REVIEW ARTICLE

Periodic lattice distortions and charge density waves in
one- and two-dimensional metals

R H Friend† and D Jérôme‡

† Cavendish Laboratory, Madingley Road, Cambridge, UK

‡ Laboratoire de Physique des Solides, Université Paris-Sud, 91405 Orsay, France

Received 3 November 1978

**Abstract.** At low temperatures, low-dimensional metals are commonly found to exhibit superlattices incommensurate with the primitive lattice (periodic lattice distortion coupled to charge density waves, PLD-CDW). Transitions to such states are driven by strong electron-phonon coupling and favoured by the particular forms of the Fermi surface of these metals. Experimental investigations of the structural distortions, and of the changes in electronic structure around the Fermi level produced by the PLD-CDW state, will be discussed for the layered dichalcogenides, the linear chain platinum complex KCP, and the organic charge transfer salts.

Contents

1. Introduction

2. Theory of low-dimensional metals

2.1. Broken symmetries

2.2. Basic theory of the electron-phonon-driven structural instability

2.3. Screening of the structural instability

2.4. Finite-temperature properties

2.5. Fluctuations and three-dimensional coupling

2.6. Superlattice-primitive lattice commensurability effects

3. PLD-CDW formation in some low-dimensional conductors

3.1. Characteristics of low-dimensional conductors

3.2. $K_2Pt(CN)_4 \cdot Br_{0.3} \cdot 3H_2O$

3.3. Organic charge transfer salts (TTF-TCNQ)

3.4. Layered transition metal dichalcogenides

3.4.1. 1T polytypes

3.4.2. 2H polytypes

3.4.3. $4H_2$ polytypes

4. Further aspects of PLD-CDW formation in low-dimensional conductors

4.1. Three-dimensional coupling and transverse ordering

4.2. High pressure and the prevention of the PLD-CDW state

4.3. Electron-phonon coupling versus electron-electron repulsion

4.4. The PLD-CDW state and superconductivity

4.5. Disorder and the PLD-CDW

5. Conclusion
# 1. Introduction

In the last decade there has been a large amount of interest in metallic systems that are loosely described as ‘one’ or ‘two’ dimensional. The discovery of low-temperature superlattice phases that are incommensurate with the primitive lattice has been interpreted as the occurrence of the well known Fermi-surface instability of a one-dimensional metal, the Peierls distortion (Peierls 1955, Fröhlich 1955). Whilst part of the underlying physics is similar for ‘one’ and ‘two’ dimensional conductors, they have been studied by different groups and their development, both theoretical and experimental, has tended to be independent. One of the reasons for this state of affairs is the unique theoretical position of a one-dimensional metal, in which any phase transition driven by short-range forces is suppressed at finite temperatures by one-dimensional thermodynamical fluctuations (Landau and Lifshitz 1958). Sophisticated solutions of the one-dimensional electron gas, predicting the existence of various one-dimensional ground states (charge-density wave (CDW), spin-density wave (SDW), singlet and triplet superconductivity) (Bychkov et al 1966, Solyom 1973, Gorkov and Dzyaloshinskii 1974, Luther and Emery 1974) have also influenced workers on the ‘one’ dimensional conductors. Interest in the ‘two’ dimensional conductors that show incommensurate superstructures, the transition metal dichalcogenides, was first directed towards the remarkable superconducting properties of these materials when intercalated by organic bases (Meyer et al 1975, Schwall et al 1976). As will be discussed later, CDW formation and its behaviour in intercalated materials can explain many of these properties, and there is no evidence for the excitonic superconductivity proposed by Little (Davies et al 1976), the search for which provided a strong stimulus for research in both ‘one’ and ‘two’ dimensional systems.

The Orsay group has been involved in a systematic high-pressure investigation of these systems. The pressure-induced increases in dimensionality have considerably widened the range of properties exhibited by each group of materials, sometimes to the extent of creating overlap between them. The purpose of this paper is therefore to survey some of the physical arguments common to all of the low-dimensional materials. In doing so we are aware that this paper will be incomplete, especially for the 1D conductors, since we shall for the most part restrict our treatment of the phase transitions to mean-field theory†. In §2 we shall review the basic theoretical developments, in §3 we shall describe in turn the basic physical properties and superlattice formation of the three groups of materials that have been most extensively investigated, the linear chain platinum complexes (KCP), the organic charge-transfer salts (TTF-TCNQ) and the layered transition metal dichalcogenides. Further aspects of the superlattice formation that may be common to all these systems will be considered in §4.

There have been several reviews in this field of low-dimensional conductors. Yoffe (1976) includes both one- and two-dimensional metals. One-dimensional systems are covered by Shchegolev (1972), Bulaevskii (1975), Berlinsky (1976) and Toombs (1978), and are the subject of two summer school proceedings (Keller 1975, 1977). Conference proceedings include Schüster (1975) and Pal et al (1977). Much of the work on two-dimensional systems is presented in Wilson et al (1975), and reviews include those by Williams (1976) and Di Salvo (1977).

<footnote>† An extended article on the *Electronic properties of quasi-one-dimensional conductors* by D Jérôme will be published in *Adv. Phys.* 1979.</footnote>
## 2. Theory of structural transitions in low-dimensional conductors

### 2.1. Broken symmetries

We assume the existence of a ground state for a conductor where, besides the lattice potential invariant under the symmetry operations of the primitive lattice, the electrons (of spin $\sigma_z = \sigma$) also see a potential

$$V_{q, \sigma}(r) = \alpha \cos(q \cdot r + \sigma\phi). \quad (1)$$

If a state characterised by $V_{q,\sigma}(r)$ is stable, the symmetry of the crystal is broken whenever $2\pi/q$ is not commensurate with a lattice spacing. When $\phi = \pi/2$ in equation (1), electrons of opposite spin see potentials out of phase. This situation describes a state characterised by a spin-density wave (SDW). Such a state can be stabilised against the Hartree-Fock paramagnetic ground state by the exchange interaction which favours energetically local alignment of spins. These ideas were proposed by Overhauser (1960, 1962) in relation to the itinerant antiferromagnetism of chromium in which a SDW state is stable below 315 K (Rice et al 1969).

Overhauser proposed an extension of the theory to the case $\phi = 0$ in equation (1) representing a charge-density wave (CDW) ground state of wavelength $2\pi/q$ (Overhauser 1968). Overhauser had argued that energy could be gained through correlation energy in a CDW state, opposed by a smaller Coulomb repulsion in the inhomogeneous electronic charge. However, now that many systems have been found which do show CDW formation, it is generally believed that an electron-phonon interaction, introducing a periodic lattice distortion (PLD), is always required to overcome the Coulomb repulsion (Chan and Heine 1973, Friedel 1977).

### 2.2. Basic theory of the electron-phonon-driven structural instability

Given an electron gas characterised by its Hartree Hamiltonian $\sum_k \epsilon_k a_k^\dagger a_k$ (where $\epsilon_k$ is the energy of Bloch state $k$ and $a_k^\dagger, a_k$ are creation and annihilation operators for state $k$), in a lattice described by the phonon Hamiltonian $\sum_q (\omega_q b_q^\dagger b_q$ (where $\omega_q$ is the frequency of phonon $q$ and $b_q^\dagger, b_q$ are creation and annihilation operators for phonon $q$), any non-zero Fourier component ($q$) of a lattice potential $v(r)$ will couple electron states $|k\rangle$ and $|k+q\rangle$ and therefore open gaps in the electron energy spectrum at $G_i \pm q/2$ ($G_i$ is a reciprocal lattice vector). The second-order perturbation theory energy change of electron state $|k\rangle$ by the perturbation $v_q \cos q \cdot r$ becomes (Friedel 1977)

$$\Delta E_k(q) = \frac{|\langle k | v_q \exp(i q \cdot r) | k + q \rangle|^2}{\epsilon_k - \epsilon_{k+q}} + \frac{|\langle k | v_q \exp(-i q \cdot r) | k - q \rangle|^2}{\epsilon_k - \epsilon_{k-q}}. \quad (2)$$

The total energy change for the electron system as a result of the potential $v_q$ is

$$\delta E_q = \sum_{k \text{ occupied}} \Delta E_k(q) = -|v_q|^2 \chi_q \quad (3)$$

where $\chi_q$, the non-interacting electron susceptibility, is defined as

$$\chi_q = \sum_k \frac{n_{k+q} - n_k}{\epsilon_k - \epsilon_{k+q}} \quad (4)$$

with $n_k$ the Fermi function for state $k$.
We consider the situation where the potential $v_q$ is derived from a static displacement $u_q$ of phonon mode $q$ coupled to the electron system through a Hamiltonian of the form:

$$H_{e-p} = N^{-1/2} \sum_q \sum_k g_q a_{k+q}^{\dagger} a_k (b_{-q} + b_q^{\dagger}). \quad (5)$$

In this case,

$$v_q = g_q u_q (2M\omega_q/\hbar)^{1/2} \quad (6)$$

where $M$ is the ionic mass. The distortion energy of the phonon mode $q$ is given by

$$E_q = \frac{1}{2} M \omega_q^2 u_q^2 \quad (7)$$

and if $\delta E_q + E_q < 0$, the ground state of the electron-phonon system is that with a finite, static amplitude in the phonon mode $q$. Using equations (3), (6) and (7), this stability criterion becomes

$$4g_q^2/\hbar\omega_q > 1/\chi_q. \quad (8)$$

Equation (8) is the basic instability condition for a state characterised by a periodic lattice distortion (PLD) of wavevector $q$. Besides the interaction energy $g_q$, a crucial function is the non-interacting electron-gas susceptibility $\chi_q$. Large values of $\chi_q$ arise for systems where at a particular value of $q$ there are many occupied electron states separated from many unoccupied states by $q$ for which the denominator in equation (4) is close to zero. This corresponds to coupling across the Fermi surface, and when there are large areas of Fermi surface separated by a single wavevector, they are said to 'nest'. In the case of a one-dimensional metal, where the Fermi surface consists of planes at $-k_F$

<figure><img src="image_1.png" /><figcaption>Figure 1. $\chi_q$ for free-electron Fermi-surfaces in one, two, and three dimensions, showing the increasing anomaly at $q = 2k_F$ with decreasing dimension. Also shown are examples of nesting of three-dimensional Fermi surfaces.</figcaption></figure>
and $+k_F$, nesting is perfect, and $\chi_q = 2k_F$ diverges with falling temperature. Following Rice and Strässler (1973a)

$$ \chi_q = 2k_F \sim \frac{1}{2} N(E_F) \ln(E_F/kT) \quad (9) $$

where $N(E_F)$ is the density of states at the Fermi level. Figure 1 shows $\chi_q$ for one-, two- and three-dimensional free-electron gases, corresponding to Fermi-surface nesting for surfaces separated by $q$, with respectively two, one and zero radii of curvature in common.

Whilst the low-temperature divergence in $\chi_q$ in one dimension will always produce a distortion even for small values of $g_{q'}$, as is the case for KCP and TTF-TCNQ, in the two-dimensional layer compounds, calculations of $\chi_q$ by Myron et al (1977) and Doran et al (1978) based on band structure calculations by Myron and Freeman (1975) and Wexler and Woolley (1976) respectively, show only weak nesting, and as stressed by Doran et al (1977), it is the strong electron-phonon coupling in these materials that is important in satisfying equation (8).

### 2.3. Screening of the structural instability

In this model the energy gap in the electronic spectrum is produced by a PLD. The conduction electrons will attempt to screen the periodic potential set up by the atomic displacements, creating a CDW in the conduction electron density. As emphasised by Friedel (1977), the CDW stiffens the lattice and makes its instability less easy to produce. Friedel considers the two extreme cases of nearly-free electrons and tight binding.

For nearly-free electrons, if $v_q$ is the potential due to a phonon, the total potential acting on the electrons is

$$ v'_{q} = v_{q} + v_{q}(\text{int}) \quad (10) $$

where $v_q(\text{int})$ is the potential set up by the CDW. Defining the dielectric constant $\epsilon_q$ such that $v'_q\epsilon_q = v_q$,

$$ \epsilon_q = 1 - (4e^2/|q|^2)\chi_q. \quad (11) $$

In the second-order terms in $v_q$ for $\delta E(q)$ in equation (3) one must distinguish between the contributions of $v_q$ and $v_q(\text{int})$, and avoid counting twice the Coulomb interactions of $\delta\rho$ with itself. As a result $|v_q v'_q|^2$ should be replaced by $|v_q v'_q| = \epsilon_q^{-1} |v_q|^2$.

For tight binding the effect of the lattice displacements can be visualised as a local modulation of the bandwidth, broadening it in regions of lattice contraction and narrowing it in regions of lattice expansion. Setting the Fermi level constant throughout the crystal will require movement of electrons towards the contracted regions of the lattice if the band is less than half full, and away from these regions if the band is more than half full. For the particular case of a half-filled band there will be no CDW.

The short-range Coulomb interaction between conduction electrons, which is minimised by a uniform electron density, will also oppose the setting up of a CDW. Chan and Heine (1973) derive as the criterion for PLD-CDW formation

$$ \frac{4g_q^2}{\hbar\omega_q} - (2U_q - V_q) > \frac{1}{\chi_q} \quad (12) $$

where $U_q$ is the average Coulomb interaction at wavevector $q$, and $V_q$ is the screened
exchange interaction at wavevector $q$. Their criterion for a SDW, driven by $V_q$, which does not affect the electron density and therefore is not opposed by $U_q$, is

$$V_q > 1/\chi_q' \qquad (13)$$

Therefore if a PLD-CDW, rather than a SDW, is to be formed,

$$4g_q^2/\hbar\omega_q > 2U_q' \qquad (14)$$

Equation (14) indicates the necessity for an electron-phonon interaction, creating a PLD, in order to set up a CDW.

## 2.4. Finite-temperature properties

In the previous sections we considered the nature of the ground state of the electron-phonon system at $T = 0$, although the finite-temperature behaviour is already taken into account in the temperature dependence of the Fermi functions in $\chi_q'$, equation (4). As already mentioned for the 1D case, singularities in $\chi_q'$ resulting from nesting are reduced at finite temperatures. Even when the PLD-CDW criterion is not fulfilled, the effect of the electron-phonon interaction is to renormalise the phonon frequencies. The appearance of Fermi-surface related structure in the phonon dispersion curves was predicted by Kohn (1959), and such structures are known as Kohn anomalies. Following Chan and Heine (1973)

$$\omega_q'^2 = \omega_q^2[1 - (4g_q^2/\hbar\omega_q)\chi_q]. \qquad (15)$$

If we consider a 1D metal, where $\chi_q'$ follows equation (9), then starting from high temperatures, the phonon dispersion curves will show softening at $2k_F$, which will increase with falling temperature. When $\omega_q' = 0$, at the Peierls transition temperature $T_p$, a static distortion will begin to build up. Equations (9) and (15) give a relation of the form

$$kT_p \sim E_F \exp(-1/\lambda) \qquad (16)$$

where $\lambda$, the dimensionless electron-phonon coupling constant is given by

$$\lambda = g_q^2 N(E_F)/\hbar\omega_q' \qquad (17)$$

Below $T_p$ a gap in the electronic energy spectrum, the Peierls gap ($2\Delta$), builds up. Following Rice and Strässler (1973a) and Toombs (1978), the coupling of states $k$ and $k + q$ modifies the electron energies to

$$E_k = E_{q/2} \pm [(E_k - E_{q/2})^2 + \Delta^2]^{1/2} \qquad (18)$$

introducing a gap $2\Delta$ in the spectrum at $q/2$. The behaviour of the Peierls gap is governed by the same gap equation that appears in the BCS theory of superconductivity, and $\Delta$ follows a BCS-type temperature dependence, finally reaching a zero-temperature value, $\Delta(0)$ given by

$$\Delta(0) = 1.76 T_p \qquad (19)$$

In this model of the Peierls distortion, the zero-temperature Peierls gap is destroyed at high temperatures by thermal excitation of electrons across the gap, giving the relationship between $\Delta$ and $T_p$ in equation (19). For the layer compounds, where this relation is not obeyed (the factor of 1.76 in equation (19) is found to be much larger), McMillan (1977) has developed a short-coherence-length model ($\xi$, the coherence length
~ superlattice period) consistent with the poorly defined peaks calculated in $\chi_q$ and the very broad Kohn anomaly in the phonon spectrum extending over ~1/3 Brillouin zone (Moncton et al 1975, 1977). Since many modes are involved, the phonon entropy dominates the electron entropy at finite temperature and determines the value of $T_p$.

## 2.5. Fluctuations and three-dimensional coupling

The mean-field treatment of the Peierls distortion outlined above produces a phase transition at finite temperature in spite of the theorem that one-dimensional systems without long-range forces cannot undergo phase transitions at finite temperatures. This is because, by considering only one phonon mode, the short-range electron-phonon interaction is effectively turned into a long-range sinusoidal force of wavevector $2k_F$. If all other Fourier components are included, the short-range character of the force is restored and the phase transition is suppressed. This has been treated by Lee et al (1973) who applied the functional integral results of Scalapino et al (1972) to a Ginzburg-Landau expansion of the force energy for an incommensurate CDW, including terms up to fourth order in the order parameter and up to second order in its spatial derivative. Although the correlation length is only infinite at $T=0$, Lee et al find that it becomes very long as the temperature is reduced below the mean-field Peierls transition temperature, $T_p^{MF}$. Below $T \approx \frac{1}{4}T_p^{MF}$ the correlation length diverges and the spin susceptibility falls rapidly towards zero, and Lee et al suggest that this is the critical temperature for 3D ordering.

One-dimensional fluctuations are very sensitive to interchain, three-dimensional coupling, and the effectiveness of interchain coupling in suppressing fluctuations has been considered by Rice and Strässler (1973b) and Horovitz et al (1975). Representing the transverse coupling by an anisotropic electron dispersion of the form

$$E(k) = E(k_z) + \eta E_F(\cos k_x a + \cos k_y a) \quad (20)$$

where $a$ is the distance between adjacent chains and $\eta$ is a measure of the interchain coupling, Horovitz et al find a characteristic temperature $T_2 \sim \frac{1}{4}\eta T_F$ at which the inverse transverse correlation length crosses the Brillouin zone boundary. If $T_p^{MF}$ is smaller than $T_2$ the thermodynamic fluctuations will have a small effect and $T_p$ will be only slightly depressed from $T_p^{MF}$. Coulomb coupling between CDW's on adjacent chains has also been proposed as a means of establishing three-dimensional long-range order (Saub et al 1976). One of the interesting possibilities if Coulomb coupling is the dominant interchain interaction is that while it couples adjacent CDW's, it has no effect on SDW's, and may favour condensation of a CDW in preference to a SDW (Lee et al 1977). Of the three series of compounds discussed later, only in KCP is behaviour strongly influenced by fluctuations, although their role in TTF-TCNQ remains controversial (Jérôme 1979a).

## 2.6. Superlattice-primitive lattice commensurability effects

One of the interesting aspects of the Peierls distortion is that the superlattice wave-vector is determined by the details of the Fermi surface and may bear no simple relation to the lattice periodicity.

When the superlattice period $\Lambda$ is equal to $na$, where $a$ is the lattice periodicity and $n$ integer, the PLD-CDW is termed commensurate, and will have preferred positions in the
lattice, arranging itself so that regions of charge accumulation sit close to the ion positions. The PLD-CDW will be unable to move without surmounting an energy barrier, and it is said to be pinned to the lattice. Clearly this energy barrier is largest for small values of *n*, and Lee *et al* (1974) find that this energy varies as

$$E_{\text{pinning}} \sim (\Delta^2/E_F)(\Delta/W)^{n-2} \quad (21)$$

where $\Delta$ is the Peierls gap and $W$ the bandwidth. Pinning is particularly strong for $n=2$.

When $\Lambda \neq na$, the PLD-CDW is termed incommensurate, and if the PLD-CDW is a simple sinusoidal distortion, then it has no preferred position in the lattice. This situation is equivalent to the jellium model for the ion background, in which the PLD-CDW has arbitrary phase with respect to the lattice, and will be free to move and thereby carry current. This is the collective mode envisaged by Fröhlich (1955) as a mechanism for superconductivity. Besides commensurability with the lattice, impurities and defects can also pin the PLD-CDW to the lattice and drive what would have been a zero-frequency phase mode of the PLD-CDW to finite frequency. Low-frequency peaks in the optical conductivities of KCP and TTF-TCNQ at low temperatures, at typically $15 \text{ cm}^{-1}$, have been interpreted in terms of this optically-active pinned Fröhlich mode (Brüesch *et al* 1975, Heeger 1977). Any contribution to the DC conductivity from the Fröhlich mode in these materials appears to be small (Thomas *et al* 1976). However, measurements of non-linear conductivity in the quasi one-dimensional system NbSe$_3$ (Ong and Monceau 1977), in which an incommensurate superlattice has recently been discovered (Tsutsumi *et al* 1977) have been interpreted in terms of unpinning of the CDW by the applied field, and Fröhlich-mode conductivity.

When $\Lambda$ is close to, but not equal to $na$ there will be competition between the electronic band structure energy, which favours the best nesting wavevector, and the commensurability energy gained when the PLD-CDW sits in preferred positions in the lattice. The commensurate superlattice is favoured for large values of the distortion amplitude, and the sequence of behaviour of first a transition from the undistorted state to an incommensurate superlattice, in which phase the superlattice period moves towards the commensurate value at low temperatures, followed by a transition to a commensurate superlattice, is commonly observed in the layered compounds, where the incommensurate superlattice is close to commensurability and the weak Fermi-surface nesting reduces the band structure energy penalty in moving away from the best nesting wavevector. For the quasi-one-dimensional systems where nesting is better, and the distortion amplitude smaller, a lock-in to the commensurate superlattice is less likely, although it has recently been observed in TTF-TCNQ under pressure (Friend *et al* 1978c). This sequence of phase transitions has been modelled using a Landau expansion of the free energy by McMillan (1975). The commensurate-incommensurate transition is described in terms of the competition between the 'spring constant' of the incommensurate CDW and Umklapp lock-in free energy terms.

As an alternative to the incommensurate phase characterised by a simple sinusoidal distortion, McMillan (1976) has proposed that when $\Lambda$ is close to $na$, the PLD-CDW may dissociate into regions of locally commensurate superlattice and regions where the phase changes rapidly with respect to the lattice, introducing harmonics of the superlattice period. The regions of rapidly changing phase, termed discommensurations by McMillan, were shown by Bak and Emery (1976) to be solutions of the one-dimensional Sine-Gordon equation, and are solitons. This soliton model for the incommensurate superlattice has also been considered by Bulaevskii (1978), Nakanishi and Shiba (1978) and
Bak (1978). Evidence for the existence of discommensurations in the layer compounds will be discussed later.

### 3. PLD-CDW formation in some low-dimensional conductors

#### 3.1. Characteristics of low-dimensional conductors

The materials which are found to possess one- or two-dimensional metallic properties, and hence develop Peierls distortions at low temperatures, are three-dimensional solids held together by the normal cohesive forces (Madelung, Van der Waals, covalent bonding) and generally do not show a comparable anisotropy in their mechanical properties.

<table><caption>**Table 1.** Comparison of axial compressibilities of several low-dimensional conductors. * denotes chain axis for 1D systems, $k_a$ is the in-plane compressibility and $k_c$ the out-of-plane compressibility for NbSe$_2$.</caption><thead><tr><th></th><th>$k_a$</th><th>$k_b$</th><th>$k_c$ (10<sup>-3</sup> kbar<sup>-1</sup>)</th></tr></thead><tbody><tr><td>TTF-TCNQ<sup>(a)</sup></td><td>2.7</td><td>4.7*</td><td>3.2</td></tr><tr><td>KCP<sup>(b)</sup></td><td>1.7</td><td>1.7</td><td>1.8*</td></tr><tr><td>(SN)<sub>x</sub><sup>(c)</sup></td><td>5.5</td><td>2.2*</td><td>5.0</td></tr><tr><td>2H NbSe<sub>2</sub><sup>(d)</sup></td><td>0.41</td><td></td><td>1.6</td></tr></tbody></table>

(a) Debray *et al* (1977)

(c) Clarke (1978)

(b) Decker and Debray (1978)

(d) Jones *et al* (1972)

Table 1 lists compressibilities for several of these materials. In the one-dimensional conductors KCP and TTF-TCNQ it is the directionality of the atomic or molecular orbitals involved in the formation of the conduction band, as well as the difference in intra-chain and inter-chain atomic separations, that is responsible for the electronic anisotropy, and these materials have either isotropic compressibilities (KCP), or are more compressible along the chain axis (TTF-TCNQ). The three-dimensional structure of the transition metal dichalcogenides is of stacks of $MX_2$ sandwiches, with weak bonding forces, usually considered to be Van der Waals, between each sandwich. This does result in a compressibility within the layers some four times smaller than that perpendicular to them.

Also included in table 1 are data for polymeric sulphur nitride, $(SN)_x$. This unusual metallic polymer attracted great interest when it was found to go superconducting at 0.3 K (Greene *et al* 1975), and has sometimes been regarded as a one-dimensional conductor. Its properties are reviewed by Yoffe (1976) and Greene and Street (1977). It is in fact the inverse of the KCP and TTF-TCNQ type one-dimensional conductors in that its mechanical anisotropy is larger than its electronic anisotropy, and partly as a result of the nature of the polymerisation process, it is impossible to grow crystals with a coherence length of more than a few hundred Å transverse to the polymer chain. The large measured electrical anisotropy is a consequence, therefore, of tunnelling-type conductivity between fibres perpendicular to the chain axis.

Table 2 lists electrical conductivities and anisotropy ratios, together with estimates of the parallel and transverse tight-binding overlap integrals $t$ (related to the bandwidth $W$, by $4t = W$). For the situation where transverse conduction is incoherent, Soda *et al*
<table><caption>**Table 2.** Room-temperature conductivity anisotropy ratio between the high-conductivity axis $\sigma_{\parallel}$ and the transverse conductivity $\sigma_{\perp}$; the room-temperature values of $\sigma_{\parallel}$; and estimates of the parallel and transverse overlap integrals for several low-dimensional conductors. The high value of $\sigma_{\parallel}/\sigma_{\perp}$ for (SN)$_x$ is more a consequence of the fibrous, polycrystalline nature of (SN)$_x$ than a measure of the intrinsic anisotropy, which is certainly much smaller than 300.</caption><thead><tr><th></th><th>σ<sub>‖</sub>/σ<sub>⊥</sub></th><th>σ<sub>‖</sub> (Ω cm)<sup>-1</sup></th><th>t<sub>‖</sub> eV</th><th>t<sub>⊥</sub> eV</th></tr></thead><tbody><tr><td>KCP</td><td>5 × 10<sup>4(a)</sup></td><td>~ 10<sup>2(a)</sup></td><td></td><td></td></tr><tr><td>TTF-TCNQ</td><td>~ 300<sup>(b, c)</sup></td><td>4 × 10<sup>2(c)</sup></td><td>0.12<sup>(d, e)</sup></td><td>0.002<sup>(e, f, g)</sup></td></tr><tr><td>HMTSF-TCNQ</td><td>30<sup>(c)</sup></td><td>1.8 × 10<sup>3(c, h)</sup></td><td>~ 0.12</td><td>0.01<sup>(f)</sup></td></tr><tr><td>(SN)<sub>x</sub></td><td>(300)<sup>(i)</sup></td><td>4 × 10<sup>3(i)</sup></td><td>0.6<sup>(i)</sup></td><td>0.12<sup>(i)</sup></td></tr><tr><td>NbSe<sub>2</sub></td><td>30<sup>(f)</sup></td><td>10<sup>4(k)</sup></td><td>0.3<sup>(l)</sup></td><td>0.1<sup>(l)</sup></td></tr></tbody></table><table><tbody><tr><td>(a) Zeller (1975)</td><td>(g) Soda et al (1977)</td></tr><tr><td>(b) Heeger and Garito (1975)</td><td>(h) Cooper et al (1976)</td></tr><tr><td>(c) Cooper et al (1977)</td><td>(i) Greene and Street (1977)</td></tr><tr><td>(d) Herman et al (1977)</td><td>(j) Edwards and Frindt (1971)</td></tr><tr><td>(e) Berlinsky et al (1974)</td><td>(k) Wilson et al (1975)</td></tr><tr><td>(f) Jérôme and Weger (1977)</td><td>(l) Mattheiss (1973), Wexler and Wooley (1976)</td></tr></tbody></table>

(1977) show that the anisotropy ratio $\sigma_{\parallel}/\sigma_{\perp}$ is proportional to $(t_{\parallel}/t_{\perp})^2$. From the large anisotropy ratio, it is evident that KCP is the most one-dimensional system. For TTF-TCNQ $t_{\perp}$ is probably smaller than $kT_p(T_p = 52\,K)$ so that the three-dimensionality of the Fermi surface has not yet developed at $T_p$, and the one-dimensional, perfect nesting theory of the Peierls distortion is applicable. The criterion for $t_{\perp}$ to interfere with the one-dimensional Fermi-surface nesting to the extent of reducing $T_p$ is treated more thoroughly by Horovitz et al (1975) who derive the following criterion for $T_p$ to be determined by the one-dimensional mean-field theory outlined in the previous section:

$$4T_p/E_F < t_{\perp}/t_{\parallel} < 3(T_p/E_F)^{1/2} \qquad (22)$$

The lower limit on $t_{\perp}/t_{\parallel}$ is that sufficient for coupling by three-dimensional tunnelling to prevent one-dimensional fluctuations. HMTSF-TCNQ which is listed in table 2 is interesting in that it is one of the most three-dimensional of the TTF-TCNQ-type organic metals, and in this compound $t_{\perp}$ is large enough to reduce Fermi-surface nesting. With estimates of $t_{\perp} \sim 0.12\,eV$ and $t_{\parallel}/t_{\perp}$ of only 5, $(SN)_x$ is clearly far from being a one-dimensional conductor. For the layer compounds, the calculated out-of-plane energy dispersion of the bands is also large, and, as will be discussed further, it is the strength of the electron-phonon coupling that is important in driving the distortions in these materials.

A Peierls distortion, or PLD-CDW will be detectable both through the new lattice periodicity in diffraction experiments, and in physical properties sensitive to the nature of the band-structure at the Fermi level. In diffraction experiments it is the PLD that is detected with superlattice spots appearing as satellites around the parent Bragg spots at $G_i + q$ (where $G_i$ is a reciprocal lattice vector, and $q$ is the superlattice wavevector). For small-amplitude distortions the satellite intensity relative to the parent Bragg spot is proportional to $|(G_i \pm q) \cdot u_q|^2$ (where $|u_q|$ is the distortion amplitude) and ranges from around $10^{-2}$ to $10^{-5}$ (Wilson et al 1975, Williams 1976, Comès 1977). For the one-dimensional metals, the opening up of an energy gap at the Fermi level clearly drives the system semiconducting. For the two-dimensional systems where only part of the Fermi surface is destroyed, the change in conductivity, which is determined by changes in
carrier concentration and mobility, can be quite small, although changes in, for example, Hall coefficient, susceptibility, and specific heat are all quite evident. Techniques which probe the local field environment of one of the atomic species present such as NMR and XPS, can give direct observation of the CDW.

### 3.2. $K_2Pt(CN)_4Br_{0.3} \cdot 3H_2O$

$K_2Pt(CN)_4Br_{0.3} \cdot 3H_2O$, usually abbreviated to KCP, is one of a family of non-integral oxidation-state platinum complexes of cyanide or oxalate investigated by Krogmann and co-workers (Krogmann 1969).

<figure><img src="image_2.png" /><figcaption>**Figure 2.** Idealised sketches of a $Pt(CN)_4^{2-}$ group and of the stacking of $Pt(CN)_4^{2-}$ groups in a KCP crystal, showing the overlap of the $d_{z^2}$ atomic orbitals based on the Pt atoms.</figcaption></figure>

A much simplified sketch of the backbone of the structure is shown in figure 2. The planar arrangement of cyanide anions around the Pt cations allows stacking along the c axis and overlap between the atomic $d_{z^2}$ platinum orbitals on adjacent ions. The intra-chain Pt spacing is 2.89 Å, which is comparable with that for metallic Pt, whereas the interchain Pt spacing is 9.88 Å. The positions of the $K^+$, $Br^-$ and $H_2O$ are not shown in figure 2; however the presence of 0.30 $Br^-$ ions per formula unit is crucial for the conduction properties, as they take electrons out of what would have been the highest filled band, of mainly $d_{z^2}$ character, and leave it with only 1.7 electrons per Pt ion. Thus, in the absence of any distortion, the material should be metallic along the chain axis. At room temperature, the conductivity is indeed several hundred ($\Omega$ cm)$^{-1}$ along the chains. For light polarised parallel to the conducting axis, the reflectivity shows a characteristic metallic Drude edge (Zeller 1975). From the observed plasma frequency, assuming 1.7 electrons per Pt ion, an effective mass equal to the free-electron mass was deduced. This free-electron behaviour from a primarily $d_{z^2}$ derived conduction band is surprising. However Messmer and Salahub (1975) have derived an almost free-electron conduction band in a two-band model using $6p_z$ and s-d hybridised orbitals. The 6s-5 $d_{z^2}$ hybridisation is forced by the chemical effects of the ligands. Messmer and Salahub also attempt to explain why the take-up of Br is 0.3, by considering the balance between the energy gained by removing electrons from the band (i.e. bonding orbitals occupied, antibonding orbitals unoccupied) and the Coulomb repulsion between the charged ions left behind.

Below room temperature, the conductivity falls rapidly, as is shown in figure 3. The energy gap consistent with the low-temperature activated conductivity is about 0.15 eV.
<figure><img src="image_3.png" /><figcaption>Figure 3. Conductivity parallel and perpendicular to the chains for KCP, after Zeller (1975).</figcaption></figure>

Diffraction studies with x-rays and neutrons indicate the appearance of an incom-
mensurate superlattice, with a chain axis repeat of 6·7 lattice units, which is the value of
2k~F~ expected for the 1·7 electron filled band (Renker and Comès 1975, Renker et al 1975).
A large Kohn anomaly can be seen in the LA phonon branch at room temperature, shown
in figure 4. Elastic neutron scattering is also observed at all directions studied which
have a 2k~F~ component of the wavevector parallel to the chains, demonstrating the
presence of a PLD together with the Kohn anomaly. At high temperatures there is no
transverse order in the PLD, but below 140 K, the intensity of the superlattice point
(π/d⊥, π/d⊥, 2k~F~) increases rapidly and then saturates at around 80 K. The intensities of
the other 2k~F~ points, such as (0, 0, 2k~F~), decrease in this temperature regime, but do not
reach zero. This indicates that whilst the coherence length of the distortion with trans-
verse period 2d⊥ increases rapidly below 140 K, the system never attains long-range

<figure><img src="image_4.png" /><figcaption>Figure 4. Acoustic phonon branches in KCP at room temperature (Renker and Comès 1975). An enhanced Kohn anomaly appears along the line q = 2k~F~ = constant.</figcaption></figure>
order. At low temperatures, neutron scattering results give the transverse coherence length as ~33 Å, or some three interchain distances, and x-ray measurements set a lower limit of 60 Pt-Pt distances for the intrachain coherence length. The distortion amplitude at 7 K was estimated to be 0.5% c by Eagen *et al* (1975). The choice of (π/d⊥, π/d⊥, 2k<sub>F</sub>) suits both Coulomb (Saub *et al* 1976) and tunnelling (Horovitz *et al* 1975) interchain energy considerations.

Measurements of the infrared reflectivity show what is interpreted as the Peierls gap, at an energy of ~0.2 eV (Zeller 1975). While this gap is most apparent at low temperatures, it is still present at room temperature. The value of 0.2 eV implies a mean-field

<figure><img src="image_5.png" /><figcaption>**Figure 5.** <sup>195</sup>Pt Knight shift in KCP as a function of temperature at ambient pressure (Launois and Niedoba 1975) and at 20 kbar (Müller *et al* 1975). Note the sharpening of the change of regime at ~120 K under pressure.</figcaption></figure>

transition of some 800 K, from equation (19), and it is evident that KCP is a system where the transition temperature is depressed from its mean-field value by one-dimensional fluctuations. Thus at room temperature there is a pseudo Peierls gap present, and the apparent 'phase transition' at around 100 K is where the pseudo-gap increases rapidly and the system tends to a three-dimensionally ordered distorted state. The growth of the pseudo-gap is seen as the disappearance of the Knight shift in the <sup>195</sup>Pt NMR line, shown in figure 5. Launois and Niedoba (1975) fit the temperature dependence to the treatment of one-dimensional fluctuations by Lee *et al* (1973), discussed in §2.5, with a mean-field transition temperature of 625 K. Interchain coupling is ignored in this theory, but with the available data it is not possible to determine whether or not such coupling plays a significant role at atmospheric pressure.

One of the complications with KCP is that both the Br⁻ and H₂O occupy sites within the crystal randomly, and the random potential they set up probably prevents the formation of a long-range ordered superlattice at low temperatures. However it is interesting to note that under pressure a sharp anomaly in the resistivity as a function of temperature is observed (Thielemans *et al* 1976). The anomaly is similar to that seen at the three-dimensional ordering transitions in the organic charge transfer salts (see figure 7), and indicates that there is a transition to a three-dimensionally ordered state
at low temperatures at pressures in excess of 20 kbar. This is also seen as a sharpening of the Knight shift transition at 20 kbar, shown in figure 5 (Müller *et al* 1975). One problem in understanding the high-pressure behaviour is that the interchain Coulomb interaction, which is generally considered to be the dominant three-dimensional coupling, is almost pressure insensitive. It depends on the product $qd_1$ (Lee *et al* 1974, Saub *et al* 1976), and for the isotropic compressibility measured by Decker and Debray (1978) this product is pressure-independent. It is possible that the randomness of the $Br^-$ and $H_2O$ site occupation is pressure-dependent.

### 3.3 Organic charge transfer salts (TTF-TCNQ)

All of the organic metals that have as yet been prepared have conduction bands derived from overlap between the $π$ electron orbitals of planar molecules stacked one on top of another. Since conduction is usually confined to the direction of the stacks they are all highly anisotropic metals and often exhibit Peierls distortions. The combination of the quasi-one-dimensional metallic properties and the novelty of a metallic system with a narrow, $π$-orbital-derived conduction band has made these materials sufficiently complicated for the interpretation of many of their physical properties to remain controversial, in particular the validity of the one-electron picture, and the nature of the scattering process that determines the metallic conductivity.

The best-studied material is TTF-TCNQ (tetrathiofulvalenium tetracyanoquino-dimethanide), and we shall for the most part restrict attention to this system, although we shall mention some of the closely related derivatives which are interesting because they show stronger three-dimensional characteristics. A selection of the molecules that are used to make these materials is shown in figure 6(a). TTF-TCNQ is an ionic salt formed from TTF which is an electron donor, and TCNQ which is an electron acceptor. The three-dimensional structure is shown schematically in figure 6(b); TTF and TCNQ molecules are arranged in segregated stacks along the *b*-axis, and there is partial charge transfer from the TTF stack to the TCNQ stack (determined from the value of the Peierls distortion wavevector to be 0.59 electrons per formula unit). In a one-electron one-dimensional band model, the Fermi wavevector, and therefore the charge transfer, is determined by the value of $k_F$ at which the bonding TCNQ band crosses the anti-

<figure><img src="image_6.png" /><figcaption>Figure 6. (a) Structures of some donor and acceptor molecules used to construct the organic charge transfer salts. (b) Schematic representation of the TTF-TCNQ structure, showing the tilted stacks of TTF and TCNQ molecules along the *b* axis.</figcaption></figure>
bonding TTF band. The transfer of charge puts 0.59 electrons into the TCNQ band, leaving the TTF band with 1.41 electrons, and with both bands partially filled, both stacks contribute to the metallic conductivity along the *b* axis, at room temperature some 500 (Ω cm⁻¹).

As shown in table 2, coupling between stacks is weak, and the conductivity anisotropy is of the order of 300 at room temperature. The tunnelling integral between stacks has been measured directly from the NMR interstack relaxation time to be a few meV (Jérôme and Weger 1977, Soda et al 1977), and over the whole of the temperature domain where TTF-TCNQ is metallic (above ~50 K), thermal motion and the short intrastack carrier lifetime prevent the coherent build up of an electron wavefunction over more than one stack, and thus effectively localise the electron Bloch states to one stack (Soda et al 1977). By this criterion, TTF-TCNQ is a one-dimensional conductor. Metallic TTF-TCNQ derivatives, for example obtained by replacement of S by Se in TSF-TCNQ, all possess the feature of segregated stacks of donor and acceptor molecules, although there are important differences in the structure transverse to the stacking axis. The TTF-TCNQ structure, with alternating donor and acceptor sheets along the *a* axis (Kistemacher et al 1974) also adopted by TSF-TCNQ, gives very poor contact between stacks, whereas the hexamethylene derivatives HMTTF-TCNQ and HMTSF-TCNQ stack with alternating donor and acceptor molecules in both transverse directions and have much larger interstack contact (Greene et al 1976, Phillips et al 1976). In HMTSF-TCNQ the larger transverse coupling gives three-dimensionally-coherent electron wavefunctions below about 200 K, and the material is a highly anisotropic three-dimensional metal, with a three-dimensional Fermi surface, in contrast to TTF-TCNQ (Jérôme and Weger 1977).

The conductivity of TTF-TCNQ along the *b* axis increases on cooling, reaching a maximum for most crystals of between 10 and 25 times the room-temperature value at around 56 K (Thomas et al 1976). Below 53 K the conductivity falls rapidly and becomes activated, as is shown in figure 7, which also contains conductivity data for several other quasi-one-dimensional conductors. Diffraction experiments, discussed below, confirm that this metal-insulator transition is indeed a Peierls transition. Reports of conductivity increases of up to 500 times the room-temperature value just before the transition, which were initially attributed to fluctuations towards BCS superconductivity (Coleman et al 1973) prompted the suggestion by Bardeen (1973) that this paraconductivity was due to the Fröhlich collective mode (Fröhlich 1955). These measurements have been criticised by Schafer et al (1974), and such large increases in conductivity have tended not to be reproduced by other groups. Thomas et al (1976) collate measurements made in many different laboratories and conclude that there is no need to invoke the Fröhlich mode to explain the magnitude of the conductivity of most crystals. The Pennsylannia group restate the case for collective-mode conductivity in a reply to this paper (Cohen et al 1976).

x-ray and neutron scattering measurements of the Peierls distortion have been hampered by the small size of crystals available. The elegant measurement of the structural changes taking place are reviewed by Comès (1977). The details of the crystal structure of TTF-TCNQ complicate the superlattice formation in several ways. Firstly the tilt of the TTF and TCNQ molecules with respect to the *b* axis, to give the herringbone structure in figure 6(b), allows the TA phonon mode polarised parallel to *c* to couple to the electrons by modulating the intrastack transfer integrals, and drive the Peierls distortion (Comès 1977). Secondly the two distinct conducting stacks may tend to distort at different temperatures, and depending on the strength of the interstack coupling, the stacks may distort
<figure><img src="image_7.png" /><figcaption>**Figure 7.** Temperature dependence of the resistivity along the high conductivity axis for several 1D conductors. Note the sharp transition in KCP at 32 kbar, similar to those observed in the organic systems.</figcaption></figure>

independently. In TTF-TCNQ three phase transitions have been observed at 52 K, 49 K and 38 K, associated with varying stages of three-dimensional ordering on the two stacks.

Diffuse x-ray scattering techniques (Denoyer *et al* 1975) were the first to reveal the Peierls distortion and Kohn anomaly. Figure 8 shows more recent measurements by Khanna *et al* (1977). Phonon softening at 0.295 *b*\* is interpreted as the 2*k*<sub>F</sub> phonon. Scattering at twice this wavevector (0.59 *b*\* or 0.41 *b*\*) in the reduced zone), first reported by Kagoshima *et al* (1976) and Pouget *et al* (1976) is also seen in figure 8. This 4*k*<sub>F</sub> scattering has a different temperature dependence from the 2*k*<sub>F</sub>, being relatively stronger at room temperature, and it cannot be considered as a harmonic of the 2*k*<sub>F</sub> distortion.

The sequence of low-temperature distortions is best seen in elastic neutron scattering (Comès *et al* 1976, Ellensen *et al* 1977). At 53 K the 2*k*<sub>F</sub> scattering condenses to give a superlattice of (2*a*, 3·4*b*, *c*). Many measurements, including ESR (Tomkiewicz *et al* 1976, 1977a,b) and ¹³C NMR (Rybaczewski *et al* 1976) indicate that this corresponds to the formation of a Peierls distortion on the TCNQ stacks. The transverse period of 2*a* can leave the TTF stacks undistorted (Bak and Emery 1976) and it is only below 49 K that the distortion builds up on them and the transverse period increases from 2*a*. Finally
<figure><img src="image_8.png" /><figcaption>Figure 8. Microdensitometer readings of a series of x-ray patterns between 60 K and room temperature for TTF-TCNQ. Note the different temperature dependences of the $2k_F$ and $4k_F$ scattering (Khanna et al 1977).</figcaption></figure>

at 38 K there is a first-order transition where the transverse period locks to a value of 4a. Inelastic neutron scattering (Shirane et al 1976; Shapiro et al 1977) shows the Kohn anomaly to be in the TA phonon branch polarised parallel to c. The TA phonon condenses at 52 K, and the $4k_F$ diffuse scattering, together with a $2k_F$ LA mode, condense at a lower temperature, probably 49 K (Khanna et al 1977). The sequence of phase transitions in TTF-TCNQ will be discussed in relation to three-dimensional coupling in §4.1.

A controversial issue for TTF-TCNQ is the validity of the one-electron model. The maximum calculated values of the bandwidths of either stack are only 1/2 eV (Herman et al 1977), whereas the estimated value of the on-site Coulomb repulsion, $U$, is ten times larger (Hubbard 1978). Explanations for the $4k_F$ scattering are usually based on a large $U$ model for one or both stacks (Pouget et al 1976; Torrance 1977; Emery 1970; Hubbard 1978). Further discussion of the role of a large $U$ is deferred until §4.3.

Derivatives of TTF-TCNQ mostly show similar metal-insulator transitions, see figure 7, except for HMTSF-TCNQ which exhibits only a weak rise in resistivity below about 30 K (Bloch et al 1975). Diffuse x-ray scattering at room temperature indicates a $2k_F$ soft phonon, and charge transfer of 0.74 electrons (Weyl et al 1976). A phase transition sets in at 24 K (Miljak et al 1978) seen by transport measurements (see also Bloch (1977) for reports of anomalies in specific heat and susceptibility) and confirmed by the condensation of the diffuse x-ray lines, (Pouget et al 1979). The properties of the HM structure materials will be further discussed in §4.2.

At ambient pressure none of the TTF-TCNQ type salts have a value of $2k_F$ close enough to a commensurate superstructure for there to be a longitudinal lock-in transition. However, recent high-pressure measurements of the b-axis conductivity of TTF-TCNQ (Friend et al 1978c) indicate that at 20 kbar, charge transfer has increased from
<figure><img src="image_9.png" /><figcaption>Figure 9. Cooling (broken curves) and warming (full curves) b-axis resistivity curves for TTF-TCNQ at 1 bar and at 4 kbar. The room temperature conductivity is normalised to 400 (Ω cm)⁻¹. Transitions at 52, 49 and 38 K are visible at 1 bar (Friend et al 1978c).</figcaption></figure>

0.59 to $\frac{2}{3}$, giving a superlattice with period $3b$. Some of the resistivity data used to assemble the phase diagram under pressure are shown in figure 9; the phase diagram is shown in figure 10. The three phase transitions at ambient pressure are all visible in figure 9, thus the 52 K transition only removes the contribution to the conductivity from the TCNQ stacks, and only below 49 K do the TTF stacks develop a distortion, which strengthens

<figure><img src="image_10.png" /><figcaption>Figure 10. Pressure dependence of the phase transitions in TTF-TCNQ (Friend et al 1978c).</figcaption></figure>
sharply at 38 K. The effect of pressure on the Peierls distortion will be discussed in more detail in §4.2; of interest here is the pressure regime above 15 kbar where there is only one phase transition. Around 20 kbar the transition temperature reaches a maximum, and the transition, which is second order, or weakly first order both above and below this pressure regime, becomes strongly first order, with a temperature hysteresis of one degree. Within this pressure regime, of some 4 kbar, the incommensurate PLD-CDW is close enough to the 3b superlattice for the lock-in Umklapp energy to drive the CDW commensurate. For the 3b superlattice, the Umklapp free-energy term is cubic, thus accounting for the first-order character of the transition to the commensurate PLD-CDW. The range over which the PLD-CDW is commensurate is limited to $\Delta k/k < 2\%$, which is consistent with the almost perfect one-dimensional Fermi surface, and contrasts with the much larger commensurability ranges seen in the layer compounds.

### 3.4. Layered transition metal dichalcogenides

The group-V transition metal dichalcogenides $MX_2(M = V, Nb, Ta; X = S, Se)$ are layered structure materials, composed of close-packed sheets of metal atoms sandwiched by layers of chalcogen atoms. Bonding between chalcogen layers in neighbouring sandwiches is weak, and these compounds, which are d-band metals, show two-dimensional metallic properties, although the anisotropy between the in-plane and out-of-plane conductivities is typically only 30 (Edwards and Frindt 1971).

Coordination of the metal by the chalcogens can be either octahedral or trigonal prismatic; many ways of stacking the sandwiches, of either or both coordination types, are possible, and there are many polytypes found. Figure 11 shows some of the more common of them, the simplest octahedral coordination polytype, 1T, the simplest trigonal prismatic polytype, with two layers per unit cell, 2H, and the $4H_b$ polytype with alternating layers of octahedral and trigonal prismatic coordination.

Physical properties are mainly determined by the coordination type of the layer, and in fact the band structure differences between the two coordination types are sufficiently large for their properties to be very different. We shall therefore discuss the properties of the 1T and 2H polytypes in turn; the $4H_b$ polytype is interesting in that the two types of layer appear to behave independently and it exhibits the properties of both 1T and 2H polytypes.

<figure><img src="image_11.png" /><figcaption>Figure 11. Structures of some of the group-V dichalcogenide polytypes.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 12.** Fermi surfaces proposed for the 1T and 2H polytypes of TaS₂ (Wilson *et al* 1975).</figcaption></figure>

Band-structure calculations for layers of both coordination types have been made using a number of different methods, including KKR, APW and the layer method (Myron and Freeman 1975, Mattheiss 1973, Wexler and Woolley 1976, Woolley and Wexler 1977). Both coordination types are d¹ metals, with the filled chalcogen p band lying below the d bands. The 1T polytypes have a fairly simple Fermi surface with electron-like cylinders at the centre of the hexagonal Brillouin zone edges. The 2H polytypes are more complicated; the Fermi level lies in the middle of a d sub-band of $d_{z^2}$, $d_{xy}$ and $d_{x^2-y^2}$ character, with a hybridisation gap of about 1 eV between it and the rest of the d band. This results in a high $N(E_F)$ and a Fermi surface with hole pockets at the zone centre and edges. Figure 12 shows Fermi surfaces for 1T and 2H TaS₂ proposed by Wilson *et al* (1975) and based on the bandstructure calculation by Mattheiss (1973), and details of the PLD-CDW wavevectors. It gives some indication of the extent to which these materials are two-dimensional metals, and of possible Fermi-surface nesting.

Structural distortions, not necessarily related to Fermi-surface nesting, are a general feature of this class of materials; all the Te compounds are distorted from the hexagonal layer structure (Wilson and Yoffe 1969), and even the group-IV material TiSe₂, which has only a low carrier density (~10²⁰–10²¹ cm⁻³) from a small p-d band overlap, exhibits a low-temperature superstructure in which the in-plane and out-of-plane cell constants are doubled (Di Salvo *et al* 1976a). The common feature for these systems is strong electron-phonon coupling seen for instance in the high infrared effective charge in TiSe₂ (White and Lucovsky 1977). Nevertheless the S and Se group-V compounds show a series of incommensurate and commensurate distortions for which it does seem to be possible to relate the superlattice period to the weak Fermi-surface nesting in the calculated band structures (Rath *et al* 1977, Doran *et al* 1978).

Resistivity versus temperature curves for both 1T and 2H polytypes are shown in figure 13. The resistivity anomalies, both continuous and discontinuous, all correspond to transitions in the PLD-CDW structure. Since not all the Fermi surface is destroyed by PLD-CDW formation they remain metallic at all temperatures, in contrast to KCP and TTF-TCNQ (the rise in the low-temperature resistivity of 1T TaS₂ is the result of falling carrier mobility from impurity-induced localisation, rather than from falling carrier concentration (Di Salvo and Graebner 1977)).
<figure><img src="image_13.png" /><figcaption>**Figure 13.** The electrical resistivity parallel to the layers of several layered compounds (Di Salvo 1977).</figcaption></figure>

### 3.4.1. 1T polytypes.
The two 1T polytypes of Ta, $TaS_2$, and $TaSe_2$, both exhibit strong distortions, with both commensurate and incommensurate structures at high and low temperatures. Figure 14 shows electron diffraction pictures of 1T $TaS_2$ at various temperatures, and shows the three regimes of superstructure, which are bounded by the two resistivity anomalies in figure 13, one at 350 K and the other centred at 200 K with

<figure><img src="image_14.png" /><figcaption>**Figure 14.** Transmission electron diffraction patterns for the three superlattice phases of 1T $TaS_2$ (Williams *et al* 1974).</figcaption></figure>
~20 K hysteresis. There is no sharp transition from the undistorted state to the incommensurate superlattice; the transition is at a high temperature (~600 K) and the search for it is hampered by the tendency for the metastable 1T polymorph to transform irreversibly to the 2H polymorph at high temperatures. Above 350 K, in the 1T₁ phase, an incommensurate structure with superlattice spots in line with the principal spots is seen. As for all the layer compounds, with trigonal or hexagonal symmetry in the layer, a triple PLD-CDW is observed, with three plane-wave PLD-CDW's at 120° to one another. Between 350 K and 200 K, in the 1T₂ phase, the superlattice reflections have rotated away from the line of the principal reflections. Although it is the tendency to achieve commensurability that must be the driving force for the rotation of the superlattice in this regime, the superlattice is not commensurate. The commensurate structure is only achieved below 200 K, by rotation of the superlattice by 13° 54', so that the axes of the hexagonal supercell are 3*a* + *b* (where *a* and *b* are the in-plane lattice vectors) or √(13)|*a*|. In addition to the rotation, the superlattice wavevector length shrinks by ~4% in going from the 1T₁ incommensurate phase to the 1T₂ commensurate phase. This large change in superlattice period indicates that Fermi-surface nesting is far from ideal, although the lock-in is favoured by the very large distortion amplitude in this material. 1T TaSe₂ does not exhibit the 1T₂ phase, and the lock-in from a similar 1T₁-type phase to the √(13)|*a*| superlattice occurs at 473 K (Wilson *et al* 1975).

The distortions of the 1T polytypes are large, with atomic displacements of ~0.25 Å (Brouwer and Jellinek 1976), and changes in the infrared reflectivity through the transitions in 1T TaS₂ indicate sizable alterations in the band structure over a range of 1/2 eV (Barker *et al* 1975). With a substantial fraction of the band being affected by the distortion, it is clear that details of the band structure in the close vicinity of the Fermi surfaces are not of great importance in determining the superlattice periodicity, and it is misleading to call these distortions Fermi-surface driven. The most elegant demonstration that the superlattice period is, however, determined by the degree of band filling and the dimensions of the Fermi surface, comes from experiments on Ti-doped TaS₂ (Wilson *et al* 1975). In a rigid-band model, alloying TiS₂ and TaS₂ merely changes the degree of band filling, with each Ti in TiₓTa₁₋ₓS₂ removing one electron from the band. As

<figure><img src="image_15.png" /><figcaption>**Figure 15.** *q/a*\*, the wavelength of the PLD-CDW in the incommensurate state divided by the reciprocal lattice vector shown versus *x* for 1T Ta₁₋ₓMₓS₂ where *M* = Ti, Nb or V (Wilson *et al* 1975). Full curve: electron diffraction experiment (Ti).</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Figure 16.** xps from Ta $4f_{7/2}$ levels in 1T TaS$_2$ taken during a cooling and heating cycle, showing hysteresis at the 200 K transition. The splitting between the two peaks at the lowest temperatures is ~0.6 eV (Hughes and Pollak 1976).</figcaption></figure>

discussed in §4.5, the effect of alloying is to reduce the strength of the distortions, but in Ti-substituted TaS$_2$, the incommensurate phase is stable over the whole range, and the reduction of the superlattice wavevector with increasing $x$ in figure 15 correlates with the expected shrinking of the electron pockets in figure 12. Note that doping with the other group-V elements, V and Nb, does not affect the superlattice period.

One of the more spectacular indications of the changes taking place is x-ray photo-emission from the Ta $4f_{5/2}$ and $4f_{7/2}$ levels. Changes in the charge density around the Ta sites create splittings in the energy levels of up to 0.6 eV. Wertheim et al (1975) estimate that such a large splitting corresponds to a charge redistribution between sites of as much as one electron. This very large CDW amplitude is large even compared with the amplitude of the atomic displacements, and it is a further indication of very strong electron-phonon coupling. Figure 16 shows measurements at various temperatures through the two transitions in 1T TaS$_2$. In the $1T_3$ phase, Hughes and Pollak (1976) argue that the 13 metal sites in the supercell are distributed into three inequivalent types in the ratio 6:6:1, and that the two peaks arise from the two more common sites. What is intriguing is that even in the $1T_2$ phase, when the superlattice is incommensurate, the two peaks are still resolved. This must indicate that the superlattice is locally commensurate and invites a description similar to the discommensuration picture developed by McMillan (1976) for the 2H polytypes. This is consistent with the finding of harmonics of the distortion in x-ray measurements (Yamada and Takatera 1977, Nakanishi et al 1977) in the $1T_2$ phase.

The only other 1T polytype that exists, VSe$_2$, shows very different behaviour, developing a 4 × 4 incommensurate superlattice at around 110 K and a commensurate 4 × 4 superlattice at around 75 K (Williams 1976). The weak PLD-CDW formation is considered to be the consequence of competition between the electron-phonon coupling driving the PLD-CDW and Coulomb repulsion in the narrow 3d conduction band (Friend et al 1978b). VSe$_2$ will be further discussed in §4.3.

### 3.4.2. 2H polytypes.
PLD-CDW formation in the 2H polytypes is weaker than in the 1T polytypes, and the characteristic commensurate superlattice is 3a × 3b. Three 2H
polytypes exhibit PLD-CDW formation: TaS₂, TaSe₂, and NbSe₂; and 2H NbS₂ is the only group-V dichalcogenide that does not apparently develop a superlattice. The failure of 2H NbS₂ to distort is consistent with the chemical trends in those 2H polytypes that do (Thompson 1975). Single crystals of 2H TaSe₂ and NbSe₂ can be grown sufficiently large for neutron diffraction work, and NbSe₂ is suitable for NMR experiments, so these two polytypes have been most studied.

The neutron diffraction investigation by Moncton *et al* (1975, 1977) of TaSe₂ and NbSe₂ shows that both compounds undergo a distortion to an incommensurate structure with $q = \frac{1}{3}a^*(1 - \delta)$ and $\delta \sim 2\%$, TaSe₂ at 122 K and NbSe at 33 K. Below these temperatures, $\delta$ decreases, and at 90 K, TaSe₂ locks into the commensurate phase. NbSe₂ however remains 1% incommensurate at liquid-helium temperatures. In the incommensurate regime of TaSe₂ Moncton *et al* noticed in addition to the satellites at $\frac{1}{3}a^*(1 - \delta)$ weaker satellites at $\frac{1}{3}a^*(1 + 2\delta)$, as is shown in figure 17. The presence of harmonics in the PLD-CDW amplitude in the incommensurate regime is consistent with the discommensuration picture for this regime (McMillan 1976), although Moncton *et al* developed a Landau treatment where lock-in Umklapp energy in the incommensurate phase can come from combinations of the order parameters of the $(1 - \delta)$ and $(1 + 2\delta)$ distortions which have the symmetry of the undistorted phase. At 4·2 K the displacements of the Ta and Se atoms are ~0.1 Å and ~0.05 Å respectively.

⁹³Nb is suitable for NMR ($I = \frac{9}{2}$), and the formation of the PLD-CDW in 2H NbSe₂ produces changes in both Knight shift and the electric field gradient (Berthier *et al* 1978). Figure 18 shows the $(\frac{1}{2}, -\frac{1}{2})$ central line at various temperatures. The broadening of the line below the transition at 33 K is compared with the calculated lineshape for a triple incommensurate CDW (shown as the full line). Berthier *et al* consider the general agreement with the experimental line to be good, but the peak in the experimental line at the high-field edge is not consistent with the step that is calculated, and may be an indication of a tendency towards local commensurability in this incommensurate phase. Berthier *et al* estimate the change in charge density below the transition to be less than 10%, a much smaller change than in the 1T polytypes.

Whereas the strongest changes in the PLD-CDW structure in the 1T polytypes are the lock-in transitions to the commensurate superstructure (the commensurability energy is very large for these large-amplitude distortions and should be visualised in terms of changes in local chemical bonding (Di Salvo 1977)), in the 2H polytypes the onset transition to the incommensurate superstructure is stronger and produces most change in the transport properties. Even so, only a small fraction of the carriers are removed by the distortion, and conductivity often improves below the onset transition temperature shown in figure 13.

Several attempts to model the PLD-CDW formation in the 2H polytypes have been made, both semi-empirical and from first principles. As an alternative to the Fermi-surface nesting model, Rice and Scott (1975) suggested that coupling between saddle points in the band structure which lie close to the Fermi level in the calculation of Mattheiss (1973) can also produce a divergence in $\chi_q$, though the saddle points lie well away from $E_F$ in the calculation of Wexler and Woolley (1976). Doran *et al* (1977) have calculated parameters for the stability criterion (equation 8) from the Wexler and Woolley band structure, and have also estimated the strength of the electron-phonon coupling from the superconducting properties of these materials (all 2H polytypes superconduct; NbSe₂ has the highest $T_c$, of around 7 K. They conclude that there is no difficulty in satisfying this criterion; the electron-phonon coupling term on the LHS is large and although $\chi_q$ does not peak sharply at $q \sim \frac{1}{3}a^*$, its overall magnitude is large because of the
<figure><img src="image_17.png" /><figcaption>**Figure 17.** Elastic neutron scattering measurements of the satellite peak intensity and position in 2H TaSe₂ through the incommensurate and commensurate PLD-CDW transitions (Moncton *et al* 1975).</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Figure 18.** <sup>93</sup>Nb NMR in 2H NbSe<sub>2</sub> showing evolution of the central line (1/2, -1/2) below 33 K. The full curve is the calculated lineshape corresponding to a Knight shift distribution associated with a triple incommensurate CDW (Berthier *et al* 1978).</figcaption></figure>
high density of states in the narrow d sub-band. As discussed in § 2.4, McMillan (1977) has developed a short-coherence-length model in which phonon entropy dominates the finite-temperature properties in order to explain the broad Kohn anomaly, over about one-third of the Brillouin zone (Moncton *et al* 1975, 1977) and the large PLD-CDW energy gap seen in the infrared reflectivity of order ~0.25 eV (Barker *et al* 1975).

### 3.4.3. $4H_b$ polytypes.
As shown in figure 11, the $4H_b$ polytype consists of alternating layers of octahedral and trigonal prismatic coordination layers. The $4H_b$ polytypes of $TaS_2$ and $TaSe_2$ have both been studied and they show apparently independent PLD-CDW formation in each type of layer, with the octahedral layers developing the √13 superlattice and the trigonal prismatic layers the 3 × 3 incommensurate superlattice (DiSalvo *et al* 1976b, Tatlock 1976). Details of PLD-CDW formation for all polytypes are summarised in table 3.

<table><caption>**Table 3.** PLD-CDW superlattices and transition temperatures for the layer compounds. $T_o$ is the onset transition temperature from the normal state to the incommensurate PLD-CDW, and $T_a$ is the commensurate-incommensurate superlattice transition temperature. The transitions in the two types of layer in the $4H_b$ polytypes have been grouped according to the coordination of the layer involved in the transition.</caption><thead><tr><th></th><th></th><th>Commensurate superlattice</th><th>% Incommensurate at ~ T<sub>o</sub></th><th>T<sub>o</sub></th><th>T<sub>d</sub></th><th>Reference</th></tr></thead><tbody><tr><td rowspan="5">Octahedral</td><td>1 T TaS<sub>2</sub></td><td>√13</td><td>+3.8</td><td>~600</td><td>350<br>200</td><td>a, b</td></tr><tr><td>1 T TaSe<sub>2</sub></td><td>√13</td><td>2.6</td><td>~600</td><td>473</td><td>a</td></tr><tr><td>4 H<sub>b</sub>TaS<sub>2</sub></td><td>√13</td><td>-5.2</td><td>~600</td><td>315</td><td>a, c</td></tr><tr><td>4 H<sub>b</sub>TaSe<sub>2</sub></td><td>√13</td><td>-4.3</td><td>~600</td><td>410</td><td>d</td></tr><tr><td>1 T VSe<sub>2</sub></td><td>4 × 4</td><td>2-6</td><td>110</td><td>~75</td><td>e</td></tr><tr><td rowspan="5">Trigonal prismatic</td><td>2 H NbSe<sub>2</sub></td><td>(3 × 3)</td><td>-2.5</td><td>33</td><td>-</td><td>f</td></tr><tr><td>2 H TaS<sub>2</sub></td><td>3 × 3</td><td>small</td><td>75</td><td>75?</td><td>g</td></tr><tr><td>2 H TaSe<sub>2</sub></td><td>3 × 3</td><td>-2.5</td><td>122</td><td>90</td><td>f</td></tr><tr><td>4 H<sub>b</sub>TaS<sub>2</sub></td><td>(3 × 3)</td><td>+7</td><td>22</td><td>—</td><td>c</td></tr><tr><td>4 H<sub>b</sub>TaSe<sub>2</sub></td><td>(3 × 3)</td><td>+4</td><td>75</td><td>—</td><td>d</td></tr></tbody></table>

(a) Wilson *et al* (1975)

(b) Williams *et al* (1974)

(c) Tatlock (1976)

(d) Di Salvo *et al* (1976b)

(e) Williams (1976) (Tatlock has reported larger values of incommensurability, see Friend *et al* 1978b)

(f) Moncton *et al* (1975, 1977)

(g) Tidman *et al* (1974)

## 4. Further aspects of PLD-CDW formation in low-dimensional conductors

### 4.1. Three-dimensional coupling and transverse ordering

The establishment of long-range order in the distorted phases of low-dimensional conductors requires coupling between chains or layers. This transverse coupling can be tunnelling coupling, reflected in the three-dimensionality of the Fermi-surface, Coulombic coupling between CDW's on adjacent stacks or layers, or strain coupling where, for example, the transverse displacement of the chalcogen atoms in the layer compounds allows certain phases of the PLD-CDW in adjacent layers to minimise the expansion of the c-axis lattice parameter.

As discussed in § 3.2, the $2d_⊥$ transverse period which KCP attempts to attain below 140 K suits both the interchain Coulomb and tunnelling energies. The short coherence
length of the transverse period reflects more the presence of Br⁻ and H₂O disorder than the weakness of the transverse coupling (see § 2.5).

Interchain Coulomb coupling has been used to explain the transverse period in the various distorted phases of TTF-TCNQ (Bjeliš and Barišić 1977). Between 52 K and 49 K, in which regime only the TCNQ stacks are distorted, the transverse period of 2a minimises the Coulomb interaction between the TCNQ stacks, separated by TTF stacks. Below 49 K, where the TTF stacks develop a distortion, the transverse period is that which minimises the TCNQ-TCNQ, TTF-TTF and TTF-TCNQ interactions, and the period grows from 2a towards 4a as the TTF stack distortion amplitude grows, and the TTF-TCNQ interaction dominates. This interaction would favour a period of a, and the transition to the period of 4a at 38 K gives extra lock-in energy terms in a Landau theory (Bak and Emery 1976). The physical reason for the lock-in to 4a remains obscure in the simple Coulomb interchain interaction model, although it does have a natural explanation in the libron model for the TTF-TCNQ distortions, where the Peierls superlattice results from condensation of libronic rotation modes of the TTF and TCNQ molecules (Weger and Friedel 1977).

The transverse period of the 2H polytype layer compounds, with two layers per unit cell in the undistorted phase, is unchanged in the superlattice phases (Moncton et al 1975, 1977). The transverse period in the incommensurate phases of 1T TaS₂ is 3c, and this period minimises the Coulomb interaction between nearest and next-nearest layers (Di Salvo 1977). In the commensurate 1T₃ phase, where the CDW origin is no longer arbitrary in the unit cell, the overall Coulomb interaction is minimised by translation of the CDW origin by 2a between layers, so that the origin of the CDW is repeated after 13 layers (the true unit cell is triclinic with c' = c + 2a, Williams et al 1974). Nearest-neighbour and next-nearest-neighbour interactions are required to establish long-range order, and in 1T Ta₁-xZrₓSe₂, the random Zr impurity potential can dominate the next-nearest-neighbour interactions, producing a random stacking (Moncton et al 1976).

While Coulombic interlayer coupling has generally been assumed to be effective, Woolley and Wexler (1977) find that the 3c transverse period gives the best nesting in 1T TaS₂, and Bulaevskii and Khomskii (1978b) consider that tunnelling coupling is stronger. The large buckling of the chalcogen sheets out of the layers in both 1T and 2H polytypes (Brouwer and Jellinek 1976, Moncton et al 1977), gives rise to large jumps in the transverse lattice parameter at the PLD-CDW transitions (Robbins and Marseglia 1978) and strain coupling may also be strong in the layer compounds. The strong interlayer coupling in the layer compounds may determine the nature of the intralayer superlattice. Thus the 3c repeat in 1T₂ TaS₂ which gives best interlayer Coulomb coupling will help to stabilise the 1T₂ phase against the 1T₃ phase with its less satisfactory transverse order (Robbins and Marseglia 1978).

The 4H₂ polytypes, in which coupling between similar coordination layers is screened by the other coordination layer, provide an interesting comparison to the single coordination type materials. Di Salvo et al (1976b) find that in 4H₂ TaSe₂ at low temperatures the strongly distorted octahedral layers are three-dimensionally ordered, whereas the more weakly distorted trigonal prismatic layers are uncorrelated.

## 4.2. *High pressure and the prevention of the PLD-CDW state*

The effect of pressure on KCP, of inducing a sharp metal-insulator transition similar to those seen in the organic charge-transfer salts, has already been discussed in §3.2.
Pressure has been used in the organic charge-transfer salts to attempt to stabilise the metallic state in these materials at low temperatures. Pressure may suppress the Peierls transition by increasing the transverse transfer integral, $t_⊥$, only if $t_⊥$ is sufficiently large. The criterion for $t_⊥$ to reduce $T_p$ from the mean-field one-dimensional value by inhibiting Fermi-surface nesting is given in equation (22) and only in the HM structure salts is this criterion satisfied. In the more one-dimensional materials the Peierls transition temperature is observed to rise under pressure, reaching 70 K at 30 kbar in TTF-TCNQ (see figure 10), and rising from 29 K at 1 atmosphere to 45 K at 10 kbar in TSF-TCNQ (Cooper *et al* 1977). There are several possible explanations for these increases, including an increase in the electron-phonon coupling, an increase in the intrachain carrier lifetime, or a reduction in the effect of the onsite Coulomb repulsion (see §4.3); however it is not clear which is the dominant effect (Friend *et al* 1978a).

HMTSF-TCNQ shows semimetallic properties below about 200 K, with a very low carrier concentration (Jérôme and Weger 1977). In the bandstructure model of Weger (1976) it is the covalency gap introduced over most of the Fermi surface by the inter-chain transfer integral that removes the carriers. Under pressure any increase in $t_⊥$ will further reduce the carrier concentration, and the likelihood of a Peierls distortion. Figure 19 shows resistivity curves for HMTSF-TCNQ and HMTTF-TCNQ under pressure showing the reduced low-temperature resistivity increase at high pressure. HMTTF-TCNQ shows two transitions in its resistivity at ambient pressure, at 49 K and 43 K (Greene *et al* 1976) and is strongly insulating at low temperature as is shown in figure 7. However above 3 kbar when only one transition remains, the low-temperature conductivity remains high below the transition. Interpretation in terms of only one type of stack distorting (Friend *et al* 1978a) is complicated by the recent x-ray diffraction study in which an incommensurate transverse period of 2.38 Å is found (Megtert *et al* 1978).

<figure><img src="image_19.png" /><figcaption>**Figure 19.** Resistivity versus temperature for the HM organic charge transfer salts at pressures sufficient to stabilise semimetallic behaviour at low temperatures (Cooper *et al* 1976, Friend *et al* 1978a). Recent measurements on HMTSF-TCNQ (Miljak *et al* 1978) reveal high-pressure behaviour closer to that in HMTTF-TCNQ, with a transition that is only gradually suppressed; the apparent removal of the transition at 14 kbar may be due to an impurity concentration comparable with the intrinsic carrier concentration (~$10^{19}$ cm$^{-3}$).</figcaption></figure>
Recent measurements on HMTSF-TCNQ (Miljak *et al* 1978) find high-pressure behaviour significantly different from the behaviour observed by Cooper *et al* (1976). In recent samples a well defined transition occurs at 24 K and one atmosphere. This transition is removed down to 16 K by 6 kbar, and then rises again gently up to 20 K at 30 kbar. Therefore, the apparent total removal of the transition previously reported by Cooper *et al* (1976) may be due to an impurity concentration comparable with the intrinsic carrier concentration (~$10^{19}$ cm$^{-3}$).

In general the effect of pressure on the PLD-CDW transitions in the layer compounds is large for the 1 T polytypes and small for the 2H polytypes. For the octahedrally coordinated TaS₂ and TaSe₂, in the 1T and 4H_b polytypes, the lock-in transition to the commensurate superstructure falls at between 3 and 5 K per kbar (Grant *et al* 1974, Di Salvo *et al* 1974, Friend *et al* 1977). It is not clear whether impaired Fermi-surface nesting or reduced electron-phonon coupling is the dominant effect.

In the 2H polytypes the onset transition to the incommensurate PLD-CDW is only weakly pressure dependent. Chu *et al* (1976) observed the 122 K transition in 2H TaSe₂ to increase by less than 2.5 K at 20 kbar, and the onset transition temperatures in the other trigonal prismatically coordinated materials are all reduced under pressure, 2H TaS₂ at -0.17 K per kbar (Delaplace *et al* 1976) 2H NbSe₂ at -0.36 K per kbar (Berthier *et al* 1976) and 4H_b TaS₂ at -0.8 K per kbar (Friend *et al* 1977). Since Fermi-surface nesting in these materials is not good (Doran *et al* 1978), increasing the interlayer coupling with pressure cannot further significantly impair it, and pressure-induced changes in the band structure and phonon spectrum affect the transition through the electron-phonon coupling strength and the density of states at the Fermi level. Comparison of the pressure dependences of the superconducting transition temperatures of these materials (see section §4.4) with that of T₀ confirms that changes in Fermi-surface nesting under pressure are not important.

1T VSe₂ is exceptional in that the onset transition temperature of 110 K, which is very low for an octahedral-coordination material, rises at 0.8 K per kbar (Friend *et al* 1978b). This stabilisation of the PLD-CDW under pressure is considered to be related to the effect of electron-electron repulsion in the narrow 3d conduction band, and is discussed in §4.3.

### 4.3. Electron-phonon coupling versus electron-electron repulsion

The establishment of a CDW in the conduction electrons necessarily brings the electrons closer together on average, and increases the Coulombic interaction between them. Thus the Coulomb repulsion between electrons opposes CDW formation, and in the mean-field treatment of Chan and Heine (1973), the Coulomb repulsion energy, *U*, is directly subtracted from the electron-phonon interaction in equation (12). Since many of the metals which develop PLD-CDW's have very narrow conduction bands, the effect of *U* on the PLD-CDW formation is expected to be significant. In both the layer compounds and the organic charge-transfer salts there are systems where these effects are evident.

PLD-CDW formation in 1T VSe₂ is, as discussed in §§3.4 and 4.2, much weaker than expected for a 1T polytype, and has the unusual feature of being strengthened under pressure. Angular-resolved photoemission measurements show 1T VSe₂ to have an occupied d bandwidth of only 0.3 eV (Hughes *et al* 1978) compared with the values of 1.5 and 1.15 eV found for 1T TaS₂ and 1T TaSe₂ (Traum *et al* 1974). This very small bandwidth produces a high *N*(*E*<sub>F</sub>) which increases 1/*χ*<sub>g</sub> on the RHS of equation (12) and is good for PLD-CDW formation, but it also reduces the value of the electron-phonon
coupling constant on the LHS of equation (12), with $g_q \propto$ band width (Barisic 1972). Typical values of U for d-band metals are at least comparable with the estimates of the electron-phonon term in equation (12), and in $VSe_2$ competition between the two terms reduces the strength of the PLD-CDW formation. Under pressure the increase in bandwidth brings $VSe_2$ closer to the 1T Ta polytypes where the effects of Coulomb repulsion are less significant.

TTF-TCNQ is in many ways an anomalous member of this series of charge-transfer salts, and many of its properties are difficult to explain on a free-electron band model. The large paramagnetic value of the susceptibility requires a very narrow band width if due only to a Pauli contribution, and is more physically explained if enhanced by Coulomb correlations (Torrance *et al* 1977). Explanations for the $4k_F$ diffuse x-ray scattering (see figure 8) are usually based on a large on-site Coulomb repulsion (the libron model of Weger and Friedel (1977) is the only exception). In a Hubbard model, when $U > 4t_\parallel$, single occupation of k-states will double the value of the Fermi wavevector for a one-dimensional band and produce a Peierls instability at $4k_F$. This is the model used by Fletcher and Toombs (1977) who assume that this behaviour is only one type of stack (TTF), and that the $2k_F$ condensation occurs on the other (TCNQ), less correlated stack. Torrance (1977, 1978) identifies the $4k_F$ scattering in the same way, but suggests that the $2k_F$ scattering is due to a spin-Peierls instability on a correlated stack. The model of Emery (1977) can be viewed as the weak-interaction limit of these models (Toombs 1978). An alternative approach is to view the electron gas as condensed to a Wigner lattice, with the electrons on the lattice sites arranged to minimise the Coulomb repulsion between them. Local distortions of the lattice from this arrangement of electrons will produce a $4k_F$ anomaly in diffuse x-ray scattering (Kondo and Yamaji 1977, Lee *et al* 1977, Hubbard 1978). Kondo and Yamaji argue that in a strongly Coulomb correlated electron system the electronic motion associated with band formation can be slower than the phonon motion, to give the necessary displacement in lattice positions for the $4k_F$ scattering to be observed. Lee *et al* (1977) apply renormalisation-group methods to a set of coupled chains with repulsive intrachain interactions, and find that there is still a three-dimensional phase transition into the CDW state.

ESR measurements can decompose the magnetic susceptibility into the individual contributions from the two types of stack (Tomkiewicz *et al* 1976, 1977a, b). Only in TTF-TCNQ is there any significant enhancement of the susceptibility, and this occurs on the TTF stacks. This is consistent with the absence of $4k_F$ scattering in TSF-TCNQ and HMTTF-TCNQ (Weyl *et al* 1976, Megtert *et al* 1978). Caron *et al* (1978) have modelled the pressure dependence of the lowest transition in TTF-TCNQ, associated with the TTF stacks, (figure 10) with a repulsive intrachain interaction at ambient pressure crossing over to an attractive interaction above 4 kbar, as the bandwidth is increased under pressure. The rapid rise in the transition temperature above 4 kbar is due to the rapid growth of the overall attractive interaction in this pressure regime.

### 4.4. The PLD-CDW state and superconductivity

The only materials that exhibit PLD-CDW formation and which are also superconductors are the 2H polytype layer compounds. In these materials only a small fraction of the conduction electrons are removed by PLD-CDW formation, and $N(E_F)$ is still large enough for them to become superconducting below the PLD-CDW transition. One of the interesting properties of the layer compounds is their ability to take in electron donors between the layers. Intercalation by organic bases has been particularly well studied because
of the substantial enhancement of the superconducting transition temperature, $T_c$, that is sometimes observed, particularly in $TaS_2 \cdot 2H TaS_2$ superconducts at 0·8 K and $4H_b TaS_2$ below 1·1 K, but when intercalated with, for example, ammonia, methyl-amine or pyridine, $T_c$ for both materials rises to between 3 and 6 K (Meyer *et al* 1975). The superconducting properties of these intercalates are often extremely anisotropic, for example in $2H TaS_2$ (pyridine)₁/₂ the separation of the layers is larger than the Ginzburg-Landau coherence length perpendicular to the layer (Schwall *et al* 1976).

Before the discovery of the Fermi-surface-driven instabilities in these materials, speculation as to the reason for the increase in $T_c$ with intercalation included enhancement through the excitonic mechanism in view of the proximity of the metal layers to the polarisable amine (Phillips 1972). However Thompson *et al* (1972) correlated the increase in $T_c$ in $2H TaS_2$ when intercalated with pyridine to the suppression of the PLD-CDW seen as the removal of the anomalies at 75 K in resistivity and susceptibility. Conclusive evidence that the excitonic mechanism had not been observed was obtained from measurements of hydrogen-intercalated $TaS_2$, where the interlayer separation is unchanged, and 10% H intercalation increases $T_c$ for most of the sample to above 4 K (Murphy *et al* 1975).

The correlation between the enhancement in $T_c$ with the suppression of the PLD-CDW state is also seen in high-pressure experiments. Molinié *et al* (1974) found that $T_c$ for $2H NbS_2$, which does not distort, is pressure independent, whereas $T_c$ for $2H NbSe_2$ is strongly enhanced under pressure. Experiments to higher pressure by Smith *et al* (1972) showed that above 36 kbar, by which pressure $T_c$ has increased from 7·2 K to ~8·2 K, the pressure dependence was as weak as in $2H NbS_2$. Berthier *et al* (1976) found that 36 kbar was the pressure required to entirely suppress the resistivity anomaly associated with the PLD-CDW transition (33 K at 1 bar). As discussed by Friedel (1975), it appears that the increase in $T_c$ is due to the increase in $N(E_F)$ as the PLD-CDW is suppressed and the Fermi surface restored to its undistorted state. That $T_c$ does not reach a maximum at the critical pressure required to suppress the distortion indicates that the softening of the Kohn anomaly phonon is of less importance.

<figure><img src="image_20.png" /><figcaption>**Figure 20.** Correlation between $T_{CDW}$, the PLD-CDW onset transition, and $T_c$ the superconducting transition in $2H$ and $4H_b TaS_2$ under pressure. Data for $T_c$ of $2H$ are from Smith *et al* (1975), for $T_{CDW}$ of $2H$ from Delaplace *et al* (1976), and for $4H_b$ from Friend *et al* (1977).</figcaption></figure>
Figure 20 shows similar results for the pressure dependence of $T_c$ in 2H TaS$_2$ and 4H$_b$ TaS$_2$ and of the PLD-CDW onset transitions in the trigonal prismatic layers in the two materials. $T_0$ is only slowly suppressed in 2H TaS$_2$, but has disappeared by 8.5 kbar in 4H$_b$ TaS$_2$, by which pressure $T_c$ has risen to 4.3 K—a very similar temperature to the values obtained with intercalation by organic amines and hydrogen. The primary effect of intercalation is therefore to suppress the PLD-CDW, although the mechanism for this process is not clear. There is probably transfer of charge from the intercalate to the metal conduction band, but the changes in band filling and $N(E_F)$ that this brings about are small. The details of the phase diagram in figure 20 where $T_0$ crosses $T_c$ must be more complicated than as shown since only two or three phase regions can meet at one point. Probably the 'quadruple' point splits into two triple points close to each other.

### 4.5. Disorder and the PLD-CDW state

Since the Peierls instability depends on the existence of well-defined momentum states for the electrons, disorder, in the form of finite chain length, impurities, radiation induced damage etc, which localises the carriers and smears out their momentum distribution, will quickly suppress the Peierls instability. As discussed by Bulaevskii (1975) the influence of lattice disorder on the Peierls transition and the influence of magnetic impurities on the superconducting transition are equivalent. Zittartz (1967) has also considered impurity scattering for the excitonic insulator model which is analogous to the Peierls distortion. Using a tight-binding model, Bulaevskii expresses the criterion for the suppression of the Peierls transition in terms of a critical localisation length, $l_c$

$$l > l_c \approx 2t/kT_p^0 \qquad (23)$$

where $t$ is the transfer integral, and $T_p^0$ the Peierls transition temperature in the absence of disorder. Chiang et al (1977) and Zuppiroli and Bouffard (1978) show the suppression of $T_p$ with increasing radiation damage in TTF-TCNQ and TSF-TCNQ. Along with reducing $T_p$, the presence of disorder introduces a non-vanishing density of states in the range of the Peierls gap (Bulaevskii 1975) and a dramatic effect of radiation damage on TTF-TCNQ and particularly TSF-TCNQ is the rapid increase in low-temperature conductivity (Zuppiroli and Bouffard 1978). Disorder has been seen as a route to the establishment of the metallic state at low temperatures in the organic conductors. One example is tetrathiotetracene iodide, (TTT)$I_x$, in which the planar TTT molecules stack to form a quasi-one-dimensional conduction band, and the presence of disorder in the iodine chains prevents a Peierls distortion, allowing conductivities of over $100 (\Omega \text{ cm})^{-1}$ at 4.2 K (Kaminskii et al 1977, Mihaly et al 1977).

Studies of disorder in the layer compounds have been made primarily by controlled doping. Di Salvo et al (1975) present a comprehensive study of doping in the 1T polytypes. The most noticeable effect of doping in these polytypes is to favour the incommensurate superlattice over the commensurate superlattice, and all the lock-in transitions are rapidly suppressed. McMillan (1975) explains this as due to the ability of the incommensurate superlattice to adjust its phase with respect to the lattice to best accommodate the impurity potential. The incommensurate superlattice in the 1T polytype is often still present for high dopant levels, hence the feasibility of obtaining the data in figure 15; the weaker 2H polytype distortions are more readily suppressed (Huntley and Frindt 1974).
Above the PLD-CDW transition, the incipient instability in the dielectric response of the electrons in the layer compounds causes impurities to be ineffectively screened. Each impurity attracts a giant Friedel oscillation (or CDW) around it, and the large scattering cross-section for the impurity site can result in Anderson localisation of the conduction electrons by the random impurity potential at relatively low levels of dopant. This is the explanation proposed by Di Salvo et al (1976c) to explain the very large increases in the low-temperature resistivity of Fe, Co and Ni doped 1T TaS₂ and 1T TaSe₂. Di Salvo and Graebner (1977) even explain the low-temperature insulating state of 1T TaS₂ as due to defects in the pure material. (An alternative explanation by Tossatti and Fazekas (1976) is Mott localisation as a consequence of cluster formation in the commensurate superlattice.)

The temperature-dependent relaxation time in both TTF-TCNQ and the layer compounds is very short, typically ~10⁻¹⁵ s at room temperature. (This is a manifestation of the strong electron-phonon coupling in the layer compounds, but the nature of the scattering mechanism in TTF-TCNQ remains controversial (Jérôme and Weger 1977, Jérôme 1977, Cooper 1978)—estimates of λ (equation 17) of ~0.2 (Cooper et al 1976, Jérôme 1979b) rule out electron-acoustic phonon scattering.) Huntley et al (1978) consider that this rapid scattering rate determines the effective temperature scale (~ħ/κτ) for the PLD-CDW transitions in the 2H polytype layer compounds, where in 2H TaSe₂ for example, ħ/κτ is ~1000 K at the 122 K transition. Similar considerations should be significant in TTF-TCNQ where ħ/τ ~ kT at Tₚ.

### 5. Conclusions

We have reviewed some of the properties of those low-dimensional conductors which show Fermi-surface-driven instabilities and develop low-temperature superstructures which have been investigated during the past few years. Many of their unusual properties have been satisfactorily explained within the framework of the Peierls distortion, although many problems remain, particularly in the organic metals where the novelty of the narrow π conduction band and the molecular structure, with the possibility of coupling to many types of phonon mode and intramolecular vibrations, are serious complications. The initial search for high conductivity, or superconductivity possibly driven by the excitonic mechanism, provided the impetus for much of the research, although this aspect has been disappointing. The properties of these materials have however proved to be a rich testing ground for many theoretical models of one-dimensional systems, and phase transitions, for example the commensurate-incommensurate superstructures and the soliton model of the lock-in transition, and the Fröhlich collective-mode conductivity. The latter phenomenon has been very elusive, although the measurements of non-linear conductivity in NbSe₃ offer encouragement that it can be observed.

Finally we emphasise that this review sets out to be representative, rather than inclusive, and we apologise for the omission of many interesting and important aspects of this field of low-dimensional conductors.

### Acknowledgments

We thank J Friedel, M Weger and A D Yoffe for many helpful discussions.
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