# Electrical conductivity in polymeric sulphur nitride at high pressures

Denis Jerome, Richard H Friend, S. Rehmatullah, A.D. Yoffe

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## To cite this version:

Denis Jerome, Richard H Friend, S. Rehmatullah, A.D. Yoffe. Electrical conductivity in polymeric sulphur nitride at high pressures. J. Phys. C: Solid State Phys, 1977. hal-05000898

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# Electrical conductivity in polymeric sulphur nitride at high pressures

R H Friend†, D Jérôme‡, S Rehmatullah† and A D Yoffe†

† Cavendish Laboratory, Madingley Road, Cambridge, UK

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

Received 23 September 1976

**Abstract.** The resistivity along the chain axis in polymeric sulphur nitride has been followed under pressure using a variety of hydrostatic and quasi hydrostatic techniques. The large increase in conductivity with pressure at room temperature previously reported up to 9 kbar is confirmed, and is seen to level off above 30 kbar at a conductivity ratio $\sigma(p)/\sigma(0) \sim 10$. The $T^2$ dependence of the resistivity seen at atmospheric pressure, associated with electron-electron scattering, has been followed with pressure. At moderate pressures this dependence is more complicated; there are two regimes with different $T^2$ coefficients which both saturate below room temperature, pointing to a semimetallic band structure with small pockets of electrons and holes present. The high-pressure limit of $\sigma(p)/\sigma(0)$ is seen to correspond to the level of electron-phonon scattering that can be expected from the superconductivity data.

## 1. Introduction

There has been considerable interest recently in polymeric sulphur nitride, $(SN)_x$ for its highly anisotropic metallic properties. It forms as crystals of parallel helical strands of alternating S and N, and is conducting along the chain (b) axis. Whilst it exhibits very anisotropic electrical, optical and mechanical properties, it shows no evidence of a Peierls transition which is found in other 'one-dimensional' metallic systems such as the mixed valence planar platinum complex $K_2Pt(CN)_4Br_{0.3}H_2O$ and the organic charge transfer salt TTF-TCNQ. For a review of $(SN)_x$ and a comparison with these systems see Yoffe (1976). $(SN)_x$ goes to a superconducting state around 0.3 K (Greene et al 1975) and this temperature rises under pressure to 0.54 K at 9 kbar (Gill et al 1975). Several band structure calculations have been made, largely to investigate the absence of the Peierls distortion, and they fall into two groups. Kamimura et al (1975, 1976) use a tight-binding approach to derive a structure with two overlapping conduction bands so that there are two sheet-like Fermi surfaces, one biconcave and localed near $\Gamma$, the other biconvex and located near Z. $(SN)_x$ in this picture is a quasi one-dimensional metal according to the definition of Rudge and Grant (1975). Berlinsky (1976) has pointed out the tendency of this system to form a two times superlattice, since $k_{F1} + k_{F2} = \pi/b$, where $k_{F1}$ and $k_{F2}$ are the wavevectors to the two Fermi sheets. In the band structures calculated by tight binding by Bright and Soven (1976) and Friesen et al (1975), using the OPW method by Rudge and Grant (1975) and the pseudopotential method by Schlüter et al (1975), interchain interactions cause the Fermi surface to be semimetallic with pockets of electrons and holes, and sufficiently three dimensional to make a Peierls distortion unlikely. Chiang et al (1976)
find a strong $T^2$ dependence of the resistivity between 15 K and 200 K and suggest that it is caused by electron-electron scattering.

High pressure is a useful means of varying the properties of anisotropic materials, tending to make them more three dimensional. In the case of $(SN)_x$, where we expect a greater reduction in interchain than intrachain separation, its properties, particularly the anisotropy of the band structure and Fermi surface, are believed to depend sensitively on the degree of interchain interaction, and we expect more 'simple metal' behaviour under pressure. We present a detailed high-pressure investigation of the resistivity along the chain axis, using a variety of hydrostatic and quasihydrostatic techniques, and discuss the behaviour of the $T^2$ term in the light of the proposed band structures. Experiments using the hydrostatic helium gas and the isopentane-isoamyl alcohol equipment were performed at Orsay, whilst experiments with the quasi hydrostatic opposed anvils and tetrahedral anvils equipment were done at the SRC High Pressure Faculty at Standard Telecommunications Laboratories at Harlow.

## 2. Experimental

### 2.1. *Crystal preparation*

Crystals of $(SN)_x$ were grown by the solid state polymerization of $S_2N_2$ formed by catalytic decomposition of $S_4N_4$ over Ag wire. Crystals used in these experiments were taken primarily from two batches grown in slightly different conditions. For the first the method described by Street *et al* (1975) was used. Both the growth and the polymerization of $S_2N_2$ were carried out at 0°C over a period of five days. Tabular shaped crystals of size between 1 and 3 mm were obtained. The second technique was a slight variation of the method described by Mikulski *et al* (1975). $S_4N_4$ was purified by repeated recrystallization from benzene followed by vacuum sublimation. The crystals of $S_2N_2$ were grown at a controlled temperature of -5°C for three days. The temperature was then raised to 20°C at a rate of 0.5 K h$^{-1}$ and maintained at that level for 24 h. Unpolymerized $S_2N_2$ was removed by keeping the crystals under a vacuum of $10^{-5}$ Torr at this temperature for 48 h. Needle shaped crystals of length up to 1.5 cm and diameter 0.5 mm were obtained.

### 2.2. *Measurements at atmospheric pressure*

Contacts were made with Dupont 4929 silver paint with a mixture of butyl and hexyl acetates as solvent. Crystals used for resistivity measurements along the *b* axis were chosen or cleaved to be long in this direction. Current contacts were lapped over the ends of the crystal and voltage probes were painted in rings between the current contacts as discussed for example by Cohen *et al* (1974). Sample resistances were typically of the order of a few ohms at room temperature, as were contact resistances. In crystals as anisotropic as $(SN)_x$ (conductivity ratio, $\sigma_\parallel/\sigma_\perp \sim 10^3$ at 300 K) crystal defects or poor contacts can lead to unexpected current distributions, so crystals were rejected unless 'unnested' voltages, were less than 0.1% of the sample voltage at room temperature. Measurements were made at DC and also using a low-frequency (30 Hz) phaselock technique, and the results were indistinguishable. Samples were supported only by the fine 20 µm gold lead wires to prevent stain when cooling. Values of resistivity at room temperature were hard to estimate accurately, but most crystals showed $\rho < 1000$ µΩ cm. However the resistance relative to its value at 300 K behaved similarly for all samples. This was also found
to be true at high pressures so from here on we shall express all resistivities relative to room temperature and atmospheric pressure.

We define

$$ \rho'_{p,T} = \frac{\rho_{p,T} - \rho_{p,T=0}}{\rho_{1\text{bar, 300K}}} $$

Samples showed ratios of $\rho_{300K}/\rho_{4.2K}$ of between 20 and 100, those with the lower ratios showing resistance minima around 20 K. Samples from the two batches displayed similar ratios. Figure 1 shows typical data. Variations in resistivity at room temperature from crystal to crystal will be caused by different connectivities of the (SN)x fibres in the crystal,

<figure><img src="image_3.png" /><figcaption>Figure 1. Resistivity of (SN)x at atmospheric pressure, showing $\rho/\rho_{300K}$ and $\rho'/T = [(\rho - \rho_{\text{residual}})/\rho_{300K}]/T$ against T.</figcaption></figure>

and we feel that the highest values of conductivity found by Chiang et al (1976) will be closest to the intrinsic behaviour of the material. The $T^2$ dependence of $\rho$ can be seen in figure 1 where $\rho'/T$ is plotted against T. A deviation from $T^2$ above 200 K is seen in agreement with Chiang et al (1976).

### 2.3. Helium gas, $p \to 8$ kbar

Three samples were studied up to 8 kbar using the high-pressure system described by Malfait and Jérôme (1969) and cooling runs down to 4.2 K performed at 3, 5 and 8 kbar. Contacts were made in the same way as for the atmospheric pressure experiments. At room temperature the conductivity rises very quickly with pressure, as can be seen in figure 2 where the data shown are from experiments to higher pressures. Following Gill et al (1975) we have plotted $\ln \sigma(p)/\sigma(0)$ against $p$ and our results are in good agreement up to the 9 kbar limit of their investigation. At moderate pressures $d\rho/dp$ is large, so pressure stability is important if reliable $\rho$ against $T$ curves are to be obtained. Pressure which was measured with a Manganin coil in the intensifying cell which remained at room temperature, was adjusted continuously when cooling and was constant within 100 bar until the helium gas froze at 60 K and 8 kbar and correspondingly lower temperatures at lower pressures. Pressure loss at freezing, caused by solidification of the helium in the capillary high-pressure tubing before complete freezing in the pressure cell, was
less than 250 bar. This could be estimated using Simon's formula (Simon *et al* 1929) to find the pressure at the helium melting point by noting the temperature at which the helium melted on warming. Temperature was measured inside the high-pressure cell

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

**Figure 2.** Conductivity of (SN)x parallel and perpendicular to the b axis relative to atmospheric pressure as a function of pressure at 300K. The broken line shows hydrostatic data parallel to b, the full lines show data from Bridgman opposed anvil experiments. The error bar shows variation between samples.

close to the sample with a copper-constantan thermocouple. Data are shown in figure 3. There was no sign of the resistance minimum seen in poorer crystals at atmospheric pressure, although residual resistances were sometimes quite high. ρ - ρ<sub>residual</sub> however did not depend on the size of the residual term over a range of ρ<sub>4·2K</sub>, ρ/ρ<sub>300K</sub>, 1 bar from

<figure><img src="image_5.png" /></figure>

**Figure 3.** ρ/ρ<sub>300K, 1 bar</sub> against T for (SN)x at 3, 5 and 8 kbar.

20 to 120. The resistivity no longer obeys a simple T² law as can be seen in figure 4. The choice of the ρ'/T against T plot will be discussed later, but it can be seen that the regime where the coefficient of T² is large saturates at a low temperature, and that by room temperature ρ is varying linearly with T. This behaviour was found in all samples.
<figure><img src="image_6.png" /><figcaption>Figure 4. ρ'/T = [(ρ - ρ<sub>residual</sub>)/ρ<sub>300K, 1 bar</sub>]/T against T for (SN)ₗ at 3, 5 and 8 kbar.</figcaption></figure>

2.4. *High-pressure fluid, p → 37 kbar*

The range of hydrostatic pressure was extended to 37 kbar using a high-pressure intensifying cell described by Delaplace *et al* (1976). The high-pressure cell was cooled to 4·2 K at 24 kbar and to 2 K at 37 kbar. For the 24 kbar run isopentane was used as the high-pressure fluid, though to reach 37 kbar a 1:1 mixture of isopentane and isoamyl alcohol was used, as this mixture remains liquid at room temperature at this pressure. Temperature was measured with a copper-constantan thermocouple inside the cell down to 4·2 K, and below by the vapour pressure of the liquid helium around the cell. Contacts were made in the same way as those for the atmospheric pressure experiments. There was no evidence of either crystal or contact deterioration in either the isopentane or the

<figure><img src="image_7.png" /><figcaption>Figure 5. ρ/ρ<sub>300K, 1 bar</sub> against T for (SN)ₗ at 24, and 37 kbar.</figcaption></figure>
isoamyl-alcohol-isopentane mixture, indeed the atmospheric pressure data shown in figure 1 were measured on the crystal taken to 37 kbar after the pressure experiment. $\rho$ was monitored continuously on an X-Y recorder and at no time was there any discontinuity in $\rho$ or $d\rho/dT$ that might be evidence of a phase transition. At room temperature the rapid increase in conductivity seen at lower pressures levels off (see figure 2). Data from the cooling runs at 24 and 37 kbar are shown in figure 5. At 37 kbar and 2 K there was no sign of superconductivity. Gill et al (1975) fit a quadratic pressure dependence to $T_c$ data up to 9 kbar which extrapolates to 3.3 K at this pressure, although a linear extrapolation of their data gives 1.2 K. We discuss this further in §3.4. At 2 K $\rho/\rho_{300K, 1bar} \sim 10^{-3}$ so taking the highest value of the conductivity found by Chiang et al (1976), of 3300 (\Omega cm)¹ we estimate a resistivity as low as 0.3 \mu\Omega cm at 2 K and 37 kbar. The $T^2$ dependence of $\rho$ seen at atmospheric pressure up to 200 K is now confined to a much smaller temperature regime, extending from 2 K to 35 K at 37 kbar and to 40 K

<figure><img src="image_8.png" /><figcaption>**Figure 6.** $\rho'/T = [(\rho - \rho_{\text{residual}})/\rho_{300\text{K, 1 bar}}] (\times 1/T)$ against $T$ for (SN)$_x$ at 24 and 37 kbar.</figcaption></figure>

at 24 kbar, as can be seen in figure 6. At higher temperatures an empirical relation of the form $\rho \propto (T - \theta)$ gives a good fit to the data, with $\theta = 55$ K at 24 kbar and 58 K at 37 kbar.

## 2.5. Bridgman anvils, $p \to 80$ kbar

The range of the high-pressure investigation was extended to 80 kbar using the quasi-hydrostatic Bridgman opposed anvils apparatus described by Pitt (1968). Contacts to the (SN)$_x$ crystals were made with colloidal silver loaded epoxy which was cured at 100°C for 12 h; the sample was then potted in the MgO/epoxy ring with epoxy resin and cured for a further 12 h at 100°C. Surprisingly there was no appreciable crystal deterioration or reaction with the epoxy. Also embedded in the ring was a chromel-alumel thermocouple, and the sample could be cooled to 100 K by blowing liquid nitrogen onto the anvils. Pressure within the ring is only hydrostatic at pressures greater than 25 kbar; also, pressure calibration is difficult up to this pressure, but as can be seen in figure 2 the agreement up to 37 kbar with the hydrostatic results is good. This agreement was seen with most samples investigated. Above 40 kbar there is very little further increase in conductivity. Cooling to 100K at 40, 60 and 80 kbar produced a linear temperature dependence of $\rho$, though the extrapolated residual term was larger than in the hydrostatic experiments. This indicates that the measured resistivity contains a transverse term caused by crystal damage from non-hydrostatic stresses.
A least-squares fit to the data gives

$$ \rho_{40\text{kbar}} = \rho_{300\text{K, 1bar}} \{-9.1 \times 10^{-4} + 1.57 \times 10^{-4} T\} $$

$$ \rho_{60\text{kbar}} = \rho_{300\text{K, 1bar}} \{3.7 \times 10^{-4} + 1.54 \times 10^{-4} T\} $$

$$ \rho_{80\text{kbar}} = \rho_{300\text{K, 1bar}} \{23.7 \times 10^{-4} + 1.53 \times 10^{-4} T\}. $$

This sample showed a larger increase in $\sigma$ than the other crystals. There is less than 2% change in the linear term between 40 and 80 kbar. Also shown in figure 2 is the transverse conductivity under pressure, and this again is in fair agreement with the data of Gill et al (1975) up to 9 kbar. On cooling to 130 K $\rho_{\perp}$ decreased slowly, and at least-squares fit gives

$$ \rho_{\perp 80\text{kbar}} = \rho_{\perp 300\text{K, 1bar}} \{2.12 \times 10^{-2} + 4.55 \times 10^{-5} T\}. $$

It is not clear however whether the measured transverse conductivity is an intrinsic property of (SN)x or limited by the fibrous disordered nature of the crystals perpendicular to the chain axis. On releasing pressure some of the samples exploded, though causing little damage; this was probably initiated by slipping of the ring between the anvils producing local heating sufficient to detonate the (SN)x.

### 2.6. Tetrahedral anvils, $p \to 40$ kbar

Samples of (SN)x were investigated up to 40 kbar and to over 600 K using the 1¼ inch tetrahedral anvil apparatus discussed by Lees and McCartney (1968). The crystal contacts were again silver epoxy, cured at 100°C for 12 h. The crystal was then plotted in an epoxy cylinder with a chromel-alumel thermocouple, and surrounded by a heater coil of nichrome wire. This was placed inside a pyrophyllite tetrahedron in the edge-to-edge configuration, which Lees et al (1968) find minimizes non-hydrostatic stresses. $\rho$ showed discontinuities when pressure was applied while the gaskets were forming around the tetrahedron, and $d\rho/dT$ indicated that both parallel and perpendicular components of $\rho$ were present. However it was possible to see to what temperature electrical contacts could be kept on the (SN)x and metallic behaviour observed. At 30 kbar contact was lost at 493 K and at 40 kbar at 618 K, at which temperature the epoxy surrounding the sample was destroyed. These temperatures must be regarded as lower limits of the stability of (SN)x at these pressures.

## 3. Discussion

The principle experimental findings to be explained are the strong $T^2$ dependence of the resistivity and its very rapid change when pressure is applied and interchain interactions are increased. The rise by a factor of ten in the conductivity when pressure is applied at room temperature cannot be accounted for by lattice stiffening, the principle effect in simple metals, and Gill et al (1975) have invoked band structure changes to explain this anomalous increase.

### 3.1. Relaxation times

That the room temperature atmospheric pressure resistivity has large contributions that are not electron-phonon in origin can be seen by relating the superconductivity data to
the optically or electrically determined relaxation time $\tau$ through the treatment of Hopfield (1970, 1972) who obtains

$$ \lambda = \frac{\hbar}{2\pi} \left\langle \frac{1}{\tau} \right\rangle \frac{1}{k_B T} $$

where $\lambda$ is the superconducting coupling constant defined as

$$ \lambda = N(E_F)\langle I^2 \rangle / M\langle \omega^2 \rangle. $$

$I$, the electron-phonon matrix element and $\omega$ are appropriately averaged. $\tau$ can be obtained either from the optical scattering time from an analysis of the Drude edge or from the electrical resistivity in the regime above the Debye temperature where $\rho \propto T$, together with the plasma frequency:

$$ \rho = m^*/ne^2\tau $$

$$ \omega_p^2 = ne^2/\epsilon_0 m^*. $$

A Drude fit to optical data using polarized light ($E \parallel b$) on a single crystal of $(SN)_x$ (Greene et al 1975b) gives $\hbar\omega_p = 4.6$ eV and $\tau/h = 1.6$ eV$^{-1}$. Grant et al (1975) estimate $\lambda = 0.31$, so we estimate that the contribution to the room temperature resistivity due to the electron-phonon processes that are involved in the superconductivity.

$$ \rho_{\text{electron-phonon 300K}} = 17 \mu\Omega \text{ cm.} $$

The optical scattering time gives

$$ \rho_{\text{optical, 300K}} = 35 \mu\Omega \text{ cm} $$

whilst the lowest measured DC resistivity of Chiang et al (1976) is 270 $\mu\Omega$ cm. Hopfield's treatment involves several approximations but he still obtains agreement to within 10% for Al and Sn. In the case of $(SN)_x$ the anisotropy of the Fermi surface might be expected to cause the agreement to be poorer, but not by a factor of ten. We conclude that most of the resistivity measured at atmospheric pressure and room temperature is caused by scattering processes that are not electron-phonon in origin. It is surprising that $\sigma_{\text{optical}}$ is not closer to $\rho_{\text{DC}}$, however the Drude fit does not extend to low energies and may overestimate $\tau$ as a result; also it is not yet clear whether the best measured value of $\rho_{\text{DC}}$ can be reduced by improved growth techniques, and the value of 270 $\mu\Omega$ cm must be taken to be an upper limit.

It is worth emphasizing here that the enormous discrepancy between the electron-phonon resistivity estimated from the superconductivity data and the measured DC resistivity makes it very unlikely that an unusual phonon spectrum in $SN_x$ will be responsible for the $T^2$ dependence of the resistivity.

Furthermore, the high-pressure asymptote of the resistivity in figure 2, of around 25 $\mu\Omega$ cm, is close to this expected electron-phonon term. Thus at high pressures the dominant scattering processes are electron-phonon in origin. There can be no phonon spectrum that can be so quickly altered by the application of pressure as to produce the rapid pressure dependence of the resistivity seen in figure 2; once again we are forced to discard an unusual phonon spectrum model. It is for this reason that the suggestion of electron-electron scattering made by Chiang et al (1976) has been followed up at length.

### 3.2. Electron-electron scattering

A large $T^2$ term in the resistivity, often associated with electron-electron scattering, has
been seen in many quasi one- and two-dimensional systems which are almost unstable to a Peierls distortion, for example TTF-TCNQ above 58 K (Seidon and Cabib 1976), the related compound HMTSeF-TCNQ (Cooper *et al* 1976) between 0.2 and 2 K at pressures sufficient to suppress the transition, and the layer compound TiS₂ (Thomson 1975) from 30 to 400 K. Thomson speculates that the electron-electron scattering may be enhanced by fluctuations into the distorted state, termed paraphasons. In the case of (SN)ₓ Chiang *et al* (1976) propose electron-electron Umklapp processes between pockets or sheets of Fermi surface separated by half a reciprocal-lattice wavevector (G/2) as likely to be effective.

The T² dependence of electron-electron scattering arises from two (kT/E<sub>F</sub>) factors which come from the Pauli restriction on allowed scattering events (Ziman 1958). The atmospheric pressure data in figure 1 shows that ρ deviates below T² above 200 K indicating that the electron or hole pockets or sheets responsible for scattering are becoming non-degenerate. The degeneracy temperature, T<sub>F</sub> = E<sub>F</sub>/k, of a pocket of electrons in a semimetal is the energy from the Fermi level to the local minimum of that conduction band. We arbitrarily define T<sub>F</sub> as the temperature at which ρ deviates 5% below T², in this instance 260 K. This is a very much lower temperature than that in for example the transition metals where electron-electron scattering is seen at low temperatures below the T⁵ electron-phonon regime (Mott and Jones 1936). Perhaps it is because the (kT/E<sub>F</sub>) factors are so much larger than the coefficient of T² is large enough to dominate the electron-phonon scattering in (SN)ₓ up to room temperature at atmospheric pressure. Chiang *et al* (1976) find ρ = ρ₀ + AT² with A = 5.6 × 10⁻⁹ Ω cm K⁻²; our data fitted to ρ = ρ₀ + αρ<sub>300K, 1 bar</sub> T² gives α = 12.6 × 10⁻⁶ K⁻², see table 1.

<table><caption>Table 1. Coefficients of the T² dependence of the resistivity and degeneracy temperatures as defined in § 3.3.</caption><thead><tr><th>Pressure</th><th>α<sub>1</sub><br>(K<sup>-2</sup> × 10<sup>-6</sup>)</th><th>α<sub>2</sub><br>(K<sup>-2</sup> × 10<sup>-6</sup>)</th><th>T<sub>F1</sub><br>(K)</th><th>T<sub>F2</sub><br>(K)</th><th>α<br>(K<sup>-2</sup> × 10<sup>-6</sup>)</th><th>T<sub>F</sub><br>(K)</th></tr></thead><tbody><tr><td>1 bar</td><td></td><td></td><td></td><td></td><td>12.6</td><td>260</td></tr><tr><td>3 kbar</td><td>1.9</td><td>4.5</td><td>78</td><td>211</td><td></td><td></td></tr><tr><td>5 kbar</td><td>7.0</td><td>2.8</td><td>31</td><td>206</td><td></td><td></td></tr><tr><td>8 kbar</td><td>8.4</td><td>1.3</td><td>35</td><td>179</td><td></td><td></td></tr><tr><td>24 kbar</td><td></td><td></td><td></td><td></td><td>4.9</td><td>40</td></tr><tr><td>37 kbar</td><td></td><td></td><td></td><td></td><td>3.9</td><td>35</td></tr></tbody></table>

All of the band structures mentioned can quite feasibly give rise to effective electron-electron scattering. Kamimura (1976) has pointed out that the two Fermi surface sheets in his model are separated by G/2 along ΓZ and can satisfy the criterion for U processes which are necessary for transfer of momentum to the lattice. He has also modified his earlier calculations (Kamimura *et al* 1975, 1976) so that the lower band is now closer to Z and its degeneracy temperature is much smaller. The semimetallic band structures all have sufficient numbers of pockets in the Brillouin zone for it to be likely that the criterion for U processes will be fulfilled. When pressure is applied we can expect the size and positions of these pockets in the Brillouin zone to shift as they will sensitively depend on the strength of interchain interactions and bond angles within the chain. The lower band in the structure by Kamimura (1976) will be very sensitive to interchain spacing and it is
likely that several pockets of electrons and holes will be formed at high pressure, a requirement we shall find necessary to explain the behaviour of the resistivity under pressure. It would be valuable to have band structure calculations that will give more detailed information on the shapes of the pockets in the Brillouin zone and their behaviour with pressure. Anomalous thermal expansion coefficients seen in polyoxymethylene (White et al 1976), a polymer with a similar structure to (SN)ₓ, suggest that band structure changes with temperature may be important.

### 3.3. Medium-pressure regime

In the pressure range between 3 and 8 kbar followed with the helium gas apparatus, the $T^2$ behaviour becomes more complex, see figure 4. We note that the strong $T^2$ dependence seen at low temperature saturates at much lower temperatures than at 1 bar, indicating that the Fermi energy of the pockets that are causing the scattering is falling, consequently so are the numbers of carriers in them. As the scattering is still effective judged by the size of the $T^2$ coefficient in this region, we posulate that these small pockets are scattering to larger pockets of Fermi surface with higher Fermi energies which are carrying most of the current. In this situation where the two pockets participating will go degenerate at different temperatures, we expect that above the degeneracy temperature of the smaller pocket there will be a linear temperature dependence as there will now be only one $(iT/E_F)$ factor. It is for this reason that we have plotted $\rho'/T$ against $T$ in figures 1, 4 and 6. Further examination of figure 4 shows that there are in fact two regimes of $T^2$. We suggest that at low temperatures the large $T^2$ coefficient comes from scattering from at least two small pockets, one of which goes degenerate at a temperature $T_{F1}$. Above $T_{F1}$ there is both a quadratic and a linear term in $\rho(T)$, hence the non-zero intercept of $\rho'/T$ extrapolated from this region to $T=0$, and the smaller $T^2$ coefficient. At a higher temperature $T_{F2}$ the second small pocket goes degenerate and we now expect a linear $\rho(T)$. We define these three temperature regimes as follows:

$$ \text{I} \qquad T < T_{F1} \qquad \rho' = (\alpha_1 + \alpha_2)T^2 $$

$$ \text{II} \qquad T_{F1} < T < T_{F2} \qquad \rho' = (\alpha_1 T_{F1}) + \alpha_2 T^2 $$

$$ \text{III} \qquad T > T_{F2} \qquad \rho' = (\alpha_1 T_{F1} + \alpha_2 T_{F2})T. $$

The two degeneracy temperatures $T_{F1}$ and $T_{F2}$, are defined by the intercepts of the lines characterizing these three regions on a plot of $\rho'/T$ against $T$, and data for three pressures are shown in table 1. $\rho_{\text{residual}}$ was determined by a least-squares fit of $\rho$ against $T^2$ in regime I. The two $T^2$ regimes are only just resolvable at 3 kbar, and at atmospheric pressure there is only one. Inspection of table 1 suggests that the pocket characterized by $\alpha_2$ and $T_{F2}$ is responsible for the $T^2$ term at this pressure; with falling pressure both $T_{F2}$ and $\alpha_2$ increase towards the 1 bar data whilst $\alpha_1$ is falling. These ideas can only be tested when band structure calculations have been made for different chain separations.

### 3.4. High-pressure regime

At 24 and 37 kbar there is only one regime of $T^2$; see figure 6 and table 1. It is not clear how this $T^2$ term relates to the $T^2$ behaviour at lower pressures. It is most likely that it
comes from the $\alpha_2$, $T_{F2}$ pocket, as $T_{F2}$ extrapolates to the low degeneracy temperature seen at these higher pressures. Above $T_F$, $\rho$ does not vary so simply and we must consider how much of $\rho$ can be attributed to electron-phonon scattering. $\lambda$ will increase with pressure as $T_c$ rises and using the estimate by Gill et al (1975) of $d\ln\theta_D/dp = 1.2\% \text{ kbar}^{-1}$ we find $\lambda = 0.39$ at 9 kbar; it will presumably be larger at 24 and 37 kbar. Taking $\rho_{300K, 1bar} = 270 \mu\Omega \text{ cm}$ from Chiang et al (1976) therefore estimate

$$ \frac{\rho_{\text{electron-phonon}}}{\rho_{300\text{K, 1 bar}}} \sim 7.9 \times 10^{-2} \quad \text{and} \quad \frac{\rho_{\text{electron-phonon}}}{\rho_{300\text{K, 1 bar}}} \times \frac{1}{T} \sim 2.7 \times 10^{-4} \text{ K}^{-1} $$

at room temperature. The Debye temperature of 170 K from the specific heat data of Greene et al (1975a) will rise to 250 K at 37 kbar, so phonon scattering will not be fully proportional to $T$ at room temperature. The contribution to $\rho'/T$ from the electron-electron scattering, $\alpha T_F = 1.4 \times 10^{-4} \text{ K}^{-1}$ added to the electron-phonon estimate is equal to $4.1 \times 10^{-4} \text{ K}^{-1}$ and compares well with the measured value of $3.7 \times 10^{-4} \text{ K}^{-1}$ at 290 K. A similar electron-phonon term must be considered for the 3, 5 and 8 kbar data; its neglect will overestimate $\alpha_2$ and $T_{F2}$ in table 1, particularly at 8 kbar. We have not attempted to adjust these values.

Between 40 and 80 kbar, in the Bridgman opposed anvils apparatus, $\rho$ and $d\rho/dT$ change very little, although we would expect to see significant lattice stiffening between these two pressures. The electron-phonon interaction is reduced as the lattice stiffens, see Paul (1963) for a review, and above $\theta_D$, where $\rho \propto (T/\theta_D)^2$, $d\ln\rho/dp = -2d\ln\theta_D/dp$. The estimate of $d\ln\theta_D/dp = 1.2\% \text{ kbar}^{-1}$ by Gill et al (1975) if used over this pressure range, requires a reduction in $\rho$ by a factor of two going from 40 to 80 kbar. It is not clear therefore why the conductivity of (SN)ₚ should be so little changed by pressure in this regime.

## 4. Conclusions

We have followed the conductivity along the chains in (SN)ₚ and find increasingly 'normal metal' behaviour as pressure is applied. The large coefficient of the $T^2$ dependence of the resistivity at atmospheric pressure is observed to change very rapidly. Between 3 and 8 kbar the presence of two regimes of $T^2$ dependence, both of which have saturated by room temperature, is most naturally explained if (SN)ₚ has a semimetallic band structure. At higher pressures electron-phonon scattering dominates the resistivity. A parallel series of high-pressure optical experiments would be interesting, as would experiments to follow the superconductivity to pressures higher than so far reported.

## Acknowledgments

We thank G Delplanque and G Malfait at Orsay and J Penfold at Harlow for technical assistance. We have had many fruitful discussions with Professor R F Frindt, Dr A J Grant, Professor H Kamimura and Dr G D Pitt. We thank Professor Sir Nevill Mott and Professor M Weger for pointing out mechanisms for electron-electron scattering. We thank the Science Research Council for a grant to the laboratory and for a studentship to RHF.
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