Organic conductors and superconductors

Denis Jerome, H.J. Schulz

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Denis Jerome, H.J. Schulz. Organic conductors and superconductors. Advances in Physics, 2006, 31 (4), pp.299-490. 10.1080/00018738200101398. hal-04962081

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Organic conductors and superconductors*

D. JÉROME† and H. J. SCHULZ‡

†Laboratoire de Physique des Solides, Université Paris-Sud, 92405 Orsay, France

‡Institut Max von Laue-Paul Langevin, 156X, 38042 Grenoble Cedex, France

[Received 14 April 1982]

Abstract

This review attempts to present the most salient developments of research on organic conductors and superconductors during the past 10 years. A theoretical introduction treats instabilities of quasi-one-dimensional electron systems and associated precursor effects which are relevant to the experimental results on organic conductors. We then describe the characterization of quasi-one-dimensional organic conductors by their transport, optical and magnetic properties. Finally, two sections are devoted to the experimental investigation of the low temperature instabilities: lattice instability in TTF-TCNQ and related compounds, superconducting or antiferromagnetic instabilities in the (TMTSF)₂X series. The importance of one-dimensional fluctuations is emphasized in both lattice and superconducting instabilities.

Contents

PAGE

<table><tr><td>1. Introduction</td><td>294</td></tr><tr><td>Bibliography</td><td>295</td></tr><tr><td>2. What is special about physics in one dimension? Some theory</td><td>296</td></tr><tr><td>2.1. The Peierls instability</td><td>297</td></tr><tr><td>2.1.1. Zero-temperature ground state</td><td>297</td></tr><tr><td>2.1.2. Excited states</td><td>299</td></tr><tr><td>2.1.3. Finite temperature</td><td>301</td></tr><tr><td>2.2. Competing instabilities in a one-dimensional interacting electron gas</td><td>306</td></tr><tr><td>2.2.1. Mean-field theory</td><td>306</td></tr><tr><td>2.2.2. Beyond mean-field theory</td><td>309</td></tr><tr><td>2.3. Interchain coupling</td><td>313</td></tr><tr><td>2.4. Ginzburg-Landau theory</td><td>319</td></tr><tr><td>2.4.1. A single chain</td><td>319</td></tr><tr><td>2.4.2. Coupled chains</td><td>324</td></tr><tr><td>2.4.3. Effects of disorder</td><td>327</td></tr><tr><td>2.4.4. Response to external fields</td><td>329</td></tr><tr><td>References to section 2§</td><td>334</td></tr><tr><td>3. One-dimensional organic conductors: electrical properties</td><td>337</td></tr><tr><td>3.1. Basis</td><td>337</td></tr></table>

<footnote>* This article was originally published in *Advances in Physics*, volume 31, 1982. It had attracted 663 citations by October 2001, and is ranked 23 in the index of articles attracting more than 100 citations.</footnote>

<footnote>§ References are given separately within each section and will be found at the end of the corresponding section.</footnote>
<table>
  <tbody>
    <tr>
      <td>3.1.1. Elementary chemical considerations</td>
      <td>337</td>
    </tr>
    <tr>
      <td>3.1.2. Basic crystal structures</td>
      <td>343</td>
    </tr>
    <tr>
      <td>3.1.3. Three-dimensional lattice properties: elastic constants</td>
      <td>344</td>
    </tr>
    <tr>
      <td>3.1.4. Band formation</td>
      <td>346</td>
    </tr>
    <tr>
      <td>3.2. Physical properties related to one-dimensional energy bands</td>
      <td>350</td>
    </tr>
    <tr>
      <td>3.2.1. Transport properties</td>
      <td>350</td>
    </tr>
    <tr>
      <td>3.2.2. Optical properties</td>
      <td>353</td>
    </tr>
    <tr>
      <td>3.2.3. Spin susceptibility</td>
      <td>356</td>
    </tr>
    <tr>
      <td>3.2.4. Nuclear spin dynamics</td>
      <td>359</td>
    </tr>
    <tr>
      <td>3.3. Electronic properties including the interchain coupling</td>
      <td>361</td>
    </tr>
    <tr>
      <td>3.3.1. The three-dimensional band structure</td>
      <td>361</td>
    </tr>
    <tr>
      <td>3.3.2. The three-dimensional escape time τ⊥</td>
      <td>364</td>
    </tr>
    <tr>
      <td>3.3.3. NMR measurement of τ⊥</td>
      <td>366</td>
    </tr>
    <tr>
      <td>3.3.4. HMTSF-TCNQ: a special case</td>
      <td>368</td>
    </tr>
    <tr>
      <td>3.3.5. Electron spin resonance in quasi-one-dimensional conductors</td>
      <td>372</td>
    </tr>
    <tr>
      <td>3.4. Pressure effects</td>
      <td>375</td>
    </tr>
    <tr>
      <td>3.4.1. The one electron bandwidth</td>
      <td>375</td>
    </tr>
    <tr>
      <td>3.4.2. Transport and magnetic properties</td>
      <td>376</td>
    </tr>
    <tr>
      <td>References to section 3§</td>
      <td>384</td>
    </tr>
    <tr>
      <td>4. Charge density wave order in TTF-TCNQ and its family</td>
      <td>388</td>
    </tr>
    <tr>
      <td>4.1. The ordered state</td>
      <td>388</td>
    </tr>
    <tr>
      <td>4.1.1. TTF-TCNQ</td>
      <td>388</td>
    </tr>
    <tr>
      <td>4.1.2. Some members of the family: TSF-TCNQ, HMTTF-TCNQ and HMTSF-TCNQ</td>
      <td>398</td>
    </tr>
    <tr>
      <td>4.1.3. Landau theory of phase transition</td>
      <td>402</td>
    </tr>
    <tr>
      <td>4.2. The precursor effects</td>
      <td>407</td>
    </tr>
    <tr>
      <td>4.2.1. X-ray diffuse scattering</td>
      <td>407</td>
    </tr>
    <tr>
      <td>4.2.2. Fluctuating Fröhlich conductivity</td>
      <td>410</td>
    </tr>
    <tr>
      <td>4.3. Origin of phase transitions</td>
      <td>421</td>
    </tr>
    <tr>
      <td>4.4. A special case: TMTSF-DMTCNQ</td>
      <td>424</td>
    </tr>
    <tr>
      <td>References to section 4§</td>
      <td>428</td>
    </tr>
    <tr>
      <td>5. Superconductivity versus antiferromagnetism or anion ordering: (TMTSF)<sub>2</sub>X</td>
      <td>432</td>
    </tr>
    <tr>
      <td>5.1. Low-temperature insulating state</td>
      <td>433</td>
    </tr>
    <tr>
      <td>5.2. The superconducting state</td>
      <td>438</td>
    </tr>
    <tr>
      <td>5.3. Superconducting fluctuations</td>
      <td>450</td>
    </tr>
    <tr>
      <td>5.3.1. Density of states and conductivity</td>
      <td>450</td>
    </tr>
    <tr>
      <td>5.3.2. Effects of irradiation</td>
      <td>464</td>
    </tr>
    <tr>
      <td>5.3.3. The fluctuating regime under high magnetic field</td>
      <td>467</td>
    </tr>
    <tr>
      <td>5.4. Origin of phase transitions</td>
      <td>472</td>
    </tr>
    <tr>
      <td>References to section 5§</td>
      <td>475</td>
    </tr>
    <tr>
      <td>6. Concluding remarks</td>
      <td>478</td>
    </tr>
    <tr>
      <td>Acknowledgements</td>
      <td>479</td>
    </tr>
  </tbody>
</table>

# 1. Introduction

This article attempts to give an overview on the recent and rapidly expanding field of organic conductors and superconductors. Instead of giving an exhaustive picture of the whole domain we have chosen to focus on the behaviour of a few extensively studied and fairly well characterized systems such as those pertaining either to the TTF-TCNQ or to the (TMTSF)<sub>2</sub>X series. Actually these systems have constituted the main interest of the authors at Orsay for the past 10 years.

§ References are given separately within each section and will be found at the end of the corresponding section.
For specific information we refer the reader to the more specialized literature
which is given in the bibliographical list of this introduction.

The first section presents the origin of theoretical results with the hope that the non-specialist reader can gain some interest in the field. Most of the theoretical aspects which are discussed in this section are based on relatively simple theoretical concepts such as mean-field theory or Ginzburg-Landau like models.

The second section deals mainly with the impact of the one-dimensionality encountered in organic conductors on the metallic properties deriving from band formation. Special emphasis is put on the experimental method probing the one-dimensional character, for example conductivity and optical properties anisotropy, magnetic field dependence of the NMR relaxation and temperature dependence of the EPR linewidth. Also, this chapter presents the effect of interchain motion on several physical properties (NMR, Fermi surface, transport coefficients ...) and the unusually large dependence of the electron properties on the application of hydrostatic pressure.

The third and fourth sections consider phase transition driven by electronic instabilities with periodic lattice distortion ground states observed in TTF-TCNQ like systems and spin density wave or superconducting ground states observed in the (TMTSF)₂X family. The main characteristics of the ordered states are discussed in some detail. Precursor effects of the low temperature ordering, related to the one-dimensional nature of the electronic spectrum are directly observed in TTF-TCNQ by X-ray methods. The application of hydrostatic pressure constituted a convincing way of showing the influence of precursor effects on transport properties in these compounds. Although precursor effects cannot be as easily visualized in the (TMTSF)₂X series, there exists a large amount of experimental data suggesting the existence of one-dimensional superconducting short range order up to about 30 K.

Bibliography

Conference proceedings

*International Conference on Organic Superconductivity*, edited by W. A. Little, 1969, *J. Polymer Sci. part 2*, **29** (1969).

*Conf. on One-dimensional Conductors*, Saarbrücken, edited by H. G. Schuster, 1975, Lecture Notes in Physics No. 34 (Springer-Verlag).

*Organic Conductors and Semiconductors*, Siofok, edited by L. Pal, G. Grüner, A. Janossy and J. Solyom, 1977, Lecture Notes in Physics No. 65 (Springer-Verlag).

*Synthesis and Properties of Low Dimensional Materials*, edited by J. S. Miller and A. J. Epstein, 1978, *Ann. N.Y. Acad. Sci.*, **313**, 1-828.

*Quasi-One-Dimensional Conductors*, I and II, Dubrovnik, edited by S. Barisic, A. Bjeliš, J. R. Cooper and B. Leontić, 1979, Lecture Notes in Physics Nos 95 and 96 (Springer-Verlag), (1979).

*Int. Conf. on Low Dimensional Synthetic Metals*, Helsingør, 1981, *Chem. scripta*, **17**, 1-230.

*Int. Conf. on Low-Dimensional Conductors* (Boulder Conference), edited by A. J. Epstein and E. M. Conwell, 1981-1982, *Molec. Crystals liq. Crystals*, **77, 79, 81, 83, 85, 86**.

Summer schools or advanced studies institutes

*Low-Dimensional Cooperative Phenomena*, 1975, edited by H. H. Keller (New York: Plenum).

*Chemistry and Physics of One-Dimensional Metals*, 1976, edited by H. J. Keller (New York: Plenum).
*Physics and Chemistry of Low-Dimensional Solids*, 1980, edited by L. Alcacer (Dordrecht, The Netherlands: D. Reidel).

*Molecular Metals*, 1979, edited by W. E. Hatfield (Plenum).

*Physics in One Dimension*, 1981, edited by J. Bernasconi and T. Schneider (Springer-Verlag).

### Books

DEVREESE, J. T., and VAN DOREN, V. E. (eds), 1979, *Highly Conducting One-Dimensional Solids* (Plenum).

GUTMANN, F., and LYONS, L. E., 1967, *Organic Semiconductors* (New York: Wiley).

KAGOSHIMA, A., SAMBONGI, T., and NAGASAWA, H., 1982, *One-Dimensional Conductors* (Tokyo: Syokabo), in Japanese.

MILLER, J. S. (ed.), 1982, *Extended Linear Chain Compounds*, Vols 1 to 4 (Plenum).

### Some review articles

ANDRÉ, J. J., BIEBER, A., and GAUTIER, F., 1976, Physical properties of highly anisotropic systems: radical-ion salts and charge transfer complexes. *Ann. Phys.*, **1**, 145.

BECHGAARD, K., and JÉROME, D., 1982, Organic superconductors *Scient. Am.*, July, p. 52.

BERLINSKY, A. J., 1976, Organic metals. *Contemp. Phys.*, **17**, 331–354.

BULAEVSKII, L. N., 1975, Peierls structure transition in quasi-one dimensional crystals. *Soviet Phys. Usp.*, **18**, 131.

ENGLER, E. M., 1976, Organic metals. *Chem. Tech.*, p. 274.

EPSTEIN, A. J., and MILLER, J. S., 1979, Linear chain conductors. *Scient. Am.*, October, p. 52.

FRIEDEL, J., and JÉROME, D., 1982, Organic superconductors the (TMTSF)₂X family. *Contemp. Phys.*, **23** (in press).

GARITO, A. F., and HEEGER, A. J., 1974, The design and synthesis of organic metals. *Accts Chem. Res.*, **7**, 232–240.

KHIDEKEL, M. L., and ZHILYAEVA, E. I., 1981, Organic metals. *Synthet. Metals*, **4**, 1–34.

NARITA, M., and PITTMAN, C. H., Jr., Preparation of tetrathiofulvalenes (TTF) and their selenium analogs–tetraselenefulvalenes (TSF). *Synthesis*, pp. 489–514.

SHCHEGOLEV, I. F., 1972, Electrical and magnetic properties of linear conducting chains. *Phys. Stat. sol.* (a), **12**, 9.

SOLOYM, J., 1979, The Fermi gas model of one-dimensional conductors. *Adv. Phys.*, **28**, 201.

TOMBS, G. A., 1978, Quasi one dimensional conductors. *Physics Rep.* C, **40**, 181–240.

TORRANCE, J. B., 1979, The difference between metallic and insulating salts of tetracyanoquinodimethane (TCNQ): how to design an organic metal. *Accts Chem. Res.*, **12**, 79–86.

## 2. What is special about physics in one dimension? Some theory

In Little's proposal for an organic high-temperature superconductor [1] one-dimensionality arose from purely geometric considerations: electrons moving along a polymer backbone are expected to interact strongly with highly polarizable side-chain molecules. The indirect electron-electron interaction mediated by the polarization of the side-chain is then strongly attractive and therefore leads to a high (BCS-type) [2] superconducting transition temperature. Thus, in the Little mechanism (which, unfortunately, has not been realized experimentally so far) one-dimensionality merely serves to optimize an important microscopic parameter of superconductivity. However, a one-dimensional electron system differs from a three-dimensional one in several more fundamental aspects:

(i) In the presence of interactions the groundstate is unstable against the formation of charge- [3] or spin-density waves [4] (Peierls instability). This instability in general competes with the superconducting BCS-instability [5].

(ii) In a one-dimensional system with short-range interactions thermal fluctuations destroy long-range order at any non-zero temperature [6, 7].
(iii) An arbitrarily small amount of disorder localizes all the electrons and therefore transforms a one-dimensional metal into an insulator [8].

In the following we describe the physical origin of the above anomalies and some of their consequences in quasi-one-dimensional conductors. The Peierls instability and the properties of the low-temperature charge-density wave (CDW) state are described in section 2.1, whereas section 2.2 is concerned with the competition between Peierls- and BCS-type instabilities. In both sections destruction of long-range order by fluctuations is shown. In real quasi-one-dimensional substances, to be discussed in the following sections, there is always a finite amount of coupling between adjacent chains. The different kinds of such coupling and their effects (mainly the possibility of long-range order at finite temperature) are considered in section 4.3. Finally, in section 4.4 we discuss Ginzburg-Landau-type theories which allow a clear semiquantitative description of many properties both near the ordering temperature and in the (often large) temperature region where fluctuation effects are important. We do not consider one-dimensional localization, a subject of great current theoretical [9, 10] and experimental [11] interest, which has however been of very little importance in the field of organic conductors, probably because of both the quite low temperatures necessary to observe this phenomenon [9, 11] and its sensitivity to three-dimensional coupling.

A broader overview over physics in one dimension can be found in the classical book by Lieb and Mattis [12] and in recent conference proceedings [13].

## 2.1. The Peierls instability

### 2.1.1. Zero-temperature ground state

Peierls [3] and Fröhlich [3] showed that a one-dimensional electron system in a deformable lattice is unstable against a modulation of the lattice of wavenumber $2k_F$ ($k_F$ is the Fermi wavenumber). Such a system is described by the second quantized hamiltonian

$$H = \sum_{ks} \epsilon_k a_{ks}^{\dagger} a_{ks} + \sum_{q} \omega_q b_{q}^{\dagger} b_{q} + \frac{g}{\sqrt{L}} \sum_{kqs} a_{k+q,s}^{\dagger} a_{ks}(b_{q} + b_{-q}^{\dagger}). \quad (2.1)$$

Here $a_{ks}^\dagger$ and $a_{ks}$ are creation and destruction operators for electrons in Bloch state $k$ with spin $s$ and energy $\epsilon_k$ ($\epsilon_{k_F} = 0$), and $b_q^\dagger$ and $b_q$ are creation and destruction operators for phonons of wavenumber $q$ and energy $\omega_q$. Units are chosen so that $\hbar = 1$, and $L$ is the length of the system. The first two terms describe non-interacting electron and phonon systems, respectively. In the third (interaction-) term, the electron-phonon coupling constant $g$ is generally a function both of $k$ and $q$, however in the following electrons with $|k| \approx k_F$, phonons with $|q| \approx 2k_F$ are the most important ones, so that the wavenumber-dependence of $g$ can be neglected. In (2.1), all wavenumber summations are over the first Brillouin zone.

The lattice displacement field $u$ is given in terms of the phonon operators as

$$u(x) = \sum_q \frac{1}{\sqrt{(2L\omega_q)}} (b_q + b_{-q}^\dagger) \exp(iqx) \quad (2.2)$$

so that a static finite amplitude lattice modulation of wavenumber $2k_F$ is described by $\langle b_{2k_F} \rangle = \langle b_{-2k_F}^\dagger \rangle \propto \sqrt{L}$, i.e. phonon models with wavenumbers $\pm k_F$ are macroscopically occupied. In the mean-field approximation one neglects fluctuations
around these mean values as well as all the other (not macroscopically occupied) phonon modes. The system is then described by the reduced hamiltonian

$$H = \sum_{ks} \varepsilon_k a_{ks}^{\dagger} a_{ks} + 2\omega_{2k_F} |\langle b_{2k_F} \rangle|^2 + \frac{2g}{\sqrt{L}} \sum_{ks} (a_{k+2k_F,s}^{\dagger} a_{ks} \langle b_{2k_F} \rangle + a_{k-2k_F,s}^{\dagger} a_{ks} \langle b_{2k_F}^{\dagger} \rangle), \quad (2.3)$$

i.e. the lattice modulation mixes states at $k$ with states at $k \pm 2k_F$. Most important is the mixing between the nearly degenerate states with $|k| \approx k_F$ which for a sufficiently small amplitude of the modulation can be treated in the nearly free-electron approximation, leading to

$$H = \sum_{ks} E_k c_{ks}^{\dagger} c_{ks} + L \frac{\omega_{2k_F} |\Delta|^2}{2g^2}, \quad (2.4a)$$

$$E_k = \text{sign}(|k| - k_F)[v_F^2(|k| - k_F)^2 + |\Delta|^2]^{1/2}. \quad (2.4b)$$

Here we have linearized $\varepsilon_k$ around the Fermi points, i.e. $\varepsilon_k = v_F(|k| - k_F)$ ($v_F$ is the Fermi velocity), $c_{ks}^\dagger$ and $c_{ks}$ are creation and destruction operators for electrons in states which are linear combinations of the old states at $k$ and $k \pm 2k_F$, and we have defined the order parameter

$$\Delta = \frac{2g \langle b_{2k_F} \rangle}{\sqrt{L}}. \quad (2.5)$$

As shown by (2.4) the lattice modulation leads to the opening of a gap of width $2|\Delta|$ in the electronic spectrum at the Fermi level (figure 1). The origin of this gap is the same as in the band theory of solids: Bragg reflection of the electrons by the periodic lattice modulation. Obviously, the energy of the electrons is decreased by the opening of the gap. On the other hand, the modulation of the lattice leads to an increase of elastic energy, described by the second term in (2.4a). To assess the net effect, consider the total energy of the electronic system:

$$\begin{align*}
E_{\text{el}}(\Delta) &= 2 \sum_{k=-k_F}^{k_F} E_k \\
&= -\frac{L}{\pi} \int_{-k_F}^{k_F} [v_F^2(|k| - k_F)^2 + |\Delta|^2]^{1/2} \mathrm{d}k \\
&= -\frac{L}{\pi v_F} \left[ E_F \sqrt{(E_F^2 + |\Delta|^2)} + |\Delta|^2 \operatorname{arcsinh} \frac{E_F}{|\Delta|} \right] \\
&= -\frac{LnE_F}{2} \left[ 1 + \frac{|\Delta|^2}{E_F^2} \ln\left(\frac{2E_F}{|\Delta|}\right) + O\left(\frac{|\Delta|^2}{E_F^2}\right) \right]. \tag{2.6}
\end{align*}$$

<figure><img src="image_2.png" /><figcaption>Figure 1. Electron dispersion of the undistorted (left) and the Peierls modulated (right) one-dimensional electron-phonon system. States with $|k| \lesssim k_F$ gain energy by the distortion.</figcaption></figure>
Here $E_F = v_F k_F$ is the Fermi energy measured from the bottom of the band and $n$ is the total electron density. From (2.6) the energy gained by the electrons is (for small $|\Delta|$) proportional to $-|\Delta|^2 \ln |\Delta|$, always larger than the loss in elastic energy (only proportional to $|\Delta|^2$), so we conclude that within the mean-field approximation the system is unstable against a lattice modulation for arbitrarily weak electron-phonon coupling. To obtain the actual value of the gap (and the modulation amplitude, see (2.5)) we minimize the total energy, leading to

$$|\Delta| = 2E_F \exp(-1/\lambda), \quad \lambda = \frac{2g^2}{\pi v_F \omega_{2k_F}}. \qquad (2.7)$$

Note that the logarithmic term in (2.6) only appears if the gap is opened exactly at $\pm k_F$, i.e. the wavelength of the lattice modulation is determined by the filling of the electron band and is equal to $\pi/k_F$. The modulation is

$$\langle u(x) \rangle = \sqrt{\left(\frac{2}{\omega_{2k_F}}\right) \frac{|\Delta|}{g}} \cos(2k_F x + \phi), \qquad (2.8)$$

where $\langle \dots \rangle$ denotes the groundstate expectation value and $\phi$ is the phase of $\Delta(= |\Delta| \exp(i\phi))$. Connected with the lattice modulation is a modulation of the electron density, usually termed a charge density wave (CDW).

### 2.1.2. Excited states

What are the excited states of the system? First note that the energy of the system (equations (2.2) and (2.6)) depends only on the amplitude of $\Delta$, not on its phase. A change of the phase corresponds to a translation of the lattice modulation and the CDW relative to the laboratory frame (see (2.8)), i.e. the phase-independence of the energy reflects the translational invariance of the system. Translational invariance holds if the wavelength of the modulation and the underlying lattice constant $a$ are incommensurate. We shall first consider this case in the following. (Strictly speaking, a commensurate situation is realized whenever one can write $n\pi/k_F = ma$ with $m, n$ integer. However, effects of commensurability decrease exponentially with increasing $m$ [14] and therefore need only be considered for small order $m$.) A spatially constant change of the phase does not require energy. Correspondingly, a long-wavelength ($|q| \ll 2k_F$) modulation of the phase (i.e. $\phi \equiv \phi(x) \propto \exp(iqx)$) costs very little energy, i.e. the frequency of long-wavelength oscillations of the phase ('phase mode' or 'phasons') is very low and has been calculated as [14]

$$\omega_{\phi}(q) = \sqrt{\left(\frac{m}{m^*}\right) v_F |q|}, \qquad (2.9)$$

where the CDW effective mass [3, 14] is

$$m^* = m \left( 1 + \frac{4|\Delta|^2}{\lambda \omega_{2k_F}^2} \right) \qquad (2.10)$$

and $m$ is the electron band mass. The phason is the Goldstone mode arising from the broken translational invariance of the original hamiltonian (2.1). Beside the phase, also the amplitude of the modulation may oscillate. However, there is a well-defined equilibrium value of $|\Delta|$, so that there is a finite restoring force acting against deviations from the equilibrium value and the corresponding oscillation frequency of the amplitude mode [14]
$$ \omega_A(q) = \left[ \lambda \omega_{2k_F}^2 = \frac{4m}{3m^*} v_F^2 q^2 \right]^{1/2} \qquad (2.11) $$

has a finite value to $q \to 0$. Both amplitude and phase mode constitute modulations of the underlying lattice of wavenumber $2k_F + q$. Thus the phonon spectrum of the modulated state is schematically given by figure 2. Note, however, that the phase and amplitude modes are not pure lattice modes: in both cases not only the lattice oscillates, but also the phase or amplitude of the CDW, i.e. we have collective oscillations of both the lattice and the electron density. In fact it is just the collective nature of the excitations which leads to the strong deviation of the $2k_F$-phonon frequencies from their bare ($g = 0$) values.

The translational invariance of the system has an interesting consequence, originally found by Fröhlich [3]: the coupled lattice-modulation-CDW system can move with constant velocity $v(\phi = vt)$ through the crystal. Such a state carries a d.c. current, i.e. in spite of the gap in the electronic spectrum there is a metal-like (ideally infinite) conductivity ('Fröhlich-conductivity'). Originally, Fröhlich suggested this mechanism as an explanation for superconductivity, due to the impossibility of decay of the current-carrying state into electron-hole pairs. It is by now however clear that the possibility to move the CDW freely is suppressed in real crystals by various mechanisms, which introduce phase-dependent terms in the energy of the system.

(i) Impurities locally favour a certain value of the phase (impurity pinning) [14–16].

(ii) In a commensurate system the phase has a preferred value, the CDW 'locks' into the underlying lattice [14, 17].

(iii) In real quasi-one-dimensional substances an incommensurate situation is often due to partial charge transfer from a donor to an acceptor chain, so that there are positively and negatively charged CDWs on donor and acceptor chains, respectively. Coulomb forces between the two types of chain then make the motion of the CDWs in opposite directions, necessary to obtain a net current, impossible [14].

<figure><img src="image_3.png" /><figcaption>Figure 2. Phonon spectrum of the Peierls modulated system in an extended zone scheme (full line). The dashed line indicates the phonon dispersion of the non-interacting system. $k_F$ is the Brillouin zone boundary of the modulated state.</figcaption></figure>
In all of the above cases the phase mode frequency goes to a finite value in the limit $q \to 0$ (for the impurity pinning case there is a distribution of frequencies, coming from the disorder), which corresponds to oscillations of the phase around a preferred value. Instead of the Fröhlich d.c. conductivity there is a maximum of the a.c. conductivity at that frequency, and a large zero-frequency dielectric constant.

Apart from the above described collective excitations also single-electron excitations across the gap are expected to exist, as in an ordinary semiconductor. For a fixed constant modulation these are stable excited states. However, the solution $\Delta = \text{const.} \neq 0$ was obtained self-consistently above requiring that all states below the gap are filled, all states above the gap are empty. A change of the occupation of the electron states therefore should also lead to a change in the lattice modulation. A detailed analysis of this effect [18–20] (valid in the adiabatic limit $m^*/m \gg 1$) shows that an electron above the gap is indeed unstable and decays into a localized state, very similar to small-polaron formation [21, 22]. The energy of this state is $2|\Delta|/\pi$, and the lattice modulation takes the form

$$ \langle u(x) \rangle = \sqrt{\left(\frac{2}{\omega_{2k_F}}\right) \frac{|\Delta|}{g} \tanh\left(\frac{x}{\xi}\right) \cos(2k_F x)}, \quad \xi = \frac{v_F}{|\Delta|}, \qquad (2.12) $$

i.e. the amplitude changes sign in a localized region of approximate length $2\xi$. This state is therefore called an amplitude soliton. The excited electron is localized to within $2\xi$, however, its additional charge density is completely compensated by change of the charge density of the electrons in states below the gap, i.e. the state has charge 0 and spin $\frac{1}{2}$. This result is valid for a single incommensurate chain. In a real system interchain coupling changes the properties of the state considerably and specifically may lead to a finite local charge [18]. We also remark that for a second order commensurate system ($k_F = \pi/2a$) states with charge $\pm e$ and 0 are also possible [18, 20, 23].

### 2.1.3. Finite temperature

The above description of the Peierls-ordered state, given for $T=0$, remains qualitatively valid (in a mean-field description) if one goes to small finite temperatures as long as the long-range order is not destroyed. However, at sufficiently high temperatures, complete destruction of the order is expected: thermally excited electrons above the gap have energy $\approx |\Delta|$, thus these electrons gain energy by decreasing the gap. A decrease of $|\Delta|$ facilitates excitation of further electrons across the gap, and by this mechanism one expects the gap and the long-range order to disappear completely above a certain temperature.

To obtain more insight into the properties of the system, we consider free (Bloch) electrons in a weak external potential

$$ \begin{aligned} H &= \sum_{ks} \varepsilon_k a_{ks}^{\dagger} a_{ks} + \int dx V(x, t) n(x) \\ &= \sum_{ks} \varepsilon_k a_{ks}^{\dagger} a_{ks} + \frac{1}{L} \sum_{kqs} V(q, t) a_{k+q,s}^{\dagger} a_{ks}, \end{aligned} \qquad (2.13) $$

where in the second line we have used the Fourier representations of the potential $V$ and the density $n(x)$:
$$V(x, t) = \frac{1}{2\pi} \int dq V(q, t) \exp(iqx) = \frac{1}{(2\pi)^2} \int dq \, d\omega V(q, \omega) \exp[i(qx - \omega t)], \quad (2.14 a)$$

$$n(x) = \frac{1}{L} \sum_{kqs} a_{k-q,s}^{+} a_{ks} \exp(iqx) = \frac{1}{2\pi} \int dq n_q \exp(iqx). \quad (2.14 b)$$

Strictly speaking, for Bloch electrons instead of the simple exponential function a product of Bloch functions should appear in (2.14 b). This modification is however unimportant for the following.

To lowest order the change of electronic density due to the potential is obtained from linear response theory [24] as

$$\begin{aligned} \delta\langle n(x, t) \rangle &= \frac{1}{(2\pi)^2} \int d\omega dq \langle n_{q\omega} \rangle \exp[i(qx - \omega t)] \\ \langle n_{q\omega} \rangle &= \Pi(q, \omega) V(Q, \omega) \end{aligned} \quad (2.15)$$

where the complex polarizability is

$$\begin{aligned} \Pi(q, \omega) &= -i \int dx dt \exp[-i(qx - \omega t)] \theta(t) \langle [n(x, t), n(0, 0)] \rangle \\ &= \frac{2}{L} \sum_k \frac{f(\varepsilon_k) - f(\varepsilon_{k-q})}{\varepsilon_k - \varepsilon_{k-q} - \omega + i0^+}, \end{aligned} \quad (2.16)$$

$f(x) = [\exp(x/T) + 1]^{-1}$ is the Fermi distribution function, and $\theta(t)$ is the unit step function. As shown in the first part of this section, at $T = 0$ the electron-phonon system is unstable against a modulation of wavenumber $2k_F$. For the vicinity of this value, one obtains from (2.16)

$$\Pi(\pm 2k_F + q, \omega) = -\frac{1}{\pi v_F} \left\{ \ln\left(\frac{2C}{\pi} \frac{E_F}{T}\right) + \psi\left(\frac{1}{2}\right) - \frac{1}{2} \left[ \psi\left(\frac{1}{2} + \frac{i(v_F q - \omega)}{4\pi T}\right) + \psi\left(\frac{1}{2} - \frac{i(v_F q + \omega)}{4\pi T}\right) \right] \right\}, \quad (2.17)$$

valid for $|v_F q|, \omega, T \ll E_F$ and obtained from a linearized electron energy. $\psi(x)$ is the digamma function [25] and $C = 1.781$. We remark that $\Pi(\pm 2k_F, 0)$ diverges for $T \to 0$, i.e. even a very small potential of wavenumber $2k_F$ creates a large modulation of the density. The origin of the divergence is quite clear from (2.16): for $q = 2k_F$, $\omega = 0$, all states near the Fermi level give rise to a very small denominator in the summation. This is not so for two- or three-dimensional systems (apart from special cases, see section 1.3). Thus, the reason for the divergence of $\Pi$ is the same as the origin of the Peierls instability: all states near the Fermi level differing by $2k_F$ in wavenumber are nearly degenerate.

We now look at the finite temperature lattice dynamical properties of our original model (2.1). The equation of motion for a Fourier component of the displacement is

$$\left(-\frac{\partial^2}{\partial t^2} - \omega_q^2\right) [b_q(t) + b_{-q}^{+}(t)] = \frac{2g\omega_q}{\sqrt{L}} n_q(t), \quad (2.18 a)$$

or, in Fourier representation

$$(\omega^2 - \omega_q^2) [b_{q\omega} + b_{-q\omega}^{+}] = \frac{2g\omega}{\sqrt{L}} n_{q\omega}, \quad (2.18 b)$$
i.e. the electronic density acts as a driving force on the lattice. On the other hand, of course the electronic density depends on the displacement: note that setting

$$V(q, t) = g\sqrt{L}[b_q(t) + b_{-q}^{+}(t)] \quad (2.19)$$

the last terms in (2.1) and (2.13) become identical, i.e. the displacement acts as an external potential on the electrons. From our above analysis, to lowest order the relation between displacement and density is then given by (cf. (2.15))

$$n_{q\omega} = g\sqrt{L\Pi(q, \omega)}[b_{q\omega} + b_{-q\omega}^{+}]. \quad (2.20)$$

Inserting into (2.18 b) we obtain a closed equation:

$$[\omega^2 - \omega_q^2 - 2g^2\omega_q\Pi(q, \omega)][b_{q\omega} + b_{-q\omega}^{+}] = 0. \quad (2.21)$$

This procedure corresponds to the random-phase approximation. The phonon frequencies of our system can now be obtained from the zeros at the first factor in (2.21). Looking at the vicinity of the unstable (at $T = 0$) mode $k = 2k_F$ one obtains the renormalized frequencies $\tilde{\omega}_k$ as [26, 27]

$$\tilde{\omega}_{2k_F+q} = -i\frac{\gamma}{2} \pm \left[ \tilde{\omega}_0^2(2k_F+q) - \frac{\gamma^2}{4} \right]^{1/2}, \\ 
\tilde{\omega}_0^2(2k_F+q) = \omega_{2k_F}^2 \left\{ 1 + \lambda \left[ \ln\left(\frac{\pi T}{2CE_F}\right) + \frac{7\zeta(3)\nu_F^2 q^2}{16\pi^2 T^2} \right] \right\}, \\ 
\gamma = \lambda\omega_{2k_F}^2 \frac{\pi}{8T} \quad (2.22)$$

valid for $|\omega|, |v_F q| \ll \pi T$. The dispersion of the bare phonons around $2k_F$ is usually small and has therefore been neglected, and $\zeta(x)$ is Riemann's zeta function [25]. Owing to the possibility of decay into electron-hole pairs, the phonons now have a finite lifetime $\gamma^{-1}$. More remarkable is the development of a sharp dip of the phonon frequency at wavenumber $2k_F$ (figure 3). This is the well-known Kohn anomaly [28], especially strong in one dimension [29] due to the possibility of exciting all electrons from one side of the Fermi distribution to the other with a single wave-vector and very little energy required. In this context one should notice that the phonons around wavenumber $2k_F$ do not only involve lattice displacements, but should rather be considered as strongly coupled collective oscillations of both the lattice and the electron density (cf. (2.20)). It is just this strong coupling which is responsible for the dip in the phonon dispersion. With decreasing temperature the dip becomes deeper and deeper, and at a critical temperature $T_c^\circ$ one finally has $\tilde{\omega}_{2k_F} = 0$, indicating the instability of the homogeneous system against a modulation of wavenumber $2k_F$ (complete lattice softening). From (2.22) one obtains

$$T_c^\circ = \frac{2C}{\pi} E_F \exp\left(-\frac{1}{\lambda}\right) \approx 1.13 E_F \exp\left(-\frac{1}{\lambda}\right). \quad (2.23)$$

Of course the same result can also be obtained looking at the destabilization of the Peierls ordered state by electron excitation across the gap [26, 30, 31].

What do impurities do to the Peierls transition? We add a new term

$$H_{\text{imp}} = \frac{1}{L} \sum_{kqls} U_q a_{k+q,s}^{\dagger} a_{ks} \exp(-iqR_l)p_l \quad (2.24)$$
<figure><img src="image_4.png" /><figcaption>Figure 3. Development of the one-dimensional Kohn anomaly with decreasing temperature ($T_3 > T_2 > T_1 > T_c^o$).</figcaption></figure>

to our original hamiltonian (2.1) which describes scattering of electrons by non-magnetic impurities. The probability of having an impurity at site $l$ is $p_l$ and $U_q$ is a Fourier component of the impurity potential. As the Peierls instability involves coupling between states at $+k_F$ and states at $-k_F$, scattering which mixes these two states, i.e. $|q| \approx 2k_F$, is expected to have an important effect. From the Born approximation the transition temperature in the presence of such 'backward scattering' is given by [27, 32]

$$ \ln \frac{T_c}{T_c^o} = \psi\left(\frac{1}{2}\right) - \psi\left(\frac{1}{2} + \frac{1}{4\pi T_c \tau}\right), \quad (2.25) $$

where the scattering time is $\tau = v_F/(n|U_{2k_F}|^2)$, $n$ is the impurity concentration and $T_c^o$ the transition temperature in the absence of impurities. Thus for weak scattering the transition temperature decreases linearly, but at sufficiently strong scattering ($\tau^{-1} > 4\pi T_c^o$) the transition is completely destroyed. On the other hand, scattering with small momentum transfer ('forward scattering') only leads to a spatial modulation of the phase of the electronic wavefunctions and therefore does not influence the transition temperature at all [33]. However, in the Peierls ordered state the phase of the modulation now is no more than a constant, but varies in space, like the phase of the electron wavefunctions.

The above described mean-field theory of the Peierls transition bears many similarities to the BCS theory [2] of superconductivity:

(i) in both cases a gap is opened on the whole Fermi surface, leading to a rapid lowering of the energy of the electrons,

(ii) in both cases the instability is signalled by a logarithmically diverging correlation function of the non-interacting system,

(iii) the results for the transition temperature and the gap at $T = 0$ only differ by a different prefactor in the two cases (coming from the cutoff in the interaction at the Debye frequency in the BCS theory), both theories give $2|\Delta(T = 0)|/T_c = 3.5$,

(iv) the effect of backscattering impurities in the Peierls system is analogous to the pairbreaking effect of magnetic impurities in a superconductor [34].

However, there is one important fact, which makes all the difference between the two cases: the BCS superconducting state is characterized by bound pairs of two
electrons in time-reversed states. This leads to off-diagonal long-range order (ODLRO) in the two-particle density matrix, which is the condition for superconductivity [35]. On the other hand, the Peierls state is formed by bound electron-hole pairs, leading to diagonal long-range order, i.e. order of crystalline type [35]. Consequently, superconductivity is stable against weak perturbations, whereas the analogue in the Peierls state, the Fröhlich conductivity, is easily suppressed by a variety of mechanisms.

The characteristic properties of the Peierls state like the lattice modulation, the electronic gap and the absence of any magnetic ordering (which should be obvious from this section) as well as the precursor lattice softening have been observed experimentally in a large number of systems, some typical examples of which will be discussed in section 3. Due to the various pinning mechanisms, the Fröhlich conductivity in these systems usually occurs at far infrared frequencies. However, there is one spectacular exception (which lies somewhat away from the subject of this article and is mentioned only briefly here): in the quasi-one-dimensional compound niobium triselenide (NbSe$_3$) the pinning forces are extremely weak and can be overcome by very small fields [36] of the order of 10 mV cm$^{-1}$, giving rise to a large (though not infinite) Fröhlich d.c. conductivity, at finite electric field.

The above discussion, based on mean-field theory, leads to the establishment of long-range order below the critical temperature, in contradiction with general theorems [6, 7] on the absence of long-range order at finite temperature in a one-dimensional system. Actually, the destruction of long-range order is inherent in our results above: at any non-zero temperature there are thermal fluctuations of the order parameter. At low temperatures the most important fluctuations come from low-energy excitations, which are, in our case, the phasons having $\omega_\phi(q) \propto |q|$ for small $|q|$ (all other excitations have a finite energy $|q| \to 0$ and are frozen out). To a good approximation the phasons can be considered as harmonic, so that the thermal average of the order parameter and its correlation function, including phase fluctuations are

$$ \langle \Delta \rangle = |\Delta| \langle \exp(i\phi) \rangle = |\Delta| \exp\left(-\frac{1}{2} \langle \phi^2 \rangle\right) \quad (2.26 a) $$

and

$$ \langle \Delta(x) \Delta^*(0) \rangle = |\Delta|^2 \exp\left\{-\frac{1}{2} \langle [\phi(x) - \phi(0)]^2 \rangle\right\}, \quad (2.26 b) $$

respectively. Now (2.9) implies that the energy of a deformation of the phase of wavenumber $k$ is proportional to $k^2$, so that one has

$$ \langle \phi^2 \rangle = AT \int \frac{dk}{k^2} \quad (2.27 a) $$

$$ \begin{aligned} \langle [\phi(x) - \phi(0)]^2 \rangle &= 2AT \int \frac{dk(1 - \cos kx)}{k^2} \\ &= 2\pi AT|x| \end{aligned} \quad (2.27 b) $$

with some (temperature independent) proportionality constant $A$. The integral (2.27 a) diverges at small $k$, i.e. there are infinite fluctuations of the phase of the order parameter, $\langle \phi^2 \rangle = \infty$. From (2.26 a) one finds $\langle \Delta \rangle = 0$, the diverging phase fluctuations have completely destroyed long-range order [7]. From (2.27 b) one sees
that the correlations decay exponentially at large distances, however at low temperatures short-range order is established on a length scale (the correlation length) increasing like $T^{-1}$.

The argument applies to incommensurate systems, where a continuous translational symmetry is broken, so that there is a Goldstone mode having $\omega_{\phi}(q) \to 0$ as $|q| \to 0$. On the other hand, in a commensurate state a discrete symmetry is broken, so that the phason has $\omega_{\phi}(q) \to 0$ for $q \to 0$ and the integral corresponding to (2.27) remains finite. However, in this case there are, for mth order commensurability, m degenerate groundstate configurations which differ only by $2\pi/m$ in their phases. A domain wall between two different structures then has a finite energy but a diverging (in the thermodynamic limit) entropy [6], i.e. at any finite temperature a large number of domain walls is created, leading to a destruction of long-range order. In this case the correlation length increases exponentially at low T. Further discussion of the effect of strong fluctuations on physical properties will be given in section 4.4.

We finally remark that in a real quasi-one-dimensional system, consisting of many parallel chains, there is always some coupling between adjacent chains, so that the denominator of the integral (2.27) is proportional to $|\mathbf{k}|^2$ (possibly with a very anisotropic prefactor), and the numerator becomes $d^3k \propto |\mathbf{k}|^2dk$. Thus $\langle\phi^2\rangle$ remains finite, i.e. long-range order is stabilized by interchain coupling (see section 4.3). The above mean-field discussion of the Peierls state should then be applicable, at least qualitatively.

## 2.2. Competing instabilities in a one-dimensional interacting electron gas

### 2.2.1. Mean-field theory

Consider a model of one-dimensional interacting fermions described by the hamiltonian [37]

$$H = H_0 + H_{\text{int}}, \qquad (2.28 a)$$

$$H_0 = \sum_{ks} v_F [(k - k_F)a_{ks}^{+}a_{ks} + (-k - k_F)b_{ks}^{+}b_{ks}] \qquad (2.28 b)$$

$$H_{\text{int}} = \frac{1}{L} \sum_{\substack{k_1,k_2,q \\ s,s'}} \left\{ \begin{aligned} & g_1 a_{k_1 s}^{+} b_{k_2 s'}^{+} a_{k_2 + 2k_F + q, s'} b_{k_1 - 2k_F - q, s} + g_2 a_{k_1 s}^{+} b_{k_2 s'}^{+} b_{k_2 + q, s'} a_{k_1 - q, s} \\ & + \frac{g_3}{2} (a_{k_1 s}^{+} a_{k_2 s'}^{+} b_{k_2 - 2k_F + q, s'} b_{k_1 - 2k_F - q, s} + b_{k_1 s}^{+} b_{k_2 s'}^{+} a_{k_2 + 2k_F + q, s'} a_{k_1 + 2k_F - q, s}) \\ & + \frac{g_4}{2} (a_{k_1 s}^{+} a_{k_2 s'}^{+} a_{k_2 + q, s'} a_{k_1 - q, s} + b_{k_1 s}^{+} b_{k_2 s'}^{+} b_{k_2 + q, s'} b_{k_1 - q, s}) \end{aligned} \right\}. \qquad (2.28 c)$$

In the free-electron term $H_0$ the electron dispersion has been linearized around $\pm k_F$, and both branches have been extended to infinity (figure 4(a)), an approximation first introduced by Luttinger [38]. The a and b operators describe electrons moving to the right and to the left, respectively. Both the linearization and the introduction of new, unphysical states ($a_{ks}$ with $k < 0$ and $b_{ks}$ with $k > 0$) change the system properties at high energies (of the order of the Fermi energy). However, for properties depending on states near the Fermi level, which we are interested in, no qualitative changes are expected. The interaction between electrons near the Fermi level is parameterized by four constants $g_1, \dots, g_4$, represented diagrammatically in figure 4(b). The $g_1$-process describes electron-electron interaction with momentum
<figure><img src="image_5.png" /><figcaption>Figure 4. (a) Linearized electron dispersion in the Luttinger approximation. (b) Diagrammatic representation of the interactions $g_1...g_4$. Full and dashed lines represent electrons near $k_F$ and $-k_F$ respectively.</figcaption></figure>

transfer $\pm 2k_F$ (backward scattering), the $g_2$- and $g_4$-processes have small momentum transfer (forward scattering) involving either electrons at both $k_F$ and $-k_F$ ($g_2$) or only on one side ($g_4$). In the $g_3$-process the total electron momentum is changed by $4k_F$ so that this process is only allowed if $4k_F$ is a reciprocal lattice vector (umklapp scattering), i.e. for a half-filled band. The interaction constants in (2.28 c) are independent of the transferred momentum. Though this is surely a reasonable approximation for the large momentum-transfer processes ($g_1$ and $g_3$), the forward scattering ($g_2$ and $g_4$) contains the long-range part of the Coulomb interaction, which is strongly wavenumber dependent. Whether this interaction is described correctly by an effective (screened) constant is not yet fully clear.

To obtain some qualitative understanding, let us use the mean-field approximation to look for ordered states of the system [39]. We look for the possibility of opening a gap at the Fermi level by a macroscopic expectation value of some operator, so that an ordered state would form, just as in section 2.1. There are actually eight such operators:

$$ (I) \qquad O_{\text{CDW}}(q) = \sum_{ks} b_{-k_F+k-q,s}^{+} a_{k_F+k,s}, \qquad (2.29 \text{ a}) $$

$$ O_{\text{SDW}\alpha}(q) = \sum_{kss'} b_{-k_F+k-q,s}^{+} \sigma_{\alpha}^{ss'} a_{k_F+k,s'} \quad (\alpha = x, y, z), \qquad (2.29 \text{ b}) $$

where $\sigma_{\alpha}^{ss'}$ is the $(s, s')$-element of the Pauli spin matrix, e.g. $\sigma_x^{+-} = 1$, $\sigma_y^{-+} = i$, $\sigma_z^{+-} = 0$, etc. The operators $O_{\text{CDW}}(q)$ and $O_{\text{SDW}}(q)$ are the $(2k_F + q)$-Fourier components of the charge- and the $\alpha$-component of the spin densities, respectively.

$$ (II) \qquad O_{\text{SS}}(q) = \sum_{ks} sb_{-k_F-k,-s} a_{k_F+k+q,s} \qquad (2.30 \text{ a}) $$

$$ O_{\text{TS}\alpha}(q) = \sum_{kss'} sb_{-k_F-k,-s} \sigma_{\alpha}^{ss'} a_{k_F+k+q,s'}, \quad (\alpha = x, y, z) \qquad (2.30 \text{ b}) $$

are the singlet ($O_{\text{SS}}$) and the three possible triplet ($O_{\text{TS}}$) Cooper pair operators, where a non-zero $q$ would characterize an inhomogeneous superconducting state.

For $g_3 = g_4 = 0$ the interaction can be written entirely in terms of the above operators (apart from a renormalization of the chemical potential):
$$H_{\text{int}} = \frac{1}{L} \sum_q \left\{ \left( \frac{1}{2} g_2 - g_1 \right) O_{\text{CDW}}^{+}(q) O_{\text{CDW}}(q) + \frac{1}{2} g_2 \sum_{\alpha} O_{\text{SDW}_{\alpha}}^{+}(q) O_{\text{SDW}_{\alpha}}(q) \right\} \quad (2.31 a)$$

$$= \frac{1}{L} \sum_q \left\{ g_1 + g_2 \right\} O_{\text{SS}}^{+}(q) O_{\text{SS}}(q) + \frac{1}{2} (g_2 - g_1) \sum_{\alpha} O_{\text{TS}_{\alpha}}^{+}(q) O_{\text{TS}_{\alpha}}(q) \quad (2.31 b)$$

Looking for an ordered state of the system, we proceed as in the previous section; such a state would be characterized by a macroscopic expectation value of one of the above operators $O_i(q)$ ($i = \text{CDW, SDW}_\alpha, \text{SS, TS}_\alpha$). Now one can write

$$\begin{aligned} O_i^+(q)O_i(q) &= [\langle O_i^+(q) \rangle + O_i^+(q) - \langle O_i^+(q) \rangle][O_i(q)\rangle + O_i(q) - \langle O_i(q) \rangle] \\ &= O_i^+(q)\langle O_i(q) \rangle + O_i(q)\langle O_i^+(q) \rangle - |\langle O_i(q) \rangle|^2 \\ &\quad + [O_i^+(q) - \langle O_i^+(q) \rangle][O_i(q) - \langle O_i(q) \rangle]. \end{aligned} \quad (2.32)$$

In the mean-field approximation one neglects fluctuations of the order parameter, i.e. the last term in (2.32), as well as all operators which do not have a macroscopic expectation value. Assuming, for example, superconducting order, the hamiltonian is then reduced to

$$H = H_0 + \frac{g_1 + g_2}{2L} \sum_k \left\{ (b_{-k+q\downarrow}^{+} a_{k\uparrow}^{+} + a_{-k+q\downarrow}^{+} b_{k\uparrow}^{+}) \langle O_{\text{SS}}(q) \rangle \right. \\ \left. + (a_{k\uparrow}^{+} b_{-k+q\downarrow} + b_{k\uparrow} a_{-k+q\downarrow}) \langle O_{\text{SS}}^{+}(q) \rangle \right\} - \frac{g_1 + g_2}{2L} |\langle O_{\text{SS}}(q) \rangle|^2. \quad (2.33)$$

This form is very similar to the mean-field Peierls hamiltonian (2.3), and can be diagonalized by a Bogolyubov transformation in the same way. A gap $|\Delta_S|$ at the Fermi surface is opened for $q = 0$, leading for small $|\Delta_S|$ to a logarithmic energy gain, as in (2.6). Minimizing the energy one finds

$$\begin{aligned} |\Delta_{\text{SS}}| &= \frac{1}{2L} |(g_1 + g_2) \langle O_{\text{S}}(0) \rangle| = 2E_{\text{F}} \exp\left(-\frac{1}{\lambda_{\text{SS}}}\right), \\ \lambda_{\text{SS}} &= \frac{g_1 + g_2}{2\pi v_{\text{F}}}, \end{aligned} \quad (2.34)$$

for $g_1 + g_2 < 0$. For $g_1 + g_2 > 0$ no self-consistent solution with a finite $|\Delta_S|$ exists. As in the previous section the transition temperature is

$$T_c = \frac{2C}{\pi} E_F \exp\left(-\frac{1}{\lambda_{\text{SS}}}\right) \quad (2.35)$$

The same argument for the other possible phases leads to a similar result, the only difference being that $\lambda_{SS}$ has to be replaced by

$$\lambda_{\text{CDW}} = -\frac{2g_1 - g_2}{2\pi v_{\text{F}}}, \quad (2.36 a)$$

$$\lambda_{\text{SDW}_\alpha} = \frac{g_2}{2\pi v_F}, \quad (2.36 b)$$

$$\lambda_{\text{TS}_\alpha} = \frac{g_1 - g_2}{2\pi v_F}, \quad (2.36 c)$$
<figure><img src="image_6.png" /><figcaption>Figure 5. Phase diagram of the hamiltonian (2.28)</figcaption></figure>

for charge-density, spin-density, and triplet-superconducting order, respectively. The most stable phase is the one with the highest transition temperature, leading to the phase diagram in the $g_1-g_2$ plane given by figure 5 [39]. Apart from the Peierls-type CDW order, already obtained in the last section, all other of the above phases are realized for appropriate coupling constants. Singlet- (SS, CDW) and triplet- (TS, SDW) phases occur for $g_1 < 0$ and $g_1 > 0$, respectively, and the density-wave and superconducting order are separated by the line $g_1 = 2g_2$. Including umklapp scattering, this line is shifted to $g_1 = 2g_2 + |g_3|$, whereas a finite $g_4$ does not change the phase diagram [39]. Note that along the critical lines both adjacent phases have a finite critical temperature. $T_c = 0$ occurs only for $g_1 = g_2 + |g_3| = 0$.

### 2.2.2. Beyond mean-field theory

Unfortunately, the above picture suffers from a serious drawback: it predicts long-range order at finite temperature, whereas in a one-dimensional system fluctuations are known to destroy long-range order, as discussed in the last section. A consistent many-body treatment of the problem, including terms of higher order than mean-field theory, encounters a problem evident from the mean-field results: There is not only a (charge- or spin-) density wave instability, but also a superconducting instability. Consequently, not only the $2k_F$-density response diverges for $T \to 0$ (cf. (2.17)) but also the Cooper pair susceptibility of non-interacting electrons

$$ 
\begin{aligned}
\Delta(q, \omega) &= -i \int dx dt \exp[-i(qx - \omega t)] \theta(t) \langle [\psi_{\uparrow}(x, t) \psi_{\downarrow}(x, t), \psi_{\downarrow}^{\dagger}(0, 0) \psi_{\uparrow}^{\dagger}(0, 0)] \rangle \\
&= \frac{1}{L} \sum_{k} \frac{n(\varepsilon_k) + n(\varepsilon_{k-q}) - 1}{\varepsilon_k + \varepsilon_{k-q} - \omega - i0^+} \\
&= -\frac{1}{\pi v_F} = -\frac{1}{\pi v_F} \left\{ \ln\left(\frac{2CE_F}{\pi T}\right) + \psi\left(\frac{1}{2}\right) - \frac{1}{2} \left[ \psi\left(\frac{1}{2} - i\frac{\omega - v_F q}{4\pi T}\right) + \psi\left(\frac{1}{2} - i\frac{\omega + v_F q}{4\pi T}\right) \right] \right\}. \quad (2.37)
\end{aligned}
$$

As the density wave susceptibility at $2k_F + q$ (2.16), the pair susceptibility diverges logarithmically for $q = 0$, $\omega = 0$ and $T \to 0$ or $T = 0$, $\omega \to 0$, $q \to 0$. The zero-order contributions to these functions (equations (2.17) and (2.37)) are represented as
<figure><img src="image_7.png" /><figcaption>Figure 6. Contributions to the Cooper pair susceptibility $\Delta(q, \omega)$ and the CDW susceptibility gap $(2k_F + q, \omega)$: (a) lowest order, and (b) a typical diagram showing mixing of the two types of instabilities.</figcaption></figure>

Feynman diagrams in figure 6 (a). The two diagrams only differ in the orientation of the single particle propagators: the density response has two antiparallel lines, corresponding to electron-hole pairs, whereas the Cooper pairing is represented by parallel lines. Problems now arise if one calculates perturbatively the effect of interactions: in the expansion of the Cooper pair response divergent contributions from density fluctuations (represented by antiparallel lines) occur, and vice versa (figure 6(b)), i.e. the two types of instability become coupled. Due to the divergencies, even for very weak interaction this coupling cannot be neglected. The problem was first treated by Bychkov *et al.* [5], and now, from renormalization group methods [40] and some exactly solvable models [41, 42], a fairly complete picture of the properties of a one-dimensional interacting electron gas has emerged. These developments have been reviewed in detail by Solyom [37] and Emery [43], and we only discuss here the principal results.

As expected, these methods show that there is no long-range order at any finite temperature. However, correlation functions defined by

$$R_i(q, \omega) = -i \int dt \exp(i\omega t) \langle T[O_i(q, t) O_i^\dagger(q, 0)] \rangle \quad (2.38)$$

$(O_i(q)$ are the operators defined in (2.29) and (2.30), $T$ is the time ordering operator) are found to behave like

$$R_i(q, \omega) \propto |\omega^2 - v_F^2 q^2|^{-\alpha_i/2} \quad (T=0) \quad (2.39 a)$$

$$\propto T^{-\alpha_i} \quad (\omega=0, q=0) \quad (2.39 b)$$

with $\alpha_i$ depending on $g_1$ and $g_2$. The correlation functions diverge (i.e. $\alpha_i > 0$) in the same regions of the phase diagram where the mean-field approximation indicates long-range order of the corresponding type (figure 5). In addition, for $g_1 > 0$ one finds $\alpha_{SS} = \alpha_{TS}$ and $\alpha_{CDW} = \alpha_{SDW}$. However, a more detailed analysis of the scaling equations [40, 44] shows that for $T \to 0$ one has

$$\frac{R_{\text{TS}}(0,0)}{R_{\text{SS}}(0,0)} \approx \frac{R_{\text{SDW}}(0,0)}{R_{\text{CDW}}(0,0)} \approx \frac{T^{-\alpha_i} [|\ln (T/E_F)|]^{1/2}}{T^{-\alpha_i} [|\ln (T/E_F)|]^{3/2}} = \left[ \ln \left( \frac{T}{E_F} \right) \right]^{-1}, \quad (2.40)$$

i.e. the SS- and CDW-divergencies are logarithmically weaker than the dominant TS- and SDW-divergencies. Still the result is quite surprising: it indicates the existence of strong singlet superconducting correlations even for *purely repulsive*
interactions ($g_1 > 0, g_2 > 0$), in contradiction to the usual BCS-theory [2], which is based on *attractive* electron-electron interactions. However, in a more elaborate theory instead of the bare interactions rather the full interaction, renormalized by many-body effects, should be considered. In one dimension, due to the divergencies discussed above, many-body effects are particularly important, and a first-order renormalization calculation of the effective interaction at energy transfer $\omega$ gives [40]

$$ \tilde{g}_1(\omega) = g_1 / \left[ 1 - \frac{g_1}{\pi v_F} \ln\left(\frac{\omega}{E_F}\right) \right], \quad (2.41 a) $$

$$ \tilde{g}_2(\omega) = g_2 - \frac{1}{2}[g_1 - \tilde{g}_1(\omega)], \quad (2.41 b) $$

i.e. even for repulsive bare interaction the long-range interaction ($\tilde{g}_2$) becomes attractive at low energies for $g_1 > 2g_1$, and the short-range ($\tilde{g}_1$) repulsion becomes extremely weak. Thus the appearance of strong SS-correlations for $g_1 > 0$ may be explained by the attractive *effective* interactions, which arise from repulsive bare interactions. Similar arguments also apply to the CDW correlations coexisting with SDW for $g_1 > 0$. One should also note that for $g_1 < 0$ there is a certain region where both SS- and CDW-correlation functions diverge, however they then have *different* exponents, and only the most divergent correlation is indicated in figure 5.

The power law divergency for $q, \omega \to 0$ of the correlation functions (equation (2.39 a)) implies that the corresponding real-space correlation function decays as a power-law for $x \to \infty$ or $t \to \infty$, i.e. there is no long-range order even at $T = 0$. As in the finite temperature case discussed at the end of the last section this fact may be understood qualitatively starting from the mean-field description. We note that long-wavelength fluctuations of the phase of the (superconducting or density-wave) order-parameter have vanishing energy for vanishing wavenumber, so that the zero-point motion of the phase (i.e. quantum fluctuations) is important at low temperatures [45]. Using, as before, a harmonic approximation the motion of the phase $\phi$ can be described by the hamiltonian

$$ H_{\phi} = \int dz \left\{ \frac{1}{2\rho} \pi^2(z) + \frac{c}{2} \left( \frac{\partial \phi}{\partial z} \right)^2 \right\}, \quad (2.42) $$

where $\pi = \rho\phi$ is the momentum density, and $\rho$ and $c$ are some effective mass density and elastic constants, respectively. The order parameter correlation function is then obtained as

$$ \begin{align} R(x, t) &= -i \langle T[\Delta(x, t) \Delta^*(0, 0)] \rangle \nonumber \\ &= -i |\Delta|^2 \langle T\{\exp[i\phi(xt)] \exp[-i\phi(0, 0)]\} \rangle \nonumber \\ &= -i |\Delta|^2 \left[ \frac{\alpha^2 \pi^2 T^2}{(\alpha + iv|t|)^2 + x^2} \frac{(x^2 - v^2 t^2)}{v^2 \sinh(\pi T/v)(x - vt) \sinh(\pi T/v)(x + vt)} \right]^{\eta/2} \nonumber \\ &\quad \left( v = \sqrt{\left(\frac{c}{\rho}\right)}, \quad \eta = (2\pi\sqrt{c\rho})^{-1} \right), \tag{2.43 a} \\ &\propto |x|^{-\eta} \quad \text{for } |x| \gg \alpha, t = 0, T = 0. \tag{2.43 b} \end{align} $$

Here $\alpha$ is a short-wavelength cutoff. At finite temperature the correlations decay exponentially at large distances (equation (2.43 a)). This is due to classical thermal
fluctuations, as described at the end of the last section. However, in a purely classical picture, there are no fluctuations at $T = 0$, i.e. long-range order exists. Allowing also for quantum fluctuation this long-range order is destroyed (equation (2.43 b)), however, the decay of the correlations is now algebraic, much slower than any exponential behaviour. This situation, occasionally termed ‘quasi-long-range order’, is similar to the low-temperature phase of a two-dimensional classical XY-magnet, with which a formal relationship exists [37, 43].

Fourier transforming (2.43) one obtains

$$R(k, \omega) \propto [\omega^2 - c^2 k^2]^{\eta/2 - 1} \quad \text{at } T = 0 \qquad (2.44 a)$$

and

$$R(k = 0, \omega = 0) \propto T^{\eta-2}, \qquad (2.44 b)$$

i.e. similar to our original model, $H_φ$ produces power law divergencies with varying exponents (compare (2.39)). Actually, an approach based on (2.42) has been shown to describe the low-temperature properties of the system (2.28) for $g_1 < 0$ [45], where in the mean-field approximation an SS or CDW phase is stable and the only low-lying excitations are the phase fluctuations (for a description of the CDW state in terms of phase fluctuations see also reference [16]). Near the separation line $g_1 = 2g_2$ (figure 5) for both the SS and CDW correlations functions $\gamma = 1/2$ is obtained, in agreement with microscopic calculations [37, 43]. On the other hand, in the region $g_1 > 0$, where TS or SDW phases dominate, the description in terms of phase fluctuations is incomplete: the energy of an ordered state is independent of the orientation of the other parameter in spin space (i.e., $SDW_x$, $SDW_y$, and $SDW_z$ are degenerate, and similarly for $TS_α$), so that there is an additional low-energy mode, corresponding to rotations in spin space. Consequently, the correlation functions in general decay faster and for $g_1 \approx 2g_2$ one finds $\gamma = 1$ for all four correlation functions [37, 43]. Finally, for a Peierls system with a large CDW-effective mass (equation (2.10)) one obtains $\gamma \approx 0$ [16], which correspond to the classical limit, i.e. quantum fluctuations are unimportant.

The existence of low-energy modes ($\omega_q \propto |q|$) is related to certain continuous symmetries of the model: the phase modes of the CDW and SDW states come from translational invariance, the phase modes of SS and TS states from particle number conservation and the spin rotation mode is due to the spin independence of the interaction hamiltonian (2.28 c). If one of these continuous symmetries is broken, the corresponding mode has a finite energy for $k \to 0$, so that the long-range quantum fluctuations are suppressed and long-range order becomes possible at $T = 0$: in a commensurate CDW or SDW state there is only a discrete translational invariance, and umklapp ($g_3$) scattering is allowed. In the presence of such a term long-range CDW order may appear [37, 43]. If in addition spin-rotation invariance is broken by spin-dependent interactions, a long-range ordered SDW state is also stable [37, 43]. Of course, thermal fluctuations destroy the long-range order at any finite temperature, as explained at the end of the last section. Particle number is always conserved, so that there is always a phase mode of the SS and TS states and no long-range superconducting order can be established in the strictly one-dimensional system considered here.

Apart from the above, the microscopic treatment of the hamiltonian (2.28) leads to another interesting results: $H$ can be separated into two commuting parts, describing long-wavelength charge- and spin-density oscillations [42]. For $g_1 < 0$
the only low-energy modes in this representation are charge-density oscillations, whereas there is a gap in the spin excitations. Consequently the low temperature spin-susceptibility is activated. On the other hand, for $g_1 > 0$ there are both low-lying charge- and spin-density excitations. In this case there is a finite spin susceptibility at $T = 0$ which increases with increasing temperature [46].

We remark that both the separation into charge- and spin-density excitations and a picture based on the mean-field ordered states lead to the same number of low-frequency modes: one mode for $g_1 < 0$ (long-wavelength charge-density fluctuations and order parameter phase mode, respectively) and two modes for $g_1 > 0$ (long-wavelength charge- and spin-density fluctuations in one case, phase mode and spin rotation in the other). However, a microscopic formulation of this relationship is lacking so far.

Our discussion up to now has concerned the case of relatively weak coupling. For strong interactions, apart from the CDW, SDW, SS and TS instabilities, divergencies in response functions involving more than two particles may appear [47]. Of particular importance for the interpretation of some experimental results (see section 4) is the fact that apart from the $2k_F$-CDW a $4k_F$-CDW instability also occurs for sufficiently strong electron-electron repulsion [48, 49]. This result can be understood by considering a Hubbard model in the limit of very large repulsion $U$ [48]. In that case the strong local repulsion prevents two electrons of opposite spin from occupying the same sites, i.e. there is an 'exclusion principle' for electrons of opposite spin also. Therefore instead of the doubly (spin-) degenerate free-electron band the system can be approximated by a non-degenerate band, with all states between $-2k_F$ and $2k_F$ filled, and the CDW instability thus appears at $4k_F$ instead of $2k_F$.

The model hamiltonian (2.28) describes a system with a direct, non-retarded electron-electron interaction. Of course, similar competition between superconducting and density-wave instabilities also occur for the retarded interaction arising from electron-phonon coupling [50]. A detailed mean-field calculation [51] shows that retardation effects tend to suppress the SS phase and to favour a CDW state. A different approach going beyond mean-field theory is possible within the renormalization group method [52]: the phonon-mediated interaction is cut-off at the Debye frequency, whereas the cutoff for the direct electron-electron interaction is of the order of the Fermi energy. The low-energy properties of the model are then the same as those of model (2.28), however with renormalized interaction constants.

We finally remark that recently, in addition to the well-known exact solution of the one-dimensional Hubbard model [53], various models similar to (2.28) have been solved exactly by the Bethe ansatz technique [54–57]. The method allows an exact calculation of wavefunctions and energies both of the groundstate and of excited states; however, so far the calculation of correlation functions, which contain most of the information on the physical properties of a system, has not been possible.

## 2.3. Interchain coupling

The discussion of the preceding sections was concerned with strictly one-dimensional systems. On the other hand, in real *quasi*-one-dimensional systems, there is always some (weak) interaction between adjacent chains. In this section we want to elucidate the different types of interchain interactions and their influence on the ordering that may occur at low temperature.
We consider a square lattice of $N$ conducting chains parallel to the $z$-direction. The potential exerted by a single chain on an electron is $U(\mathbf{r})$, i.e. $U(\mathbf{r})$ is periodic in the $z$-direction and therefore allows free motion along that direction, but strongly localizes an electron in the $x$- and $y$-directions. The single-electron hamiltonian for the system of parallel chains then is, in first quantization

$$H_0 = -\frac{\Delta}{2m} + \sum_{\mathbf{l}} U(\mathbf{r} - \mathbf{R}_{\mathbf{l}}), \qquad (2.45)$$

with $\mathbf{R}_l = a(l_1, l_2, 0)$, i.e. the two-dimensional vector $\mathbf{l}$ is a chain index, and $a$ is the interchain distance. In going to a second quantized representation we use an orthonormal set of functions which are Bloch functions with respect to the longitudinal (along $z$) motion, but localized on a single chain (i.e. Wannier functions with respect to the transverse directions). The electron creation operator then is

$$\psi_s^+(\mathbf{r}) = \sum_{k_z, \mathbf{l}_s} a_{k_z, \mathbf{l}_s}^+ \psi_{k_z, \mathbf{l}}^*(\mathbf{r}), \qquad (2.46)$$

i.e. the operator $a_{k_z, \mathbf{l}_s}^+$ creates an electron of wave-vector $k_z$ and spin $s$ on the $\mathbf{l}$th chain. In (2.46) only states belonging to a single band are involved, which we will assume to be the conduction band.

One should note that the functions $\psi_{k_z, \mathbf{l}}$ are not eigenfunctions of the single-chain hamiltonian defined by

$$\left[-\frac{\Delta}{2m} + U(\mathbf{r} - \mathbf{R}_{\mathbf{l}})\right] \phi_{k\mathbf{l}}(\mathbf{r}) = \varepsilon_{k}^{\circ} \phi_{k\mathbf{l}}(\mathbf{r}). \qquad (2.47)$$

The $\phi_{k\mathbf{l}}$ are not orthogonal for $\mathbf{l} \neq \mathbf{l}'$. However, using a tight-binding approximation for the transverse motion [58], so that only states of a single energy band of the single-chain problem (2.46) are involved in forming the conduction band of the hamiltonian (2.45), by orthogonalizing the functions $\phi_{k\mathbf{l}}$ one obtains the relation

$$\psi_{k_z, \mathbf{n}}(\mathbf{r}) = \frac{1}{N} \sum_{\mathbf{k}_\perp, \mathbf{l}} C(\mathbf{k}_\perp, k_z) \exp(i\mathbf{k}_\perp \cdot \mathbf{R}_{\mathbf{l}-\mathbf{n}}) \phi_{k_z, \mathbf{l}}(\mathbf{r}), \qquad (2.48 a)$$

$$C(\mathbf{k}_{\perp}, k_z) = \left[ \sum_{\mathbf{m}} \exp(-i\mathbf{k}_{\perp}\cdot\mathbf{m}) \int \phi_{k_z, \mathbf{m}}^{*}(\mathbf{r}) \phi_{k_z, 0}(\mathbf{r}) d^3\mathbf{r} \right]^{-1/2} . \qquad (2.48 b)$$

In general the single-chain functions are expected to be strongly localized, so that all terms in (2.48 b) with $m \neq 0$ terms may be neglected, so that one obtains $\psi_{k_z, \mathbf{l}} = \phi_{k_z, \mathbf{l}}$. The second quantized form of $H_0$ becomes

$$H_0 = \sum_{k_z, \mathbf{l}} \varepsilon_{k_z} a_{k_z, \mathbf{l}}^{+} a_{k_z, \mathbf{l}} - \sum_{\substack{\mathbf{m} \\ \mathbf{m} \neq \mathbf{l}}} t_{\mathbf{m}} a_{k_z, \mathbf{l}+\mathbf{m}}^{+} a_{k_z, \mathbf{l}} \qquad (2.49 a)$$

with

$$\varepsilon_{k_z} = \varepsilon_{k_z}^{\circ} + \langle k_z | \sum_{\mathbf{m} \neq \mathbf{l}} U(\mathbf{r} - \mathbf{R}_{\mathbf{m}}) | k_z \rangle, \qquad (2.49 b)$$

$$t_{\mathbf{m}} = -\langle k_z | \mathbf{m} + \mathbf{l} | \sum_{\mathbf{n} \neq \mathbf{l}} U(\mathbf{r} - \mathbf{R}_{\mathbf{n}}) | k_z \rangle, \qquad (2.49 c)$$

and $|k_z\mathbf{l}\rangle$ is a single-chain eigenstate (cf. (2.47)).
A non-zero value of $t_m$ in (2.49 a) means that there is a finite probability for an electron to jump from chain **I** to chain **I** + **m**, i.e. transverse electron motion is possible due to the second term in (2.49 a).

We retain the largest of the transverse transfer integrals $t_m$, namely the nearest-neighbour term ($t_⊥$). Going to a Bloch representation also with respect to the transverse motion:

$$a_{k_z \mathbf{l}_s} = \frac{1}{\sqrt{N}} \sum_{\mathbf{k}_{\perp}} a_{\mathbf{k}_{\perp} s} \exp(i\mathbf{k}_{\perp} \cdot \mathbf{R}_{\mathbf{l}_s}) \quad (\mathbf{k} = (\mathbf{k}_{\perp}, k_z)) \quad (2.50)$$

one finally obtains

$$H_0 = \sum_{\mathbf{k}_s} \varepsilon(\mathbf{k}) a_{\mathbf{k}_s}^{+} a_{\mathbf{k}_s} \quad (2.51 a)$$

with

$$\varepsilon(\mathbf{k}) = \varepsilon_{k_z} + \varepsilon_{\perp}(\mathbf{k}) \approx v_F(|k_z| - k_F^{\circ}) + C(|k_z| - k_F^{\circ})^2 + \varepsilon_{\perp}(\mathbf{k}), \quad (2.51 b)$$

$$\varepsilon_{\perp}(\mathbf{k}) = -2t_{\perp}(\cos k_x a + \cos k_y a), \quad (2.51 c)$$

where in (2.51 b) we have expanded $\varepsilon_{k_z}$ around the Fermi wavenumber $k_F^{\circ}$ in the absence of the transverse motion ($t_{\perp} = 0$). Assuming a tight-binding form for the longitudinal motion $\varepsilon_{k_z} = -2t_{\parallel} \cos k_z b$ ($b$ is the longitudinal lattice constant) one has $v_F = 2t_{\parallel}b \sin \pi\rho/2$, $C = 2t_{\parallel}b^2 \cos \pi\rho/2$, and $\rho$ is the number of electrons per unit cell. Of course, in (2.51 a) as well as before, wave-vector summations are over the first Brillouin zone. The Fermi surface resulting from (2.51) is given by

$$\mathbf{k}_{F_z}(\mathbf{k}_{F_{\perp}}) = \pm \left\{ k_F^{\circ} - \frac{2\varepsilon_{\perp}(\mathbf{k}_{F_{\perp}})}{v_F} \left[ 1 + \sqrt{1 - \frac{4C\varepsilon_{\perp}(\mathbf{k}_{F_{\perp}})}{v_F^2}} \right]^{-1} \right\} \quad (2.52)$$

shown in figure 7. The planar surfaces of a strictly one-dimensional ($t_{\perp} = 0$) situation become modulated by a finite $t_{\perp}$, but the Fermi surface remains open in the transverse directions as long as $t_{\perp}$ is not comparable to the Fermi energy.

What now is the effect of transverse electron motion on the instabilities of a one-dimensional system? As discussed in the first section, the Peierls instability is triggered by the low-temperature divergency of the polarizability $\Pi(2k_F, 0)$ (equa-

<figure><img src="image_8.png" /><figcaption>Figure 7. Cut at $k_y = 0$ through the Brillouin zone of a system of weakly coupled chains. The shaded area indicates occupied electron states.</figcaption></figure>
tion (2.16)), which in turn is due to the vanishing energy denominator in (2.16) for all states at the Fermi surface. In the present situation the energies (2.51) have to be used in (2.16) (generalized to three-dimensional **q**). Then, looking for example at **q**<sub>⊥</sub> = (2k<sup>o</sup><sub>F</sub>, 0, 0), only a very small number of terms have a vanishing energy denominator, most terms have a denominator of the order *t*. Consequently, with decreasing temperature, Π(*q*, ω) ceases to diverge at *T* ≈ *t*<sub>⊥</sub>, i.e. the Peierls instability is strongly suppressed. However, as first pointed out by Horovitz et al. [59], a special situation occurs for **q**<sub>0</sub> = (2k<sup>o</sup><sub>F</sub>, π/a, π/a). Let us assume *C* = 0. In that case states at **k** on the left part of the Fermi surface (figure 7) and at **k** + **q**<sub>0</sub> on the right part have both zero energy, and more generally one has ε(**k**) = −ε(**k** + **q**<sub>0</sub>) a situation referred to as 'nesting' of the Fermi surface. Consequently, the denominator in equation (2.16) vanishes for all states at the Fermi surface. For finite C(∞ *t*<sub>‖</sub>) there are deviations from exact nesting which are of order (*t*<sub>⊥</sub>/*t*<sub>‖</sub>)<sup>2</sup> and the typical energy denominators in (2.16) are then *t*<sub>⊥</sub><sup>2</sup>/*t*<sub>‖</sub> ≪ *t*<sub>⊥</sub>. Consequently, Π(**q**<sub>0</sub>, 0) only ceases to diverge at *T* ≈ *t*<sub>⊥</sub><sup>2</sup>/*t*<sub>‖</sub>, i.e. at much lower temperatures than Π(**q**<sub>1</sub>, 0), so that a relatively weak interaction is sufficient to obtain an ordered state at low temperatures, and only for *t*<sub>⊥</sub> ≳ *v*<sub>F</sub>√(*T*<sub>c</sub><sup>o</sup>(*t*<sub>‖</sub> = 0)/*C*) is the *q*<sub>0</sub>-Peierls ordering suppressed by 'three-dimensionality' [59]. The ordered state then has modulation wave-vector (2k<sup>o</sup><sub>F</sub>, π/a, π/a), i.e. apart from the modulation in the longitudinal direction, the lattice constant is also doubled in the transverse directions, the CDWs on adjacent chains are in antiphase. We note that in the presence of deviations from nesting the electronic states coupled by the modulation lie not all exactly at the Fermi level. For a *q*<sub>0</sub>-modulation the energy difference is of order *t*<sub>⊥</sub><sup>2</sup>/*t*<sub>‖</sub>, so that the metallic Fermi surface is completely destroyed as soon as one has Δ ≈ *t*<sub>⊥</sub><sup>2</sup>/*t*<sub>‖</sub>. On the other hand, a much larger value Δ ≳ *t*<sub>⊥</sub> is necessary to destroy the Fermi surface by a *q*<sub>1</sub>-modulation.

As the Peierls instability, spin-density wave formation is due to electron-hole
bound pairs (with a different spin orientation), and the opening of a gap at the Fermi
surface. Consequently, the above discussion applies in precisely the same way to the
SDW case. Specifically, the interchain motion of the electrons leads to an antiphase
ordering of the SDW state in the transverse directions.

Though particularly easy to achieve in a quasi-one-dimensional system, nesting
may also occur in some isotropic three-dimensional cases. The antiferromagnetism
of chromium has been explained by nesting [60], and a tight-binding-type band-
structure on some cubic lattices also shows this phenomenon [61]. On the other hand,
if quasi-one-dimensional conductors consist of alternating chains with electron-
and hole-conduction, respectively, as in the TTF-TCNQ family (see section 4), the
coupling between unlike chains will usually completely destroy the nesting, i.e.
nesting is essentially restricted to conductors with only one type of conducting
chains.

Clearly, not only CDW and SDW order on individual chains are coupled by the
possibility of interchain electron motion, but also superconductivity: the possibility
of individual electron motion from one chain to the next also implies that Cooper
pairs can move coherently between chains. Actually, the coupling between two
chains is (at small *t*<sub>⊥</sub>) completely analogous to Josephson tunnelling between two
superconductors separated by an insulating barrier [62]. The coupling then favours
equal phases of the superconducting order parameters on adjacent chains. Of course,
nesting is not relevant for superconductivity: the denominator of the corresponding
susceptibility Δ(0,0) (equation (2.37)) contains the sum ε**k** + ε<sub>-**k**</sub>, which, by time
reversal symmetry, always vanishes at the Fermi level, independent of dimension-
ality. However, at sufficiently low temperatures the Peierls type divergencies are
suppressed by a finite $t_⊥$, so that the coexistence problem discussed in section 2.2
disappears and only superconducting type instabilities need to be considered, albeit
with renormalized coupling constants [63, 64, 51].

Coupling between order on adjacent chains may also come from the interaction between particles. If this interaction is given by a pair potential $V(\mathbf{r}-\mathbf{r}')$ (the Coulomb interaction in the simplest case), the interaction hamiltonian takes the form

$$
H_{\text{int}} = \frac{1}{2} \sum_{\substack{k_1, \dots, k_4 \\ \mathbf{l}_1, \dots, \mathbf{l}_4 \\ \text{ss}'}} \langle k_1 \mathbf{l}_1, k_2 \mathbf{l}_2 | V | k_3 \mathbf{l}_3, k_4 \mathbf{l}_4 \rangle a_{k_1 \mathbf{l}_1, s}^{\dagger} a_{k_2 \mathbf{l}_2, s'}^{\dagger} a_{k_3 \mathbf{l}_3, s'} a_{k_4 \mathbf{l}_4, s}, \quad (2.53 \text{ a})
$$

$$
\langle k_1 \mathbf{l}_1, k_2 \mathbf{l}_2 | V | k_3 \mathbf{l}_3, k_4 \mathbf{l}_4 \rangle = \int d^3\mathbf{r} d^3\mathbf{r}' \psi_{k_1 \mathbf{l}_1}^*(\mathbf{r}) \psi_{k_2 \mathbf{l}_2}^*(\mathbf{r}') V(\mathbf{r} - \mathbf{r}') \psi_{k_3 \mathbf{l}_3}(\mathbf{r}') \psi_{k_4 \mathbf{l}_4}(\mathbf{r}). \quad (2.53 \text{ b})
$$

The matrix elements will all four l's on the same chain are, of course, the
interaction constants considered in the last section. The coupling between orders on
different chains is given by terms with two of the l's on one chain and the other two
l's on the other chain. Further, due to the localization of the $\psi_{kl}$ on individual chains,
all interactions between further then nearest neighbours can be neglected. Then the
coupling between a chain at l = 0 and at $\hat{x} = (a, 0)$ is given by the operator (with an
obvious generalization of the notation (2.29), (2.30)).

$$
H_{\hat{x}} = \frac{1}{L} \sum_q \left\{ \left( g_{1\hat{x}} - \frac{1}{2} g_{\text{ex}} \right) [O_{\text{CDW}}^{+}(q, 0) O_{\text{CDW}}(q, \hat{x}) + \text{c.c.}] \right. \\
- \frac{1}{2} g_{\text{ex}} \left[ \sum_{\alpha} O_{\text{SDW}_{\alpha}}^{+}(q, 0) O_{\text{SDW}_{\alpha}}(q, \hat{x}) + \text{c.c.} \right] \\
- g_{\text{S}} [O_{\text{SS}}^{+}(q, 0) O_{\text{SS}}(q, \hat{x}) + \text{c.c.}] \\
\left. - g_{\text{T}} \left[ \sum_{\alpha} O_{\text{TS}_{\alpha}}^{+}(q, 0) O_{\text{TS}_{\alpha}}(q, \hat{x}) + \text{c.c.} \right] \right\} \quad (2.54)
$$

with the coupling constants

$$
g_{1\hat{x}} = \langle k_F 0, -k_F \hat{x} | V | k_F \hat{x}, -k_F 0 \rangle, \quad (2.55 \text{ a})
$$

$$
g_{\text{ex}} = \langle k_F 0, -k_F \hat{x} | V | -k_F 0, k_F \hat{x} \rangle, \quad (2.55 \text{ b})
$$

$$
g_T + g_S = \langle k_F 0, -k_F 0 | V | k_F \hat{x}, -k_F \hat{x} \rangle. \quad (2.55 c)
$$

$$
g_T - g_S = \langle k_F 0, -k_F 0 | V | k_F \hat{x}, -k_F \hat{x} \rangle. \quad (2.55 d)
$$

In (2.54) c.c. means the conjugate operator, it has been assumed that the matrix
elements are approximately constant at the single chain Fermi level, for simplicity
inversion symmetry is assumed and omitted terms do not couple directly the orders
on adjacent chains. Among the omitted terms is the long-range part of the Coulomb
interaction, which gives rise to interchain forward scattering. In this process
electrons are not transferred from $k_F$ to $-k_F$ or vice versa, and therefore the different
types of order, which involve pairing of states at $k_F$ and $-k_F$ are not coupled by
interchain forward scattering [65]. Also omitted are terms with three l's on the same
chain and only one $|$ on the neighbour. These terms give rise to coupling between orders only in second order in the corresponding matrix element.

The interchain backward scattering matrix element $g_{1\hat{x}}$ describes the Coulomb interaction between CDWs on adjacent chains. It contains components of $V(\mathbf{r})$ with longitudinal momentum transfer $\approx 2k_F$, and decreases like $\exp(-2k_F a)$ with increasing interchain distance [66]. Due to the repulsive nature of the Coulomb interaction, $g_{1\hat{x}}$ is positive. On the other hand, the interchain exchange matrix element $g_{ex}$ and the combination $g_S + g_T$ involve small longitudinal momentum transfer, which should be much larger than the $2k_F$ component, however with products of wavefunctions $\psi_{k0}^*(\mathbf{r})\psi_{k\hat{x}}(\mathbf{r})$ centred on different chains. Assuming exponential decay $(\propto \exp(-\alpha r))$ of the single-chain wavefunctions, both $g_{ex}$ and $g_S + g_T$ are then positive, and should decay like $\exp(-2\alpha a)$ [67]. Finally, the combination $g_T-g_S$ contains both $2k_F$-longitudinal momentum transfer and a rapidly decaying (with interchain distance) product of wavefunctions, and thus is expected to be negligible compared to the other terms in (2.55). Of course, a monotonic transverse decay of the wavefunctions is a gross simplification of the situation in real systems, where there are oscillations of the wavefunctions and this may change the above rough estimates drastically. Also one should include phonon-mediated interactions in a realistic description.

The operator $H_{\hat{x}}$ is easily included in the mean-field investigation of possible ordered phases given in the last section. In general the interchain interactions should be quite weak, so that the type of order is essentially determined by the interchain interactions. The interchain interactions then determine only the type of transverse ordering. In the discussion of the last paragraph all coupling constants are positive, so that one expects transverse antiphase ordering of a CDW state, and in-phase SDW, SS, and TS order. Thus for CDW, SS, and TS, $H_{\hat{x}}$ leads to the same type of order as the single-electron interchain tunnelling. On the other hand, for the SDW case tunnelling favours antiphase ordering, in competition with the interchain exchange. The coupling energy due to tunnelling is proportional to $t_{\perp}^2$, of the same order of magnitude as $g_{ex}$, so that there is no a priori reason which favours in-phase with respect to antiphase order or vice versa.

The most important consequences of the interchain coupling due to transverse electron motion (2.49) or interaction (2.54), is that now the low-frequency modes of the ordered states are coupled in the transverse direction, so that there is transverse dispersion in their spectrum, i.e.

$$ \omega(\mathbf{q}) \propto [q_z^2 + C_{\perp}(q_x^2 + q_y^2)]^{1/2} \qquad (2.56) $$

with non-zero $C_{\perp}$. The integrals like (2.27 a) are then finite, i.e. interchain coupling stabilizes long-rang order at finite temperature (though for large anisotropy (small $C_{\perp}$) the order parameter is much smaller than its mean-field value). If all types of order are coupled with equal strength, the ordered state of the coupled-chain system is the same as that of the single chain in the mean-field treatment. However, as discussed in the previous section, in a single chain several types of short-range order may coexist, and a consistent treatment of the many-chain problem with weak interchain interactions requires the same methods as the single-chain problem [63–65, 68, 69]. Then, relatively strong interchain coupling leads to long-range order of one type, even if another type of order dominates the isolated chain problem: for example, if interchain backward scattering is most important, a CDW ordered state
is realized even for $g_1 > 0$, where for a single chain an SDW state is favoured [68] (figure 5). Another interesting result is that near the crossover from one phase to another (e.g. the line $g_1 = 2g_2$ in figure 5) a maximum of the critical temperature (establishment of long-range) is obtained [64, 69].

Summarizing this section, we have shown that both single-electron interchain motion (equation (2.49)) and interchain interactions (equation (2.55)) couple orders of different types on adjacent chains. In the presence of such coupling fluctuations remain finite, and long-rang three-dimensional order is possible at finite temperature. Some properties of weakly coupled chains will be discussed in the following section.

## 2.4. Ginzburg–Landau theory

### 2.4.1. A single chain

The previous sections have shown that there are important fluctuation effects in a one-dimensional system. A (at least) qualitative understanding of the properties can be gained from a Ginzburg–Landau description, both for superconducting [70] and Peierls [71] instabilities. In that approach one considers a system with a given, slowly varying (compared to the electronic wavelength $\pi/k_F$) order parameter (or energy gap) $\Delta(z)$, which is considered as a classical quantity. The free energy for a single chain is then expanded as

$$F\{\Delta(z)\} = \int dz \left\{ a|\Delta(z)|^2 + b|\Delta(z)|^4 + c \left|\frac{\partial \Delta}{\partial z}\right|^2 \right\}, \quad (2.57)$$

and the properties of the system are obtained by averaging over all configurations $\Delta(z)$ with a Boltzmann weight factor $\exp(-\beta F\{\Delta\})$ ($\beta = 1/T$). The coefficients in (2.57) can be calculated from microscopic theory [72, 73], and for both the Peierls and singlet superconducting cases are:

$$a = N(E_F) \ln \frac{T}{T_c^\circ} \approx a' \left( \frac{T}{T_c^\circ} - 1 \right), \quad (|T - T_c^\circ| \ll T_c^\circ), \quad (2.58 a)$$

$$b = N(E_F) \frac{7\zeta(3)}{16\pi^2 T^2}, \quad (2.58 b)$$

$$c = b v_F^2, \quad (2.58 c)$$

where $N(E_F) = 1/\pi v_F$ is the density of states per spin at the Fermi level and $T_c^\circ$ the mean-field transition temperature. The above description does not directly apply to the triplet (SDW and TS) phases, where all three complex components of the order parameter have to be included in a free energy functional like (2.57). In the following we restrict ourselves to the singlet (CDW and SS) phases.

In the mean-field approximation, one neglects fluctuations of $\Delta$ about the value minimizing $F$. For $T > T_c^\circ$ one finds $\Delta = 0$ (no order), for $T < T_c^\circ$ one has $|\Delta| = (-a/2b)^{1/2}$ (long-range order), which near $T_c^\circ$ gives $|\Delta| \propto (T_c^\circ - T)^{1/2}$. Lowest order fluctuation effects above $T_c$ can be described by neglecting $b$ (the gaussian approximation). For example, one finds that the order parameter correlation function (equation (1.2 b)) decays exponentially with a correlation length (or coherence length)

$$\xi(T) = \left(\frac{c}{a}\right)^{1/2}, \quad (2.58 d)$$

which diverges as $(T - T_c^\circ)^{1/2}$ as $T_c^\circ$ is approached.
However, as discussed before, mean-field theory is inadequate in one dimension. Using functional integration methods the exact calculation of static properties like the specific heat, the amplitude of the order parameter fluctuations or the correlation length may be reduced to the solution of a one-particle Schrödinger equation with an anharmonic potential [74]. Using the numerical results of Scalapino et al. [74] we have plotted the inverse correlation length as a function of relative temperature $t = T/T_c^\circ$ for various values of the Ginzburg critical temperature region $\Delta t$ (figure 8). $\Delta t$ is the distance in relative temperature above $T_c^\circ$ where deviations from mean-field behaviour become important and in one dimension is given by [74]:

$$ \Delta t = 2 \left( \frac{b T_c^\circ}{a^{3/2} c^{1/2}} \right)^{2/3} . \qquad (2.59) $$

Of course, there is no one phase transition, however figure 8 shows that for small $\Delta t$ the correlation length becomes very large below $T_c^\circ$, whereas slightly above $T_c^\circ$ one finds essentially mean-field behaviour. Similarly, the specific heat grows rapidly inside the critical region [74]. Thus, for small $\Delta t$ the mean-field transition temperature $T_c^\circ$ is still an important parameter of the system, as it is at this temperature that various properties show marked anomalies. On the other hand, for $\Delta t \lesssim 1$ figure 8 shows that the correlation length does not show any anomalous behaviour near $T_c^\circ$ and becomes very large only near $T = 0$. Using the results of Scalapino et al. [74] a similarly smooth behaviour can be found for the other properties of the system. Thus, for $\Delta t \gtrsim 1$ the system behaves continuously through the whole temperature range and $T_c^\circ$ sets merely a temperature scale.

The change in behaviour of the systems properties near $T_c^\circ$ for small $\Delta t$ may be understood by considering the two gradient-free terms of $F(\Delta)$ which act as a local potential on the fluctuations of the order parameter (figure 9). For $T > T_c^\circ$ ($a > 0$) this potential has a single minimum at $\Delta = 0$ and the fluctuations are characterized by a local finite amplitude of $\Delta$. We shall refer to this temperature range as the

<figure><img src="image_9.png" /><figcaption>Figure 8. The inverse correlation length for various values of the critical region $\Delta t$, as obtained from the results of [7]. $\xi_0$ is the mean-field correlation length at $T = 2T_c^\circ$.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 9. Order parameter fluctuations: (a) $a > 0$ ($T > T_c^°$), the amplitude of the order parameter (heavy dot) fluctuates around $|\Delta| = 0$; (b) $a < 0$ ($T < T_c^°$), the phase of the order parameter rotates in the minimum of $F(\Delta)$, at $|\Delta| \approx \text{const.}$</figcaption></figure>

amplitude fluctuation regime. In contrast to this, for temperatures well below $T_c^°$ ($a < 0$) the potential has its minimum at $\Delta_0 = (-a/2b)^{1/2}$, independent of the phase of the order parameter. As the potential depends only on $|\Delta|$, there is no restoring force acting against variations of the phase, whereas a finite restoring force acts against deviations of the amplitude of the order parameter from $|\Delta_0|$. Therefore, as discussed before, at low temperatures there are only fluctuations of the phase at $\Delta$, whereas the amplitude defined anomalies. On the other hand, for large $\Delta t$ the amplitude fluctuations are frozen out. We call this case the phase fluctuation regime. For small $\Delta t$ the transition from the amplitude to the phase fluctuation regime takes place in a narrow temperature range just below $T_c^°$ and leads thus to the above discussed well-defined anomalies. On the other hand, for large $\Delta t$ the amplitude fluctuations remain important even well below $T_c^°$ and the phase fluctuation regime is realized only very near to $T = 0$. The transition between the two regimes is smooth in this case.

In the amplitude fluctuation regime the fluctuations are oscillations of the order parameter around $\Delta = 0$. The anharmonic term in (2.57) then renormalizes the energy of the fluctuation. A simple treatment of this renormalization can be given using a Hartree-Fock-like approximation for the fourth-order term [76-77] in (2.57): two of the four factors $\Delta(z)$ are replaced by their thermal average, i.e. one sets

$$|\Delta(z)|^4 \approx 4 \langle |\Delta(z)|^2 \rangle |\Delta(z)|^2. \qquad (2.60)$$

The resulting effective free energy is quadratic in $\Delta$, so that explicit calculation is possible, with $a$ replaced by $a + \Sigma(T) = a + 4b\langle|\Delta|^2\rangle$. The average $\langle|\Delta|^2\rangle$ has to be determined self-consistently from the effective free energy, so that the self-energy $\Sigma(T)$ is given by the equation [78]

$$\Sigma(T) = \frac{2Tb}{\sqrt{c\sqrt{[a + \Sigma(T)]}}}, \qquad (2.61)$$

which can be easily solved. The resulting squared amplitude of the order parameter fluctuations is shown in figure 10 together with the exact [74] and the mean-field
<figure><img src="image_11.png" /><figcaption>Figure 10. Order parameter fluctuations: exact results ([74], full line), Hartree-Fock calculation (long dashed line), and mean-field theory (small dashed line).</figcaption></figure>

results. This figure shows that the Hartree-Fock approximation is in excellent agreement with the exact results in the amplitude fluctuation region. Especially, it is an important improvement over the mean-field results which predict diverging fluctuations at $T_c^°$, and it allows analytical calculation whereas the exact method of Scalapino et al. [74] requires the numerical solution of a one-particle Schrödinger equation. Figure 10 was obtained assuming $\Delta t \ll 1$, so that the factors $T$ in equation (2.58) may be set equal to $T_c^°$, however equally good agreement is obtained for arbitrary $\Delta t$ for $T > T_c^°$. For example, in figure 11, the inverse square of the

<figure><img src="image_12.png" /><figcaption>Figure 11. Temperature dependence of the inverse square of the correlation length; for $\Delta t = 1$: (a) Hartree-Fock calculation, (b) exact result [74], and (c) mean-field theory.</figcaption></figure>
correlation length $\xi$ is shown for $\Delta t = 1$. For simplicity $a'$, $b$ and $c$ have been assumed temperature-independent. Again, Hartree-Fock theory yields good agreement with the exact results, down to nearly $T_c^\circ/2$. Above that temperature $\xi^{-1}(T)$ is a nearly perfect straight line, i.e. in that temperature region the fluctuation effects are well described by replacing $T_c^\circ$ in (2.58 a) by an effective $T_{eff} \approx T_c^\circ/3$.

What does the Hartree-Fock approximation mean in terms of microscopic theory? To be specific we consider the Peierls case. The order parameter is essentially the $2k_F$-phonon amplitude (equation (2.5)), and calculating the order parameter correlation function from (2.57) with $b=0$ includes the lowest order phonon self-energy corrections, discussed in section 2.1.3 (equation (2.21)), represented diagrammatically in figure 12 (a). The coefficient $b$ in (2.57) now represents an electronic diagram with two ingoing and two outgoing $2k_F$-lines. The Hartree-Fock approximation then is given by the diagrams of figure 12 (b). In addition the Bose distribution function $n(\omega) = [\exp(\omega/T) - 1]^{-1}$ is approximated by the high-temperature (classical) approximation $n(\omega) = T/\omega$. Similar considerations apply to the superconducting case.

Going to lower temperature, (2.61) never admits an ordered state: the transition to such a state requires $\alpha + \Sigma(T) = 0$, which is impossible due to the divergent right-hand side of (2.61) (apart from $T=0$, where the classical model (2.57) of course has long-range order). Thus, the Hartree-Fock approximation gives the expected vanishing transition temperature. However, figures 10 and 11 show that well below $T_c^\circ$, in the phase fluctuation region, the quantitative predictions of the Hartree-Fock theory are largely in error. As discussed above, in this region the character of the fluctuation is qualitatively different from that in the amplitude fluctuation region, and is more appropriately represented as local rotations of the phase, the amplitude remaining fixed (figure 9). It is therefore not surprising that a simple self-energy correction like equation (2.61) cannot describe this situation. A more convenient starting point then consists in considering only the phase fluctuations in a harmonic approximation, as discussed in sections 2.1 and 2.2 (cf. equation (2.42)). From such

<figure><img src="image_13.png" /><figcaption>Figure 12. Diagrammatic representation of the order parameter correlation function: (a) mean-field theory, (b) Hartree-Fock approximation. The wiggly line is the bare phonon propagator, the zig-zag line is the approximate order parameter propagation correlation function.</figcaption></figure>
an approach one obtains, for example, $\xi(T) = 2c|a|/(Tb)$, in agreement with the exact results [74, 71]. One should note, however, the form of the constants $a, b$ and $c$ is different from equations (2.58), especially there are no divergencies for $T \to 0$, i.e. $\xi(T)$ grows like $T^{-1}$ for low $T$.

For the Peierls case, separating approximately phase and amplitude degrees of freedom it has been shown [71, 79] that the properties of the system in the phase fluctuation region are very similar to those obtained from mean-field theory [14]. Especially, there are well-defined phase and amplitude nodes of the CDW, the phase mode leading to Fröhlich collective conductivity and a central peak of finite width in the structure factor (instead of satellite Bragg peaks in a long-range ordered state). The absence of long-range order also smears the gap in the electronic spectrum, however, due to the well-developed short-range order the density of states at the Fermi level is strongly reduced [72, 79, 80], the so-called 'pseudo-gap'. The pseudo-gap is also found in microscopic calculations [81].

From (2.59) the critical temperature region follows as

$$ \Delta t \approx 1.6. \qquad (2.62) $$

Such a value is actually obtained for all the phases considered in section 2.2, i.e. instabilities in a one-dimensional conductor are generally associated with a large critical region, so that no particular anomalies are to be expected at the mean-field transition temperature, as would be the case for small $\Delta t$. $T_c^\circ$ thus sets only a temperature scale for the fluctuation phenomena, which, due to the large $\Delta t$ should exist even well above $T_c^\circ$. It is interesting to note that in a three-dimensional superconductor the coefficients of the free-energy functional are very similar to those given in (2.58) [70]. However, in three dimensions, the low-frequency long-wavelength fluctuations which determine the width of the critical region have only a small phase space compared to the one-dimensional case, leading to a very small $\Delta t$ [82]. Thus the large one-dimensional critical region is not due to some peculiarities of the microscopic interactions, but rather to the large phase space of the low energy fluctuations.

The free-energy functional (2.57) is independent of the phase of the order parameter. If the CDW case is considered, this describes an incommensurate situation. For an $m$th order commensurate case, i.e. for $2mk_F = G$, umklapp processes lead to the appearance of additional terms of the form $|\Delta(x)|^m \cos m\phi$ ($\phi$ is the phase of the complex order parameter) in the free-energy functional. Thus, there is a finite energy of phase fluctuations in the long-wavelength limit, and long-range order is only destroyed by the thermal activation of domain walls, discussed at the end of section 2.1. Consequently, the correlation length diverges exponentially at low temperatures [83, 84].

### 2.4.2. Coupled chains

In a system of coupled chains, a phase transition into a low-temperature ordered state is possible. The system is described by the functional

$$ F\{\Delta_i(z)\} = \sum_i F_1[\Delta_i(z)] - \lambda \sum_{i,\delta} \int dz \, \text{Re}[\Delta_i^*(z)\Delta_{i+\delta}(z)]. \qquad (2.63) $$

Here $i$ labels the individual chains, $\delta$ counts nearest neighbours in a square lattice, $F_1$ is the single chain energy (2.57) and the last term in the interchain coupling.
Depending on the type of order considered, different processes contribute to the coupling, as discussed in section 2.3. For example, superconducting order is mainly coupled by interchain electron motion, and the coupling parameter then is, in the simplest case [78, 85],

$$ \lambda = 2ct_{\perp}^2/v_F^2, \qquad (2.64 a) $$

where $v_F$ is the single-chain Fermi velocity. Thus the transverse correlation length is

$$ \xi_{\perp}(T) = \sqrt{(\lambda/c)\xi_{\parallel}(T)} = \frac{\sqrt{2t_{\perp}d}}{v_F}\xi_{\parallel}(T), \qquad (2.64 b) $$

$d$ is the interchain distance and $\xi_{\parallel}$ the single-chain correlation length (2.58 d). For coupling between CDWs, the Coulomb interaction generally is most important, and then one has $\lambda \propto g_{1x}$.

The transition temperature in a system described by (2.63) has been calculated by Scalapino et al. [86] in the limit of small interchain coupling, using exact results [74] for the single-chain problem and a mean-field approximation for the interchain coupling. In this approach, $T_c$ is given by

$$ 1 - 4\lambda\chi_{1D}(0)|_{T=T_c} = 0, \qquad (2.65) $$

where $\chi_{1D}(q)$ is the order parameter susceptibility of a single chain:

$$ \chi_{1D}(q) = \frac{1}{T} \int dz \exp(iqz) \langle \Delta^*(z) \Delta(0) \rangle. \qquad (2.66) $$

In the phase fluctuation region one has

$$ \chi_{1D}(q) = \frac{\xi}{2c(1 + \xi^2 q^2)}, \qquad \xi = \frac{2c|a|}{Tb}, \qquad (2.67) $$

and inserting into this (2.65) we obtain

$$ T_c = \frac{2|a|}{b} \sqrt{(2c\lambda)}. \qquad (2.68) $$

The square root dependence of $T_c$ on the interchain coupling agrees with general arguments of Barisic and Uzelac [87].

If the interchain coupling becomes stronger, the above approximations fail. However, for sufficiently large value of $\lambda$, the Hartree-Fock approximation may be used to obtain the depression of the transition from its mean-field value [85]:

$$ T_c = T_c^\circ \left( 1 + \frac{b T_c^\circ}{a' \sqrt{(c\lambda)}} \right)^{-1}. \qquad (2.69) $$

(Actually, a Hartree instead of a Hartree-Fock method has been used, by Schulz [85], which amounts to the replacement $b \to b/2$ in (2.69). However, the Hartree-Fock approximation gives the correct low $b$ expansion, see Tucker and Halperin [77].)

The weak coupling result (2.68) is modified if not only the classical thermal but also quantum fluctuations of the phase are considered, as discussed in section 2.2. Then the low-temperature divergence of the single chain has an exponent different from 1, and one obtains, using a mean-field theory for the interchain coupling [45, 65]
$$T_c \propto \lambda^{1.\overline{2-\eta}}, \qquad (2.70)$$

where $\eta$ is the exponent appearing in equations (2.43) and (2.44). In many cases $\eta$ lies between 1 and 2, so that the quantum corrections become quite important.

As temperatures near the three-dimensional ordering the transverse coherence length is much bigger than the interchain distance, so that the critical behaviour is three-dimensional [88]. For example, the order parameter (the gap) grows like $(T_c - T)^\beta$, with an exponent $\beta$ determined for a three-dimensional system. At some crossover temperature $T^*$ the transverse coherence length becomes equal to the interchain distance [71, 87]. Above $T^*$ the fluctuations on neighbouring chains are essentially uncoupled and may thus be considered as one-dimensional. Treating the one-dimensional fluctuations exactly on the basis of the renormalization group results and the interchain coupling by mean-field theory it has been shown [89] that the critical region associated with the three-dimensional ordering at $T_c$ may be quite narrow.

We now briefly consider the case of a CDW which is nearly commensurate. Due to its experimental relevance (section 4), we consider the third-order case, i.e. $2k_F = G/3 + \delta$, where $G$ is the reciprocal lattice wavenumber along the chain and $\delta$ is the deviation from commensurability. Inclusion of third-order umklapp terms leads to the free-energy functional [90–93]

$$F = \int dz \left\{ a|\Delta(z)|^2 + b|\Delta(z)|^4 + c \left| \left( \frac{\partial}{\partial z} - i\delta \right) \Delta(z) \right|^2 - 2V|\Delta(z)|^3 \cos 3\phi(z) \right\}. \quad (2.71)$$

We assume that there is sufficient interchain coupling, so that fluctuation effects can be neglected, and the phase diagram is at least qualitatively obtained by minimizing $F$ [90, 91–93] (figure 13). For $\delta = 0$ the transition from the normal ($\Delta = 0$) to the

<figure><img src="image_14.png" /><figcaption>Figure 13. Schematic phase diagram (not to scale) near third order commensurability, containing a commensurate (C), an incommensurate (IC), and a normal (N) metallic phase. Full lines are first order transitions, dashed lines are second order (after [93]).</figcaption></figure>
commensurate CDW state ($|\Delta| \neq 0, \phi \equiv 0$) is of first order, due to the third-order term, with a transition temperature higher than $T_c^o$. For small finite $\delta$ there is still a direct first-order normal-commensurate transition, the loss of 'elastic energy' ($\propto c\delta^2$) being compensated by the gain in commensurability energy ($\propto V$). However, if $\delta$ exceeds the critical value $\sqrt{(bc)/V}$, the transition is of second order into an incommensurate state at $T_c^o$, and only at lower temperature does the CDW 'lock' into a commensurate state. Near incommensurate-commensurate transition the incommensurate state consists of large regions of commensurate order separated by equally spaced domains where the phase of the order parameter varies rapidly [90, 92, 94, 95], so as to give an exactly incommensurate wavelength of the structure. The commensurate-incommensurate transition is of second order if only spatial variations of the phase are allowed for [94, 95]. However, if the amplitude $|\Delta|$ is also allowed to vary, the commensurate-incommensurate transition becomes first order even for $T$ well below $T_c^o$ and only quite far away from the normal-commensurate-incommensurate triple point is there a multicritical point below which the commensurate-incommensurate transition is of second order [93].

### 2.4.3. Effects of disorder

One effect of impurities on the Peierls transition has been discussed in section 2.1: pair breaking by electron backscattering which decreases the mean-field transition temperature. There is another, more drastic effect: consider first a single impurity at site $R_0$ in a single chain, described by (2.24). Its effect in the presence of electron-phonon interactions can be described by adding a term $U_q \exp(iqR_0)$ to (2.19). Then locally around $R_0$ a finite static value of the order parameter is induced:

$$ 
\begin{aligned}
\Delta(z) &= \frac{g}{\sqrt{L}} \sum_{|q| \ll 2k_F} \langle b_{2k_F+q} + b_{-2k_F-q}^+ \rangle \exp(iqz) \\
&= -U_{2k_F} \Pi(2k_F, 0) \exp(-2ik_F R_0) \frac{\pi v_F}{2\xi(T) \ln(T/T_c^o)} \exp\left(-\frac{|z-R_0|}{\xi(T)}\right),
\end{aligned} 
\quad (2.72) $$

the lattice deformation is

$$ 
\begin{aligned}
\langle u(z) \rangle &= \sqrt{\left(\frac{2}{g^2 \omega_{2k_F}}\right)} \text{Re}[\Delta(z) \exp(2ik_F z)] \\
&\propto \exp\left(-\frac{|z-R_0|}{\xi(T)}\right) \cos 2k_F(z - R_0),
\end{aligned} 
\quad (2.73) $$

and there is similar modulation of the charge density. This of course is the Friedel oscillation induced by the impurity potential, especially strong due to the divergent one-dimensional polarizability at $2k_F$ (equation (2.16)). As $T_c^o$ is approached, the correlation length $\xi(T)$ diverges, so that the distortion due to a single impurity has longer and longer range, and will be in competition with the distortion induced by other impurities when $\xi(T)$ becomes comparable to the mean distance between impurities. To treat this situation, one includes an impurity part into the free energy functional [96]:

$$ F = F_p + \sum_i \int dz \text{Re}[h_i(z) \Delta_i^*(z)], \quad (2.74) $$
where the pure part $F_p$ is given by equation (2.63), and the random field on chain $i$ is

$$h_i(z) = 2U_{2k_F} \Pi(2k_F, 0) \sum_l \exp(-2ik_F R_{li}) \delta(z - R_{li}). \quad (2.75)$$

The constants are chosen that the effect of a single impurity in an isolated chain (equation (2.72)) is reproduced. As before all quantities varying slowly around the wave-vector $2k_F$ have been replaced by their value at $2k_F$. The $R_{li}$ are the sites of randomly distributed impurities in the $i$th chain.

Now, near $T_c^*$ (which actually should contain the pairbreaking correction according to (2.25)) the impurity-induced deformation becomes large and long-ranged, so that the fourth-order terms in $F$ become important. Using, as above, the Hartree-Fock approximation, in $\langle |\Delta|^2 \rangle$ both a thermal and an impurity average are now contained [97], leading to

$$\langle |\Delta|^2 \rangle = \langle |\Delta|^2 \rangle_{\text{pure}} + \frac{n|U_{2k_F} \Pi(2k_F, 0)|^2}{8\pi\lambda \sqrt{\{c[a + \Sigma(T)]\}}}. \quad (2.76)$$

Here $n$ is the density of impurities in a single chain. In a system of pure coupled chains the mean-square amplitude $\langle |\Delta|^2 \rangle_{\text{pure}}$ remains finite as $a + \Sigma(T) \to 0$, so that a phase transition is possible at finite temperatures, as discussed above. However, in the presence of a finite concentration of impurities the second term in (2.76) diverges for $a + \Sigma(T) \to 0$, as pointed out by Sham and Patton [97].

Retaining only this term, the Hartree-Fock equation for the self-energy reads

$$\Sigma(T) = 4b\langle |\Delta|^2 \rangle = \frac{bn|U_{2k_F} \Pi(2k_F, 0)|^2}{2\pi\lambda\{c[a + \Sigma(T)]\}}, \quad (2.77)$$

which is similar to (2.61) derived for a pure one-dimensional system. As in that case there is obviously no solution having $a + \Sigma(T) = 0$, i.e. an arbitrarily small amount of disorder suppresses the phase transition and the establishment of long-range order. However, for weak disorder (2.77) is the same as (2.61) for a system with a small Ginzburg critical region. Similar to the discussion of section 2.4.1 the sharp phase transition of the pure system will then only be smeared out over a small temperature interval, with significant anomalies of specific heat, resistivity and other properties. Only for quite large disorder anomalies of specific heat, resistivity and other properties. Only for quite large disorder will the transition become very broad.

The Hartree-Fock approximation used above loses its validity well below $T_c^*$, where the amplitude of the order parameter is nearly constant and fluctuations of the phase are most important. The impurity term in (2.74) then acts to favour locally a fixed value of the phase. For this situation it has also been shown that an arbitrarily small concentration of impurities destroys long-range order [98, 99], as above, and one finds exponentially decreasing correlations at large distances. If Coulomb interactions between adjacent chains are included, the system is less sensitive to impurities: instead of exponential decay, the order parameter correlations only decay algebraically, and power-law divergencies of the static structure factor are predicted [98, 99].

The complete suppression of the phase transition and the long-range order by weak disorder is due to the fact that a single impurity locally favours a fixed value of the order parameter. Because of the randomness a state coherent over the whole
system cannot be established. For weak interchain coupling this 'coherence breaking' is particularly easy to achieve between adjacent chains. Obviously, coherence breaking is a much stronger effect than the lowering of $T_c^\circ$ by the pair-breaking effect discussed in section 2.1.3.

The easy destruction of long-range order by the random 'external field' $h_i(z)$ is in agreement with arguments of Imry and Ma [100] (see also Fukuyama and Lee [16, 99]) demonstrating that for arbitrary weak disorder a long-range ordered system can gain energy by splitting into domains of approximately constant order parameter of linear dimension $L$: averaged over the domain size one has $\langle h \rangle = 0$, however fluctuations are of order $\langle h^2 \rangle \propto L^{-d/2}$ (assuming for example a gaussian distribution for $h(z)$, $d$ is the space dimensionality). Therefore, adjusting the value of the order parameter so as to take advantage of the fluctuations of $h$, the domain gains an energy proportional to $L^{d/2}$. On the other hand, the creation of walls between domains costs energy of order $L^{d-2}$ (for a continuous symmetry as in an incommensurate Peierls system). For $d/2 > d - 2$ or $d < 4$ it is then always energetically favourable to create domains, i.e. long-range order is lost. Of course, finite temperature increases with tendency to domain formation, due to the larger entropy of a state with domains compared to long-range order.

Our above consideration concerns impurities in a Peierls system with backscattering of electrons from one side of the Fermi surface to the other. Forward-scattering impurities have a much weaker effect [101, 102]: they do not couple linearly to the order parameter, and a finite concentration is needed to destroy long-range order.

The superconducting order parameter locally creates two electrons. Due to particle conservation, an external perturbation cannot couple linearly to the order parameter, i.e. terms like $(2.74)$ are forbidden in the superconducting case. A magnetic impurity rather acts to decrease the order parameter amplitude locally [103, 104], thus a random distribution of magnetic impurities is described by coefficient $a(z)$ in the free-energy functional (2.57) which has a component varying randomly in space. Such a term will not lead to coherence breaking, however the lowering of the transition temperature is expected to be much larger in a quasi-one-dimensional system than in the isotropic case [105].

### 2.4.4. Response to external fields

Strong fluctuations in quasi-one-dimensional systems may have a significant influence on the response to external fields. These effects can be described using the Ginzburg-Landau theory.

One of the most interesting properties of the ordered Peierls (CDW) state is the possibility (in the ideal case) of d.c. electrical conduction by a rigid sliding of the CDW. In a real system various effects suppress this mechanism, as discussed in section 2.1.2. However, as first pointed out by Bardeen and coworkers [106, 26], CDW fluctuations above the phase transition (directly observed in X-ray scattering, see section 3) should also contribute to the d.c. conductivity. Due to their finite coherence length the mechanism is much less sensitive to the pinning effects of section 2.1.2.

A theoretical description of CDW fluctuation conductivity can be given using the expression for the current density induced by a slowly varying CDW order parameter calculated from microscopic theory [107]:
$$ 
\begin{aligned}
j(z,t) &= \frac{i a_0}{2} \left[ \Delta^*(z,t) \frac{\partial}{\partial t} \Delta(z,t) - \Delta(z,t) \frac{\partial}{\partial t} \Delta^*(z,t) \right] \\
&= -a_0 |\Delta(z,t)|^2 \frac{\partial}{\partial t} \phi(z,t),
\end{aligned} 
\quad (2.78 a) $$

$$ a_0 = \frac{e\tau}{2\pi^2 T} \left[ \psi'\left(\frac{1}{2} + \frac{1}{4\pi T \tau}\right) - \frac{1}{4\pi T \tau} \psi''\left(\frac{1}{2} + \frac{1}{4\pi T \tau}\right) \right]. \quad (2.78 b) $$

Equation (2.78 b) follows from microscopic results [27, 108], $\psi'$ and $\psi''$ are derivatives of the digamma function, $\tau$ the electronic lifetime (cf. (2.25)) and $e$ the electronic charge. Equation (2.78 a) shows that a finite current density is only obtained if the fluctuating CDW has a finite phase velocity. On the other hand, a pure amplitude motion ($\partial\phi/\partial t = 0$) leads only to a redistribution of the charge density on the scale of the wavelength $\pi/k_F$ of the CDW, i.e. a short-wavelength contribution to the current density. As equation (2.78 a) describes only the long-wavelength ($\gg \pi/k_F$) part of the current density, the amplitude dynamics make no contribution.

From (2.78 a) the contribution of the fluctuation CDW to the conductivity can be calculated using the Kubo formula. We are only interested in the homogeneous conductivity, so that we have to calculate the correlation function of the total current,

$$ J(t) = \int dz j(z, t). \quad (2.79) $$

To calculate the conductivity, a knowledge of the dynamics of the order parameter is necessary. To lowest order the equation of motion of $\Delta$ is of course identical to that of the phonons (equation (2.21)), and higher order terms may be included using the Ginzburg-Landau free energy as a generating functional

$$ \frac{\partial^2 \Delta}{\partial t^2} + \gamma \frac{\partial \Delta}{\partial t} + 2g^2 \omega_{2k_F} \frac{\delta F\{\Delta(z)\}}{\delta \Delta^*(z)} = 0. \quad (2.80) $$

In the commensurate case, due to translational invariance, $F$ depends only on the modulus of $\Delta$ and of its derivatives. It is then easy to show from (2.78) and (2.80) that the total current obeys the simple equation of motion

$$ \frac{dJ}{dt} = -\gamma J, \quad (2.81) $$

even in the presence of interactions terms in equation (2.80). Thus, for vanishing damping, $J$ is a conserved quantity, due to the translational invariance of the incommensurate system. From (2.81) the contribution of sliding fluctuating CDWs to the conductivity follows as

$$ \sigma_F(\omega) = a_0^2 g^2 \omega_{2k_F} \langle |\Delta|^2 \rangle \frac{\gamma}{\gamma^2 + \omega^2}. \quad (2.82) $$

A lowest order one-dimensional calculation gives

$$ \sigma_F(\omega) = \frac{a_0^2 g^2 \omega_{2k_F} T}{2\sqrt{(ac)} \gamma^2 + \omega^2}, \quad (2.83) $$

and using the expressions (2.58) and (2.78 b) for the parameters, this agrees with microscopic calculations [27, 108] via the Azlamazov-Larkin diagram.
Near the mean-field transition temperature the expression (2.83) has a square-root divergency. This is of course a spurious effect: near $T_c^\circ$ the lowest order approximations breaks down, and higher order terms have to be taken into account. If no interchain coupling is present, $\langle|\Delta|^2\rangle$ can be calculated exactly using functional integration methods. In the amplitude fluctuation region the Hartree-Fock approximation gives a very accurate approximation to the exact value of $\langle|\Delta|^2\rangle$ (figure 10). If the interchain coupling is present, i.e. in the three-dimensional case, $\langle|\Delta|^2\rangle$ tends to a large, but finite value at $T_c$. However, in the region of one-dimensional fluctuations (small interchain coherence length) $\langle|\Delta|^2\rangle$ may still increase rapidly. One should note that $\langle|\Delta|^2\rangle$ is actually proportional to the intensity of diffuse X-rays scattering around $2k_F$, so that some idea of the temperature region where fluctuation conductivity plays a role may be obtained from scattering experiments.

In addition to the collective conductivity mechanism discussed above, the CDW fluctuations also give rise to scattering of individual electrons, thereby diminishing the single-electron part of the conductivity by an amount $\sigma_{sc}$. A quite general argument [109] as well as microscopic calculations [27, 108] show that the decrease of conductivity due to this mechanism is also proportional to $\langle|\Delta|^2\rangle$.

What now is the total contribution of the fluctuations to the conductivity? we note that the CDW fluctuations in the present problem play a role similar to that of phonons in the resistivity of ordinary metals. Then, neglecting scattering by impurities and defects, by analogy with general arguments by Peierls [110] the two contributions discussed above should nearly cancel if the lifetime of the fluctuations is only limited by decay into electron-hole pairs. A decrease of conductivity results if a direct interaction between fluctuations decreases further the fluctuation lifetime.

The near cancellation of $\sigma_F$ and $\sigma_{sc}$ is similar to the phonon-drag effect in metals, which is actually described by the same type of diagrams as $\sigma_F$ [111]. In spite of this formal similarity, there is a physical difference between phonon-drag and CDW-fluctuation conductivity: in the phonon-drag effect electrons and phonons scatter individually from each other, leading to an equilibrium between the distribution functions of electrons and phonons separately. On the other hand, in the fluctuation-conductivity mechanism, a coherent (coherence length $\xi$) simultaneous modulation of the lattice and the charge density is involved. In fact, the high probability for the creation of such a fluctuation (i.e. its low energy) is a consequence of the strong coupling between electrons and $2k_F$-phonons. The fluctuation conductivity is then due to a translation of the rigidly coupled modulations of the lattice and the charge density. In some sense, the fluctuation conductivity may be considered as 'coherent phonon drag'. The Kohn anomaly in the phonon dispersion, which reflects the existence of CDW fluctuations, develops continuously out of the phonon spectrum of the non-interacting system with decreasing temperature (figure 3) corresponding to an increasing coherence length. From this it is clear that phonon-drag and fluctuation conductivity are not two independent processes existing in parallel. Rather, with increasing coherence length the incoherent phonon drag [112] becomes more and more coherent and is then more appropriately described as CDW-fluctuation conductivity, due to the strong electron-lattice coupling in the coherent region.

Our above arguments, indicating that fluctuations do not enhance the conductivity over its (hypothetical) low-temperature value, may become invalid if higher order effects are taken into account. For example, a CDW fluctuation of coherence length $\xi$ will create a local decrease of the density of state (the pseudo-gap). Locally,
within $\xi$ this effect is stronger than the average, experimentally observable depression of the density of states. The low-density of states then leads to a smaller probability of decay into electron-hole pairs, i.e. the lifetime $\gamma^{-1}$ increases, and thus $\sigma_F$. Also a decrease of the local density of states would diminish the scattering by other lattice modes [113, 114]. To what extent these effects may increase the conductivity above its value in the absence of fluctuations is not clarified yet. On the other hand, if the coherence length becomes comparable to the mean distance between impurities, the pinning effect discussed in section 2.1.2 suppresses $\sigma_F$. In addition, near $T_c$ the buildup of local order around an impurity (equation (2.73)) enhances single-electron scattering [115], so that in that case the *resistivity* is increased by the fluctuations.

Translational invariance, which was the reason for the simplicity of (2.81), breaks down in a commensurate case. For third-order commensurability, from (2.71), (2.78 a), (2.79) and (2.80) one obtains

$$ \frac{dJ}{dt} + \gamma J = 6g^2\omega_{2k_F}Va_0 \int dz |\Delta(z)|^3 \sin 3\phi(z), \quad (2.84) $$

i.e. due to the broken translational invariance, the current is no more a conserved quantity for $\gamma = 0$. The fluctuation conductivity for finite $V$ can then be calculated as [116]

$$ \sigma_F(0) = \frac{a_0^2 g^2 \omega_{2k_F} T}{2\pi\gamma} \left[ \frac{dk}{a+ck^2} \left\{ 1 + \xi^2(T) \left(\frac{q}{2} + \delta\right)^2 \right\}^3 \right]^{-1}, \quad (2.85 a) $$

$$ \eta = \frac{(72\omega_{2k_F}^3 g^6 V^2 T)^2}{\pi \tilde{\omega}_0^{12} (2k_F) \xi^2(T)} \frac{\tilde{\omega}_0(2k_F)}{\gamma} \ln \frac{2\tilde{\omega}_0(2k_F)}{\gamma}, \quad (2.85 b) $$

and $\delta$ is the deviation from exact commensurability (equation (2.71)). Thus even a weak commensurability potential (small $V$) may reduce the fluctuation conductivity considerably as long as $\tilde{\omega}_0 \gg \gamma$. Physically this corresponds to the requirement that the fluctuations should be sufficiently long-lived to permit the commensurability pinning force to have an appreciable effect. For large deviations from commensurability ($\xi\delta > 1$), equation (2.85 a) goes over smoothly to the value for an incommensurate system. A picture of the temperature dependence of $\sigma_F$ may be obtained approximating $\tilde{\omega}_0^2(2k_F) \approx \tilde{\omega}_0'^2(T - T_c)/T_c$. Figure 14 then demonstrates the strong depression of $\sigma_F$ by commensurability. On the other hand, the contribution to single-electron scattering, $\sigma_{sc}$, is only weakly affected by commensurability [116]. The above results are obtained for $\tilde{\omega}_0 > \gamma$. A reduction of $\sigma_F$ is also found in the opposite limit $\gamma > \tilde{\omega}_0$ [117]. However, in real systems like TTF-TCNQ, over the whole temperature range where fluctuations are observed one has $\tilde{\omega}_0 > \gamma$, i.e. a weakly-damped mode (section 4).

Fluctuation effects in three-dimensional superconductors, due to creation of finite size 'droplets' of superconducting order above $T_c$ have been widely studied [118]. Due to weak interchain coupling, these effects are strongly enhanced in quasi-one-dimensional superconductors. From time-dependent Ginzburg-Landau theory the superconducting fluctuation conductivity along the chain direction is [78, 119]

$$ \sigma' = \frac{\nu_F^2}{4t_{\perp}^2 d^2} \frac{e^2 a'}{32\sqrt{ac}} \left\{ 1 - \frac{d}{[d^2 + 4\pi\xi_{\perp}^2(T)]^{1/2}} \right\}. \quad (2.86) $$
<figure><img src="image_15.png" /><figcaption>Figure 14. The CDW fluctuation conductivity (equation (2.85)) in the commensurate case for various values of the parameter $\eta(2T_c^\circ)$. $\eta = 0$ represents the incommensurate case, $\sigma_F^{ic}(2T_c^\circ)$ is the fluctuation conductivity of an incommensurate system at $2T_c^\circ$.</figcaption></figure>

In the region of three-dimensional fluctuations $|\xi_\perp \gg d|$ this has the same form as in an isotropic superconductor [120], however with an enhancement factor $v_F^2/(4t_\perp^2 d^2)$, which may be quite large in real systems (section 5). This factor is essentially the square of the coherence length anisotropy (equation (2.64 b)). It may be understood noting that, due to the small $\xi_\perp$, the superconducting droplets have quite small transverse diameters. The energy of such a droplet is proportional to its volume and to the square of the local order parameter (equation (2.57)). On the other hand the droplet is created thermally, so its energy should be of the order $T$. Then, decreasing the transverse coherence length (i.e. the volume) at constant temperature increases the order parameter inside the droplet and consequently the fluctuation conductivity, as reflected by the enhancement factor in (2.86).

Higher-order fluctuation effects, neglected so far, may be incorporated using the Hartree–Fock approximation discussed in section 2.4.1. In the amplitude fluctuation region this is a good approximation and leads essentially to the replacement of the mean field-$T_c^\circ$ by some effective temperature which for typical values of the parameters (2.58) is of the order $T_c^\circ/3$. On the other hand the behaviour well below $T_c^\circ$, in the phase fluctuation region, cannot be well described by this approach.

In addition to the Ginsburg–Landau terms, corresponding to the Azlamazov–Larkin diagram of microscopic theory, there are contributions analogous to those representing scattering of individual electrons by fluctuations in the CDW case. Unlike the CDW case, however, these Maki–Thompson [121, 122] terms also increase the conductivity, in general by an amount comparable to $\sigma'$. However a detailed investigation of this effect in quasi-one-dimensional conductors is lacking so far.

Superconducting fluctuations also lead to a diamagnetic contribution to the magnetic susceptibility, due to the generation of circular Meissner currents. In a quasi-one-dimensional superconductor, due to the weak interchain coupling, a current between chains is difficult to establish. This just compensates for the large
fluctuation amplitude inside a droplet, and consequently the expression for the fluctuation-induced diamagnetism (in perpendicular field) [78, 119]

$$ \chi_{\perp} = - \frac{T e^2}{6\pi c^2} \frac{\xi_{\parallel}(T)\xi_{\perp}(T)}{[\xi_{\perp}(T)^2 + d^2/4]^{1/2}} \quad (2.87) $$

does not show any enhancement with respect to an isotropic case (c is the speed of light).

Finally we note that Ginzburg-Landau theory can be used to calculate critical fields of a superconductor. Including a vector field in the free energy (2.63) the upper critical field (assuming type II superconductivity [70]) for a field applied along one of the perpendicular directions is [123]

$$ H_{c2}^{\perp} = \frac{\Phi_0}{2\pi\xi_{\parallel}(T)\xi_{\perp}(T)} = \frac{\Phi_0}{2\pi\xi_{\parallel}^2(T)} \frac{v_F}{\sqrt{2t_{\perp}d}} \quad (2.88) $$

$\Phi_0$ is the flux quantum. Just below $T_c$ this shows the well-known linear relation between critical field and temperature: $H_{c2}^{\perp} \propto T_c^{\circ} - T$. If there is also an anisotropy between the two perpendicular directions, from Turkevich and Klemm [123] one has

$$ \frac{H_{c2}^{x}}{H_{c2}^{y}} = \frac{t_{x} d_{x}}{t_{y} d_{y}}, \quad (2.89) $$

in obvious notations. For parallel applied fields one finds

$$ H_{c2}^{\parallel} = \frac{\Phi_0}{2\pi\xi_x(T)\xi_y(T)}, \quad (2.90) $$

so that the anisotropy is

$$ \frac{H_{c2}^{\parallel}}{H_{c2}^{x}} = \frac{\xi_{\parallel}(T)}{\xi_{x}(T)} = \frac{v_{F}}{\sqrt{2t_{x}} d_{x}} \\ = \sqrt{2} \sin \frac{\pi \rho t_{\parallel} d_{\parallel}}{2 t_{x} d_{x}}, \quad (2.91) $$

where in the last line a tight-binding band of total bandwidth $4t_{\parallel}$ has been assumed for the motion along the chain.

The above results are obtained from mean-field arguments. Fluctuation effects are expected to change these results. Especially, due to the anisotropy of the fluctuations, a change of critical field anisotropies (equations (2.89) and (2.91)) may be expected if fluctuations are strong. A detailed theory of these effects is lacking so far.

Concluding this section we wish to point out that the essential points discussed in the previous sections, like the coexistence (and competition between) of superconducting and density-wave type instabilities, large fluctuations, or the strong effect of disorder, are all of some importance in understanding experimental results on quasi-one-dimensional conductors to be discussed in the following sections.

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### 3. One-dimensional organic conductors: electronic properties

#### 3.1. Basis

##### 3.1.1. Elementary chemical considerations

The first molecular crystal exhibiting highly conducting properties was reported in 1954 [1], i.e. long before the discovery of TTF-TCNQ made this field of research very popular. When a solution of the organic molecule perylene was treated with bromine a highly conducting solid separated (figure 15).

However, the field of organic conductors really started in 1960 and 1970 with the discoveries of the molecules TCNQ (tetracyanoquinodimethane) and TTF (tetra-thiofulvalene), respectively [2, 3].
<figure><img src="image_16.png" /><figcaption>Figure 15. Chemical structure of some usual molecules which form charge transfer complexes (TCNQ-electron acceptor, Perylene, TTF, TSF, TMTSF, HMTSF-electron donors) and organic cations involved in radical anion salts, quinolinium (Qn) and NMP.</figcaption></figure>

TCNQ is a large planar molecule, consisting of a quinoid cycle. It is electrically neutral and diamagnetic with typically closed outer shells (figure 15).

The peculiarity of the TCNQ molecule is that it is an 'electron-poor' molecule which easily accepts an extra electron to become a chemically stable open shell anion (anion radical). TCNQ by itself is unable to conduct. The reason is that although the molecule will easily accept electrons from its neighbours all molecules have the same desire and finally a TCNQ crystal remains a Van der Waals solid with insulating properties.

If TCNQ is in an environment where electrons can be obtained from an electron-donating partner, however, the situation changes [4]. In solids such as quinolinium (TCNQ)₂, one electron is injected in each pair of TCNQ molecules. The charged counter-ion quinolinium thus exhibits the structure of a closed-shell (diamagnetic) ion (figure 15).

In the crystal, TCNQ molecules pile up like pancakes to form stacks (figure 16). Since the highest occupied states of the TCNQ molecule are filled on average by half an electron only, the transfer of electrons between TCNQs in a given stack is potentially possible. Systems like quinolinium (TCNQ)₂ are called radical-anion salts. They exhibit semiconducting properties. Moreover, systems such as radical-cation salts TTF-Clₓ [5] which can be considered as the cation counter part of Qn(TCNQ)₂ are highly conducting and show a metallic-like conduction in a limited temperature domain below room temperature.
<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>16 (a)</figcaption></figure>
<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>
<figure><img src="image_21.png" /><figcaption>16 (c)</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>16 (d)</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Figure 16—*concluded*. Crystal structure of some quasi one dimensional conductors. (a) TTF-TCNQ, a view normal to the *ac* plane of the crystal packing (top), a view down [100] of the crystal packing (bottom), (after reference [10] in section 3). (b) HMTSF-TCNQ, a view projected along the high conduction axis (top) and along the *b*-axis (bottom) (after reference [12] in section 3). (c) (TTF)Br<sub>0.79</sub> viewed down the *c*-axis (top) and down the *b*-axis (bottom) (after: LA PLACA, S. J., CORFIELD, P. W. R., THOMAS, R., and SCOTT, B. A., 1975, *Solid St. Commun.*, **17**, 635). (d) (TMTSF)<sub>2</sub>PF<sub>6</sub>, viewed along the *a*-axis (top) and side-viewed tilted 10° (bottom) (after reference [11] in section 3).</figcaption></figure>

TTF is a planar conjugated and ‘electron-rich’ molecule (figure 15) which can easily give a charge to another species ready to accept it. Actually TTF and TCNQ molecules combine to form a one-to-one ionic charge transfer complex according to a charge transfer reaction:

$$TTF + TCNQ \rightarrow TTF^{+\rho}TCNQ^{-\rho}$$

with an energy required for charge transfer which reads

$$\Delta E_{CT} = I - A - C < 0,$$

where *I* and *A* are the ionization potential of the donor and the electron affinity of the acceptor respectively, and *C* includes the Coulombic (Madelung), polarization and exchange energy contributions. In a complex such as TTF-TCNQ the amount of charge transfer *ρ* is unknown *a priori* by the stoichiometry of the complex. However, it can be determined very accurately from X-ray diffuse scattering experiments (see section 4).

In forming solids, the molecules TTF and TCNQ crystallize in segregated stacks of TTF and TCNQ molecules (figure 16). In the individual stacks of TTF and TCNQ molecules the π orbitals in the partially filled outer molecular shells overlap, so that the discrete levels of the individual molecules are spread out to form an energy band. The partially filled bands can thus lead to metallic-like conduction, provided the intramolecular electron repulsion (the Hubbard *U*) is not too strong compared to that bandwidth, namely *U* < *W*.

Moreover, a rather special feature of TTF-TCNQ or related compounds, as compared to ordinary metals is that the π orbitals of the molecules are strongly directional and interact mainly along the stacks, leaving only a weak interaction in
the perpendicular directions. Hence, the conductors exhibit very anisotropic conduction properties, they behave as quasi-one-dimensional conductors, although they are of course three-dimensional crystals.

The four sulphur atoms of the TTF molecule can be substituted with selenium to create the molecule TSF (tetraselenafulvalene) [6] (figure 15). Moreover, substituting the parent molecule with weakly donating alkyl groups, for instance methylene (CH₂) groups, has led to the molecule HMTSF [7] (hexamethylene/tetraselenafulvalene) or HMTTF [8] for the sulphur molecule, and methyl groups (CH₃) has led to TMTSF [9] (tetramethyltetraselenafulvalene). The present and following sections will show how these apparently minor modifications of the parent molecule significantly affect the crystal structure and the electronic properties of the molecular conductors.

### 3.1.2. Basic crystal structures

Two kinds of molecular stackings are observed for highly conducting molecular crystals.

(i) The 'skew' stackings illustrated by the structure of TTF-TCNQ [10] (figure 16), which consists in a regular stacking of molecules along a stack direction with the molecular planes tipped with respect to the stacking axis. Most conducting charge-transfer complexes exhibit this kind of stacking.

(ii) The 'zig-zag' stacking of molecules along quasi-regular chains, illustrated by the structure of the radical-cation salts (TMTSF)₂X [11] X = PF₆, AsF₆, ..., NO₃, ClO₄, BF₄, ReO₄, ... (figure 16).

TTF-TCNQ crystallizes in the monoclinic system with parallel conducting chains of TTF and TCNQ separately stacked along the crystallographic *b*-axis (figure 16). The tilt of the normal to the molecular plane with respect to the *b*-axis is 24.5 and 34.0° for TTF and TCNQ, respectively, and the interplanar distances are 3.47 and 3.17 Å for TTF and TCNQ chains, respectively (see table 1 for the crystal data of TTF-TCNQ).

A view of the crystal normal to the *a*-*c* plane reveals sheets of cations and anions parallel to the *b*-axis (figure 16).

The charge transfer complex HMTSF-TCNQ, in which the donor molecule is significantly longer than the TTF molecule, adopts a different arrangement [12]. The monoclinic (nearly orthorhombic) structure of HMTSF-TCNQ consists of segregated stacks of HMTSF cations and TCNQ anions. However, a view of the structure along the conducting axis (figure 16), labelled *c* by the crystallographers, shows an arrangement of HMTSF and TCNQ chains which alternates along the *b* and *a*-directions (a chessboard pattern in the *a*-*b* plane).

The salient feature of the HMTSF-TCNQ crystal structure is the relatively short selenium-nitrogen interchain distance, Se···N = 3.10 Å instead of 3.50 Å for the sum of Se and N Van der Waals radii. This fact suggests that significant interchain π bonding could arise from the particular structure, increasing substantially the interchain coupling along the *b* direction.

The basic feature of the structure of the isomorphous (TMSF)₂X salts [11] (X = PF₆, ..., NO₃, ClO₄, ..., BF₄, ...) is the zig-zag stack of the nearly planar TMTSF molecules parallel to *a* (figure 16). In this triclinic structure the stacks are arranged in sheets with fairly close contacts of Se atoms in neighbouring stacks. The shortest Se···Se distance is 3.87 Å, whereas the Van der Waals distance is 4 Å.
<table><caption>Table 1. Crystallographic data.</caption><thead><tr><th>Crystallographic system</th><th>TTF-TCNQ<sup>d</sup><br>Monoclinic</th><th>TSF-TCNQ<sup>b</sup><br>Monoclinic</th><th>HMTSF-TCNQ<sup>e</sup><br>Monoclinic</th><th>TMTSF-DMTCNQ<sup>d</sup><br>Triclinic</th><th>(TMTSF)<sub>2</sub>PF<sub>6</sub><sup>e</sup><br>Triclinic</th></tr></thead><tbody><tr><td>Space group</td><td>P2<sub>1</sub>/c</td><td>P2<sub>1</sub>/c</td><td>C2/m</td><td>P<span style="text-decoration: overline">I</span></td><td>P<span style="text-decoration: overline">I</span></td></tr><tr><td>a</td><td>12.298</td><td>12.505</td><td>21.999</td><td>3.938</td><td>7.29</td></tr><tr><td>b</td><td>3.819</td><td>3.876</td><td>12.573</td><td>8.035</td><td>7.71</td></tr><tr><td>c</td><td>18.468</td><td>18.514</td><td>3.890</td><td>18.956</td><td>13.52</td></tr><tr><td>α</td><td>—</td><td>—</td><td>—</td><td>97.31</td><td>83.39</td></tr><tr><td>β</td><td>104.46</td><td>—</td><td>90.29</td><td>98.12</td><td>86.27</td></tr><tr><td>γ</td><td>—</td><td>—</td><td>—</td><td>91.37</td><td>71.01</td></tr><tr><td>V (unit cell) Å<sup>3</sup></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Density (from structure)</td><td>839.9</td><td></td><td>1075</td><td>592.0</td><td>714.3</td></tr><tr><td>g cm<sup>-3</sup></td><td>1.615</td><td></td><td>2.09</td><td>1.841</td><td>2.42</td></tr></tbody></table>

Within a stack the units repeat by an inversion operation leading to an alteration of the average interplanar distances, 3.63 and 3.66 Å. (TMTSF)<sub>2</sub>PF<sub>6</sub> therefore presents a slight dimerization of the structure along *a*. This dimerization, already mentioned in the study of the isomorphous (TMTTF)<sub>2</sub>X family [13] is specific to the (TMTSF)<sub>2</sub>X structure. It opens a gap in the electronic band structure at a wave-vector which is twice the Fermi wave-vector (see section 3.1.4). In the particular structure of the (TMTSF)<sub>2</sub>X salts the anions reside at an inversion centre of the three-dimensional structure. Consequently, non-centrosymmetric anions such as tetrahedral ClO<sub>4</sub><sup>-</sup> or planar NO<sub>3</sub><sup>-</sup> can occupy either two equivalent positions statistically (they are therefore disordered) or are ordered with a possible change in the one-dimensional lattice periodicity. When they order, two different situations are encountered: one which leads to a doubling of the one-dimensional lattice periodicity (alternating as for the ReO<sub>4</sub><sup>-</sup> salt) and another which does not affect the lattice periodicity (uniform as for the ClO<sub>4</sub><sup>-</sup> salt at low temperature) (see section 5.4). Although these closed-shell and widely spaced anions play no role in the conductivity, the possible ordering may have important consequences on the electronic properties at low temperature (see section 5.4.1).

The one-dimensional character of the electronic properties of the charge transfer complexes and salts which derive from the directionality of the electron wave-functions forming the conduction band should not be confused with the three-dimensional nature of the lattice frame.

### 3.1.3. *Three-dimensional lattice properties: elastic constants*

Elastic properties, which are closely related to the crystal cohesion, have provided valuable information about the nature of intermolecular forces. The effect of pressure on the structure of TTF-TCNQ has been studied up to 4.6 kbar using single-crystal diffraction [14]. The axial compressibility coefficients thus obtained at

<footnote><sup>a</sup> KISTENMACHER, T. J., PHILLIPS, T. E., and COWAN, D. O., 1974, *Acta crystallogr.* B, **30**, 763.</footnote>

<footnote><sup>b</sup> SCHULTZ, T. D., and CRAVEN, R. A., 1979, *Highly Conducting One-Dimensional Solids*, edited by J. T. DEVREESE (New York: Plenum), p. 147; CORFIELD, P., and LA PLACA, S., unpublished results.</footnote>

<footnote><sup>c</sup> PHILLIPS, T. E., KISTENMACHER, T. J., BLOCH, A. N., and COWAN, D. O., 1976, *Chem. Commun.*, 334.</footnote>

<footnote><sup>d</sup> ANDERSEN, J. R., BECHGAARD, K., JACOBSEN, C. S., RINDORF, G., SOLING, H., and THORUP, N., 1978, *Acta crystallogr.* B, **34**, 1901.</footnote>

<footnote><sup>e</sup> THORUP, N., RINDORF, G., SOLING, H., and BECHGAARD, K., 1981, *Acta crystallogr.* B, **37**, 1236.</footnote>
room temperature and ambient pressure are $k_a = 0.18\% \text{ kbar}^{-1}$, $k_b = 0.59\% \text{ kbar}^{-1}$ and $k_c = 0.13\% \text{ kbar}^{-1}$. In the 4–6 kbar range, transverse compressibilities are only weakly pressure dependent, whereas the compressibility along the chains show an important decrease of ≈ 8% kbar$^{-1}$. Therefore, above 10 kbar the crystal would become nearly isotropic, from the strictly lattice point of view. Single-crystal diffraction patterns under pressure are in fair agreement with a re-examination of earlier powder data obtained up to ≈ 15 kbar [15], indicating $k_a = 0.1\% \text{ kbar}^{-1}$, $k_b = 0.18\% \text{ kbar}^{-1}$ and $k_c = 0.11\% \text{ kbar}^{-1}$ at P = 15 kbar. Subsequent single-crystal high-pressure studies [16] have shown that the compressibility along the stacking direction is strongly temperature dependent, dropping by a factor of about 2 at 10 K and P = 1 atm. The change of compressibilities is however much less severe for the transverse directions. Moreover, the elastic constants obtained from high-pressure crystallography agree fairly well with the values derived from the initial slope of the dispersion relation for LA phonons propagating along the chain axis [17]. One can notice that conducting molecular crystals of the TTF–TCNQ series appear stiffer than the vast majority of insulating Van der Waals molecular solids; for example k = 2.9, 1.64 and 1.42% kbar$^{-1}$ in solid neon, naphthalene and anthracene, respectively.

Furthermore, conducting molecular crystals appear slightly softer along the conducting direction than along the transverse directions. The linear compressibility along the *b* direction is similar to that observed for soft metal like sodium (*k* = 0.45% kbar$^{-1}$). Along the *a* and *c* directions TTF–TCNQ resembles ionic solids and the compressibilities are actually very cose to the values typically found in inorganic ionic crystals (*k* = 0.24% kbar$^{-1}$ in CsI). The intrachain bonding is of the metallic type and the equilibrium intermolecular distance along the chain direction results from the interplay between an attractive contribution (band formation) and a repulsive Coulomb contribution between similarly charged molecules in a chain. The relatively large stacking distance of the molecules in the chains and the resulting small bandwidth make the variation of the cohesive energy with the *b* distance weak, resulting in a relatively high compressibility. The binding along the *a*-axis direction is due to the Coulomb attraction between oppositely charged donor and acceptor molecules. An appreciable proportion of the charge transfer presumably occurs between the chalcogen atoms and the nitrogen atoms of the donor and acceptor, respectively [18]. Therefore, the stability of the structure is given by the short N–S distance, of the order of the sum of the Van der Waals radii, with attractive Coulomb forces and Van der Waals forces balancing repulsive forces of the usual origin. The packing of the molecules in the (*b*, *c*) planes is slightly more subtle since there exist structures in which alternation of donor and acceptor molecules occurs along the *c* direction (the chess-board structure of HMTTF–TCNQ) and structures in which the (*b*, *c*) planes consist of sheets of similarly charged molecules (the TTF–TCNQ structure (figure 16)). Obviously, Coulomb model prefers charge alternation along the *c*-axis as found in the HMTSF–TCNQ structure. Indeed the calculation of the Madelung energy made by various authors [19] shows that this energy is insufficient by itself to stabilize the TTF structure. Consequently, we have to invoke other contributions to the binding energy in order to explain both the stability of the conducting charge transfer salts and the differences of structure [15]. Polarization energy may bring a dominant contribution to the cohesive energy. This contribution results from dynamically induced dipolar attraction (Van der Waals forces) between highly polarizable neighbouring molecules [20]. It can possibly overwhelm the monopole Coulombic repulsion existing
for example in the (b, c) layers of the TTF structure. The electrostatic interaction between the dynamically induced dipoles $M_A$ and $M_D$ is $(M_A^2 + M_D^2)/R^6$ and $2M_A M_D/R^6$ for the TTF and the HM structure, respectively, provided the distance of separation R between molecules is larger than the length of individual molecules [15]. The TTF structure becomes more stable than the HM structure if the gain of cohesive energy including Madelung and polarization energy going from the TTF to the HM structure $(M_D - M_A)^2/R^6 - E_M > 0$ ($E_M$ is necessarily a positive quantity). Following this argument, the existence of donor molecules long compared to the TCNQ molecule should make R large and favour the HM structure. Actually, we do not expect $(M_D - M_A)^2$ to vary much, if at all, going from TTF-TCNQ to HMTTF-TCNQ.

The difference of structure is not reflected in the c-axis conduction, for which absolute values of $\sigma_c$ are nearly similar throughout various compounds, while very different values are noticed along the b and a directions (section 3.2.1 and table 2).

The thermal expansion of TTF-TCNQ has been studied by X-ray diffraction [21], capacitance dilatometry [22] and single-crystal neutron diffraction [14]. The neutron-diffraction data lead to thermal expansion coefficients at room temperature $\alpha_a = 0.37 \times 10^{-4}$, $\alpha_b = 1.47 \times 10^{-4}$, $\alpha_c = 0.27 \times 10^{-4}$ and $\alpha_{vol} = 1.5 \times 10^{-4}$ (K$^{-1}$). The thermal expansion anisotropy is of the same order of magnitude as the compressibility anisotropy. The thermal expansion along the chain axis is particularly sensitive to the temperature since it drops to about $0.5 \times 10^{-4}$ K$^{-1}$ at liquid nitrogen temperature. The lattice contraction along the stacking axis is 2.5% between 300 and 10 K for TMTSF-DMTCNQ [23], i.e. very similar to values reported for TTF-TCNQ of 2.4% [21]. As far as the organic superconductor (TMTSF)$_2$PF$_6$ is concerned [23] a significantly larger effect of the temperature on the stacking axis distance is observed since cooling from 300 to 10 K amounts to a contraction of 3.7%.

A major result of the study of the effect of pressure and temperature on the TTF-TCNQ structure is the following: a hydrostatic pressure of about 5 kbar is essentially equivalent as far as the structure is concerned to the effect of a temperature decrease from 300 to about 77 K. This important remark accounts for the modification of the overlap integral $t_\parallel$ and thus of the bandwidth occurring upon cooling. This feature explains why the charge transfer, equal to 0.55 at room temperature, increases to 0.59 at 59 K as measured in accurate X-ray diffuse scattering experiments [24]. Moreover, this equivalence between pressure and temperature makes great care necessary when investigating the temperature dependence of physical properties which are also strongly volume dependent. As first emphasized for TTF-TCNQ, the constant volume temperature dependence of the resistance [25, 26] is qualitatively different from the data obtained under constant pressure (the most usual experimental situation). Obviously this remark extends to all organic conductors. Finally the study of the anisotropy of compressibility and thermal expansion [14] shows that the intermolecular forces in TTF-TCNQ are strongly anharmonic in the b-direction and much less in the (ac) plane (the volume Grüneisen constant is about 2.3–2.7). The main contribution to the anharmonicity is provided by the intrachain intermolecular forces.

### 3.1.4. Band formation

The piling up of flat molecules in conducting molecular crystals seems to be mandatory for achieving strong overlap between electron clouds of neighbouring molecules and consequently charge delocalization. An approximation description of
<table><caption>Table 2. Conduction and anisotropy at 300 K (Ω cm)⁻¹.</caption><thead><tr><th></th><th>σ<sub>‖</sub>(Ω cm)<sup>-1</sup></th><th>σ<sub>‖</sub>/σ<sub>a</sub></th><th>σ<sub>‖</sub>/σ<sub>c</sub></th></tr></thead><tbody><tr><td>Qn(TCNQ)<sub>2</sub></td><td>70–80</td><td>—</td><td>—</td></tr><tr><td>TTF–TCNQ</td><td>{ 400<sup>b,c</sup><br>300–500<sup>d</sup></td><td>{ 350<sup>b</sup><br>500<sup>c</sup></td><td>120<sup>c</sup></td></tr><tr><td>TSF–TCNQ</td><td>800<sup>e,b</sup></td><td>300<sup>b</sup></td><td>280<sup>b</sup></td></tr><tr><td>HMTTF–TCNQ</td><td>{ 550<sup>b</sup><br>400<sup>g</sup></td><td>80<sup>b</sup></td><td>160<sup>f</sup></td></tr><tr><td>HMTSF–TCNQ</td><td>1800<sup>b,h</sup></td><td>30<sup>b</sup></td><td>450<sup>b</sup></td></tr></tbody><thead><tr><th></th><th>σ<sub>‖</sub>(Ω cm)<sup>-1</sup></th><th>σ<sub>‖</sub>/σ<sub>b</sub></th><th>σ<sub>‖</sub>/σ<sub>c</sub></th></tr></thead><tbody><tr><td>TMTSF–DMTCNQ</td><td>400–600<sup>i</sup></td><td>200–300<sup>i</sup></td><td>200–300<sup>i</sup></td></tr><tr><td>(TMTSF)<sub>2</sub>PF<sub>6</sub></td><td>540<sup>j</sup></td><td>{ 200<sup>k</sup><br>360<sup>j</sup><br>50<sup>l</sup></td><td>30 000<sup>k</sup></td></tr><tr><td>(TMTTF)<sub>2</sub>PF<sub>6</sub></td><td>40<sup>m</sup></td><td></td><td></td></tr><tr><td>(TMTTF)<sub>2</sub>Br</td><td>240<sup>m</sup></td><td></td><td></td></tr></tbody></table>

the electron energy spectrum along the chains has been obtained by the extension to molecular orbitals obtained by the extended Hückel method [27] of the tight-binding approximation in metals [28].

Bloch wavefunctions of the one-dimensional electron gas are formed from linear combinations of molecular orbitals, namely

$$ \psi_{\mathbf{k}}(r) = \sum_{R_i} \phi(r - R_i) \exp(i\mathbf{k} \cdot \mathbf{R}_i), $$

where $-\pi/a < k < \pi/a$, and a is the lattice spacing along the chain. In this approximation, the energy related to the state $\psi_{\mathbf{k}}$ is given by:

$$ \varepsilon(\mathbf{k}) = 2t_{\parallel} \cos ka, \qquad (3.1) $$

<footnote><sup>a</sup> BURAROV, L. I., FEDUTIN, D. N., and SCHEGOLEV, I. F., 1971, *Sov. Phys. JETP*, **32**, 612; WEYL, C., JÉROME, D., CHAIKIN, P. M., and STRZELECKA, H., 1981, *Solid St. Commun.*, **38**, 1143.</footnote>

<footnote><sup>b</sup> COOPER, J. R., JÉROME, D., ETEMAD, S., and ENGLER, E. M., 1977, *Solid. St. Commun.*, **22**, 257.</footnote>

<footnote><sup>c</sup> COHEN, J., COLEMAN, L. B., GARITO, A. F., and HEEGER, A. J., 1974, *Phys. Rev. B*, **10**, 1298.</footnote>

<footnote><sup>d</sup> FERRARIS, J. P., and FINNEGAN, T. F., 1976, *Solid St. Commun.*, **18**, 1169.</footnote>

<footnote><sup>e</sup> ETAMAD, S., PENNEY, T., ENGLER, E. M., SCOTT, B. A., and SEIDEN, P. E., 1975, *Phys. Rev. Lett.*, **34**, 741.</footnote>

<footnote><sup>f</sup> FRIEND, R. H., JÉROME, D., FABRE, J. M., GIRAL, L., and BECHGAARD, K., 1978, *J. Phys. C*, **11**, 263.</footnote>

<footnote><sup>g</sup> GREENE, R. L., MAYERLE, J. J., SCHUMAKER, R., CASTRO, G., CHAIKIN, P. M., ETEMAD, S., and LA PLACA, S. J., 1976, *Solid St. Commun.*, **20**, 943.</footnote>

<footnote><sup>h</sup> BLOCH, A. N., COWAN, D. O., BECHGAARD, K., PYLE, R. E., BANKS, R. H., and POEHLER, T. O., 1975, *Phys. Rev. Lett.*, **34**, 1561.</footnote>

<footnote><sup>i</sup> JACOBSEN, C. S., MORTENSEN, K., ANDERSEN, J. R., and BECHGAARD, K., 1978, *Phys. Rev. B*, **18**, 905.</footnote>

<footnote><sup>j</sup> BECHGAARD, K., JACOBSEN, C. S., MORTENSEN, K., PEDERSEN, H. J., and THORUP, N., 1980, *Solid St. Commun.*, **33**, 1119.</footnote>

<footnote><sup>k</sup> JACOBSEN, C. S., MORTENSEN, K., WEGER, M., and BECHGAARD, K., 1981, *Solid St. Commun.*, **38**, 423.</footnote>

<footnote><sup>l</sup> GREENE, R. L., and ENGLER, E. M., 1980, *Phys. Rev. Lett.*, **45**, 1588.</footnote>

<footnote><sup>m</sup> COULON, C., DELHAES, P., and FLANDROIS, S.</footnote>
<figure><img src="image_24.png" /><figcaption>Figure 17. Electron energy dispersion for a one-dimensional conductor. In the tight-binding band model the bandwidth is $4t_‖$, with $t_‖ < 0$ in the present situation.</figcaption></figure>

where $t_‖$ (the longitudinal transfer integral) is proportional to the ionization potential and to the nearest neighbour overlap of molecular orbitals (figure 17). The relevant molecular orbitals of planar molecules such as TTF or TCNQ are the $π$-molecular orbitals formed from the linear combination of atomic $p_z$ functions. Figure 18 displays the highest occupied states molecular orbitals of the singly charged radical ions TTF$^+$ and TCNQ$^-$. The wavefunctions of TCNQ$^-$ are odd under reflection through the molecular plane. Consequently, a bonding interaction between two nearest neighbour TCNQ molecules is obtained only if, along a chain, a molecule is displaced laterally with respect to its nearest neighbour. Along a

<figure><img src="image_25.png" /><figcaption>Figure 18. Conduction band molecular orbitals for TCNQ$^-$ and TTF$^+$. Numbers in circles are amplitudes of $p_z$ atomic orbitals. Dashed lines indicate approximate positions of nodal surfaces (after reference [28] of section 3).</figcaption></figure>
direction perpendicular to the plane of the molecules the C=C double bond lies at the centre of the six carbon quinoid ring. For such a stacking, the lowest electron energy occurs when all molecular orbitals are 'in phase', i.e. the $k = 0$ state of the tight-binding band is the state of lowest energy, namely ($t_{\parallel}^Q < 0$), as shown in figure 17. On the other hand the band structure of the TTF chain is inverted, namely $t_{\parallel}^F > 0$, with an energy maximum at the Brillouin zone centre. The extended Hückel model [28] gives rise to band parameters $t_{\parallel} = 0.11$ eV, 0.05 eV for TCNQ and TTF stacks, respectively. The TCNQ conduction bandwidth is about 0.4–0.5 eV, and the TTF band is probably half as wide as the TCNQ band. Moreover, the bandwidths transverse to the molecular stacking are about two orders of magnitude smaller than those along the stacks. The latter bandwidths compare favourably with the calculation of either the splitting between the highest occupied states of TTF dimers, 0.12 eV or the 'bandwidth' for the trimer situation of 0.2 eV, leading to infinite stack bandwidth of about 0.3 eV [29]. In TTF-TCNQ, 2 electrons per formula unit (coming from the TTF molecule) must be accommodated in the bands deriving from TTF and TCNQ. In the charge transfer crystal the electron charge $\rho$ is transferred from TTF to TCNQ. Since the overall two-chain conductor is neutral the two bands intersect at a single Fermi wave-vector related to $\rho$ by the equation:

$$2k_F = \frac{\pi}{a}\rho. \qquad (3.2)$$

In compounds like TTF-TCNQ (with 2 electrons per unit cell) all k-states between $-\pi/a$ and $\pi/a$ are occupied, with the restriction that between $-k_F$ and $+k_F$, the occupied states belong to the TCNQ band whereas outside this domain they pertain to the TTF band (figure 19). NMP-TCNQ is a system which deserves some comments since it has been considered for a long time as a radical-ion salt in which the diamagnetic NMP$^+$ ions in TCNQ$^-$ NMP$^+$ play no role in the conducting properties [30]. Extensive studies [30] have been performed substituting the phenazine molecule for the NMP molecule. The phenazine is of similar size, shape and polarizability to NMP$^+$ but is neutral (closed shell) and symmetric. Consequently, the number of conduction electrons per unit cell to be shared among the donors and TCNQ is equal to the fraction $x$ of NMP in the alloy (NMP)$_x$(Phen)$_{1-x}$ TCNQ. An X-ray diffuse scattering study of the solid solutions with $0.5 < x < 1$

<figure><img src="image_26.png" /><figcaption>Figure 19. Electron energy dispersion for TTF-TCNQ (left) assuming no tunnelling interchain coupling and for NMP-TCNQ (right) showing the existence of two sets of Fermi planes at $\pm k_F^Q$ and $\pm k_F^P$.</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>Figure 20. Number of conduction electrons per TCNQ (Δ) and per donor molecule (·) versus NMP concentration in the alloys (NMP)$_x$(Phen)$_{1-x}$ TCNQ. The solid line gives the total conduction electron concentration available. The upper part of the figure shows that with decreasing x, three different electronic instability regimes are successively observed on the TCNQ chains: (i) the $2k_F$ instability for $2/3 \le x \le 1$, (ii) both $2k_F$ and $4k_F$ instabilities for $0.57 \le x \le 2/3$, and (iii) the $4k_F$ instability alone for x close to 1/2. Note the appearance of the $4k_F$ instability as the half-filled band situation is approached for the acceptor (TCNQ) band (after POUGET, J. P., COMÈS, R., EPSTEIN, A. J., and MILLER, J. S., 1982, *Molec. Crystals* **85**, 203).</figcaption></figure>

leads to the conclusion of electronic conduction on both the donor and acceptor chains and has led to the determination of the respective density of carriers (figure 20). Even in pure NMP-TCNQ there exists a back charge transfer from TCNQ on to the NMP molecules (about 1/3 electron per NMP molecule) [30]. As there is only one electron per NMP-TCNQ unit to fill both the TCNQ and the NMP bands, only half of the *k* states between $-\pi/a$ and $+\pi/a$ are occupied. NMP-TCNQ exhibits two sheets of Fermi surfaces, one for the TCNQ chain with $k_F^{\text{TCNQ}} = \pi/3a$ and another one for the NMP chain with $k_F^{\text{NMP}} = 5\pi/6a$ (figure 19).

## 3.2. *Physical properties related to one-dimensional energy bands*

### 3.2.1. *Transport properties*

The highly conducting character of some charge transfer complexes or salts is summarized in table 2. The longitudinal conductivity of sulphur-based compounds does not exceed $\approx 600 (\Omega \text{ cm})^{-1}$, whereas much higher values have been observed in selenium-based compounds, in particular for HMTSF-TCNQ which displays a conductivity of $\approx 1800 (\Omega \text{ cm})^{-1}$ at room temperature. The uncertainty existing in the absolute value determination of the conductivity is usually related to the inaccuracy of the sample size measurement. However, both temperature or pressure
dependences can be reliably obtained with an accuracy of four digits. Furthermore, the temperature dependence of the conductivity on cooling shows typical metallic properties [31, 32]. Cooling from 300 to 60 K, the conductivity increases by factors about 15–25, 12 and 4, for TTF-TCNQ [33–35], TSF-TCNQ [36] and HMTSF-TCNQ [37], respectively (figure 21). In the liquid nitrogen temperature range there seems to be not much difference between the absolute values of the conductivities for the various compounds listed in table 2 as long as the band-filling is incommensurate.

In the case of commensurate band filling two situations are encountered:

(i) a slow decrease of the conductivity at low temperature as shown by the behaviour of Qn (TCNQ) [38, 4] or

(ii) a sharp increase of the conductivity which is characteristic of the (TMTSF)₂X family [39] in which superconductivity has been found at low temperature (see section 5).

The thermopower of most compounds is seen to be rather small at room temperature and takes the sign of the carriers which are dominant in the conduction process. In a two-chain conductor the total thermopower is given by the individual thermopower coefficients weighted by the respective conductivities [40, 41]:

$$S = \frac{\sigma_1 S_1 + \sigma_2 S_2}{\sigma_1 + \sigma_2}. \qquad (3.3)$$

Even for a single band thermoelectric powers involve the energy dependence of state density and relaxation times. If, however, one ascribes the quasi-linear temperature dependence of the TTF-TCNQ negative thermopower between 100 and 300 K [40]

<figure><img src="image_28.png" /><figcaption>Figure 21. Temperature dependence of the resistivity in some representative organic conductors. A Peierls transition is observed at 53, 29 and 24 K for TTF-TCNQ, TSF-TCNQ and HMTSF-TCNQ respectively. The conductor-semiconductor transition at 12 K in (TMTSF)₂PF₆ is due to the formation of a spin density wave (SDW) state. The resistivity of the radical-anion salt Qn(TNCQ)₂ does not exhibit any sharp transition on cooling.</figcaption></figure>
to the electrons in a one-dimensional tight-binding band of the TCNQ chain, the relation (3.3) becomes:

$$S = - \frac{\pi^2 k_B T}{3|e|} \frac{\cos \pi \rho / 2}{2|t| \sin^2 \pi \rho / 2} \quad (3.4)$$

According to the data in figure 22, the TCNQ band filled by 0.59 electron mol⁻¹ leads to a bandwidth $4t_‖ = 0.54$ eV.

Although the energy dependence of the relaxation time also contributes to the coefficient of T in the thermopower it seems likely that the contributions coming from the band structure, equation (3.4), is dominant. The electron mean free path is small at room temperature and therefore it is conceivable that the energy dependence of the scattering is sufficiently smeared out that the scattering dependent term is negligible with respect to the band structure term. The large thermopower commonly observed in organic conductors (about one order of magnitude larger than observed in typical metals) is due to the existence of small bandwidths, less than 1 eV or so.

The study of the isostructural series (TSF)ₓ(TTF)₁₋ₓ – TCNQ [42], 0 < x < 1, has provided a good illustration of (3.3) which becomes, for a two-chain conductor within the tight binding approximation:

$$S = \frac{\pi^2 k_B^2 T}{6|e|} \left| \frac{\tau_1 - \tau_2) \cos \pi \rho / 2}{(|t_1|\tau_1 + |t_2|\tau_2) \sin^2 \pi \rho / 2} \right| \quad (3.5)$$

<figure><img src="image_29.png" /><figcaption>Figure 22. Temperature dependence of the absolute thermopower coefficients in a series of charge transfer complexes and radical- (cation or anion) salts.</figcaption></figure>
where $\tau_1$ and $\tau_2$ are the scattering times. The room temperature thermopower varies from $-28 \ \mu V \text{K}^{-1}$ for pure TTF-TCNQ to $+3 \ \mu V \text{K}^{-1}$ for TSF-TCNQ. Furthermore upon doping the donor stack, disorder increases on the stack, making its contribution on the total conductivity smaller: the disorder introduced by doping shortens the electron scattering time of the $(\text{TSF})_x(\text{TTF})_{1-x}$ stack. The decrease of the conduction on the donor stack enables the thermopower to reach its maximum (negative) value in the most disordered situation, namely $x \approx 0.5$. It has been concluded that the conductivity is predominantly on the TCNQ chain in TTF-TCNQ ($\sigma_Q \approx 5\sigma_F$) but about equally balanced between both chains for the selenium compound [41].

The thermopower of single-chain conductors, for example the $(\text{TMSF})_2\text{X}$ series ranging around $+20 \ \mu V \text{K}^{-1}$ at room temperature follows a quasi-linear temperature dependence [39], and extrapolates to a value near zero at 0 K (figure 22). The use of the tight-binding formula (equation (3.4)) for the thermopower in a three-quarter filled band leads to an overall bandwidth (i.e. neglecting the small dimerization of the TMTSF molecules along the *a*-axis) between 0.8 and 1.1 eV.

$\text{Qn}(\text{TCNQ})_2$, another single chain conductor, exhibits a fairly large Seebeck coefficient [38], close to $-60 \ \mu V \text{K}^{-1}$ at 300 K and almost temperature independent down to 100 K (figure 22). It has been suggested by Chaikin and Beni [43] that the value $-60 \ \text{V K}^{-1}$ may be associated with the entropy of a free spin, namely $k_B \ln 2$. All highly conducting 1:2 TCNQ salts show a thermopower which saturates to this value at room temperature [38]. This has been taken as strong evidence for sizeable correlations in these systems.

The measurement of the Hall coefficient also reveals the metal-like conducting nature of these molecular crystals. In TTF-TCNQ, the d.c. Hall constant (figure 23) is about $-4.2 \times 10^{-11} \text{ V cm A}^{-1} \text{Oe}^{-1}$, at room temperature [4], and is also largely independent of temperature down to 100 K. The Hall effect results confirm the predominance of the TCNQ carriers in the conduction of TTF-TCNQ and the value of the charge-transfer, $0.58 \text{ electron mol}^{-1}$, derived more accurately from X-ray scattering results (see section 4). The Hall constant of HMTSF-TCNQ [45] is however positive at room temperature, indicating that the carriers on the HMTSF stack are dominant in the conduction process. The magnitude $R_H \approx 2 \times 10^{-11} \text{ V cm A}^{-1} \text{Oe}^{-1}$ suggests that like TTF-TCNQ there is about 0.5–1 carrier mole$^{-1}$ contributing to the conduction in fair agreement with the X-ray determination of the charge-transfer $\rho = 0.74$ [46].

### 3.2.2. *Optical properties*

The metallic character of TTF-TCNQ and related compounds is also revealed by the optical spectra obtained in the visible and near-infrared region [47–50]. A typical Drude-like edge is observed in the 1 eV range for polarization of the light parallel to the highly conducting direction of the single crystals (figure 24). For a transverse polarization the light is not observed in the visible and near-infrared domains. This property reflects the very anisotropic nature of the electronic structure. However, the Drude-like edge observed at low temperature in $(\text{TMTSF})_2\text{PF}_6$ [50] (figure 25) for polarization of the light along the transverse *b* direction suggests the existence of a somewhat stronger interchain coupling along the same transverse direction, which is corroborated by an inspection of the crystal structure [11]. The interchain Se···Se distance which is particularly small in the $(\text{TMTSF})_2\text{X}$ structure, makes the
<figure><img src="image_30.png" /></figure>

Figure 23. Temperature dependence of the Hall constant for TTF-TCNQ compared with that of HMTSF-TCNQ (after COOPER, J. R., MILJAK, M., DELPLANQUE, G., JÉROME, D., Weger, M., FABRE, J. M., and GIRAL, L., 977, *J. Phys., Paris*, **38**, 1097; and COOPER, J. R., Weger, M., DELPLANQUE, G., JÉROME, D., and BECHGAARD, K., 1976, *J. Phys. Lett., Paris*, **37**, L349).

<figure><img src="image_31.png" /></figure>

Figure 24. The polarized reflectance of TTF-TCNQ in the infrared (after TANNER, D. B., JACOBSEN, C. S., GARITO, A. F., and HEEGER, A. J., 1976, *Phys. Rev. B*, **13**, 3381).
<figure><img src="image_32.png" /><figcaption>Figure 25. The polarized reflectance of (TMTSF)₂PF₆ at various temperatures. The rise of a plasma edge is observed at low temperature near 1000 cm⁻¹ for E||b' (after reference [50] of section 3).</figcaption></figure>

interchain electron hopping much easier than in the TTF-TCNQ or TSF-TCNQ systems. A Drude interpretation of the reflectance data, taking

$$R = \frac{1 + |\epsilon| - [2(|\epsilon| + \epsilon_1)]^{1/2}}{1 + |\epsilon| + [2(|\epsilon| + \epsilon_1)]^{1/2}}, \quad (3.6)$$

with $\epsilon(\omega) = \epsilon_{\infty} - \omega_p^2/[\omega(\omega + i\tau_{sp}^{-1})] = \epsilon_1 + i\epsilon_2$ and $\omega_p^2 = 4\pi n e^2/m^*$ ($\epsilon_{\infty}$ is the high-frequency dielectric constant arising from core polarizability) leads to the results summarized in table 3 for TTF-TCNQ [47, 48], TSF-TCNQ [49] and (TMTSF)₂PF₆ [50]. In the conducting regime of TTF-TCNQ, almost no temperature dependence of $\omega_p$ has been observed (the very small increase reported by Grant et al. [47] at low
<table><caption>Table 3. Optical data.</caption><thead><tr><th></th><th>ω<sub>p</sub> (eV)</th><th>τ (s<sup>-1</sup>)</th><th>ε<sub>0</sub></th></tr></thead><tbody><tr><td>TTF-TCNQ (300 K)</td><td>1.38<sup>a</sup></td><td>2.3 × 10<sup>-15</sup> <sup>a</sup></td><td>3.3<sup>a</sup></td></tr><tr><td></td><td>1.2<sup>b</sup></td><td>3 × 10<sup>-15</sup> <sup>b</sup></td><td>2.4<sup>b</sup></td></tr><tr><td>TTF-TCNQ (60 K)</td><td>1.46<sup>a</sup></td><td>3 × 10<sup>-15</sup> <sup>a</sup></td><td>3<sup>a</sup></td></tr><tr><td></td><td>1.2<sup>b</sup></td><td>6 × 10<sup>-15</sup> <sup>b</sup></td><td>2.15<sup>b</sup></td></tr><tr><td>TSF-TCNQ</td><td>1.4<sup>c</sup></td><td>4.2 × 10<sup>-15</sup> <sup>c</sup></td><td>2.25<sup>c</sup></td></tr><tr><td>(TMTSF)<sub>2</sub>PF<sub>6</sub> (300 K)</td><td>1.39<sup>d</sup></td><td>3 × 10<sup>-15</sup> <sup>d</sup></td><td>2.6<sup>d</sup></td></tr><tr><td>(TMTSF)<sub>2</sub>PF<sub>6</sub> (25 K)</td><td>1.41<sup>d</sup></td><td>9.9 × 10<sup>-15</sup> <sup>d</sup></td><td>2.9<sup>d</sup></td></tr></tbody></table>

temperature may be ascribed to the thermal contraction of the unit cell, increasing the charge transfer in a two-chain conductor (see X-ray scattering data [24]).

The electron scattering time definitely changes with temperature increasing by about a factor 2 between room temperature and 60 K, but this moderate increase is in striking contrast with that of the conductivity which peaks at 15–25 times the room temperature conductivity around 60 K; but it agrees in order of magnitude with the temperature dependence of the transverse conductivity [51]. These phenomena are understandable in the framework of the one-dimensional Golden Rule for transverse electron motion (see section 3.3.2) and fluctuating longitudinal conduction (see section 4).

(TMTSF)<sub>2</sub>PF<sub>6</sub> provides another striking example in which optical and conductivity electron scattering times display very different temperature dependences [50]. Between 300 and 25 K τ<sub>opt</sub> increases only by a factor 3.5 (table 3), whereas in the same temperature interval the conductivity is multiplied by about 100 [39]. This feature has been taken as the signature of precursor effects of the superconductivity which increase drastically the d.c. conductivity at low temperature, though leaving the single-particle scattering time unaffected. Polarized light optical reflectance data have also been obtained in the conducting radical-cation salt (TTF)Br<sub>0.79</sub> [52] leading to an electronic bandwidth (in a simple one-dimensional tight-binding model) 4t<sub>||</sub> = 1.13 ± 0.1 eV. This value is the largest yet reported for any sulphur-based organic conductor. However, the possibility of large intrachain overlap integrals is not unexpected in view of the eclipsed overlap of the uniformly stacked TTF molecules in the crystal structure (figure 16). The bandwidth derived from an extended Huckel calculation [29] namely 1 eV is in fair agreement with the optical data.

### 3.2.3. Spin susceptibility

Besides optical reflectance, the measurement of the spin susceptibility of a metal usually provides, a very good characterization of the electronic properties (bandwidth, role of electron-electron interactions, etc. ...). Spin susceptibility has been extensively studied in organic conductors, using either the classical Faraday method [53] or the amplitude of the E.P.R. signal [54, 55], since most quasi-one-dimensional conductors exhibit relatively narrow E.P.R. lines visible up to room temperature.

<footnote><sup>a</sup> GRANT, P. M., GREENE, R. L., WRIGHTON, G. C., and CASTRO, G., 1973, *Phys. Rev. Lett.*, **31**, 1311.</footnote>

<footnote><sup>b</sup> BRIGHT, A. A., GARITO, A. F., and HEEGER, A. J., 1974, *Phys. Rev. B*, **10**, 1328.</footnote>

<footnote><sup>c</sup> WELBER, B., SEIDEN, P. E., and GRANT, P. M., 1978, *Phys. Rev. B*, **18**, 2692.</footnote>

<footnote><sup>d</sup> JACOBSEN, C. S., TANNER, D. B., and BECHGAARD, K., 1981, *Phys. Rev. Lett.*, **46**, 1142.</footnote>
<figure><img src="image_33.png" /><figcaption>Figure 26. Temperature dependence of the static magnetic susceptibilities of TTF-TCNQ, HMTSF-TCNQ and (TMTSF)₂AsF₆. As far as (TMTSF)₂AsF₆ is concerned, the data represent single crystal measurements in a field along the b*-direction, smaller than the spin-flop field of the SDW state (after reference [64] of section 2).</figcaption></figure>

The spin susceptibility of the TTF-TCNQ and related compounds (figure 26) listed in table 4 are very anomalous in several respects:

(i) The magnitude at room temperature (6 × 10⁻⁴ e.m.u. mol⁻¹ for TTF-TCNQ) [53] is about two to three times the value which can be inferred from the Pauli susceptibility of a one-dimensional non-interacting electron gas at T ≪ T<sub>F</sub>, namely (per molecule)

$$ \chi_P = 2N(E_F)\mu_B^2, \quad (3.7) $$

where the density of states at Fermi level per molecule and one spin direction reads

$$ N(E_F) = \left( 2\pi t_{\parallel} \sin \frac{\pi\rho}{2} \right)^{-1}. \quad (3.8) $$

The discrepancy may however be slightly reduced if the electron correlations are included via a Stoner enhancement factor of the non-interacting susceptibility.

(ii) Another approach [56] assumes the correlations between electrons as very strong and treats the susceptibility in the nearly localized limit. If the Coulomb repulsive interaction is described by a very short-range potential, the Hubbard model leading to a constant value U of its Fourier transform, then in the limit U/t ≫ 1 the susceptibility may be expressed in the form of a one-dimensional antiferromagnetic chain of electrons, the Bonner-Fischer model [57] which at T = 0 reads:

$$ \chi = \frac{2\mu_B^2\rho}{\pi^2 J_{\text{eff}}}. \quad (3.9) $$

The only parameter in (3.9) is the exchange parameter J<sub>eff</sub> given by [58]:

$$ J_{\text{eff}} = \frac{2t^2}{U} \rho \left( 1 - \frac{\sin 2\pi\rho}{2\pi\rho} \right). \quad (3.10) $$
<table><caption>Table 4. Susceptibility.</caption><thead><tr><th></th><th>χ (300 K)–χ (0) (10<sup>4</sup> e.m.u. mol<sup>-1</sup>)</th><th>χ<sub>D</sub></th><th>χ<sub>A</sub></th></tr></thead><tbody><tr><td>TTF–TCNQ</td><td>6<sup>a</sup></td><td>4.20<sup>b</sup><br>3.48<sup>c</sup></td><td>1.80<sup>b</sup><br>2.52<sup>c</sup></td></tr><tr><td>TMTTF–TCNQ</td><td>4.8<sup>a</sup></td><td>2.6<sup>d</sup></td><td>2.2<sup>d</sup></td></tr><tr><td>HMTTF–TCNQ</td><td>2.4<sup>d</sup></td><td>0.7<sup>d</sup></td><td>1.7<sup>d</sup></td></tr><tr><td>TSF–TCNQ</td><td>3.3<sup>e</sup></td><td>0.7<sup>d</sup></td><td>2.6<sup>d</sup></td></tr><tr><td>TMTSF–DMTCNQ</td><td>5.5<sup>d</sup></td><td>1.65<sup>d</sup></td><td>3.85<sup>d</sup></td></tr><tr><td>TMTSF–TCNQ</td><td>3.3<sup>f</sup></td><td></td><td></td></tr><tr><td>HMTSF–TCNQ</td><td>3.8<sup>f,g</sup></td><td></td><td></td></tr><tr><td>(TMTSF)<sub>2</sub>PF<sub>6</sub></td><td>1.2(χ (300 K)–χ (20 K)<sup>h</sup><br>1.25(χ (20 K)–χ (4.2 K)<sup>i</sup></td><td>2.5<sup>h,i</sup></td><td></td></tr></tbody></table>

The fit of the experimental data of TTF–TCNQ at 300 K, by the atomic limit of the susceptibility, leads to $J_{\text{eff}} = 1.3t^2/U = 200$ K (using $\rho = 0.59$), i.e. $U/4t_{\parallel} \approx 2$ if $4t_{\parallel} = 0.4$ eV [56].

(iii) Furthermore, none of the previous approaches can reproduce the large temperature dependence observed at low temperature in the highly conducting regime, $\chi(60\text{K}) \approx \frac{1}{2}\chi(350\text{K})$, though the Bonner–Fischer and the $g_1 - g_2$ models (section 2.2) give rise to a susceptibility increasing with temperature. However one must be aware that since the spin susceptibility is strongly volume dependent in TTF–TCNQ and TMTTF–TCNQ [59] there exists a very significant difference between the constant volume (the way $\chi_S$ is usually given in a theory) and constant pressure (measured in the experiments) temperature dependences (see section 3.4.2). Moreover, the total susceptibility $\chi = \chi_D + \chi_A$ of a two-chain conductor is divided between donor and acceptor chains. The decomposition of the susceptibility requires a study of the EPR g-factor weighted by the respective spin susceptibilities of the two chains [60] or NMR experiments on isotopically enriched molecules [61]. As far as TTF–TCNQ is concerned, the ratio $\chi_F/\chi_Q$ has not yet been determined with certainty as the g-factor of individual chains may present some temperature dependence. However a value $\chi_F/\chi_Q \approx \frac{3}{2}$ in the conducting regime of TTF–TCNQ seems to be reasonable. The susceptibility is usually smaller for selenium compounds

<footnote><sup>a</sup> SCOTT, J. C., GARITO, A. F., and HEEGER, A. J., 1974, *Phys. Rev. B*, **10**, 3131.</footnote>

<footnote><sup>b</sup> Decomposition of χ, using TCNQ(C<sup>13</sup>)NMR. RYBACZEWSKI, E. F., SMITH, L. S., GARITO, A. F., HEEGER, A. J., and SILBERNAGEL, B., 1976, *Phys. Rev. B*, **14**, 2746.</footnote>

<footnote><sup>c</sup> Decomposition of χ, using the *g*-decomposition. TOMKIEWICZ, Y., TARANKO, A. R., and ENGLER, E. M., 1976, *Phys. Rev. Lett.*, **37**, 1705.</footnote>

<footnote><sup>d</sup> TOMKIEWICZ, K., ANDERSEN, J. R., and TARANKO, A. R., 1978, *Phys. Rev. B*, **17**, 1579.</footnote>

<footnote><sup>e</sup> SCOTT, J. C., ETEMAD, S., and ENGLER, E. M., unpublished.</footnote>

<footnote><sup>f</sup> BLOCH, A. N., CARRUTHERS, T. F., POEHLER, T. O., and COWAN, D. O., 1977, *Chemistry and Physics of One-Dimensional Metals*, edited by H. Keller (New York: Plenum), p. 47.</footnote>

<footnote><sup>g</sup> In HMTSF–TCNQ Δχ includes a diamagnetic contribution arising from Landau–Peierls orbital magnetism below 100 K. The spin susceptibility derived from NMR experiments is about 1.5 × 10<sup>-4</sup> e.m.u. mol<sup>-1</sup> at 300 K, see SODA, G., JÉROME, D., WEGER, M., BECHGAARD, K., and PEDERSEN, E., 1976, *Solid St. Commun.*, **20**, 107.</footnote>

<footnote><sup>h</sup> In (TMTSF)<sub>2</sub>PF<sub>6</sub> a spin density wave occurs below 12 K. The spin susceptibility of a powder sample is therefore finite at 0 K, see SCOTT, J. C., PEDERSEN, H. J., and BECHGAARD, K., 1980, *Phys. Rev. Lett.*, **45**, 2125.</footnote>

<footnote><sup>i</sup> When studying the susceptibility of a (TMTSF)<sub>2</sub>AsF<sub>6</sub> single crystal in a field of 3 kOe aligned with the *b*<sup>*</sup> axis, the drop of susceptibility between 12 and 4.2 K amounts to ≈ 1.25 × 10<sup>-4</sup> e.m.u. mol<sup>-1</sup> (MORTENSEN, K., TOMKIEWICZ, Y., and BECHGAARD, K., 1982, *Phys. Rev. B*, **25**, 3319). The total spin susceptibility at 300 K amounts to ≈ 2.5 × 10<sup>-4</sup> e.m.u. mol<sup>-1</sup>.</footnote>
(table 4). In this respect HMTSF-TCNQ is an interesting example since the spin susceptibility of $1.5 \times 10^{-4}$ e.m.u. mol$^{-1}$ at room temperature suggests practically no enhancement due to correlations [62]. TMTSF-DMTCNQ is a notorious exception among selenium compounds, the total spin susceptibility amounts to $5.5 \times 10^{-4}$ e.m.u. mol$^{-1}$ at room temperature [63]. However a g-factor decomposition suggests that the major contribution to the susceptibility is provided by the narrow band of the DMTCNQ chain [62] ($\chi_Q = 2.85$ e.m.u. mol$^{-1}$) at room temperature. The susceptibility of the TMTSF chain $1.9 \times 10^{-4}$ e.m.u. mol$^{-1}$ is about equal to the contribution found in the single-chain (TMTSF)$_2$X series, $1.65 \times 10^{-4}$ e.m.u. mol$^{-1}$ in (TMTSF)$_2$PF$_6$ [64].

### 3.2.4. Nuclear spin dynamics

A very direct proof of the one-dimensional character of the electronic motion is given by NMR properties. The spin-lattice relaxation rate of a nuclear spin I in a conductor is due to the modulation of hyperfine interaction $AI \cdot S$ by the electronic spin motion [65]. The relaxation rate is given by the relation:

$$ \frac{A^2}{T_1 T} \propto F(\omega_e) = \int_0^\infty \cos(\omega_e t) \phi(t) \, dt, \qquad (3.11) $$

where $\phi(t)$ is the electron spin S autocorrelation function

$$ \phi(t) = \sum_{\mathbf{q}} \phi_{\mathbf{q}}(t) \qquad (3.12) $$

with $\phi_q(t) = \langle [S_q^+(t), S_{-q}^-(0)] \rangle$ where $S_q$ is the spatial Fourier transform of the electron spin density, for example $S_q^+ = \sum_k a_{k+q, \uparrow}^* a_{k, \downarrow}$ and $\omega_e$ is the electron Larmor frequency.

In a one-dimensional system, electrons travelling along a chain experience two kinds of scatterings which limit their contribution to the conduction. There is first the scattering in which the electron momentum is changed by $2k_F$ (backward scattering) and secondly the $q \approx 0$ scattering (forward scattering) which hardly affects the momentum. As far as the forward scattering is concerned the power spectrum of the small $q$ spin excitations is diffusive ($q \ll \Lambda^{-1}$), where $\Lambda$ is the electron mean free path and given by [66]:

$$ \frac{\chi''(\mathbf{q}, \omega)}{\omega} = \chi_S \frac{Dq^2}{(Dq^2)^2 + \omega^2}, \qquad (3.13) $$

where $D$ in (3.13) is the diffusion constant of the long-wavelength spin excitations and $\chi_S$ the spin susceptibility. The corresponding autocorrelation function reads [67, 68]

$$ \phi_q(t) \approx \exp(-Dq^2t) \qquad (3.14) $$

and thus, from (3.11) and (3.12) the contribution of the forward scattering to the nuclear relaxation becomes

$$ (T_1 T)_{q \approx 0}^{-1} \approx A^2 \chi_S (D \omega_e)^{-1/2}. \qquad (3.15) $$

Therefore, the Korringa law which sets $T_1^{-1}$ independent of the magnetic field breaks down in a one-dimensional conductor. This anomalous behaviour of $1/T_1$ is due to the always finite probability for electrons making random walks in one dimension to
come back through the origin at large time [69]. The short-wavelength spin excitations ($q \approx 2k_F$) would give rise to a coherent propagation of the spin excitation ($2k_F > \Lambda^{-1}$) and consequently to a field independent relaxation rate [70, 71]:

$$ (T_1 T)_{q=2k_F}^{-1} \approx A^2 \chi_S^2. \qquad (3.16) $$

Both kinds of scattering (equations (3.15) and (3.16)) lead to a relaxation rate varying as

$$ (T_1 T)^{-1} \approx A^2 \left[ \chi_S^2 + \frac{\chi_S}{\sqrt{D\omega_e}} \right]. \qquad (3.17) $$

However $(T_1 T)^{-1}$ departs from the $\omega_e^{-1/2}$ law (3.17) for small enough $\omega_e$ for which it saturates. This is an effect of the interchain coupling which is discussed later in section 3.3.3. Besides the contact (scalar) hyperfine interaction giving rise to a relaxation such as (3.17), the dipolar hyperfine interaction may contribute as well with terms $I^+S_z$ or $I^-S_z$ in the hamiltonian [70, 72]. However the time scale now becomes the nuclear Larmor period $\omega_n^{-1}$ instead of $\omega_e^{-1}$ as previously. Therefore, at fields attainable in the laboratory (less than 100 kOe or so) the dipolar contribution to the nuclear relaxation is saturated (due to the interchain electron tunnelling, see section 3.3.3) and consequently does not perturb the field dependence of (3.17).

Figure 27 shows the field dependence of the TTF proton relaxation rate in TTF-TCNQ(D₄) plotted versus the reciprocal square root of the applied field. According to the data on figure 27 the slope of $d(T_1^{-1}T)/d(H_0^{-1/2}) \propto \chi_S/\sqrt{D}$ drops by a factor 6 between 300 and 150 K. Since, in the same temperature interval, the spin susceptibility drops by about a factor 2 (see section 3.2.3) the spin diffusion constant increases by a factor 9. If a random phase approximation with moderate Coulomb correlations is used for the derivation of the diffusion constant [70], the data in figure 27 correspond to an electron single-particle scattering time increasing by about a

<figure><img src="image_34.png" /><figcaption>Figure 27. $(T_1 T)^{-1}$ versus $H_0^{-1/2}$ for TTF-TCNQ(D₄) at several temperatures ○ (296 K), △ (280 K), × (260 K), ● (240 K), ▲ (210 K), ▽ (180 K), □ (150 K), ■ (110 K). The solid lines are the theoretical curves, equation (3.31) using the parameters $\tau_⊥$, $C_1$ and $C_2$ from the experiment (after reference [70] of section 3).</figcaption></figure>