., Paris* (to be published)), (c) (TMTSF)₂PF₆ (after reference [105] of section 3).</figcaption></figure>

mismatch between the energy $qV_s$ of a phonon and the excitation energy $qV_F$ of an electron-hole pair. Therefore scattering involving $\Delta q \approx 0$ (forward scattering around $q=0$ for one-dimensional band structures) are inefficient for the spin reversal. On the other hand, backward scattering involving $\Delta q = 2k_F$ are also very inefficient for one-dimensional conductors since the hermitian spin-orbit operator $O^+ = O$ invariant through time reversal operation $KOK^{-1} = O$ has no matrix element between Kramers conjugate states such as $|+k_F, \uparrow\rangle$ and $|-k_F, \downarrow\rangle$ [98].
Hence, Elliot's relation does not hold for one-dimensional conductors. Spin reversal mechanisms are considerably weakened by dimensionality effects [101]. However the finite interchain coupling $t_⊥$ admixes some three-dimensional character to the originally flat Fermi surface specified by the ratio $\tau_‖/\tau_⊥$ (see section 3.3.2). Weger and coworkers [78] have suggested taking the deviation from the one-dimensional behaviour into account in the spin relaxation mechanism. The linewidth then becomes:

$$ \Delta H \sim \frac{(\Delta g)^2 \tau_‖}{\tau_‖ \tau_⊥} = (\Delta g)^2 \tau_⊥^{-1}. \quad (3.33) $$

$\Delta H$ and $\tau_⊥^{-1}$ (derived from NMR in TTF-TCNQ) behave similarly in the conducting regime (figures 31 and 36) whatever the reason for the change of $\tau_⊥^{-1}$, i.e. temperature or pressure. The drop of the linewidth observed in the Peierls state ($T < 54$ K) is due to the vanishing of the density of states at the Fermi level which is required for electron scatterings. An extensive study of the isostructural solid solutions (TSF)$_x$(TTF)$_{1-x}$TCNQ has been performed by the IBM group [102]. As the spin-orbit coupling is strongly dependent on the nature of the heavy heteroatom of the donor molecule some kind of normalization must be performed in the attempt to make comparative studies over the whole series $0 < x < 1$. The study of the normalized ESR linewidth $\Delta H/(\Delta g)^2$ versus $x$ reveals an increase going from TTF to TSF-TCNQ (figure 37). The normalized linewidth is 16 times larger at $x=1$ than at $x=0$. According to (3.33) $\Delta H/(\Delta g)^2$ should follow $\tau_⊥^{-1}$ and therefore $|t_⊥|^2\tau_‖$. Consequently, EPR studies of the isostructural series suggest an increase of the interchain coupling $t_⊥$ by a factor 3 or so between TTF-TCNQ and TSF-TCNQ. Some increase in the interchain coupling may also be inferred from the crystal structure [103]. Furthermore it is particularly interesting to notice that HMTSF-TCNQ, which is believed to show the strongest interchain coupling among two-chain conductors, does not exhibit any ESR signal at room temperature [76]. The width is probably larger than 4000 Oe or so. This result corroborates the absence of field dependence of the proton relaxation time up to a field of 60 kOE [62], also indicating strong interchain coupling. At the other extreme lies TMTSF-DMTCNQ, where the observation of a particularly narrow ($\Delta H \sim 70$ Oe at 300 K) ESR line for a selenium compound has been taken as evidence for weak interchain coupling [63].

<figure><img src="image_44.png" /><figcaption>Figure 36. ESR linewidth $\Gamma$ and normalized linewidth $\Gamma/(\Delta g_F)^2$ vs $x$ in (TSF)$_x$(TTF)$_{1-x}$TCNQ, where $\Delta g_F$ is the deviation of the donor stack $g$ value from the free electron value following the procedure of TOMKIEWICZ, Y., ENGLER, E. M., and SCHULTZ, T. D., 1975, *Phys. Rev. Lett.*, **35**, 456.</figcaption></figure>
<figure><img src="image_45.png" /><figcaption>Figure 37. Polarized light reflectance data in TTF-TCNQ under pressure (after reference [49] of section 3).</figcaption></figure>

A still very challenging question is understanding the EPR properties of single-chain or pseudo-single-chain conductors [104] such as (TMTSF)₂X, Qn(TCNQ)₂ or NMP-TCNQ and TTT₂I₃. (TMTSF)₂PF₆ and related systems [105] are particularly interesting as ΔH decreases steadily towards a low temperature (figure 35), whereas in the same temperature domain σ⊥ increases significantly [106]. The spin relaxation of these compounds may be influenced by the development of the superconducting one-dimensional divergence at low temperature. As far as the non-superconducting single-chain systems are concerned, for example (Qn(TCNQ)₂), strong one-dimensionality could also favour the establishment of a one-dimensional pseudo-gap (Peierls-like) at high temperature with a concomitant reduction of the density of states.

### 3.4. Pressure effects

The study of organic conductors under high pressure had been initiated at Orsay with the idea that a pressure-enhanced interchain coupling could help to stabilize a conducting state at 0 K [75]. It has provided one of the most rewarding contributions towards understanding the electronic properties of these quasi-one-dimensional conductors. The unusually large effect of pressure on the conductivity of TTF-TCNQ at room temperature has been detected very early [107, 108] in the study of TTF-TCNQ, and attributed to a possibly large pressure dependence of the bandwidth [107]. However, subsequent band-structure calculations [28, 29, 109] and high-pressure investigations pointed out very clearly the contrast existing between the moderate coefficients of the 1 electron bandwidth and the unusually large pressure dependences exhibited by all Fermi-level properties [59].

#### 3.4.1. *The one electron bandwidth*

Among the various measurements which are usually performed in the laboratory in the attempt to characterize the band structure of a conductor, the optical reflectance spectrum allows a fairly good derivation of the 1 electron bandwidth.
High-pressure optical studies on TTF-TCNQ and TSF-TCNQ [49] reveal an increase of the reflectance under pressure in the energy range $\nu > 0.6$ eV (figure 37). The shift of the plasma edge towards high energy and its sharpening under pressure is obvious from figure 37. The plasma frequency of TTF-TCNQ increases by about 15% in 20 kbar. Since the band formation in a one-dimensional conductor results essentially from the overlap of molecular wavefunctions along the stacking direction it seems reasonable to relate the change of $\omega_p$ to the change of the b-axis spacing. Consequently, data on figure 37 and axial compressibility measurements of TTF-TCNQ [14, 15] lead to $\partial \ln \omega_p / \partial \ln b = -2.5$. As the plasma frequency is related to the bandwidth in a one-dimensional tight-binding band model by $\omega_p^2 \alpha t_\parallel$, we derive in order of magnitude a bandwidth pressure coefficient of:

$$ \frac{\partial \ln t_\parallel}{\partial \ln b} \approx 5 \qquad (3.34) $$

neglecting a smaller contribution to the pressure dependence of $\omega_p$ coming from the pressure dependence of the charge transfer. The experimental pressure coefficient of the overlap integral is in fair agreement with the calculated b-axis dependence of the bandwidths [109] (figure 38). Near ambient pressure $\partial \ln t_\parallel / \partial \ln b$ coefficients are -7 and -3.5 for the TCNQ and TTF bands, respectively.

Furthermore, in a two-chain conductor, the charge transfer (or band filling) increases under pressure as shown by various experimental data in TTF-TCNQ (phase diagram showing $\times 3$ longitudinal commensurability [110], anomaly of the sound velocity related to the 53 K Peierls transition [111], direct determination by neutron scattering under pressure [112]). The pressure coefficient of the charge transfer derived from high-pressure data becomes:

$$ \frac{\partial \ln k_F / b^*}{\partial \ln b} = -1.3 \qquad (3.35) $$

and we are able to assume that the pressure dependence of the 1 electron bandwidth is reasonably well understood with a tight-binding model [113].

### 3.4.2. Transport and magnetic properties

The conductivity of a large number of highly conducting compounds has been studied under pressure [107, 108, 114] (figure 39). The largest pressure dependence among highly conducting compounds in figure 40 is observed for TTF-TCNQ with

<figure><img src="image_46.png" /><figcaption>Figure 38. Calculated bandwidth vs b-axis parameter in TTF-TCNQ and TSF-TCNQ (after reference [109] of section 3).</figcaption></figure>
<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /><figcaption>Figure 39. Pressure dependence of the conductivity of TTF-TCNQ at 300 K (a) and of other compounds together with the pressure dependence of the plasma edge (b).</figcaption></figure>
<figure><img src="image_49.png" /><figcaption>Figure 40. Right hand scale: resistivity versus temperature at constant pressure for TTF-TCNQ. Left hand scale: pressure coefficient of the conductivity versus temperature for various TTF-TCNQ samples (after reference [116] of section 3).</figcaption></figure>

$\partial \ln \sigma / \partial P \approx 28\% \text{ kbar}^{-1}$ at room temperature. The piezoresistance of Qn(TCNQ) is even larger since $(\partial \ln \sigma / \partial P)_{Qn(TCNQ)_2} \approx 40\% \text{ kbar}^{-1}$ [115]. Furthermore figure 39 seems to suggest a similar asymptotic value of the conductivity (at $6-7 \times 10^3 \text{ }\Omega \text{cm})^{-1}$ for all conductors under very high pressure. In terms of b-axis dependence the pressure coefficients of the conductivity can be derived using the compressibility data [14]. It amounts, for TTF-TCNQ to:

$$ \partial \ln \sigma / \partial \ln b = -46, \qquad (3.36) $$

i.e. more than one order of magnitude larger than the pressure coefficient of the plasma edge (see figure 37).

In a very crude model plasma frequency and d.c. conductivity are related by the Drude relation:

$$ \sigma_{\text{d.c.}} = \omega_p^2 \tau_{\parallel}. \qquad (3.37) $$

Since the P-dependence of $\omega_p$ is less than, say +1% kbar$^{-1}$ we are forced to conclude that most of the pressure dependence of $\sigma_{\text{d.c.}}$ must be attributed to the scattering time $\tau_{\parallel}$, i.e.

$$ \partial \ln \tau_{\parallel} / \partial \ln b \approx -41. \qquad (3.38) $$

Such a large volume dependence implies significant consequences for the temperature dependence of the resistivity [25, 26].
Most theories provide constant volume results, whereas observations are
performed at constant pressure. The constant volume correction can be made if
the pressure coefficient of the resistivity is known. The correction should be small
whenever the pressure coefficient is small, but it can become an extremely important
effect for the volume dependence of the conductivity observed in TTF-TCNQ at
high temperature [116] and displayed in figure 40.

Direct evidence that most of the volume dependence of the resistivity is
determined by the lattice parameter of the highly conducting direction (b-axis)
comes from the measurement of the resistivity of TTF-TCNQ under uniaxial strain
[108]. A strain applied along the b-axis has an order of magnitude larger effect on the
conductivity than a strain along a transverse axis. Furthermore the transverse
compressibilities are more than three times smaller than the longitudinal compress-
ibility [14] (see section 2.3).

Constant *b*-axis resistivity versus temperature curves shown in figure 41 were
obtained by calculating at each temperature the pressure required to bring the *b*
lattice parameter to its value at 60 K and correcting the measured value of the
resistivity to its value at that pressure using the data for thermal expansion [21, 22]
and the pressure dependence of the conductivity displayed in figure 40 [116]. Thus a
reduction in *b* by 2.34% at 300 K requires a pressure of 5 kbar, increasing the
conductivity to 2.4 times its ambient pressure value. The temperature dependence of
the constant *b* resistivity is close to a linear *T* law above 140 K, contrasting
significantly with the *T*<sup>2.3</sup> law observed under ambient pressure (figure 41).

As emphasized by Cooper [25] a successful model for the scattering mechanism
that determines the resistivity must be able to account for two conditions:

(i) the strong *b*-axis dependence,

(ii) the quasilinear temperature dependence in nearly the entire temperature
domain where the conduction is metallic-like.

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

Figure 41. Resistivity versus temperature for two TTF-TCNQ crystals with *b*-lattice parameter constant at its 60 K value, ρ(300 K, 1 bar) is normalized to 2.5 × 10<sup>-3</sup> Ω cm for both samples (after reference [116] of section 3).
If one assumes the Boltzman formalism to be an adequate description for one-dimensional conduction, scattering by acoustic phonons could lead to a linear $T$ dependence for $T > \theta_D$, and a pressure dependence such as $\partial \ln \tau_\parallel / \partial \ln V = -2\gamma_G$ ($\gamma_G$ is the Gruneisen constant) could be expected. Since $\gamma_G \approx 1.5$ as derived from the volume dependence of the compressibility [14], it is obvious that the scattering against acoustic phonons is unable to explain a volume coefficient $\partial \ln \tau_\parallel / \partial \ln V \approx -30$. Several phonon scattering processes have been claimed to explain the resistivity of TTF-TCNQ: acoustic phonon scattering [22], acoustic and intramolecular phonon scattering [117], second order scattering with librations [118]. Other approaches have also been proposed, for example, electron-molecular vibration interactions leading to the formation of polaron bands [25] and strong electron-electron correlations [119] which constitute the basis of an electron-spin excitation collision model [26]. The latter model gives a linear temperature dependence and the enhanced scattering cross section is very sensitive to small variations in $UN(E_F)$ (due to changes of the bare bandwidth) when this quantity is close to unity. However we do not wish to spend here more time with the resistivity of TTF-TCNQ since, for reasons which will be detailed in section 4, a large fraction of the conductivity below 150 K is due to fluctuating sliding CDWs. Therefore, when discussing the origin of the single-particle conductivity the sliding conductivity contribution must be first subtracted out of the measured conductivity before any attempt is made to understand the origin of the strongly volume dependent single-particle scattering. The transverse conductivity of TTF-TCNQ or TSF-TCNQ has been found to increase under pressure with the same rate as the longitudinal conductivity in the pressure domain $P \lesssim 10$ kbar [120] (figure 42). This pressure independence of the conductivity anisotropy is, however, in agreement with the interpretation of section 3.3.2 since $\sigma_\parallel / \sigma_\perp$ depends on the ratio $(t_\parallel / t_\perp)$. As shown above, the longitudinal transfer integral is only weakly pressure dependent and furthermore it has been suggested [25] that the pressure dependence of the transverse tight binding transfer integral may be similar to that of $t_\parallel$ since the effect of larger Slater parameters is practically compensated by the smaller transverse compressibilities (see section 3.1.3). The pressure insensitivity of the conduction anisotropy has also been observed in the single-chain conductors (TMTSF)$_2$X [121] although other studies [122] suggested a decrease of this quantity under pressure. One can confidently expect that in all organic conductors so far studied at high pressure the overlap integrals along all three axes show very similar percentage shifts with pressure. In short, pressure does not significantly change the anisotropy parameter $\eta$ for the band structure of quasi-one-dimensional conductors [81].

Other transport coefficients exhibit interesting pressure dependences. This is the case of the thermopower and Hall effect coefficients of TTF-TCNQ. The large changes under pressure observed for both quantities are related to a change of the relative conductivities on the two chains. Thermopower [123] and Hall effect [44] studies suggest that the conduction on the TCNQ chain is dominant under atmospheric pressure, with $\sigma_Q \sim 5\sigma_F$, and therefore makes the conductor look like a single-carrier system. Under a pressure of 8 kbar the ratio $\sigma_Q / \sigma_F$ is reduced to about 2 or 2.5 assuming comparable individual thermopower (and Hall constants) for both chains. Similarly, the large pressure dependence of the TMTSF-DMTCNQ thermopower indicates that the DMTCNQ chain contributes more to the conductivity in the compound under pressure [124] than at ambient pressure [125]. In contrast to the behaviour of the thermopower under pressure previously discussed in two-chain
<figure><img src="image_51.png" /></figure>

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

Figure 42. (a) Pressure dependence of the conductivity in TSF-TCNQ, direct method for $\sigma_b$ and $\sigma_a$, Montgomery method for $\sigma_b$ and $\sigma_c$. (b) Pressure dependence of the conductivity in TTF-TCNQ, direct method for $\sigma_b$ and $\sigma_a$ (after reference [126] of section 3).
conductors, the thermopower of single-chain conductors behaves quite differently. $Q_n(\text{TCNQ})_2$ has a thermopower reduction of about 20% in 15 kbar [115] and almost no effect of pressure can be detected in $(\text{TMTSF})_2\text{PF}_6$ up to 12 kbar [126]. As far as $Q_n(\text{TCNQ})_2$ is concerned, no large changes in the thermopower are expected in a model of strongly correlated electrons where the spin entropy ($k_B \ln 2$) is dominant in the electron entropy [115]. For the single-chain conductor $(\text{TMTSF})_2\text{PF}_6$, the thermopower remaining at 12 kbar within 15% of its value at ambient pressure corroborates the optical data of TTF-TCNQ or TSF-TCNQ under pressure [49]: the 1 electron bandwidth is increased by about 15% under a pressure of 10 kbar. High-pressure studies have contributed recently to the understanding of thermal conductivity properties in an organic conductor $(\text{TMTSF})_2\text{PF}_6$ [127]. The interpretation of the pressure dependence of the thermal conductivity at room temperature leads to lattice and electronic contributions of 13 and 8 mW cm$^{-1}$ K$^{-1}$, respectively, in fair agreement with earlier data obtained with TTF-TCNQ [128].

The NMR properties of TTF-TCNQ reveal strong pressure dependences in agreement with the model discussed in section 3.3.3. The hopping frequency $1/\tau_\perp$ increases under pressure like the transverse conduction [70] (figure 43) and the electron scattering time derived from the interpretation of the field dependence of the nuclear relaxation follows very closely the d.c. conductivity under pressure. Moreover, the pressure dependence of the proton relaxation rate (HMTSF-TCNQ at 300 K $\approx$ -8% kbar$^{-1}$) can be reasonably well attributed to the pressure broadening of the band in a weakly correlated electron gas [59]. The dependence of -23% kbar$^{-1}$ measured for $1/T_1$ in TTF-TCNQ could imply a dominant role of many body effects [59], for instance a large value of the Coulomb repulsion to bandwidth ratio.

The spin susceptibility of organic conductors has also been studied by EPR techniques under pressure whenever the EPR linewidth was small enough to allow EPR experiments in low fields [59, 129]. The EPR linewidths show a strong pressure dependence similar to that of the single particle electron scattering time in agreement

<figure><img src="image_53.png" /><figcaption>Figure 43. Pressure dependence of the interchain hopping rate in TTF-TCNQ($D_4$) (●) and in TTF-TCNQ (■). Normalized transverse conductivity is shown by the dashed line (after reference [70] of section 3).</figcaption></figure>
with (3.33). The spin susceptibility derived from the integration of the ESR signal leads to a pressure dependence which once again is too big to be interpreted by 1 electron band broadening.

As all Fermi level properties of the TTF-TCNQ series are strongly volume dependent, the argument put forward concerning the intrinsic temperature dependence of the resistivity should also apply to the spin susceptibility. If a crude reconstruction of the constant b-axis temperature dependence of the susceptibility is performed in TMTTF-TCNQ starting from the data of the EPR study under pressure [59], the temperature dependence of the susceptibility is substantially reduced since $\chi_s(300\text{ K})/\chi_s(60\text{ K}) \approx 1.5$ keeping the b parameter equal to its 60 K value. Admittedly, the constant b axis temperature dependence of the susceptibility is in qualitative agreement with the one-dimensional Hubbard model when the ratio of the Coulomb repulsion to the total width of the band is about unity [130].

Closing this section, we summarize the salient electronic properties of organic conductors around room temperature.

(i) The highly conducting behaviour of the charge-transfer compounds of the TTF-TCNQ series ($\sigma(300\text{ K}) > 500 (\Omega\text{ cm})^{-1}$) is due on the one hand to the ability of these molecules to form stable open shell cation or anion radicals and on the other hand to the particular packing of molecules in segregated stacks which allows a strong overlap of $\pi$-molecular wave functions within each stack and favours charge delocalization.

(ii) The one-dimensional properties of the conduction are a direct consequence of the planar shape of the molecules and of their columnar arrangement.

(iii) Compressibility and thermal expansion coefficients exhibit a much smaller anisotropy, the stacking axis being usually the direction showing the largest pressure and temperature dependences.

(iv) Optical properties are reasonably well understood in terms of a one-dimensional tight binding band model, with bandwidths of the order 0.5–1 V, increasing at a rate of a few percent per kilobar under hydrostatic pressure.

(v) The special $t^{-1/2}$ time dependence of the spin autocorrelation function in one dimension makes the nuclear spin relaxation by mobile electrons field dependent $1/T_1 \propto H^{-1/2}$.

(vi) The divergence of the relaxation at low fields is, however, limited below a cut-off frequency by the possibility for the electrons to jump on to a neighbouring chain. The field dependence of $1/T_1$ allows a direct experimental determination of the interchain hopping frequency $\tau_{\perp}^{-1}$.

(vii) In quasi-one-dimensional conductors the interchain hopping rate is related to the broadening of the one-dimensional Fermi surface due to the finite electron lifetime ($\tau_{\perp}^{-1}\alpha|t_{\perp}|^2\tau_\parallel$). This is the one-dimensional Fermi Golden Rule, which applies when the transverse motion of the carriers is diffusive, i.e. $t_{\perp}\tau_\parallel < \hbar$.

(viii) The finite (although small) interchain overlap integrals modify the shape of the Fermi surface whenever the interchain coupling $t_{\perp}$ and the temperature are such that the condition $t_{\perp}\tau_\parallel(T) > \hbar$ is fulfilled.

(ix) This situation is encountered in the nearly two-dimensional conductor HMTSF-TCNQ, where magnetotransport data and Landau-Peierls diamagnetic susceptibility show the existence of a semimetallic Fermi surface with a small number of carriers at low temperature.
(x) All quantities related to Fermi level properties (conductivity, spin susceptibility, nuclear relaxation, etc...) show a pressure dependence which is much too large to be understood by the pressure enhancement of the bare bandwidths. We suggest the possibility of a dominant role played by many body effects, possibly correlations, in the TTF-TCNQ family.

(xi) It is pointed out that the change of volume due to thermal expansion must be taken into account when studying the temperature of all Fermi level properties.

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# 4. Charge density wave order: TTF-TCNQ and its family

## 4.1. The ordered state

### 4.1.1. TTF-TCNQ

Phase transitions in the organic conductor TTF-TCNQ at ambient pressure were first discovered by anomalies in the longitudinal (b-axis) d.c. conductivity at the different temperatures $T_H = 54$ K [1, 2] and $T_L = 38$ K [3, 4]. Subsequent structural investigations by diffuse X-ray [5, 6] and elastic neutron scattering [7] have shown that at $T_H$ the lattice becomes modulated with period $2a \times 3.4b \times c$ ($a$ and $c$ are the transverse lattice constants). With decreasing temperature below $T_I = 49$ K the period in the $a$ direction increases and jumps discontinuously to $4a$ at $T_L$. The decrease of the conductivity and the lattice modulation are in general believed to be due to the Peierls mechanism which leads to the appearance of a charge density wave (CDW), lattice modulation of wavelength $\pi/k_F$ and an energy gap $2\Delta(T)$ at the Fermi surface. Due to the gap the temperature dependence of the resistivity has the form

$$ \rho(T) = \rho_0 \exp\left[\frac{\Delta(T)}{k_B T}\right], \quad (4.1) $$

so that at $T = T_c$, when $\Delta(T)$ starts to increase from zero with an infinite slope (in general one expects $\Delta(T) \propto (T_c - T)^\beta$ with $0 < \beta < 1$, so that $\partial\Delta/\partial T = \infty$ at $T = T_c$); the magnitude of $\partial\log\rho/\partial T$ has a sharp maximum [8].

The behaviour of $\partial\log\rho/\partial T$ in TTF-TCNQ at different pressures is displayed in figure 44. For pressures lower than $\approx 15$ kbar the peaking of $\partial\log\rho/\partial T$ occurs at two distinct temperatures $T_H$ (high) [1, 2] and $T_L$ (low) [3, 4]. It is clear, however, that the strongest peaking is always related to $T_L$ and above 15 kbar the two transitions merge into a single one. The transition at $T_L$ is first order under atmospheric pressure as shown by a temperature hysteresis of 1 K observed at $T_L$ between cooling and warming runs. Under atmospheric pressure, some hysteresis is also observed in the temperature domain ($T_L - T_I$) by transport measurements [9] and neutron diffraction patterns [10].

Above 15 kbar only a single transition occurs. A sharp peaking of the transition temperature is observed at 19 kbar [11–13]. At that pressure, the transition temperature reaches 74 K (figure 45). The width of the peaked region in the phase diagram amounts to a domain of $\approx 4$ kbar in pressure units. In that pressure domain, the phase transition exhibits a first-order character as shown by a one degree hysteresis observed at the transition [11]. The TTF-TCNQ phase diagram above 15 kbar has been understood in terms of commensurability between periodic lattice distortion and underlying lattice. Assuming a charge transfer 0.66 the wavelength of the CDW is three times the lattice constant in the chain direction. In this case third-order invariants may appear in the Landau expansion of the free energy. As
<figure><img src="image_54.png" /><figcaption>Figure 44. Plot of d log R/dt versus temperature for TTF-TCNQ at various pressures, after JEROME, D., and SCHULZ, H. J., 1982, *Extended Linear Chain Compounds*, Vol. 2, edited by J. S. Miller (New York: Plenum Press), p. 159.</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>Figure 45. Phase diagram of TTF-TCNQ derived from transport properties data after JEROME, D., and SCHULZ, H. J., 1982, *Extended Linear Chain Compounds*, Vol. 2, edited by J. S. Miller (New York: Plenum Press), p. 159.</figcaption></figure>
discussed in the first chapter the third-order terms makes the transition from the metallic state into the commensurate phase first order with a higher transition temperature $T_{c1}$ than the second order transition into the incommensurate state at $T_{c2}$ [14, 15]. From figure 45 a rough estimate can be obtained: $T_{c1} - T_{c2} \approx 10$ K. If the charge transfer deviates slightly from its commensurate value 0.66 there is a competition between the third-order terms which favour the commensurate state and the elastic energy of the CDW which drives the wavelength of the CDW to the incommensurate value $\pi/k_F$. For small deviations from commensurability the commensurate state is still favoured.

Third-order commensurability in TTF-TCNQ ($\rho = \frac{2}{3}$, so that $2k_F = b^*/3$) is suggested by the peaking and the first-order character of the transition at 19 kbar. The possibility of an enhancement of the charge transfer under pressure is a prerequisite for the observation of a commensurate state at 19 kbar. Such an increase of the charge transfer can reasonably be expected from the broadening of tight binding bands under pressure. Initial neutron scattering studies up to 6 kbar [16, 17] have indeed shown an increase of charge transfer with pressure consistent with commensurability at 19 kbar, and more recent experiments [18] show that $\rho = \frac{2}{3}$ is reached at 14.5 kbar. The difference of this value from the 19 kbar deduced from resistivity is attributed to the use of deuterated samples in the neutron experiments. It is worth mentioning that the pressure dependence of $k_F/b^*$ is moderate since it only amounts to $(\partial \ln k_F/b^* / (\partial \ln b)) \approx -3$. Similar ‘weak’ volume dependences are also found for all properties dealing with the bare bandwidths (see, for example, the plasma edges [19]), but much larger volume dependences are observed for quantities related to the Fermi level properties [20, 21].

Since both chains, TTF and TCNQ, are likely to contribute to the conduction, the change of conductivity at $T_H$ or $T_L$ cannot point to the nature of the chain on which a distortion sets in. TTF-TCNQ remains a fairly good conductor between $T_H$ and $T_L$ and only half of the charge carriers are lost at $T_H$ (figure 46) under atmospheric pressure. Fortunately, measurements of the thermoelectric power

<figure><img src="image_56.png" /><figcaption>Figure 46. Resistivity of TTF-TCNQ between $T_H$ and $T_L$ in the low-pressure regime ($P < 15$ kbar) showing the existence of a low resistivity state at $T_L < T < T_H$ (after reference [11] of section 4).</figcaption></figure>
coefficient yield additional information [22]. The absolute TEP coefficient of a two-chain conductor is expressed in terms of the individual TEP coefficients of each chain $S_Q$ and $S_F$ by [23]:

$$S = \frac{\sigma_Q S_Q + \sigma_F S_F}{\sigma_Q + \sigma_F} \qquad (4.2)$$

For a single one-dimensional band the absolute TEP coefficient is given by

$$S = - \frac{\pi^2 k_B^2 T}{3|e|} \left[ \frac{\cos \pi\rho/2}{2|t_\parallel| \sin^2 \pi\rho/2} + \frac{\tau'(\varepsilon)}{\tau(\varepsilon)} \right]_{E_F} \qquad (4.3)$$

where $\tau(\varepsilon)$ is the energy dependent electron scattering time, $\rho$ the amount of charge transfer and $t_\parallel$ the longitudinal transfer integral. When the band structure contribution to the TEP dominates in (4.3), the sign of the TEP is negative for $\rho < 1$ and positive for $\rho > 1$. The charge transfer can be determined from the additional X-ray scattering due to the charge-density wave, using the related $k_F = \rho/b$ (b is the lattice constant along the chain), and is 0.55 electrons under ambient conditions, so that we may assign $S_Q$ negative and $S_F$ positive.

Figure 47 summarizes the behaviour of the temperature dependence of $S$ at several pressures [24]. At low pressures the TEP becomes more positive below $T_H$, whereas at elevated pressures ($P > 8$ kbar) the change of $S$, accompanying the transition at $T_H$, clearly shows a tendency to become more negative. Consequently, following (4.2), (4.3) and figure 47 the ambient and low pressure regime can be understood in terms of an energy gap occurring on the TCNQ chain at $T_H$. Above 8 kbar however the interpretation of figure 47 suggests that the loss of carriers at $T_H$

<figure><img src="image_57.png" /><figcaption>Figure 47. Absolute thermopower of TTF-TCNQ showing the sign reversal around 6 kbar, related to the opening of gaps in the energy dispersion at $T_H$, after WEYL, C., JÉROME, D., CHAIKIN, P. M., and BECHGAARD, K., 1982, J. Phys., Paris, **43**, 1167.</figcaption></figure>
predominantly affects the TTF chains. The subsequent removal, at $T_L$, of the free carriers left below $T_H$ is accompanied by a sharp increase of $S$, since in a semiconducting state according to Boltzmann theory [23]:

$$S = \frac{k_B^2}{|e|} \left( \frac{b-1}{b+1} \frac{\Delta}{k_B T} + \ln \frac{m_h}{m_e} \right), \quad (4.4)$$

where $b$ is the ratio of electron to hole mobility and $m_h$ and $m_e$ are, respectively, the effective mass of the holes and of the electrons. The sign of $S$ in the semiconducting state depends on the impurity content in as much as it is fixed by the position of the Fermi level in the energy gap. The interplay between the contribution to the thermopower given by impurities in the gap of the distorted chains and the contribution due to the remaining carrier on the non- (or weakly) distorted chain does not allow a simple interpretation of the low temperature thermopower data (figure 47). However we believe that changes in the Seebeck coefficient occurring at $T_H$, where impurities should play a minor role, can be related to the intrinsic properties of TTF-TCNQ. It can be inferred from figure 47 that the change of regime occurring at $T_H$ under pressure is continuous, i.e. that the loss of carriers at $T_H$ is continuously transferred from the TCNQ chain to the TTF chain as pressure is increased.

From the conductivity and TEP investigation on TTF-TCNQ under pressure it follows that both chains order at $T_H$. Thermopower shows conclusively that in the low pressure limit the gaps which open at $T_H$ on the TCNQ chains on the TTF chains are, respectively, larger and smaller than $k_B T_H$, the converse becoming true above ~8 kbar. Such a picture is corroborated by recent high-accuracy measurements, at ambient pressure, of EPR and static spin susceptibility of TTF-TCNQ [25], showing a small but finite drop of the TTF-chain susceptibility occurring at $T_H$. $T_I = 49$ K is the temperature below which the transverse 'phase sliding' region begins, but no peaking of $\partial \log \rho / \partial T$ has been detected [11]. Following a resistivity reinvestigation of the phase transition at 1 bar [9] we believe that 49 K may represent the temperature below which the gap on the TTF chains starts to rise slowly above $k_B T$.

The phase diagram of TTF-TCNQ derived from the transport properties investigation is displayed in figure 45. The salient feature of this diagram is the existence of a temperature domain between $T_H$ and $T_L$ in which the system is neither a conductor, since $\partial \sigma / \partial T > 0$, nor an insulator since $\sigma > 10^3$ (Ω cm)$^{-1}$. As already suggested in this section, the three-dimensional ordered distortion is presumably large on only one of the two chains between $T_L$ and $T_H$. A determination of the density of states at the Fermi level of the TCNQ chain performed through Knight shift measurements on TTF-TCNQ(C$^{13}$) [26] shows conclusively that a sudden lowering of $N(E_F)$ occurs at $T_H$. An EPR decomposition [27] of the susceptibility of TTF-TCNQ also indicates that the $T_H$ transition affects primarily the TCNQ chain under ambient pressure.

The determination of the phase transition temperatures by diffraction techniques has been reviewed in several articles [28]. Therefore, we shall give here only a brief outline of the salient results both under atmospheric pressure and under pressure. The main complexity of TTF-TCNQ is the existence of periodic lattice distortions with wave-vector $2k_F$ and $4k_F$. Moreover the $2k_F$ distortion exhibits two different polarizations; either longitudinal along $b^*[2k_F L(b^*)]$ or transverse along
<figure><img src="image_58.png" /></figure>

Figure 48. X-ray pattern of TTF-TCNQ at 50 K in the upper modulated phase. Note the presence of well defined $2k_F$ satellite reflections with wave vectors $(0.5a^*, 0.29b^*, 0c^*)$. In this phase, the $4k_F$ scattering keeps its one-dimensional character (no $4k_F$ satellite reflections). The white arrow points towards the $2k_F$ (3.5, 3.295, 3) satellite. The sample is oriented in the reciprocal equatorial phase (0, k, l), the angle between the incident X-ray beam and the b*-axis being 124° (after reference [29] of section 4).

$c^*[2k_F T(c^*)]$. The latter distortion sets in at $T_H$ [29, 30], leading to the $2a \times 3.4b \times c$ superstructure (figure 48), but the former at $T_I$ only. Below $T_I$ a $4k_F$ distortion is also observed [29]. Though this point has not yet been clarified, the fact that the $4k_F$ scattering near $T_I$ is of an intensity comparable to that of the longitudinal $2k_F$ scattering [29] seems to indicate that there is an independent ordering of the high temperature $4k_F$ diffuse scattering [31, 32] at $T_I$, i.e. the $4k_F$ distortion is not merely a secondary effect of the $2k_F$ ordering. Below $T_I$ the modulation period in the *a* direction starts to increase continuously [10, 33], and at $T_L$ a first-order phase transition with an hysteresis of about 1 K takes place and the superlattice locks in its $4a \times 3.4b \times c$ modulation without any further changes at lower temperatures (figure 49).

The structure of the different low-temperature phases of TTF-TCNQ under pressure has been investigated by elastic neutron diffraction techniques [16–18, 34]. Figure 50 displays the phase diagram derived from these studies and shows several interesting features:

(i) the $T_I$ transition observed at 49 K at ambient pressure splits into two transitions at $T_{I1}$ and $T_{I2}$, which increase and decrease with increasing pressure, respectively. Between $T_H$ and $T_{I1}$ the modulation has only a transverse polarization (phase IV), whereas below $T_{I2}$ a longitudinal component is also observed (phase I). The modulation periodicity does not change at $T_I$.

(ii) The temperature $T_{I2}$ of the onset of a continuously varying transverse modulation (phase II) decreases rapidly with pressure and merges into $T_L$ at $P \approx 2.5$ kbar.

(iii) The $4a$-modulated phase below $T_L$ (phase III) exists only up to $P \approx 3$ kbar, above this pressure the transverse modulation period decreases continuously
<figure><img src="image_59.png" /><figcaption>Figure 49. X-ray pattern of TTF-TCNQ at 30 K in the lower modulated phase. Both $2k_F$ and $4k_F$ scatterings are condensed into satellite reflections at the wave vectors $(0.25a^*, 0.295b^*, 0c^*)$ and $(0.5a^*, 0.41b^*, 0c^*)$ respectively. White arrows point towards the $2k_F(4.25, 2.295, 1)$ and $(3.75, 2.705, 0)$ satellites, black arrows point towards the $4k_F(3.5, 2.41, 1)$ and $(4.5, 2.59, 0)$ satellites (after reference [29] of section 4).</figcaption></figure>

(phase V), and at 4.6 kbar only the $2a$-modulated phase is observed between $T_H$ and the lowest investigated temperature.

(iv) At 12 kbar and above, especially in the pressure region of longitudinal commensurability, there is no transverse modulation. Between ambient pressure and about 12 kbar the longitudinal period decreases linearly, corresponding to an increase of charge transfer (figure 51). At 14.5 kbar the longitudinal period locks into a third-order commensurate state (modulation $a \times 3b \times c$), corresponding to charge transfer $\rho = \frac{2}{3}$.

At pressures below 5 kbar the transitions of $T_H$ and $T_L$ are clearly observed in transport measurements, as discussed above (see figure 45), whereas at most, very weak anomalies are associated with $T_{I1}$ and $T_{I2}$. At first sight, the disappearance of the $T_L$-transition in the neutron experiments at 4.6 kbar disagrees with the observation of transport anomalies at 30 K at the same pressure. However, as pointed out by Megtert, the use of a deuterated crystal in the neutron scattering may lead to a shift in pressure scale with respect to the samples used in transport measurements, so that in the transport measurements $T_L$ should disappear in the pressure region where there is a gap in the $T_L$-line in figure 45. No detailed transport investigations have been done so far in that region. The lower transition line in figure 45 between 10 and 15 kbar most probably corresponds to the transition between phases I and VI implied by figure 50, which happens at pressures not investigated by diffraction experiments so far.

As stated above, below $T_H$ TTF-TCNQ remains a fairly good conductor as a substantial gap (i.e. gap energy $E_g > k_B T$) is only opened on the TCNQ-chains. Only below $T_L$ is a gap opened on the TTF chains, so that TTF-TCNQ becomes a small-gap semiconductor. The activation energy of the resistivity below 38 K is [8] $\Delta_\rho \approx 230$ K, and a similar value is found in far-infrared measurements [35]. This leads to an energy gap $E_g = 2\Delta_\rho = 460$ K. In disagreement with this value, however, the activation energy of the magnetic susceptibility is found to be $\Delta_\chi \approx 120-150$ K.
<figure><img src="image_60.png" /><figcaption>Figure 50. Phase diagram of deuterated TTF-TCNQ derived from elastic neutron scattering experiments (after reference [18] of section 4 and MEGTERT, S., COMÈS, R., VETTIER, C., PYNN, R., and GARITO, A. F., 1982, *Molec. Crystals liq. Crystals*, **85**, 159, and reference 17 of section 4). The signature of each transition is given by the transverse period along the *a*-axis (I and IV, 2a; II, *x*(*T*, *P*)*a*; III, 4a; V, *x*'((*T*, *P*)*a* ≈ 3a; VI, 0*a*). The *T*<sub>H</sub> phase line is supported by 'transverse satellites' in the (013) zone whereas the 'longitudinal satellites' (no component along *c*<sup>+</sup> of the scattering vector) are not present above the *T*<sub>M</sub> upper phase line. The intensity decreases from full to open symbols. The sharp points are for zero intensity (background).</figcaption></figure>

[36, 37], implying different energies for charge and spin degrees of freedom. As noted by Tomkiewicz *et al.* [36] different energies for charge and spin excitations occur in the one-dimensional Hubbard model [38], and are actually a quite general property of one-dimensional interacting fermion systems [39, 40], however, so far it has not been clarified to what degree this effect also occurs in coupled-chain systems like TTF-TCNQ. Krumhansl *et al.* [41] proposed another explanation: amplitude solitons of the coupled electron-lattice system (see section 2) have an energy below the single-electron gap and carry spin 1/2 and zero charge [42, 43]. Such excitations would therefore contribute to the spin susceptibility, but not to the conductivity.
<figure><img src="image_61.png" /><figcaption>Figure 51. Pressure dependence of the $2k_F$ Fermi wave vector of TTF-TCNQ derived from elastic neutron scattering experiments (after reference [18] of section 4).</figcaption></figure>

Under pressure the gap initially decreases [44], in agreement with the decrease of $T_L$ (figure 45). Around the pressure of longitudinal commensurability ($P \approx 19$ kbar) there is a sharp increase of the gap [11], indicating a width of the low-temperature commensurability region of 4 kbar as also seen in the neutron diffraction experiments. At longitudinal commensurability as well as far away from it the thermopower is proportional to $T^{-1}$, as expected for a semiconductor. On the other hand, in the commensurate-incommensurate transition region the low temperature thermopower is approximately constant [13], as expected for classical particles. Noting that near the commensurate-incommensurate transition the wavenumber of the Peierls lattice modulation deviates slightly from $2k_F$ (see section 2) a possible explanation [13] for this phenomenon is that the carriers remaining above the gap form due to their coupling to the lattice, charged solitons [42, 45] which should behave as classical charged particles.

The possibility of translating the CDW rigidly is a unique feature of the Peierls state, compared to an ordinary semiconductor. As discussed in section 2, pinning in a coupled-chain system leads to an infrared-active mode at low but finite frequency. Initial studies [46, 47] could not unambiguously identify such a mode, however, recent far-infrared experiments [48] have shown a sharp mode at $\omega = 40$ cm$^{-1}$ at 25 K. This feature weakens with increasing temperature and appears at zero frequency above the $T_H$ phase transition, as expected for the CDW phase mode. From a sum rule the effective mass of the phase mode is determined as
$M = 60m = 20m^*$, *m* and *m*\* being the free electron and the TTF-TCNQ band mass, respectively.

The effect of defects on the phase transitions has been investigated both by irradiation of the samples [49–51] and by isostructural doping [52]. Deuteron irradiation experiments by Chiang *et al*. [49] show a rapid decrease of both the $T_H$ and $T_L$ transition temperature and an increasing width of these transitions. At a concentration of some tenths of a percent the transitions are completely washed out [49, 50]. At the same time there is a considerable increase in low-temperature conductivity. The large effect of a small concentration of defects is due to the pinning of the phase of the CDW to the defect. This inhibits the build-up of long-range interchain correlations which are necessary for a three-dimensionally ordered state (see section 2). Zuppiroli and Bouffard [50] estimate that a single defect actually leads to an important distortion of the CDW over a region of approximately 50 Å diameter. The distortion leads to a finite density of states at the Fermi surface, i.e. the conductivity is locally nearly metallic, so that the conductivity increases with radiation dose. Higher irradiation doses lead to a decreasing conductivity with activated temperature dependence [51]. This is explained by an increasing number of breaks in the chains, so that electrons are localized within segments of chains and conduction involves activated hops between parallel segments [51]. Very similar behaviour is found in many organic solids [51]. Measurements of the low-temperature susceptibility [53] seem to agree with this picture: in irradiated samples with some percent defects (as well as intrinsically disordered organic conductors) one has $\chi \propto T^{-\alpha}$, $\alpha \approx 0.7$ instead of a Curie law. This power law is typical of one-dimensional disordered Heisenberg models [54], where in the present case the breaks in the chains lead to localization of the spins within a segment.

Another way of introducing controlled disorder in TTF-TCNQ is doping with the selenium analogue of TTF-TSF. The whole alloy series (TSF)$_x$(TTF)$_{1-x}$ TCNQ has been synthesized leading to the following behaviour [52]: low doping levels depress the $T_L$ transition sharply, and at $x = 0.03$ this transition disappears, confirming again that this transition is driven by the TTF chains. On the other hand the (TCNQ-driven) $T_H$ transition only decreases slightly up to doping levels larger than $x = 0.5$. However, at such doping the transition is very broad. As in the irradiation case, this is due to the pinning of the CDW and the corresponding loss of interchain phase coherence and long-range order.

Closing this subsection we summarize the principal low-temperature properties of TTF-TCNQ:

(i) At ambient pressure there are three phase transitions at $T_H = 54$ K, $T_I = 49$ K and $T_L = 38$ K. The $T_H$ transition is driven by the CDW (Peierls) instability of the TCNQ chains, the other two transitions are driven by the ordering of the TTF chains.

(ii) Between $T_H$ and $T_I$ there is a $2a \times 3.4b \times c$ superstructure, between $T_I$ and $T_L$, the period in the *a* direction varies continuously and below $T_L$ there is a $4a \times 3.4b \times c$ superstructure.

(iii) For $T_H > T > T_L$ TTF-TCNQ remains a fairly good conductor as the TTF chains remain nearly metallic, and only below $T_L$ is the system a Peierls-type semi-conductor with a band-gap of $E_g \approx 460$ K. There is a so far incompletely understood difference between the conductivity and susceptibility. The phase mode of the Peierls state has been observed at $\omega \approx 40$ cm$^{-1}$.
(iv) Under applied pressure the modulation periods both in the *a* and *b* directions vary. Most notably, at *P* ≈ 19 kbar the superstructure becomes commensurate in the *b* direction with periodicity *a* × 3*b* × *c*, and the transition temperature peaks at 74 K.

(v) The phase transitions are extremely sensitive to defects, because of the easy destruction of the interchain coherence necessary for three-dimensional long range order.

We finally remark that no sign of magnetic ordering has been observed in TTF-TCNQ: the NMR spin-lattice relaxation time $T_1$ increases markedly at the phase transitions [55], and no anisotropy of the magnetic susceptibility has been found [37]. This is in contrast to the (TMTSF)₂X family, to be considered in the following section where a sharp decrease of $T_1$ and an anisotropic susceptibility have provided evidence for an antiferromagnetic insulating state, in contrast to the CDW state of TTF-TCNQ.

### 4.1.2. Some members of the family: TSF-TCNQ, HMTTF-TCNQ and HMTSF-TCNQ

Replacing TTF by its selenium analogue TSeF gives rise to the organic metal TSF-TCNQ, isostructural to TTF-TCNQ. Measurements of the conductivity [56] and thermopower [23] show a single transition into a semiconducting state at $T_c = 29$ K. Below this transition an incommensurate $2a \times 3.15b \times c$ superstructure is observed [57], giving a value $\rho = 0.63$ for the charge transfer. High-temperature ($T = 50$ K) precursor scattering (section 4.2) shows that there are both longitudinal and $c^*$-transverse components of the displacements [58]. In contrast to the behaviour of TTF-TCNQ, no further phase changes occur down to the lowest observed temperatures, and the electronic gap is obtained consistently from resistivity [8], susceptibility [59] and infrared [60] measurements as $E = 250$ K, showing that both TSeF and TCNQ stacks order at the same temperature, $T_c = 29$ K.

Under hydrostatic pressure both the transition temperature and the low-temperature gap increase [44], and at $P \approx 6$ kbar a maximum of the $T_c$ versus $P$ curve (figure 52) indicates, in analogy to TTF-TCNQ, that a third-order longitudinal commensurability is achieved at that pressure [61]. The lower commensurability pressure with respect to TTF-TCNQ is easily explained by noting that the ambient pressure charge transfer of 0.63 is considerably nearer to the commensurate value $\frac{2}{3}$ than in TTF-TCNQ ($\rho_{TFF-TCNQ} = 0.59$). Although no structural investigations on TSF-TCNQ under pressure have been performed, an $a \times 3b \times c$ is also expected in this case.

Irradiation-induced defects have effects similar to those in TTF-TCNQ, most notably a rapid decrease of the low-temperature resistivity with some tenth of a percent defects [50]). Interesting effects are produced by doping TSF-TCNQ with small amounts of TTF. The single phase transition of the pure system splits into two considerably broadened transitions [62], and the upper transition temperature ($T_H$) increases rapidly with increasing doping [52] (figure 53). On the other hand, the lower transition, observed only up to some percent TTF doping level, decreases with increasing doping [62]. From the g-shift there is some evidence that the $T_H$-transition involves mainly the TCNQ stacks [62], and with increasing TTF doping this transition smoothly goes over into the (TCNQ-driven) transition of pure TTF-
<figure><img src="image_62.png" /><figcaption>Figure 52. Phase diagram of TSF-TCNQ suggesting the existence of a third order longitudinal commensurability around 6 kbar (after reference [6] of section 4).</figcaption></figure>

<figure><img src="image_63.png" /><figcaption>Figure 53. The dependence of the upper transition $T_H$ of TTF-TCNQ upon doping the donor stack with TSF molecules (after reference [52] of section 4).</figcaption></figure>

TCNQ [52]. These results show a high sensitivity of the coupling between TSF and TCNQ stacks to the defects.

The difference between TSF-TCNQ and TTF-TCNQ may be partly ascribed to the larger orbitals of the highest occupied level in selenium with respect to sulphur. This leads to larger molecular orbits in TSF and consequently to larger intermolecular overlap integrals. The increased longitudinal overlap leads to an increased bandwidth, i.e. more efficient screening of the electron-electron interactions. This may explain the equal susceptibility and resistivity-gaps, provided the origin of their difference in TTF-TCNQ is indeed the electron-electron interaction. However, as pointed out by Weger [63], the 4a-transverse period facilitates soliton creation in TTF-TCNQ compared to TSF-TCNQ (2a-period), so that the different behaviour of susceptibility and resistivity-gaps may also be explained by the soliton mechanism.

The increase of bandwidth of the TSF stacks with respect to TTF has also been invoked to explain the lower transition temperature of TSF-TCNQ [64]. In this
picture the increased bandwidth leads to a larger electron-phonon coupling [65], so that the intrinsic Peierls transition temperatures of the TSF and TCNQ stacks should be nearly comparable, contrary to the case of TTF-TCNQ. The competition between TSF and TCNQ instabilities leads then to a lower transition temperature. Doping of the TSF chains diminishes the competition leading to the observed sharp increase of the transition temperature. At the same time the defects in the TSF chains lead to a random field acting on the TCNQ chains, which explains the rather broad transition.

It has also been argued [66] that the increased *transverse* overlap should lead to a more effective coupling between TSF and TCNQ chains via an increased covalency gap (see section 3). One should note however that even in pure TSF-TCNQ the transverse conductivity is diffusive, indicating that the TSF and TCNQ hybridization is quite unimportant. It seems unlikely that the small increase of the single-particle scattering time by 1-2% doping will then have an important additional effect on the effective hybridization, which however would be necessary in this picture to explain the splitting of the transition.

Whereas in TTF and TSF-TCNQ the TTF (or TSF) and TCNQ stacks form sheets which alternate along the *a* direction, in the compound HMTSF-TCNQ the HMTSF and TCNQ stacks are ordered in a chessboard-like pattern [67]. Like TTF-TCNQ, this compound shows a resistivity minimum at about 40 K [68]. However, below this temperature the resistivity only increases to about twice its room temperature value, i.e. there is no semiconducting low-temperature state. Under hydrostatic pressure the temperature of the resistance minimum decreases, and was originally observed to vanish at pressures above 4 kbar [69]. More recent experiments on high-purity samples [70, 71] indicate that the resistivity minimum is associated with a phase transition at $T_c = 24$ K (ambient pressure) and that $T_c$ decreases under hydrostatic pressure, but never vanishes (figure 54). This 24 K resistivity anomaly arises from the formation of a three-dimensional superstructure [72] of incommensurate period $a \times 2.7b \times c$, which is the condensation of the precursor scattering observed at high temperature [57, 58]. The longitudinal modulation period corresponds to charge transfer $\rho = 0.74$.

<figure><img src="image_64.png" /><figcaption>Figure 54. Pressure dependence of the Peierls phase transition of HMTSF-TCNQ for samples behaving as those of figure 33 (*b*) (after MILJAK, M., unpublished work at Orsay).</figcaption></figure>
As observed in the first resistivity measurements, the state below the phase transition always has a resistivity of metallic size and never becomes semiconducting. Moreover, a large magnetoresistance [69, 71] and the observation of Shubnikov-de Haas oscillations [73] show that this state is adequately described as a three-dimensional semimetal. Semimetallic properties are also observed above $T_c$ (see section 3). An explanation for this behaviour has been given by Weger [74], who showed that in HMTSF-TCNQ the relatively large interchain transfer integrals should indeed lead to a semimetallic Fermi surface. Due to the important deviations from planarity of the Fermi surface the Peierls instability can only partly destroy it, so that the rest of the Fermi surface retains its semimetallic character even below the transition, as experimentally observed. In this picture a possible explanation for the decrease of $T_c$ with pressure is a pressure-induced increase of the interchain transfer integrals, which increases the non-planarity of the Fermi surface so that the Peierls instability can only affect a smaller portion of the Fermi surface and therefore occurs at lower temperature (see section 2). One should note that due to the existence of two chains with different bandstructures (section 3) nesting of Fermi surfaces does not occur, so that the Peierls instability is much more sensitive to changes in the interchain transfer integrals than for a single chain system with nesting [75]. The same remark also applies to HMTTF-TCNQ under pressure, which is discussed in the following.

The sulphur analogue of HMTSF-TCNQ is HMTTF-TCNQ, with a structure similar to that of HMTSF-TCNQ, possessing, especially, the chessboard-type order of HMTTF and TCNQ stacks. Resistivity measurements [76] indicate two phase transitions at $T_H = 48$ K and $T_L = 43$ K, similar to the $T_H$ and $T_L$ transitions of TTF-TCNQ and anomalies at these temperatures are also found in the specific heat [77]. X-ray scattering studies [78, 58] show that below $T_H$ a short-range ordered three-dimensional superstructure (period $2.38a \times 2.78b \times c$) appears and that only below $T_L$ is long-range three-dimensional order of the modulation achieved, together with a rapid increase of the modulation (figure 55). However, the modulation period is constant at all temperatures below $T_H$ and no symmetry changes occur at $T_L$. In contrast to the behaviour of TTF-TCNQ only a longitudinal component of the displacement is observed, and from the b-axis modulation period the charge transfer is $\rho = 0.72$.

An explanation for the discrepancy between transport and thermodynamic results on one side (indicating two phase transitions) and the observation of only one transition in the X-ray studies has been given by Bjelis and Barisic [79]. These authors show that in an incommensurate (in the *a* direction) structure like HMTTF-TCNQ the onset of the intrinsic instability of the second set of chains leads only to a non-singular increase of the modulation amplitude, although this increase may be very steep. Such a steep increase would lead to the observed anomalies at $T_I$ in HMTTF-TCNQ, whereas a phase transition (i.e. a symmetry change associated with thermodynamic singularities) occurs only at $T_H$.

The low-temperature single-electron gap, determined from resistivity, is $E_g = 480$ K [76]. Under hydrostatic pressure the gap decreases rapidly to $E_g = 20$ K at 8 kbar, and at higher pressures the temperature dependence of the resistivity of HMTTF-TCNQ is very similar to that of HMTSF-TCNQ at lower pressure [80]. Between ambient pressure and 19 kbar the $T_H$ transition falls from 48 to 24 K, and the $T_L$ anomaly already becomes unobservable at 3 kbar.
<figure><img src="image_65.png" /><figcaption>Figure 55. Temperature dependence of the satellite peak intensity showing clearly the two phase transitions of HMTSF-TCNQ at $T_H = 49$ K and $T_L = 42$ K. The inset shows the temperature dependence of the satellite half width at half maximum (HWHM) along *a*<sup>+</sup> revealing the short-range interchain correlation between $T_H$ and $T_L$ (after POUGET, J. P., MEGTERT, S., and COÑES, R., 1979, *Quasi One Dimensional Conductors*, edited by S. Barisic, A. Bjelis, J. R. Cooper, and B. Leontic, Lecture Notes in Physics 95 (Berlin: Springer-Verlag), p. 14.</figcaption></figure>

The vanishing of the gap at 8 kbar and the similarity at higher pressures to HMTSF-TCNQ indicate that pressure, via an increase of the interchain electron transfer integrals, transforms the low-temperature phase of HMTTF-TCNQ from semiconducting, i.e. similar to TTF-TCNQ, to a semimetallic phase as in HMTSF-TCNQ.

Summarizing this subsection we have shown that replacing the TTF molecule in TTF-TCNQ by very similar ones, the properties of the Peierls-ordered state of TTF-TCNQ can be drastically changed:

(i) In TSF-TCNQ the coupling between the two types of stacks becomes sufficiently strong so that only one phase transitions occur, for both chains simultaneously.

(ii) In HMTSF-TCNQ interchain electron transfer is important, so that the Fermi surface is only partly destroyed by the Peierls instability, and HMTSF-TCNQ remains a semimetal at low temperatures.

(iii) Applying pressure to HMTTF-TCNQ one may interpolate between behaviour similar to TTF-TCNQ (two anomalies, associated with the two types of stacks) and semimetallic low-temperature properties, as in HMTSF-TCNQ.

### 4.1.3. Landau theory of phase transition

In the following we discuss some of the conclusions on the phase transitions in TTF-TCNQ and its relatives which have been drawn considering the symmetry of the low-temperature ordered states. The question of the microscopic origin of the instability is addressed in section 4.3.
Assuming that the two different sets of organic chains in TTF-TCNQ have different ordering temperatures (in agreement with experiment, see section 4.1.1). Bak and Emery concluded from the Landau theory of phase transitions that there should be another temperature $T_I$ between $T_H$ and $T_L$ [81]. At that temperature the transverse ordering wavenumber $q_a$ should start to fall continuously from $\pi/a$ and lock discontinuously to $\pi/2a$ at $T_L$. This transition has in fact been observed [10, 33] with $T_I = 49$ K. The temperature dependence of the transverse ordering wave number is shown in figure 56.

The physical origin of the variation of $q_a$ in the Bak-Emery model may be understood on the basis of the interaction between two charge density waves on parallel conducting chains [82]. It is given in terms of their amplitude $\rho_i, \rho_j$ and their phases $\phi_i, \phi_j$ by

$$V_{ij} = t_{ij} \rho_i \rho_j \cos(\phi_i - \phi_j). \qquad (4.5)$$

If the interaction is due to Coulomb forces $t_{ij}$ has the form [83]

$$t_{ij} = e^2 K_0(2k_F d_{ij}), \qquad (4.6)$$

where $e$ is the electron charge, $d_{ij}$ the interchain distance and $K_0$ is a modified Bessel function [84], which decays very rapidly ($\sim x^{-1/2} \exp(-x)$) for large arguments. This interaction $V_{ij}$ encourages neighbouring CDWs to be in antiphase. Tunnelling of electrons between the chains may also lead to antiphase ordering [75], however this term is usually much smaller as it depends on the square of the tunnelling integral.

Based on the interaction (4.5) a semimicroscopic model for the variation of $q_a$ has been constructed [82]. First, experiments do not indicate any transverse modulation in the $c$ direction, so that one may consider a single sheet of alternating TTF and TCNQ chains in the $a$ direction. As shown in section 4.1.1 the 54 K transition involves mainly the TCNQ stacks. One may therefore assume that the amplitude $\rho_Q$ of the CDW on the TCNQ stacks is much bigger than the TTF amplitude $\rho_F$. The most important interactions are then between nearest-neighbour TTF and TCNQ stacks ($t_{QF}$) and between the next nearest neighbour TCNQ stacks ($t_{QQ}$), whereas the TTF-TTF interaction is negligible owing to the smallness of $\rho_F$. Assuming constant amplitudes $\rho_Q$ and $\rho_F$, a constant phase difference $\theta$ between TCNQ and TTF stacks, and a linear variation of the phase in the $a$ direction ($\phi_l - q_a r_l$), the phase-dependent part of the free energy per unit cell follows from (4.5) as

$$f(q_a) = t_{QQ} \rho_Q^2 \cos q_a a + 2t_{QF} \rho_Q \rho_F \cos \theta \cos(q_a a/2). \qquad (4.7)$$

The variation of $q_a$ with temperature follows immediately: at 54 K only the TCNQ stacks order ($\rho_F = 0$), and the minimum at $f(q_a)$ is given by $q_a = \pi/a$. At $T_I = 49$ K,

<figure><img src="image_66.png" /><figcaption>Figure 56. Temperature dependence of the transverse wave number of the distortion along $a$ in $a^*$ units for TTF-TCNQ (after reference [7] of section 4).</figcaption></figure>
the TTF stacks start to order with $\rho \sim (T_I - T)^{1/2}$. For small $\rho_F$, the minimum of (4.7) is given by [82]

$$ \cos \frac{q_a a}{2} = \frac{t_{QF} \rho_F}{2 t_{QQ} \rho_Q}. \qquad (4.8) $$

From this result we obtain $(q_a - \pi/a)^2 \sim T_I - T$, in agreement with the experimental result. Owing to the rapid decrease of $t_{ij}$ with distance, $t_{QF}$ is usually much bigger than $t_{QQ}$. Thus, if $\rho_F$ becomes comparable with $\rho_Q$ the minimum of $f(q_a)$ is given by $q_a = 0$. On the other hand, experiment shows that below the $T_L$ (38 K) transition, which is mainly induced by the TTF stacks, we have $q_a = \pi/2a$ instead at the expected vanishing of $q_a$. This has been explained by strong fourth-order umklapp terms, locking $q_a$ to $\pi/2a$ [85]. An alternative explanation of the $T_L$ transition has been given including, in addition to the interchain interactions along $a$, also the nearest and next-nearest neighbour interactions along $c$ and allowing for a finite $q_c$ [86]. Then it is possible to have two distinct local minima of the free energy as function of $q_a$. With varying temperature, the parameters in the free energy expansion may lead to a first-order transition between the two minima and only a small commensurability energy is required to lock $q_a$ to its value $\pi/2a$. From symmetry reasons this model shows that there should be a small but finite $q_c$ if $q_a$ deviates from $\pi/a$, and a drastic change of the relative phases of the two adjacent TCNQ chains in the $c$ direction at the $T_L$ transition [86, 87]. These effects have so far not been confirmed experimentally.

The continuous growth of $(q_a - \pi/a)$ below $T_I$ in the model discussed above is due to the fact that only the relative phases of adjacent stacks are changed, the amplitudes remaining constant (figure 57). Therefore the energy of a single chain remains unaltered and only the interchain energy varies continuously as the phase difference between the chains is varied. Another possibility of varying $q_a$, not considered above, is modulation of the amplitude of the CDW (figure 57). For $q_a = \pi/a$ the two possibilities are equivalent, as a phase difference of $\pi$ between adjacent chains represents the same physical situation as an alternation of $\rho_Q$ and $-\rho_Q$. For $q_a = \pi/a + \delta$ with finite $\delta$ however, in the case of a sinusoidal amplitude modulation, there are large parts of the crystal where the CDW amplitude deviates considerably from the (local) optimum value. Varying $\delta$ leads only to change in the wavelength of the modulation, however; the number of chains with CDW amplitude in a given small interval being independent of $\delta$ as long as $\delta \neq 0$. Only for $\delta = 0$ is every chain allowed to have its optimum value. Consequently, it has been shown [88] that the energy of the system varies discontinuously if $\delta \neq 0$ is realized by a sinusoidal amplitude modulation, whereas phase modulation gives a continuous variation and is thus energetically favoured for small $\delta$. From a similar consideration it follows that a finite lock-in energy to $q_a = \pi/2a$ at $T_L$ is only possible if the structure below $T_L$ is amplitude modulated [85].

In the above argument on the lock-in energy at commensurability, only the fundamental Fourier component of the order parameter has been considered. If higher harmonics are taken into account the amplitude modulation is no longer purely sinusoidal, but there are large regions of commensurate order, separated by small zones where the phase of the modulation varies rapidly. These small zones are usually termed phase kinks or discommensurations. In the simplest case the effect of the higher harmonics can be described by a time-dependent Sine-Gordon equation [81, 89] which has a second-order commensurate-incommensurate transition, in
<figure><img src="image_67.png" /><figcaption>Figure 57. The two possibilities of transverse modulation: (a) phase modulation, (b) amplitude modulation (after ABRAHAMS, A., SOLYOM, J., and WOYNAROVICH, F., 1977, *Phys. Rev. B*, **16**, 5238).</figcaption></figure>

contrast to the experimental results on the $T_L$ transition. However, if a coupling between the order parameter and a strain field is taken into account, the transition may become first order [89, 90] in agreement with the observed $T_L$ transition. At the transition at $T_I = 49$ K, the incommensurate phase can be changed continuously into the commensurate state by phase modulation (figure 57(a)), so that this transition is of second order. This result is unaffected by fluctuation effects, as shown by Bak [9].

From the above discussion, the following picture of the variation at $q_a$ in TTF-TCNQ at ambient pressure emerges: at $T_H$ only the TCNQ stacks order, the repulsion between maxima at neighbouring CDWs leads to $q_a = \pi/a$. At $T_I$ the TTF stacks start to order and the competition between the $t_{QQ}$ and $t_{QF}$ terms in (4.8) leads to a temperature variation of $q_a$ [81, 82]. The first-order lock-in transition at $T_L$ is either entirely due to fourth-order Umklapp terms [85] or to a change of the respective importance of the various interchain interactions and there is only a weak lock-in energy to obtain an exactly commensurate $q_a$ [86]. In the first case coupling to a strain co-ordinate must be invoked to account for the first-order character of the transition [89, 90]. This picture has some support in experimental results, since these show an increase in slope of $\rho(T)$ at $T \approx T_I$ (figure 46) which may correspond to the onset of order on the TTF stacks. Another argument in favour of this picture comes from X-ray [29] and neutron [30] investigations, which show that at $T_H$ a transverse
mode condenses, whereas at $T_I$ there is a condensation of a second, longitudinally polarized mode. These two modes might be interpreted as corresponding to order on TCNQ and TTF stacks, respectively.

The relatively simple picture of the variation of the transverse modulation in TTF-TCNQ based on (4.8) is questioned by newer results. First, a Landau-type analysis of the phase transitions in TTF-TCNQ, including the exact symmetry [88], shows that the different types of stacks are not completely decoupled and that a finite CDW amplitude on the type of stack should always induce a (small) amplitude on the other type of stack. This effect is essentially due to the tilt of the molecules, which leads to different interactions on a CDW on one stack with its right and left neighbouring stacks. This effect has been observed experimentally in magnetic measurements, showing that below 54 K in addition to the dominant gap on the TCNQ stacks there develops also a small TTF gap [25]. More drastic effects are observed in elastic neutron scattering under pressure [16–18, 34] (figure 50), showing that under pressure the $T_I$ transition splits up into two transitions; the higher transition ($T_{I1}$) corresponds to a condensation of a longitudinally polarized mode, as observed in measurements at ambient pressure [29, 30]. $T_{I1}$ grows with increasing pressure and the transverse modulation is $q_a = \pi/a$ above as well as below $T_{I1}$. The lower of the two intermediate transitions ($T_{I2}$) corresponds to the beginning of a variation of the transverse modulation, $T_{I2}$ decreasing with increasing pressure. From a symmetry analysis, together with results of transport measurements described in section 4.1.1, Megtert et al. [92] conclude that the $T_H$ transition is due to the condensation of a transverse acoustic mode involving mainly the TCNQ stacks, whereas the $T_{I1}$ transition comes from a longitudinal optical mode involving translations of TTF molecules. A detailed analysis of the X-ray structure factor of TSF-TCNQ shows that it may be explained entirely by rigid molecular translations [93] and that any significant contribution from molecular rotations can be excluded [94]. By analogy it seems unlikely that rotations [95, 96] play a major role in the ordering of TTF-TCNQ.

From the above points it is clear that the onset of variation of $q_a$ at $T_{I2}$ is not related to the appearance of the longitudinally polarized deformation, but must be due to some other mechanism. One possibility would be the condensation at $T_{I2}$ of one of those modes investigated by Megtert et al. [92] which allows the existence of a Lifschitz invariant and could therefore lead to a variation of $q_q$, as in the Bak–Emery model. One might also consider condensation of the $4k_F$ anomaly, for which there are some indications [29], though a theory for the effect on the lattice modulations is lacking.

Another problem arises in the simple theory based on (4.7) in relation to the constant transverse modulation ($q_a = \pi/a$) in TTF-TCNQ at 5 kbar [16, 17] and in TSF-TCNQ at ambient pressure [57]. Well below the phase transition temperature there is a well-developed gap on both types of stacks in the two cases (see sections 2.1 and 4.1.2). Therefore the parameters $\rho_F$ in (4.7) are of comparable order of magnitude. Usually it may be assumed that the nearest-neighbour interaction $t_{QF}$ is much bigger than the next-nearest-neighbour interaction $t_{QQ}$ (calculations based on (4.6) give $t_{QQ}/t_{QF} = 1/60$ [82]), and therefore the minimum of (4.8) is given by $q_a = 0$ and not by the experimentally observed $q_a = \pi/a$. However, it should be noted that $t_{QQ}/t_{QF} = 1/60$ has been derived [82] neglecting the spatial extension of the molecules. Taking their extension into account the ratio is considerably increased [94]. Further it has been remarked that fourth-order interchain coupling terms may
lead to an enhanced effective coupling between like chains [64]. These effects may
explain the observation of a 2a periodicity in TTF-TCNQ and TSF-TCNQ under
various conditions.

At pressures around 19 kbar in TTF-TCNQ and around 6 kbar in TSF-TCNQ the charge transfer becomes $\rho = 2/3$ [11, 12, 18, 61]. In this case the wavelength of the lattice modulation *along* the chain directions is three times the underlying lattice constant, i.e. third-order commensurability is achieved. In this case third-order terms are allowed to appear in a Landau energy functional, provided the *transverse* wavenumber $q_a$ fulfils $3q_a = G_a$, where $G_a$ is the transverse component, i.e. one should have $q_a = 0$ or $q_a = 2\pi/3a$. Experimentally $q_a = 0$ is found [18], so that the third-order terms explain the first order transition and the increased transition temperature [14, 15]. These facts show that the energy gain due to commensurability is larger than the loss of interchain coupling energy due to $q_a = 0$ (cf. equation (4.7)), i.e. the third-order terms in the Landau expansion are quite strong.

As discussed in section 2, the deviation of the charge transfer from the exactly
commensurate value $\frac{2}{3}$ has to be larger than some critical value in order to realize an
incommensurate phase. The domain structure of the incommensurate phase near the
commensurate-incommensurate boundary is due to the possibility of an energy gain
from the third-order terms and therefore is only possible for $q_a = 0$. This probably
explains why even at 12 kbar, where TTF-TCNQ is incommensurate, $q_a = 0$ is
observed [34]. The phase diagram of TTF-TCNQ around 19 kbar (figure 45),
especially the finite width of the commensurate state and the maximum of the
transition temperature, is in good overall agreement with the simple Landau model.
However, a detailed investigation of the phase diagram, in particular near the
commensurate-incommensurate normal triple point, where interesting thermo-
dynamic effects are expected [15, 97], is still lacking.

For HMTTF-TCNQ and HMTSF-TCNQ, due to the chessboard arrangement of their molecular stacks, the Landau energy expansion takes a different form, and depending on the coefficients many different types of ordering may occur [79]. As already mentioned in section 4.1.2 the incommensurate transverse modulation in HMTTF-TCNQ [78] implies that both chains order at the same temperature, though the amplitude on one chain may be initially very small [79]. On the other hand, in HMTSF-TCNQ there is no transverse modulation ($q_q = q_c = 0$) [57, 58, 72]. This corresponds to case (i) of Bjelis and Barisic [79], again implying order on both sets of chains.

## 4.2. *The precursor effects*

4.2.1. *X-ray diffuse scattering*

The interest of one-dimensional conductors lies mainly in their providing a new
conductivity mechanism to take place at moderately high temperatures [98]. This
new channel of conduction involves the coherent sliding of all the electrons building
up an incommensurate CDW in a given chain. It is a collective process in as much as
it can give rise to conductivity even in the presence of an energy gap in the single-
particle spectrum. In the three-dimensional ordered state, CDWs lose their one-
dimensional character because of the interchain electrostatic interactions. Hence, the
free sliding of the CDWs becomes pinned by vanishingly small amounts of
impurities. Above the phase transition, however, when the transverse coherence
length remains smaller than the interchain distance, a one-dimensional fluctuating
CDW of finite coherence length can exist in a broad temperature range as demonstrated by X-ray diffuse scattering [28].

In TFF-TCNQ fluctuations related to the CDW ordering at 54 K have been observed up to 200 K [29, 32]. Between 54 and 60 K the fluctuations are three-dimensional, whereas above 60 K they are one-dimensional (figure 58) with a correlation length decreasing monotonously from 100 Å at 60 K to 5 Å at 150 K. Similarly, a wide temperature range of one-dimensional fluctuations, extending to even higher temperatures, has also been found in HMTSF- and HMTTF-TCNQ [57, 58, 99] (figure 59). Somewhat different behaviour is found in TSF-TCNQ [57,

<figure><img src="image_68.png" /></figure>

Figure 58. X-ray pattern of TTF-TCNQ at 60 K, a few degrees above the upper phase transition, showing 2k~F~ and 4k~F~ diffuse lines at 0.295b* and 0.59b* respectively from the layers of main Bragg reflections (see the joined schematic representation). The sample is oriented with the b* and c* directions in the equatorial plane, the angle between the incident X-ray beam and b* being 124°(after reference [31] of section 4).
<figure><img src="image_69.png" /><figcaption>Figure 59. X-ray pattern of HMTSF-TCNQ at room temperature, showing well-defined $2k_F$ diffuse lines (white arrows) corresponding to one-dimensional structural fluctuations. The variation of the diffuse intensity within each line is due to the structure factor of scattering molecules. The sample is oriented with the *c*-direction (chain direction) in the equatorial phase and the *b*-direction vertical (from POUGET, J. P. (unpublished results)).</figcaption></figure>

58, 100] ($T_c$ = 29 K), where fluctuations on neighbouring chains have short-range two-dimensional correlations up to about 200 K (figure 60). All these studies show a smooth behaviour of these substances in the temperature range of one-dimensional (or short-range two-dimensional in the case of TSF-TCNQ) fluctuations. This is in

<figure><img src="image_70.png" /><figcaption>Figure 60. Temperature dependence of the half width at half maximum (HWHM) of $2k_F$ diffuse sheets of TSF-TCNQ. After subtraction of the resolution, this quantity gives the inverse correlation length of the $2k_F$ CDW in the chain direction. From this data, one gets for example $\xi_\parallel$ (200 K) ~ 20 Å and $\xi_\parallel$ (50 K) ~ 100 Å (after reference [94] of section 4).</figcaption></figure>
agreement with the wide one-dimensional critical region obtained from the theory of instabilities in one-dimensional metals. Especially, no anomalies indicating a transition from the amplitude to the phase fluctuation region have been found. Thus, amplitude fluctuations are important in the whole temperature region of one-dimensional fluctuations.

The giant Kohn anomaly of the phonon spectrum at $2k_F$ has also been detected by neutron scattering of deuterated TTF-TCNQ crystals [101]. A decrease of the phonon energy from the room temperature value of 5.0-4.4 meV at 60 K is clearly observed below 120 K on the TA phonon branch polarized along the $c^*$ direction. Inelastic neutron studies have failed to show significant softening at $2k_F$ in the LA($b^*$) phonon mode. Furthermore, the $4k_F$ anomaly detected by X-ray scattering [31] and polarized along $b^*$ has not yet been revealed by neutron inelastic scattering [30].

In the region of three-dimensional fluctuations in TTF-TCNQ ($54 \text{ K} < T < 60 \text{ K}$) fluctuation effects have been found in specific heat [102, 103], resistivity [104, 105] and magnetic susceptibility [106]. From these results it has been concluded that the critical region of the three-dimensional fluctuations is quite narrow ($\Delta t_{3D} \approx 0.05-0.1$), in agreement with theory [107].

### 4.2.2. Fluctuating Fröhlich conductivity

Bardeen and coworkers [108, 109] suggested that long lifetime fluctuating CDWs of the Fröhlich type could contribute to d.c. longitudinal conductivity. In this process a CDW of finite coherence length is created by thermal fluctuations. The rigid translation of this CDW gives a positive contribution to the conductivity. The theoretical discussion of CDW fluctuation conductivity has been given in section 2, we shall summarize here the salient results for the sake of comparison with the experimental data.

In an incommensurate system consisting only of one type of chain, the fluctuation induced conductivity is [109] (see equation (2.82)):

$$ \sigma_F(\omega) \propto \langle |\Delta|^2 \rangle \frac{\gamma}{\gamma^2 + \omega^2}, \quad (4.9) $$

where $\gamma^{-1}$ is the lifetime of fluctuations due to decay into electron-hole pairs, and the missing proportionality constant depends only weakly on temperature. In a strictly one-dimensional model, a lowest order calculation gives a square-root divergence at the mean-field transition temperature, which however disappears if higher order effects are included (see figure 9). In a quasi-one-dimensional case (including interchain coupling), $\langle |\Delta|^2 \rangle$ still increases rapidly on approaching the transition temperature (as shown by X-ray scattering, section 4.2.1), but tends to a finite value at $T_c$. In lowest order, $\sigma_F(0)$ just compensates the decrease of single-particle d.c. conduction by the increased scattering of single electrons from the fluctuations, but at higher order an increase of the total conductivity seems possible, as discussed in section 2.4.4. One should also note that the Drude-like frequency dependence of the fluctuation conductivity is governed by the fluctuation lifetime $\gamma^{-1}$, which in general will be considerably longer than the single-electron lifetime (equation (2.22)).

The various mechanisms suppressing the Fröhlich conductivity in a long-range-ordered Peierls state can also suppress the fluctuation conductivity:
(i) if the longitudinal correlation length becomes comparable to the mean distance between impurities, the fluctuating CDWs become pinned;

(ii) if interchain correlations between CDWs of opposite charge become important (i.e. in the region of three-dimensional fluctuations), the Coulomb attraction suppresses the motion of fluctuating CDWs in opposite directions;

(iii) experimentally important, commensurability also strongly diminishes fluctuating CDW conductivity (equation (2.85), figure 14).

Recently, the existence of a significant contribution to the conduction of collective origin has been verified because of the possibility of achieving commensurability under pressure [12, 110]. As TTF-TCNQ is driven by high pressure towards commensurability there occurs a significant drop of conductivity (figure 61). However, the loss of conductivity becomes smaller at high temperature and practically unobservable above 250 K. High-pressure optical reflectance studies [19] allow us to discard the possibility of drastic (non-monotonous) changes in the band structure under pressure especially at commensurability. Consequently, the data shown in figure 61 can be explained in the following two ways.

(i) Pinning of the rigid translations of fluctuating CDWs by commensurability effects.

(ii) Increase of the single particle scattering mechanism and/or a decrease in the density of states at the Fermi level in the commensurability domain.

If only single-particle scattering is considered, the resistivity is independent of the phase of the CDW fluctuations but eventually depends on the amplitude. Measurements of the longitudinal conductivity alone and results such as those of figure 61 are unable to distinguish between the two previous pictures. Fortunately, the

<figure><img src="image_71.png" /><figcaption>Figure 61. Pressure dependence of the longitudinal conductivity of TTF-TCNQ at various temperatures. The ratio $\sigma(60\text{ K})/\sigma(300\text{ K})$ is 25 under ambient pressure (after reference [12] of section 4).</figcaption></figure>
transverse conductivity, whenever it is diffusive, is sensitive to both the density of states at the Fermi level and the smearing of the one-dimensional Fermi surface by the single particle scattering time, $\tau_{\parallel}^{\text{sp}}$ [111, 112]. The diffuse transverse conductivity is given by the relation:

$$ \sigma_{\perp} \sim N(E_F)\tau_{\perp}^{-1}, \qquad (4.10) $$

where $1/\tau_{\perp}$ is the interaction hopping frequency due to the existence of a finite (but small) interchain transfer integral $t_{\perp}$. In a quasi-one-dimensional conductor the hopping frequency can be derived with the one-dimensional Golden Rule [111]:

$$ 1/\tau_{\perp} = \frac{2\pi}{\hbar} |t_{\perp}|^2 \tau_{\parallel}^{\text{sp}}. \qquad (4.11) $$

From (4.10) and (4.11), $\sigma_{\perp}$ becomes

$$ \sigma_{\perp} \sim N(E_F) |t_{\perp}|^2 \tau_{\parallel}^{\text{sp}}. \qquad (4.12) $$

The relation (4.11) is true in the $\tau_{\perp} > \tau_{\parallel}^{\text{sp}}$ limit when the transverse conductivity is diffusive. This restriction does not constitute a problem in TTF-TCNQ for which experiments lead to $\tau_{\perp}/\tau_{\parallel}^{\text{sp}} \approx 10^3$ at room temperature [111]. Figure 62 displaying the pressure dependence of the transverse conductivity shows the non-existence of a significant dip in the commensurability regime. Therefore figures 61 and 62 and equation (4.12) lead to the conclusion that neither $N(E_F)$ not $\tau_{\parallel}^{\text{sp}}$ are significantly affected by commensurability. The loss of conductivity at commensurability (figure

<figure><img src="image_72.png" /><figcaption>Figure 62. Pressure dependence of the transverse conductivity of TTF-TCNQ at various temperatures (after JÉROME, D., and SCHULZ, H. J., 1982, *Extended Linear Chain Compounds*, Vol. 2, edited by J. S. Miller (New York: Plenum Press), p. 159).</figcaption></figure>
61) is thus a one-dimensional effect. It can be attributed to the pinning by a commensurability potential of a fluctuating sliding conductivity, as discussed above.

Quantitative values of the parameters of our theory for experimentally interesting systems such as TTF-TCNQ are difficult to obtain due to the lack of experimental data at high pressure. An order of magnitude estimate can be obtained by noting that a term like (4.14) leads to a first-order phase transition with a higher $T_c^1$ than in the incommensurate case [15]:

$$ \frac{T_c^1 - T_c}{T_c} = \frac{V^2}{a'b}. \qquad (4.13) $$

A conservative estimate would be to set $T_c^1 - T_c$ equal to the observed one degree hysteresis and to attribute the rest of the increase of the transition temperature near $T_c$ to the reduced effect of fluctuations in a commensurate system. One then obtains $V^2/(a'b) \approx 0.02$, with (2.22) and (2.58), making (as in figure 14) the approximation $\tilde{\omega}_0^2(2k_F) \approx \tilde{\omega}_0'^2(T - T_c)/T$, and using $v_F \approx 0.7$ eV Å, $\xi(2T_c) \approx 30$ Å (obtained from X-ray scattering [29]) and $T_c = 65$ K one finds:

$$ \left. \frac{72\omega_{2k_F g^6 V^2 T}}{\tilde{\omega}_0^6(2k_F)\xi(T)} \right|_{T=2T_c} = 0.25. \qquad (4.14) $$

This value is sufficiently small for the single-particle scattering to be essentially unaffected by commensurability [113]. In agreement with this, measurements of the transverse conductivity (figure 62), which is proportional to the longitudinal single-particle conductivity, as discussed above, do not show any significant features near commensurability. On the other hand, neutron scattering [101] gives $\tilde{\omega}_0(2k_F)|_{\approx 2T_c} \approx 5$ meV, and infrared measurements (see below) result in $\gamma \approx 0.5$ meV, so that from $(2.85(b))$ one finds:

$$ \eta(2T_c) \approx 0.7, \qquad (4.15) $$

which leads to a strong reduction of the fluctuating CDW conductivity (figure 57). Thus we can conclude that commensurability pinning of fluctuation conductivity provides a reasonable explanation of the drop of conductivity observed in TTF-TCNQ (and TSF-TCNQ, see below) at commensurability.

An order of magnitude estimate of the contribution of the Fröhlich mode of the fluctuations to the observed $\sigma_\parallel$ can be made from the data on figure 61. For example, at 80 K, the difference of conductivity at 19 kbar between the interpolated value from the 10 and 22 kbar data) and the measured quantity is about $4500 (\Omega \text{ cm})^{-1}$. According to the previous interpretation of the pinning of the Fröhlich mode by a commensurability potential, that contribution represents a lower limit value to the fluctuating conduction since a significant part but not the whole sliding conduction is pinned by the commensuration at high temperature (see figure 14). The derivation of the Fröhlich mode contribution, taking advantage of the commensurability pinning, is in very good agreement with other measurements, using different properties (namely impurity pinning or far-infrared reflectance studies, see below). Let us now comment briefly on recent pressure data obtained on TSF-TCNQ. At atmospheric pressure TSF-TCNQ is closer to third-order commensurability than TTF-TCNQ since the charge transfer amounts to 0.63 [57] instead of 0.59. Consequently, commensurability is expected in the vicinity of 7 kbar assuming similar compressibility for both sulphur and selenium compounds [114]. In some experiments a peaking of very small amplitude is observed near 6 kbar in the pressure dependence
of $T_c$ [61] (figure 52). The conclusive evidence for commensurability comes from a loss of conductivity occurring around 6.25 kbar in the constant temperature pressure dependence of $\sigma_\parallel$ [61] (figure 63) in a way somewhat similar to TTF-TCNQ at 19 kbar. However the loss of conductivity is clearly observed only in a limited range of temperatures, namely, in the vicinity of 130 K. This fact can presumably be related to the following features.

(i) The existence of two-dimensional couplings which have been observed between CDWs of neighbouring chains by X-ray diffuse scattering up to 200 K or so [57, 58, 100]. According to the above discussion the fluctuating conductivity is thus destroyed by interchain coupling below, say, 100 K.

(ii) At elevated temperatures (around 300 K) the CDW coherence length becomes too small to give rise to substantial fluctuating conductivity.

Moreover, the relative amplitude of the loss of conductivity for TSF-TCNQ, 10% at 125 K, agrees fairly well with the data for TTF-TCNQ at the same relative temperature, namely 210 K, see figure 61. It is also interesting to notice that the

<figure><img src="image_73.png" /></figure>

Figure 63. Pressure dependence of the TSF-TCNQ longitudinal conductivity at various temperatures for several samples showing the weak depression of conductivity associated with the 2/3 commensurability around 6 kbar. The bottom curves display the pressure dependence of the transition temperature for the respective samples (after reference [6] of section 4).
conduction dip at commensurability is independent of the presence or absence of $4k_F$ scattering (no $4k_F$ scattering in TSF-TCNQ [57]). This rules out the explanation of the dip in terms of $4k_F$ scattering increasing the single particle rate, $1/\tau_\parallel$, since when $2k_F = b^*/3$, $2k_F$ and $4k_F$ scattering differ only by a reciprocal lattice vector [115].

Additional evidence for an important contribution of CDW fluctuations to the longitudinal conductivity comes from experiments on irradiated samples [116] showing a suppression of the 19 kbar conduction dip (figure 64). Defects, as introduced by irradiation should very effectively pin the free motion of CDW fluctuations, thus suppressing the CDW-fluctuation conductivity even in the incommensurate situation. Correspondingly, no significant drop of conductivity has been observed in the irradiated samples near commensurability. Were the drop of conductivity due to changes in the single-particle scattering, one would expect (from Mathiessen's rule) a similar drop to appear in the irradiated samples. This is not observed. We thus conclude that an important part of the metallic conductivity of (incommensurate, unirradiated) TTF-TCNQ is due to the CDW-fluctuation mechanism. Of course, near the transition temperature, where the fluctuations become more and more three-dimensional this mechanism is very easily suppressed by impurity pinning. Therefore in the region of three-dimensional fluctuations *just above* 54 K, the main effect of the fluctuations is *single-particle* scattering, leading to a *decrease* of conductivity [105]. As the data of figure 64 correspond to a concentration of about $2 \times 10^{-3}$ molecular defects, i.e. 1 molecule out of 500 affected by a defect on a given chain and since the coherence length derived from X-ray scattering data is not larger than 100 molecular spacings in the same temperature domain, one should not expect any dramatic pinning of the sliding CDWs. Considering the experimental results we are led to conclude that there is a three-dimensional effect of the damaged

<figure><img src="image_74.png" /></figure>

Figure 64. Pressure dependence of the longitudinal conductivity at constant temperature for irradiated TTF-TCNQ (continuous line). The dashed lines indicate the behaviour of pristine samples (after reference [116] of section 4).
molecules: a defect or an impurity in a quasi-one-dimensional electron gas induces a spatial oscillation of the charge density around it (of wavelength $π/k_F$). As a consequence of Coulomb interaction, coupling to a long-range distortion is also induced on neighbouring chains. The CDW phase is pinned to each of these local CDWs and when the interdefect distance and the CDW coherence length become comparable we may expect a substantial reduction of the conductivity related to the sliding of fluctuating CDWs. To explain the experimental data displayed in figure 64 we have to assume that the mean distance between damaged molecules is about 50 Å at 100 K. This value provides an efficient pinning of the charge density waves given an experimental coherence length of $\xi_\parallel$ (100 K) $\approx$ 60 Å (see section 4.1.1). Figure 64 provides a contribution $\sigma_F \approx 5000$ (Ω cm)$^{-1}$ at 100 K, in fair agreement with the decomposition derived from commensurability.

Detailed far-infrared measurements on TTF-TCNQ at low temperatures [117, 48] provide an additional estimate of the fluctuating sliding mode contribution to the conduction. Figure 65 shows the b-axis frequency-dependent conductivity of TTF-TCNQ derived from the reflectance data by a Kramers-Kronig analysis following the procedure described by Tanner *et al.* [48]. The strong peak of the conductivity around $40 \text{ cm}^{-1}$ at 25 K has been taken as the signature of the phase-mode of the pinned CDW in the Peierls state. At 60 K, the peak of conductivity has shifted to zero frequency. This phenomenon can be considered as the existence of a collective mode at zero frequency: the collective mode giving rise to the sliding of finite lifetime CDWs.

The lifetime of the collective mode may be estimated from the width of the zero-frequency peak in figure 65 as $1/\tau_c = \gamma = 4 \text{ cm}^{-1}$, or $\tau_c = 1.6 \times 10^{-12} \text{ s}$ [117]. This value is in agreement with the dominance of the collective conductivity at this temperature [118]. Moreover, this fluctuation lifetime is in between the sorter single-

<figure><img src="image_75.png" /><figcaption>Figure 65. Frequency dependent b-axis conductivity of TTF-TCNQ at three temperatures 25, 60 and 160 K (after reference [117] of section 4).</figcaption></figure>
particle lifetime $\tau_{\parallel}^{\text{sp}} \approx 10^{-14}$ s and the larger lifetime of an electron on a given chain, $\tau_{\perp} \approx 8 \times 10^{-12}$ s, as derived from NMR [111].

The existence of a conductivity mechanism of fluctuating Fröhlich origin does not prevent the more traditional single-particle contribution from being an important one. Both conduction channels add up to form the total conductivity. A particularly important task for the understanding of the electronic properties of TTF-TCNQ will be the quantitative decomposition of the conductivity into two components, namely $\sigma_{\parallel}^{\text{F}}$ and $\sigma_{\parallel}^{\text{sp}}$. A salient result of the previous section is that both conduction channels add and can therefore be written:

$$ \sigma_{\parallel} = \sigma_{\parallel}^{\text{F}} + \sigma_{\parallel}^{\text{sp}}. \qquad (4.16) $$

We propose a derivation which allows an estimation of both components of (4.16) at given pressures and temperatures. To accomplish this decomposition two assumptions are required. The first assumption is that $\sigma_{\parallel}^{\text{F}}$ (300 K) is negligibly small compared with $\sigma_{\parallel}^{\text{sp}}$ (300 K). This assumption is justified by the monotonous increase of $\sigma_{\parallel}$ through the commensurability regime around room temperature. The accuracy of the data of figure 61 shows that $\sigma_{\parallel}^{\text{F}}$ (300 K) is smaller than, say, $\sigma_{\parallel}^{\text{sp}}/10$ (19 kbar, 300 K) thus leading to $\sigma_{\parallel}^{\text{F}}$ (300 K) $\ll 250$ (Ω cm)$^{-1}$. The second assumption of the derivation is based on the use of (4.12): in the one-dimensional regime ($\tau_{\perp}/\tau_{\parallel}^{\text{sp}} \gg 1$), $\sigma_{\parallel}^{\text{sp}}$ and $\sigma_{\perp}$ exhibit the same temperature dependence according to (4.12), i.e. that of $\sigma_{\parallel}^{\text{sp}}$, since the band structure is only weakly temperature and pressure dependent. The model decomposition of the conductivity [110] is illustrated by the experimental situations of figure 66. At every pressure we have plotted the temperature dependence of $\sigma_{\parallel}$, with an initial value $\sigma_{\parallel} = 400$ (Ω cm)$^{-1}$ under ambient conditions. We have also displayed the temperature dependence of the transverse conductivity along the a direction normalized to the value of $\sigma_{\parallel}$ at room temperature.

(i) In the incommensurate regime, $P = 1$ bar and 12 kbar in figures 66 (a) and (b), respectively, the temperature dependence of $\sigma_{\parallel}$ and $\sigma_{\perp}$ appear very different. The ratio $\sigma_{\parallel}(T)/\sigma_{\parallel}$ (300 K) peaks at 20 and 6 at low temperature for 1 bar and 12 kbar, respectively, whereas the peaking of $\sigma_{\perp}(T)/\sigma_{\perp}$ (300 K) does not exceed a factor 2 or 3. Following (4.12), $\sigma_{\parallel}^{\text{sp}}$ in (4.17) is derived from $\sigma_{\perp}$ and the component of $\sigma_{\parallel}^{\text{F}}$ in (4.16) can be derived from the difference at every temperature between the $\sigma_{\parallel}$ and $\sigma_{\perp}$ curves. This procedure leads back to $\sigma_{\parallel}^{\text{F}}$ (300 K) = 0, as expected, since this was our initial assumption.

(ii) In a commensurate (or nearly commensurate) regime a similar decomposition of $\sigma_{\parallel}$ (figure 66 (c) and (d)) leads to a much smaller contribution $\sigma_{\parallel}^{\text{F}}$. Following the discussion of section 2 a finite value of $\sigma_{\parallel}^{\text{sp}}$ is expected between $T_c$ and $2T_c$ even near commensurability provided the pinning potential is not too strong.

(iii) In the pressure regime above commensurability, figure 66 (e), $P = 24$ kbar, some fluctuating contribution to the conductivity is recovered, as in the situation below commensurability (i). The above decomposition leads to a fluctuating collective contribution which represents quite a significant part of the total conductivity in incommensurate TTF-TCNQ. For a sample of conductivity peak ratio (CPR) at 59 K of 17.5 as shown in figure 66 (a), the fluctuating contribution reaches $\approx 5500$ (Ω cm)$^{-1}$ at 59 K, namely 80% of total conductivity. Moreover it appears from figure 66 (a) that $\sigma_{\parallel}^{\text{F}}$ and $\sigma_{\parallel}^{\text{sp}}$ are
<figure><img src="image_76.png" /></figure>

P = 1 bar

(a)

<figure><img src="image_77.png" /></figure>

P = 12 kbar

(b)

<figure><img src="image_78.png" /></figure>

P = 17 kbar

(c)

<figure><img src="image_79.png" /></figure>

P = 20 kbar

(d)

<figure><img src="image_80.png" /></figure>

P = 24 kbar

(e)

<figure><img src="image_81.png" /></figure>

P = 28 kbar

(f)
about equal in amplitude when $T = 150–200$ K. This is also the temperature regime where $2k_F$ reflections of one-dimensional character are first observed when cooling [29, 30]. Our analysis brings a further confirmation of the $2k_F$ CDWs ability to carry a current. The sensitivity to impurities of the fluctuating conductivity may be responsible for different CPR in different samples [116].

A major experimental difference between single-particle and collective contributions to the conductivity is the behaviour under pressure. On the one hand, the single-particle contribution increases at a rate $\partial \ln \sigma_{\parallel}^{\text{SP}}/\partial P \approx 30\% \text{ kbar}^{-1}$ around room temperature [119, 6]. On the other hand, $\sigma_{\parallel}^{\text{F}}$ is only slightly enhanced by high pressure, increasing at a rate of $\approx 2\% \text{ kbar}^{-1}$ or so at 75 K. Such a moderate volume dependence is also encountered in the behaviour of the phase transition $T_H$ under pressure (besides the commensurability peaking). It brings an additional confirmation of the tight relation between the structural properties and the fluctuating CDW contribution to the conductivity. As the effect of the large volume dependence of the conductivity on its intrinsic $T$ dependence it has been claimed that, because of the change of volume provided by the thermal expansion of the lattice, it is important to distinguish between $T$-dependence under constant volume and that under constant pressure [119, 120]. At this stage we wish to emphasize that the constant volume reduction of the conductivity must operate only on the contribution to the total conductivity which is strongly volume dependent: the single-particle contribution. After the removal of the $\sigma_{\parallel}^{\text{F}}$ contribution at every pressure and temperature, the estimation of the constant volume reduction of $\sigma_{\parallel}^{\text{SP}}$ follows a procedure similar to that used earlier for the total conductivity [119]. In the present situation the volume dependence of $\sigma_{\parallel}^{\text{SP}}$ is given by the volume dependence of $\sigma_{\perp}$ on figure 62. Temperature dependences of the single-particle resistivity are displayed in figure 67. The important aspect of figure 67 is the quasilinear temperature dependence of $\rho_{\parallel}^{\text{SP}}$ for both $V(60 \text{ K})$ and $V(300 \text{ K})$, and the positive intercept of the constant volume $T$-dependence with the $T=0 \text{ K}$ axis. This latter fact contrasts significantly with the existence of a $T=0 \text{ K}$ axis negative intercept previously derived for the total constant volume resistivity [119].

The origin of the single-particle scattering time is still a controversial matter. However we feel justified in giving here some suggestions derived from the experimental data obtained under pressure. The message given by conductivity [121] and optics under pressure [19] is the very large volume dependence of the single-particle scattering time $\partial \ln \tau_{\parallel}^{\text{SP}}/\partial \ln b = -60$, about 10 times larger than the volume dependence of the band parameters. Since the temperature dependence of $1/\tau_{\parallel}^{\text{SP}}$ (at constant volume) is quasi-linear (figure 67) there is no reason not to consider first-

Figure 66. Decomposition of the longitudinal conductivity of TTF-TCNQ into a fluctuating contribution $\sigma_{\parallel}^{\text{F}}$ and single particle contribution $\sigma_{\parallel}^{\text{SP}}$. The dashed line corresponds to $\sigma_{\parallel}^{\text{F}} = \sigma_{\parallel} - \sigma_{\parallel}^{\text{SP}}$ assuming a similar temperature dependence for $\sigma_{\parallel}^{\text{SP}}$ and $\sigma_{\perp}$ and zero fluctuation contribution at 300 K. (a) and (b) correspond to the incommensurate regime while (c) and (d) correspond to the pressure of the commensurability domain, (e) and (f) are located in the incommensurate domain. Note that the previous method of decomposition of the conductivity is probably not very relevant at high pressure (e) and (f), when $\sigma_{\parallel}^{\text{SP}}$ becomes larger than the fluctuation contribution over the whole temperature domain (after reference [110] of section 4).
<figure><img src="image_82.png" /></figure>

Figure 67. Temperature dependence of the single particle resistivity under constant pressure (full line) or constant volume at 60 K (dashed line) and at 300 K (dotted line). The dash-dotted line represents the temperature dependence of the *total* resistivity under ambient pressure (after JÉROME, D. and SCHULZ, H. J., 1982, *Extended Linear Chain Compounds*, Vol. 2, edited by J. S. Miller (New York: Plenum Press), p. 159).

order electron-phonon scattering as a possible origin for $\tau_{\parallel}^{\text{sp}}$. Such a suggestion has been put forward by Soda et al. [111] but the volume dependence of $\tau_{\parallel}^{\text{sp}}$ was ascribed to the volume dependence of the lattice modes. The argument was based on the observation of large *P* dependences of acoustic phonons in other organic crystals. However, we wish to point out that charged molecular crystals such as TTF-TCNQ are significantly less compressible than uncharged crystals like anthracene or naphthalene [114]. Moreover, the stiffening of the TA($a^+$) phonon branch due to thermal contraction between 300 and 84 K (equivalent to 5 kbar pressure) amounts to only ≈20% [122]. Thus, we feel it is rather hard to attribute the large volume dependence of $\tau_{\parallel}^{\text{sp}}$ (at all temperatures) to the lattice modes only. It has been suggested many times that electron-electron interactions play an important role in the electronic properties of organic conductors. We wish to emphasize here that *e-e* interactions are known to enhance the electron-phonon scattering (this is the electron-phonon vertex correction). The electron-phonon coupling constant can be derived from the infrared reflectivity studies [123] using the Hopfield relation $1/\tau_{\parallel}^{\text{sp}} = (2\pi/\hbar)\lambda k_B T$. A value $\lambda \approx 1.3$ has been found by Heeger and coworkers [123]. This value of $\lambda$ is far too large to agree with the bare electron-phonon coupling constant $\lambda_0$ needed for magnetic and structural properties [21], namely $\lambda_0 = 0.12$. The vertex correction of $\lambda_0$, which has been estimated within a Parquet theory, leads to an enhancement factor of 10 for TTF-TCNQ at room temperature. In the previous theoretical approach [21] the pressure dependence of the vertex correction was attributed mainly to the volume dependence of the bare *e-e* interactions. Although such a possibility cannot be ruled out we must be aware that the Parquet approximation is probably very crude whenever the *e-e* interactions are large. For
TTF-TCNQ in particular what is needed is an improved theoretical treatment which can treat properly the screening and the volume dependence of the electron-electron interactions.

Summarizing section 4.2, high-pressure experiments have helped significantly to clarify the question concerning the contribution of the CDWs to the metallic conduction of TTF-TCNQ, which had so far not received any direct and convincing answer. In fact, this problem has been particularly active and controversial in recent years [118]. During that time, several authors proposed explanations for the resistivity via single-particle scattering; namely, electron-acoustic phonon [124], electron-intramolecular phonon [125], second order electron-libron [126], interchain electron-electron [127], electron-spin fluctuations [128]. Admittedly, some of the above single-particle mechanisms found their justification in the unusually large volume dependence of the resistivity observed near ambient temperature [119]. However, we have seen in this section that most of the arguments put forward in favour of single-particle scattering must be taken with a grain of salt.

(i) A calculation of the contribution of the fluctuating CDW sliding to the conduction shows that the fluctuating Fröhlich conductivity is proportional to the average squared modulus of the CDW order parameter, namely the intensity of X-ray diffuse scattering which can thus take appreciable values for a one-dimensional electron gas in a broad temperature domain above the Peierls transition temperature.

(ii) High pressure data for the conductivity of TTF-TCNQ show an appreciable drop of the longitudinal conductivity (with no effect on $\sigma_\perp$) in the pressure domain around 19 kbar where commensurability between the CDW and the underlying lattice has been established.

(iii) An interpretation of the latter phenomenon is given in terms of the pinning by the commensurability potential of sufficiently long-lived CDW fluctuations. Other sources of pinning have been investigated; for instance defects induced by irradiation.

(iv) The quantitative derivation of $\sigma_F$ from the temperature dependences of $\sigma_\parallel$ and $\sigma_\perp$ shows that the fluctuating sliding collective mode contributes up to about 80% to the electronic conduction in incommensurate TTF-TCNQ at 60 K. This is also in fair agreement with the Kramers-Kronig analysis of the far-infrared reflectance data at 60 K.

### 4.3. Origin of phase transitions

The Peierls instability, in its original form, is based on the electron-phonon interaction (section 2.1.1) which gives an attractive effective interaction between the electrons. On the other hand, as discussed in section 2.1.2 (see in particular figure 5), purely repulsive electron-electron interactions may also give rise to a $2k_F$-CDW instability. In this situation it is of some interest to know the mutual importance of electron-phonon and electron-electron (Coulomb) interactions in the compounds discussed above.

In TTF-TCNQ there is actually evidence for an important role of both types of interactions. The existence of a superstructure in the low-temperature phase [5-7], characterized by rigid translations of the molecules [29, 30], and the precursor Kohn anomaly of a transverse branch [101] are evidences for the role of the interaction between the conduction electrons and external phonon modes (i.e. modes involving
rigid displacements of the molecules). In a tight-binding picture this interaction comes mainly from the modulation of the electron intermolecular transfer integrals with the change of intermolecular distance [129, 130]. More precisely, it has been argued [92] that the $T_H$ transition is associated with the condensation of a transverse acoustic phonon (in agreement with the observed precursor Kohn anomaly [101]), whereas at lower temperatures longitudinal optic modes are also involved. Libra-tions of the molecules have also been suggested to participate in the Peierls transition [95, 96]. However, a detailed analysis of the structure factor [93, 94] of TSF-TCNQ excludes any important contribution from such modes (though low-temperature IR spectra show some features which can be explained by librations [60]).

Apart from the external modes in molecular systems like TTF-TCNQ the coupling between conduction electrons and intramolecular vibrations also contributes to the Peierls instability [131]. This coupling mainly comes from the change of the energy of the conduction-band orbital by a deformation of the molecule. In TTF-TCNQ, the modes coupling linearly to the electrons have $a_g$ symmetry, and thus are normally not infrared active. However, as shown by Rice, in the Peierls state these modes are linearly coupled to the phase of the CDW and thus become infrared active for polarization along the chain direction [132]. The shift of the frequencies of the molecular modes with respect to the frequencies of an isolated molecule can then be used to calculate directly the corresponding electron–intramolecular-vibration coupling constant [132, 133]. The activation of normally inactive modes has been observed in TTF-TCNQ [134, 135], and from the coupling constants thus derived the contribution of the intramolecular modes to the Peierls instability is estimated to be considerably larger than the contributions from the external modes [135], in agreement with theoretical calculations [136]. A large contribution from intramolecular vibrations to the total electron–phonon coupling is not necessarily in contradiction with the fact that X-ray and neutron-scattering results are well explained by considering only rigid molecular motions [29, 30, 94]: the frequencies of intramolecular vibrations are much higher than external mode frequencies, the corresponding displacements in the Peierls state therefore much smaller, too small to be observed in scattering experiments. The importance of intramolecular modes in the Peierls transition is also in agreement with the observation of a positive isotope effect in TTF-TCNQ [137], which is expected if high-frequency phonons play a role [138].

Besides the above evidence for the role of electron–phonon interactions in the phase transitions there is considerable evidence for strong Coulomb interactions in TTF-TCNQ [139].

(i) The spin susceptibility is considerably larger than is expected from a non-interacting electron model (section 3.2.3), and Coulomb interactions comparable to the bandwidth have to be invoked to explain the enhancement.

(ii) Similar values are deduced from the enhancement of the NMR spin–lattice relaxation rate [111].

(iii) Theoretical explanations for the $4k_F$ anomaly observed up to room temperature [31, 32] usually invoke strong Coulomb correlations [140–143] (see section 2.1.2). It has been argued [144] that the $4k_F$ anomaly is the second harmonic of a short-range ordered $2k_F$ spin density wave, which may already occur for weak Coulomb interactions. One should, however, notice
that the intensity of the $4k_F$ scattering is comparable to the $2k_F$ scattering at low temperatures. This would imply a very large amplitude SDW, which seems in contradiction with the short correlation length of about $10$ Å at room temperature [29]. Also no other direct evidence of a SDW has been found so far in TTF-TCNQ

(iv) High-frequency optical spectra of TTF-TCNQ also indicate the importance of Coulomb interactions [139, 143, 145], especially on the TCNQ stack.

Though one could imagine that in TTF-TCNQ Coulomb interactions are important only on one type of molecular stack (say TTF because of the smaller bandwidth) this seems not to be the case: the enhancement factors of both the NMR relaxation rate [111] and of the spin susceptibility are comparable on both types of stacks. Also, the optical results seem to indicate important Coulomb effects on the TCNQ stacks [139, 145]. The importance of Coulomb interactions in quasi-one-dimensional conductors is often characterized by the ratio of the local electron-electron repulsion $U$ of a one-dimensional Hubbard model to the total bandwidth $4t_\parallel$. The above cited results in general imply $U/4t_\parallel = 1-1.5$ [111, 139].

The transverse coupling necessary to stabilize an ordered CDW state against fluctuations is largely due to Coulomb interactions: Bjelis and Barisic [82] estimate a Coulomb coupling constant of 60 meV between adjacent chains. On the other hand, the coupling due to transverse electron motion is of the order $t_\perp^2/t_\parallel$ [75] which is some tenth of a millielectronvolt in TTF-TCNQ (section 3.3.3). Even though, allowing for the finite extension of the molecules (neglected by Bjelis and Barisic [82]), the Coulomb coupling will be reduced [95], it is still much larger than the coupling due to electron interchain motion. One should, however, notice that the elastic interchain coupling may also play some role [82].

Now, how can one put the different interactions together to explain some of the properties of TTF-TCNQ? Lee *et al.* [146] consider a model with both intrachain ($U/4t_\parallel \approx 1.5$) and interchain Coulomb interaction, using renormalization group methods. In figure 5, the model is situated in the region $g_2 > 0$, $g_1 < 2g_2$. Correspondingly, as long as no interchain correlations develop, the SDW response function is enhanced over the CDW function. When the temperature is lowered, the interchain Coulomb interaction couples CDW fluctuations on adjacent chains (but not SDW fluctuations, section 2.3), so that there occurs a phase transition into a CDW (Peierls-like) state. The model thus reconciles the existence of the Peierls transition in TTF-TCNQ with the experimental evidence for strongly repulsive Coulomb interactions, situating this substance in the region $g_1 > 0$, $2g_2 > g_1$ of figure 5. Moreover, it qualitatively explains some properties related to the Coulomb interactions, like the $4k_F$ scattering or the increase of spin susceptibility with increasing temperature. Also, the fact that the Kohn anomaly [101] is only developing very slowly between 150 and 60 K may be related to the fact that in this model the CDW response function starts to diverge strongly only very near the transition temperature. Electron-phonon interactions can be included in the renormalization group formalism noting that they lead to an effective electron-electron attraction only for energy transfer smaller than the phonon energy, and neglecting the electron-phonon interaction at high energy transfer, as in the BCS model [147]. Consequently, one can account for the electron-phonon interactions by including the corresponding effective electron-electron interaction into the renormalization at temperatures *below* the phonon energy [148, 149]. Applying these ideas to
TTF-TCNQ, at room temperature all but the lowest intramolecular modes are already included in the total effective interaction constant, but with decreasing temperature the external lattice modes also become successively important. The inclusion of these modes then accelerates the decrease of the effective couplings, to negative values, i.e. it leads to an increase of the transition temperature. Thus, one cannot say that either the electron-electron or the electron-phonon interactions are dominant in the transition mechanism, rather both processes play an important role, though in TTF-TCNQ some properties of the metallic state seem hard to explain without taking into account the repulsive Coulomb interaction. This point of view is of course by no means in contradiction with the observation of structural effects in the low-temperature phase which are usually considered neglecting the Coulomb interactions. Actually, any CDW, irrespective of its origins, will, via the linear electron-phonon coupling, lead to a lattice modulation or the activation of normally infrared-inactive modes. One should also note that the above discussion implies that the experimentally-derived values of the Coulomb interaction already include a renormalization from intramolecular modes and may thus depend significantly on molecular structure.

How do the other compounds discussed in sections 2 and 4.2 fit into the above picture? Neither in TSF-TCNQ nor in HMTSF-TCNQ has a $4k_F$ anomaly been observed. In TSF-TCNQ the enhancement of the spin susceptibility over its free electron value is weaker than in TTF-TCNQ, and there is no such enhancement in HMTSF-TCNQ (section 3.2.3). Thus in these selenium compounds Coulomb interactions seem to play a less important role. This is not surprising because the larger selenium orbitals (compared to those of sulphur) lead to a large bandwidth and consequently, in a tight-binding picture, to stronger interaction with external phonon modes [129]. Secondly, the larger orbitals also lead to a decrease of the on-site Coulomb repulsion, so that a reduction of $U/4t_\parallel$ from its value in TTF-TCNQ results. In the selenium compounds the electron-phonon interaction thus seems to play a more important role, though the enhancement and temperature dependence of the spin susceptibility in TSF-TCNQ also indicate some importance of Coulomb effects in that compound.

An intermediate case between the selenium compounds and TTF-TCNQ seems to be given by HMTTF-TCNQ, as indicated by a weak $4k_F$ X-ray scattering [58, 59].

### 4.4. *A special case: TMTSF-DMTCNQ*

Like TTF-TCNQ, TMTSF-DMTCNQ consists of alternating sheets of TMTSF and DMTCNQ stacks [150], though the details of the structures are quite different. At $T_c = 42$ K a transition from metallic (high temperature) to semiconducting (low temperature) properties is observed in conductivity [151] and magnetic [152] measurements. Below this transition from the conductivity activation energy the gap energy is derived as $E_g = 290$ K [151]. The observation of a three-dimensionally ordered superstructure [153] of longitudinal period $4a$ (in the TMTSF-DMTCNQ structure the direction also the chains is usually denoted $a$ [150] not $b$ as in TTF-TCNQ) completes the picture of a Peierls instability as the origin of the semiconducting phase. The only somewhat unusual properties are

(i) the low charge transfer of only $\rho = 0.5$, leading to a fourth-order commensurate superstructure (figure 68) and
<figure><img src="image_83.png" /><figcaption>Figure 68. Microdensitometer reading of X-ray patterns from TMTSF-DMTCNQ at various temperatures, showing commensurate $2k_F$ and $4k_F$ scattering at reduced wave vectors $0.25a^*$ and $0.50a^*$ respectively. The $2k_F$ scattering increases in intensity and sharpens in width as the temperature is reduced. It condenses in satellite reflections below 42 K, while the $4k_F$ scattering vanishes at low temperature without giving rise to satellite reflections (after reference [153] of section 4).</figcaption></figure>

(ii) the indication both from magnetic [152] and transport measurements that the TMTSF (i.e. donor) chains drive the transition, unlike most other organic conductors (see section 4.1).

Concerning the microscopic origin of the transition, the larger bandwidth and correspondingly larger electron-phonon coupling [65] can explain why the TMTSF stacks drive the transition [152]. The observation of diffuse $4k_F$ scattering (figure 68), though less important than in TTF-TCNQ, indicates that Coulomb interactions have some importance, situating the system somewhere between TTF-TCNQ and TSF-TCNQ. However neither a detailed investigation of the importance of Coulomb interactions (via NMR and spin susceptibility measurements) nor a study of the electron-phonon coupling (by infrared measurements in the Peierls distorted phase or inelastic neutron scattering) have been made so far, so that nothing very definite can be said. Considering the high-conductivity anisotropy in the metallic region, the interchain coupling necessary to establish a three-dimensionally ordered state is probably due to interchain Coulomb interactions, as in the other TTF-TCNQ-like
compounds. As in these compounds the three-dimensional order is easily destroyed by small amounts of irradiation-induced defects [154].

More spectacular phenomena, justifying the title of this section, happen upon application of moderate hydrostatic pressure [155, 156]. Up to about 9.7 kbar the transition temperature remains nearly constant; however at higher pressures the metal-insulator transition vanishes abruptly and the system remains metallic down to the lowest investigated temperatures. The metallic state is actually quite stable: decreasing the pressure at low temperatures, from $P = 10$ kbar, the metallic state is found to exist down to at least 8.5 kbar, well below the critical pressure of about 9.7 kbar, where the metal-insulator transition is observed to vanish in cooling runs at constant pressure. In some measurements the metallic state even seems to exist down to 7 kbar. The phase diagram is shown in figure 69. The metal I-metal II transition corresponds to a small anomaly in the temperature-dependent conductivity, which is not yet understood. Apart from a very high conductivity, saturating only below 10 K (figure 70) the metallic state is characterized by a metal-like thermopower [156], a large and anisotropic magnetoresistance (the longitudinal conductivity decreases one order of magnitude in a field of 75 kG applied perpendicularly to the conducting axis, whereas for a longitudinal field the effect is much smaller [156]) and a diamagnetic susceptibility [157].

Thus, TMTSF-DMTCNQ under pressure represents the rather exceptional case of an organic conductor remaining metallic down to the lowest temperatures without any structural instability. At the time of this discovery, the only other example of such a behaviour was $(TSeT)_2Cl$ [158], though some more examples are now known as described in the following section. The explanation of the results still seems difficult: though one is tempted to attribute the sudden drop of the Peierls transition temperature to deviations from commensurability, triggered by a pressure-induced

<figure><img src="image_84.png" /><figcaption>Figure 69. TMTSF-DMTCNQ: (a) conductivity under ambient pressure and 13 kbar, (b) phase diagram showing the ≈3 kbar pressure hysteresis accompanying the suppression of the insulating state (after reference [156] of section 4).</figcaption></figure>
<figure><img src="image_85.png" /><figcaption>Figure 70. Temperature dependence of the longitudinal resistivity of TMTSF-DMTCNQ under 13 kbar in zero magnetic field and in a transverse field of 75 kOe (after reference [155] of section 4).</figcaption></figure>

change in charge transfer, this does not explain the complete disappearance of the transition. The absence of a Peierls transition is even more surprising noting that the conductivity anisotropy in TMTSF-DMTCNQ is an order of magnitude larger than in HMTSF-TCNQ and increases under pressure, so that deviations of the Fermi surface from planarity should be much smaller in TMTSF-DMTCNQ. Nevertheless, HMTSF-TCNQ has a Peierls transition, TMTSF-DMTCNQ does not. Thus, enhanced three-dimensional coupling cannot explain the metallic properties, and another, so far unknown, mechanism must be the origin. Another problem is posed by the low-temperature transport properties. Thermopower [156], magnetoresistance [156] and diamagnetism [157] are consistent with a semimetallic picture [156, 157], similar to that discussed for HMTSF-TCNQ (section 4.1.2). However, unlike HMTSF-TCNQ, the conductivity of TMSF-TCNQ increases down to 10 K ($\sigma(10\text{ K}) \approx 10^5 (\Omega \text{ cm})^{-1}$), and extremely high values of the mobility are required to explain the value of the conductivity [157]. At the same time the conductivity anisotropy also increases.

TMTSF-DMTCNQ appears more one-dimensional than HMTSF-TCNQ as far as the interchain tunnelling is concerned. Consequently, the transition from a one-dimensional to a three-dimensional Fermi surface should arise at temperature lower than 100 K in the former compound. The onset of diamagnetism below 25 K under pressure [157] could possibly be the signature of the Fermi surface dimensionality cross-over. A direct evidence of this cross-over would actually be the observation of quantum magnetoresistance oscillations at low temperature in the pressure stabilized conducting phase. However, using the arguments already developed for the understanding of HMTSF-TCNQ, the longitudinal conductivity should not vary very significantly through the cross-over region since both the density of carriers and the Fermi wave-vector are changing in about the same proportion (see section 3.3.4).
Experimentally, a comparison between the resistivity of TMTSF-DMTCNQ, figure 70, and HMTSF-TCNQ, figure 33 (a), shows clearly striking differences, namely the existence of a large drop of the resistivity below 25 K in TMTSF-DMTCNQ whereas below 100 K the resistivity of (imperfect) HMTSF-TCNQ crystals drops by no more than a factor 2 at low temperature. We are thus tempted to believe that the high conductivity of TMTSF-DMTCNQ cannot be explained solely by a possible semimetallic Fermi surface at low temperature under pressure. These difficulties are so far unsolved. However, in section 5 we shall propose a possible solution which may solve both of the above problems, namely the sudden disappearance of the Peierls transition and the origin of the low-temperature transport problems.

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## 5. Superconductivity versus antiferromagnetism or anion ordering: $(TMSF)_2X$

In 1979 Bechgaard et al. [1] found that the family of quasi-one-dimensional organic salts $(TMSF)_2X$ (X is an inorganic anion, the most common ones being $PF_6$, $AsF_6$, $SbF_6$, $TaF_6$, $NO_3$, $BF_4$, $ReO_4$ and $ClO_4$) shows metal-insulator transitions at unusually low temperatures of the order 15 K (figure 71), with conductivities as high as $10^5$ ($\Omega$ cm)$^{-1}$ preceding the transition. A short time later, with an applied pressure of about 10 kbar, the first observation of superconductivity was reported for one of the compounds of the family, $(TMSF)_2PF_6$ [2]. Now, superconductivity has also been found without pressure in $(TMSF)_2(ClO_4)$ [3].

In the following we shall first discuss the quite interesting properties of the insulating low-temperature state, and then consider the most remarkable feature of the $(TMSF)_2X$ family, the establishment of a superconducting state.
### 5.1. Low-temperature insulating state

Unlike TTF-TCNQ and similar compounds, in the $(TMTSF)_2X$ family only the $TMTSF$ molecular stacks can give rise to metallic properties, whereas the inorganic anions are completely ionized and therefore inactive, at least concerning low-energy phenomena. A structural analysis of $(TMTSF)_2PF_6$ shows that there is one $TMTSF$ stack per (triclinic) unit cell [4]. Replacing $PF_6^-$ by another anion does not significantly change the structure. The fact that there is only one type of conducting chain considerably simplifies the interpretation of the physical properties (see also section 3).

Resistivity measurements on $(TMTSF)_2PF_6$ show a phase transition from a highly conducting ($\sigma_a \approx 10^5 (\Omega \text{cm})^{-1}$, $a$ is the stack direction) to a semiconducting state around 10 K [1]. Though originally, in analogy with TTF-TCNQ, this was assumed to be a Peierls transition, involving formation of a CDW and a lattice modulation, the absence of any sign of lattice modulation in X-ray measurements [5, 6] clearly showed that another mechanism must be responsible (figure 72). As discussed in section 2, a spin-density wave (SDW) of wavenumber $2k_F$ may also appear in a one-dimensional metal and leads to the opening of a gap at the Fermi surface; this, however, would not lead to a lattice modulation (or at most a very

<figure><img src="image_86.png" /><figcaption>Figure 71. Temperature dependence of the resistivity of various $(TMTSF)_2X$ conducting salts, $X = NO_3^-$, $PF_6^-$, $AsF_6^-$, $BF_4^-$ (after reference [1]).</figcaption></figure>
<figure><img src="image_87.png" /><figcaption>Figure 72. Microdensitometer readings of (TMTSF)₂PF₆ X-ray patterns, showing at *h* = 2.5*a*\* a weak 2*k*<sub>F</sub> scattering between 125 K and 33 K (after [6]).</figcaption></figure>

weak one via spin-orbit coupling or effects of higher harmonics). The possibility of a
SDW low-temperature state was first suggested by Walsh *et al.*, based mainly on an
observed non-linearity of the electrical conductivity [7]. Though the meaning of these
non-linearities is not yet clear [8], magnetic effects are evident in susceptibility
measurements: whereas the EPR susceptibility abruptly vanishes below the metal-
semiconductor transition [9], the static (powder) susceptibility remains finite, albeit
with an anomaly in its temperature dependence [10]. More direct evidence for a
SDW comes from NMR experiments: the ⁷⁷Se NMR line of (TMTSF)₂PF₆ broadens
inhomogeneously at 12 K and becomes too broad to be detected below 11 K (figure 73)
[11, 12]. The simultaneous decrease of the spin-spin relaxation rate T₂⁻¹ (figure 74),
which is proportional to the homogeneous linewidth, demonstrates the inhomoge-
neous nature of the linewidth broadening. Since both sets of nuclei have spin 1/2,
NMR shows the existence of strong local magnetic fields below the metal-insulator
transition. More insight into the spin structure has been gained by measurements of
the anisotropy of the low-temperature static susceptibility [13]. At low fields these
measurements show a fast decrease with temperature of the susceptibility for an
applied field perpendicular to the conducting a-axis, whereas for a parallel field there
is much less temperature dependence. In addition the perpendicular susceptibility is
strongly field dependent at fields below 10 kOe, indicating the existence of a spin-flop
transition around 4.5 kOe. These results are typical for an antiferromagnet (equi-
valently a SDW state) with an easy axis in the transverse (b-c) plane [13]. For
(TMTSF)AsF₆ which is expected to have the same behaviour as the PF₆ salt, the
low-temperature, low-field spin susceptibilities obey [14] χ_c > χ_a > χ_b (figure 75), so
that the b axis is the easy axis, the (longitudinal) a axis is the intermediate axis and
the c axis is the hard axis, and the same order probably holds also in (TMTSF)₂PF₆.
We remark that in both compounds the transition temperature increases slightly
with an applied field [13, 14]. In the low-temperature phase the effect of magnetic
modes can be observed: near the transition these modes are soft and thus contribute