# Pressure-enhanced superconductivity and superlattice structures in transition metal dichalcogenide layer crystals

By P. MOLINIÉ

University of Nantes, France

D. JÉROME

University of Paris XI, Orsay, France

and A. J. GRANT

Cavendish Laboratory, Cambridge, England

[Received 12 July 1974 and in final form 19 August 1974]

## ABSTRACT

The pressure dependence of the superconducting transition temperature $T_c$ is investigated for the metallic layer crystals 2H NbS$_2$ and 2H NbSe$_2$. In 2H NbS$_2$, $T_c$ is pressure-independent, as in the 3R polytype. It is suggested that the pressure-enhanced $T_c$ in 2H NbSe$_2$ may be related to recent observations in electron diffraction and N.M.R. of a phase change to a superlattice structure in this material at low temperatures. Our results would tend to rule out the model of pressure-dependent Josephson tunnelling between layers suggested by previous authors. The anisotropy and pressure dependence of the critical field $H_{c2}$ are described.

## § 1. INTRODUCTION

The metallic compounds niobium sulphide NbS$_2$ and niobium selenide NbSe$_2$ are members of the family of transition metal dichalcogenide layer crystals. This class of compound is characterized by a pronounced anisotropy in the optical and transport properties in, and perpendicular to, the crystal layers, and these materials have been the subject of a number of recent review articles (Wilson and Yoffe 1969, Yoffe 1973, 1974).

The two common polytypes of NbS$_2$ are 2H and 3R, with a repeat distance of two and three layers respectively, and with superconducting transition temperatures $T_c$ of 6.3 K and 5.5 K (van Maaren and Schaeffer 1966). Jones, Shanks, Finnemore and Morosin (1972) found $T_c$ for 3R NbS$_2$ to be 5.9 K and pressure-independent to 10 kbar.

For 2H NbSe$_2$ (the 3R polytype is not found, although 3R Nb$_{1-x}$Se$_x$ has been grown at 900-1000°C by Kadijk (1964)) $T_c$ has been observed to increase under pressure from around 7.0 K at 1 bar (Jerome, Grant and Yoffe 1971, Jones et al. 1972, Il'ina, Itskevich and Kalyuzhnaya 1972, Yamaya and Sambongi 1972 a, b, Smith, Delong, Moodenbaugh, Geballe and Schwall 1972). The measurements of Smith et al. (1972) were made up to 140 kbar, and $T_c$ was observed to increase rapidly with pressure up to about 35 kbar, but to remain almost constant within experimental error above 35 kbar, which suggests a structural change at this pressure at low temperatures.
Anomalous behaviour has been reported in the transport properties of 2H NbSe₂ (at 1 bar) at low temperatures (Lee, Garcia, McKinzie and Wold 1970). A kink is seen in the electrical resistivity versus temperature at 145 K, with a peak in the magnetic susceptibility at 40 K and a sign change (from +ve to -ve on cooling) in the Hall coefficient $R_H$ at 26 K. Ehrenfreund, Gossard, Gamble and Geballe (1971) have carried out N.M.R. measurements on 2H NbSe₂ which demonstrate the absence of antiferromagnetic ordering and have suggested a low-temperature structural distortion below 77 K. X-ray studies by Marczio, Dernier, Menth and Hull (1972), while apparently not conclusive, tend to support the suggestion of a low-temperature structural distortion. More recently, Huntley and Frindt (1974) have quoted unpublished work by Williams and Valic (1973) in which a phase change around 25 K was found to take place in only those crystals of 2H NbSe₂ which exhibited a sign reversal of the Hall coefficient.

In this paper we report the pressure dependence of $T_c$ for 2H NbS₂ and 2H NbSe₂. The results for 2H NbS₂ are particularly significant since they resolve the question concerning the influence of the interlayer interaction (i.e. the crystal polytype) on the superconducting transition temperature and its pressure dependence. It is suggested that the pressure dependence of $T_c$ in 2H NbSe₂ is related to the presence of a low-temperature superlattice phase in this material and not to pressure-dependent Josephson coupling between layers as suggested by previous authors. We also report on the anisotropy and pressure dependence of the critical field $H_{c2}$ in these Type 2 superconductors.

## § 2. EXPERIMENTAL

The 2H NbSe₂ crystals were grown by vapour phase transport at 780°C with iodine transport agent. The polytype was confirmed by X-rays (see Brown and Beerntsen 1965) and $T_c$, obtained by extrapolation of $H_{c2}$ to zero, was 6.95 K, indicating good stoichiometry (Antonova, Medvedev and Shebalin 1970).

The crystals of 2H NbS₂ were grown from the elements by vapour phase transport with iodine transport agent at 850-900°C in the presence of a large (15%) S excess of sulphur. The crystal structure was confirmed by X-rays (Glazer 1974) and the value of $T_c$ obtained by extrapolation of $H_{c2}$ was 6.23 K.

No detailed study is known of the effect of deviations from stoichiometry in NbS₂ on the value of $T_c$, as has been made by Antonova *et al.* (1970) for NbSe₂. By analogy with NbSe₂, however, the value of $T_c$ for NbS₂ would suggest that no substantial deviation from stoichiometry occurred.

The high-pressure measurements were made in the cell described by Delplanque, Malfait, Rieux and Jérome (1970), with isopentane as the pressure fluid. The cell was cooled slowly at each pressure and the pressure monitored and adjusted as the isopentane froze, in order to obtain a known hydrostatic pressure at low temperatures. The pressure sensor was a manganin gauge located in the pressure intensifier at room temperature.

The sample was placed inside a small coil, and its susceptibility monitored by means of a susceptibility bridge (Schawlow and Devlin 1959). A copper-constantan thermocouple was present inside the pressure cell, and an iron/gold-constantan thermocouple was attached to the outside of the pressure cell.
<figure><img src="image_1.png" /><figcaption>Fig. 1</figcaption></figure>

Normalized change in oscillator frequency as a function of the applied magnetic field for 2H NbSe₂ at T=6.50 K and p=0.7 kbar. ω₀ and ω₁ correspond to the superconducting and normal states respectively.

A small piece of pure Nb metal (T_c ~ 9.2 K) was placed alongside the sample to provide a cross-check on the copper-constantan thermocouple, since the pressure dependence of T_c for Nb is accurately known (Smith 1970). The c axis of the layer crystal sample was oriented either parallel to (‖) or normal to (⊥) the magnetic axis of a superconducting magnet which surrounded the pressure cell.

At each pressure the temperature dependence of the critical field H_c2 was obtained by stabilizing the temperature at a series of values below T_c. The Schawlow-Devlin oscillator frequency was measured as a function of the applied field at each temperature. The change in the oscillator frequency was proportional to the change in the sample susceptibility. In fig. 1 we show a typical trace for 2H NbSe₂ at 6.50 K and p=0.7 kbar. The ordinate is (ω₀ - ω) / (ω₀ - ω₁), where ω₀, ω₁ are the frequencies associated with the superconducting and normal states of the sample respectively. The abscissa is the applied magnetic field, and the field H_c2 is taken at the point where the normalized frequency departs from unity. The minimum in the trace for the orientation H‖c in 2H NbSe₂ resembles the minimum observed in measurements of the normalized surface resistance in Nb and Nb₀.₉Ta₀.₁ by Gilchrist and Monceau (1968). Those authors associated the effect with flux pinning. No minimum was observed in 2H NbSe₂ by Monceau and Waysand (1974).

### § 3. RESULTS

The temperature dependence of H_c2 for 2H NbS₂ is shown in fig. 2. Data are shown for the orientation H_c2(‖c) at pressures 0, 7 and 13 kbar, and for H_c2(⊥c) at 0 kbar. H_c2 is found to vary linearly with the temperature in
<figure><img src="image_2.png" /><figcaption>Fig. 2</figcaption></figure>

The temperature dependence of $H_{c2}$ for 2H NbS$_2$.

each case, and $T_c$ is obtained by linear extrapolation to $H_{c2}=0$. The values of $T_c$ were independent of orientation, and in the case of $H_{c2}(\|c\)$, independent of pressure within experimental error. In addition, the slope of $H_{c2}(\|c\)$ versus $T$ was independent of pressure.

The temperature dependence of $H_{c2}$ for 2H NbSe$_2$ is shown in fig. 3. At each pressure the extrapolated value of $T_c$ agrees within experimental error for the two orientations $H_{c2}(\|c\)$ and $H_{c2}(\perp c)$. In this case $T_c$ is found to increase with pressure at the rate $5.3 \times 10^{-5}$ K/bar. The slope of $H_{c2}$ versus $T$ decreases with pressure for both orientations.

The temperature dependence of $H_{c2}$ was found to be linear, within experimental error, for both 2H NbSe$_2$ and 2H NbS$_2$. Curvature in the $H_{c2}$ versus $T$ plot for NbSe$_2$ has been reported by Woollam, Somoano and O'Connor (1974), but was not observed in the same material by Monceau and Waysand (1973, 1974).

In the upper part of fig. 4 is shown the anisotropy ratio $\rho = H_{c2}(\perp c)/H_{c2}(\|c\)}$ for 2H NbSe$_2$ as a function of pressure. This ratio is nearly constant, but
<figure><img src="image_3.png" /><figcaption>Fig. 3</figcaption></figure>

The temperature dependence of $H_{c2}$ for 2H NbSe$_2$.

decreases slightly under pressure. The value of this ratio for 2H NbS$_2$ was obtained only at 1 bar, and was equal to 2.93. In the lower parts of fig. 4 the coefficient $\alpha=1/T_c$. $dH_{c2}/dT$ is shown as a function of pressure and orientation for 2H NbSe$_2$ and 2H NbS$_2$. For an isotropic solid it may be shown (de Gennes 1966) that the upper critical field

$$H_{c2} = \frac{\phi_0}{2\pi\zeta^2} \quad (1)$$

where $\phi_0 = 2 \times 10^{-7} \text{ G cm}^2$ and the coherence length

$$\zeta(T) = 0.74 \zeta_0 \left( \frac{T_c}{T_c - T'} \right)^{1/2} \quad (2)$$

Substituting for

$$\zeta_0 = \frac{0.18 \hbar v_F}{k T_c} \quad (3)$$
values for $\zeta_0$ and $v_F$ for electrons in the layer planes may be calculated (corresponding to $H \parallel c$), and these are given in the table. The calculation of a single value for $v_F$ in the layers is of course a considerable oversimplification, since the value of $v_F$ will be a function of direction depending on the shape of the energy band in reciprocal space. The values of $v_F$ in the table may be compared with the average value of $62 \times 10^6$ cm sec$^{-1}$ for Nb metal calculated by Mattheiss (1970). The corresponding values of $\zeta$ and $v_F$ normal to the planes can be obtained only if the temperature dependence of $\zeta_s$, and the relationship between $\zeta_{0z}$ and $v_F(z)$ are known. We include in the table, with these reservations, values for $\zeta_{0z}$ calculated assuming a similar temperature dependence for $\zeta_s$ as for $\zeta$.

The value of $\alpha$ (not shown) for 2H NbS$_2$ at 1 bar in the orientation $H \perp c$ was $-3.7$ kG K$^{-2}$. In fig. 5 the dependence of $T_c$ on pressure is shown for 3R NbS$_2$ (after Jones *et al.* 1972), for 2H NbS$_2$ and 2H NbSe$_2$.

## § 4. DISCUSSION

### 4.1. Dependence of $H_{c2}$ on temperature and pressure

The present results on the pressure dependence of $H_{c2}$ in 2H NbSe$_2$ and 2H NbS$_2$ are of value in relation to a number of previous studies on 2H NbSe$_2$ at 1 bar. For example, our value of $H_{c2}(\parallel c)$ for 2H NbSe$_2$ at 4.2 K is 22.4 kG, in excellent agreement with the results of Muto, Toyota, Noto and Hoshi (1973), Foner and McNiff (1973), de Trey, Gygax and Jan (1973) and Monceau and

<figure><img src="image_4.png" /><figcaption>Fig. 5</figcaption></figure>

The pressure dependence of the superconducting transition temperature $T_c$ for 2H NbSe$_2$, 2H NbS$_2$ and 3R NbS$_2$ (after Jones *et al.* 1972).
Waysand (1973, 1974). (Note that these authors have referred to the directions parallel and normal to the layers, rather than the c axis as in this paper.) The anisotropy $\rho = H(\perp c)/H(\parallel c)$ for 2H NbSe$_2$ was found to be constant over the very narrow temperature range for which it was measured (i.e. H varied linearly with T for both orientations) whereas Muto et al. observed an increase in $\rho$ from 2.5 at 6-7 K to 3.1 at 4 K. A slight decrease in $\rho$ under pressure was observed, however (fig. 3, upper part).

The anisotropy in layer crystals has been treated in the effective mass approximation (Kats 1969, Lawrence and Doniach 1971, Morris, Coleman and Bhandari 1972, Monceau and Waysand 1973) which gives a good fit to the angular dependence of $H_{c2}$ in terms of a single parameter $\epsilon$, which is related to the effective mass ratio

$$ \frac{H_{c2}(\parallel c)}{H_{c2}(\perp c)} = \epsilon = \left( \frac{m(\perp c)}{m(\parallel c)} \right)^{1/2} $$

This model is valid for homogeneous anisotropic materials. At high magnetic fields for $H \perp c$, however, it is not clear that the homogeneity condition is fulfilled, since the coherence lengths normal to the layers (table) are only several times the crystal repeat distance (~12 Å). A discussion of the application of the model in describing the angular dependence of $H_{c2}$ is given by Monceau, Saint-James and Waysand (1974, to be published). Values for $m_{\parallel}/m_{\perp}$ for 2H NbSe$_2$ are 11 (Morris et al. 1972), 11 (de Trey et al. 1973), 10 (Foner and McNiff 1973), 6.3 (Muto et al. 1973, near 7 K) and 6.2 (this work). The corresponding effective mass ratio for 2H NbS$_2$ was 8.4.

These effective mass ratios may be compared with the much lower values ($m_{\parallel}/m_{\perp}$) = 1.5 determined optically by Liang (1973), in a study of the polarization dependence of the reflectivity in 2H NbSe$_2$. The free carrier plasma resonance was found to correspond approximately to a Drude formula, and would thus give an average mass for all the electrons in the band.

The effect of pressure on the coefficient $\alpha = 1/T_c$. $dH_{c2}/dT$ is of interest, since for an isotropic system it may be shown that $\alpha$ is proportional to $v_F^{-2}$ (eqn. (4)). In fig. 4 it will be seen that $\alpha$ increases under pressure in 2H NbSe$_2$ for both orientations of the magnetic field. The fractional changes $(1/\alpha)$ $(d\alpha/dp)$ are similar, $2 \times 10^{-5}$ bar$^{-1}$ ($H \parallel c$) and $2.5 \times 10^{-5}$ bar$^{-1}$ ($H \perp c$) in 2H NbSe$_2$.

The compressibility data of Jones et al. (1972) indicate a reduction in the $\alpha$ and $c$ parameters in 2H NbSe$_2$, for a pressure of 10 kbar, of 0.41% and 1.6% respectively. As a first approximation we may suppose the Fermi wavevectors $k_F$ to increase by about these amounts. Since the changes in $v_F$ deduced from the change in $\alpha$ would be of order 10% for 10 kbar (see eqn. (4)), changes in $k_F$ arising from lattice contraction would be too small to account for this. It is possible that the effect may instead be due to a decrease under pressure in the effective mass and that the fractional change is similar $\parallel$ and $\perp$ to $c$.

The pressure independence of $\alpha$ for 2H NbS$_2$ indicates that $v_F$ is constant. This is difficult to understand, since on the simplest model $v_F = \hbar k_F/m^*$, and one would expect $k_F$ to increase and $m^*$ to decrease under pressure, both contributing to an increase in $v_F$.

The zero-pressure values of $\alpha(H \parallel c)$ for 2H NbSe$_2$ and 2H NbS$_2$ are almost exactly equal, implying equal values of $v_F$ in the layer planes for each material.
The effective mass is almost certainly greater in the selenide, so that the Fermi wavevector $k_F$ should also be greater in the selenide. This is difficult to reconcile with the probable appearance of a superlattice in 2H NbSe$_2$ at low temperatures, and the consequence of a reduced Brillouin zone. It may thus be unrealistic to attach too much significance to $\alpha$ in terms of $v_F$ in these highly anisotropic solids.

### 4.2. Pressure-enhanced $T_c$ and low-temperature phase transitions

The pressure independence of $T_c$ for both 2H and 3R polytypes of NbS$_2$, compared with the pressure-enhanced $T_c$ for 2H NbSe$_2$ (fig. 5), clearly indicates that the crystal polytype is not the determining factor. In fact, it is possible that part or all of the difference in $T_c$ between 2H and 3R NbS$_2$, for example, may result from small deviations from stoichiometry, so that the stacking polytype may have little or no influence either on the value of $T_c$ or on its pressure dependence.

It is of interest to recall the results of Smith *et al.* (1972), mentioned in § 1, where $T_c$ was found to be nearly pressure-independent above 35 kbar for 2H NbSe$_2$. That result would suggest that the pressure enhancement of $T_c$ is associated with the presence of a structural distortion in NbSe$_2$, bounded by $T=26$ K at $p=0$, and by $T=9$ K at $p=35$ kbar. At higher pressures the undistorted 2H phase may be more stable, giving a pressure-independent $T_c$ as in 2H and 3R NbS$_2$. Low-temperature X-ray and electron diffraction studies are in progress on these materials to explore this hypothesis. Electron diffraction studies (Scruby, Williams and Parry 1974) have provided no evidence for a low-temperature phase transition in either 2H or 3R NbS$_2$, and the Hall coefficient in 2H NbS$_2$ remains positive down to 4.2 K (Forgan 1974).

The model of Antoniewicz and Fredricks (1973), which attempts to account for the pressure enhancement of $T_c$ in 2H NbSe$_2$ in terms of a pressure-dependent Josephson coupling between layers, does not account for Smith's data above 35 kbar, nor for the pressure independence of $T_c$ in 2H and 3R NbS$_2$. The data presented in this paper show that the directional anisotropy is similar for NbS$_2$ and NbSe$_2$, so that a similar mechanism would be expected for both solids. We would therefore rule out the model of pressure-dependent Josephson tunnelling for these unintercalated layer crystals.

The reasons for the pressure enhancement of $T_c$ in the distorted '2H$_d$' NbSe$_2$ will naturally be closely connected with the nature of the structural transformation which takes place at low temperatures. Preliminary electron diffraction evidence (Scruby *et al.* 1974) reveals the presence of additional scattered intensity below 145 K, which correlates with the resistivity kink at this temperature. The pattern suggests a possible Kohn anomaly which may lead to a charge density wave or Peierls superlattice at lower temperatures (26 K) as in 1T TaS$_2$ (Williams, Parry and Scruby 1974, Shepherd, Williams, Young and Scruby 1974).

Two mechanisms may be visualized which would lead to an enhanced $T_c$ under pressure. (See, for example, Smith 1972, Evans, Ratti and Gyorffy 1973, Allen 1974.) Either the phonon frequencies in the distorted phase may be anomalously reduced under pressure, or the density of states at the Fermi
level may increase. Neutron studies of the phonon spectrum, known to be in progress at a number of laboratories, together with low-temperature measurements of the elastic constants and the specific heat as a function of pressure, and a knowledge of the effective Gruneisen parameter, would be of considerable value in resolving this question.

A correlation between a high and often pressure-enhanced $T_c$, and a lattice instability has been found for a number of crystal structures (see Smith, Shelton and Lawson 1973 and references therein). One material whose behaviour bears a certain resemblance to that of 2H NbSe$_2$ is the A15 compound $V_3Si$. This material is known to undergo at 22 K a cubic to tetragonal phase transition which is accompanied by a softening of the sound velocity and of certain phonon modes (Weger and Goldberg 1973, Testardi 1973). The effect of pressure is to decrease the lattice transformation temperature $T_L$, but to increase $T_c$ in the distorted phase. The measurements of Chu and Testardi (1974) on this system at pressures up to 18 kbar indicate by extrapolation that $T_L$ and $T_c$ would cross at about 24 kbar, and at higher pressures a different dependence of $T_c$ on pressure in the cubic phase would presumably result.

It has also been observed in A15 materials that when the structural transformation is inhibited, a higher $T_c$ results than in a sample which undergoes the phase transition (Testardi 1971). It has been suggested that in the A15 system $T_c$ will be a maximum at the state of maximum lattice softness, when $T_c=T_L$ (Testardi 1972).

The results of Huntley and Frindt (1974) on 2H NbSe$_2$ are of interest in this regard. The data for their purest samples show that a decrease in $T_H$ (where the Hall coefficient changes sign) is accompanied by an increase in $T_c$. For less pure samples the effect no longer holds, since $T_c$ is then rapidly depressed. Yamaya and Sambongi (1972 a, b) have observed an irreversible enhancement in $T_c$ of 0.2 K in 2H NbSe$_2$ which had been subjected to pressure cycling, together with the disappearance of the sign change in $R_H$. The pressure-enhanced $T_c$ subsequently observed by these authors is then difficult to understand unless the low-temperature phase transition is only partially inhibited.

Recent band structure calculations by Mattheiss (1973) show that the Fermi level $E_F$ will lie in the relatively narrow d band which has $d_z^2$ character at $\Gamma$, and which is usually referred to as the 'd$_z^2$ band' (Wilson and Yoffe 1969). The Fermi surface expected for each of the two 'd$_z^2$' bands in 2H NbSe$_2$ has been sketched by Hughes and Liang (1974). The density of states at $E_F$ will be high because of the narrow bands, and preliminary band structure calculations by Wooley and Wexler (1974) suggest that the bands are narrower in 2H NbSe$_2$ than 2H NbS$_2$, in agreement with the specific heat measurements of Van Maaren and Haarland (1969). However, Somoano, Hadek and Rembaum (1973) have observed superconductivity in 2H MoS$_2$ intercalated with the alkali metals K, Cs and Rb, with $T_c$ between 6.1 K and 6.3 K. These values of $T_c$ are strikingly similar to the value of 6.3 K for 2H NbS$_2$, despite the fact that in the intercalated semiconductor, the electrons lie in higher d bands and the density of states at the Fermi level will be lower (for 0.3 to 0.4 alkali metal atoms per MoS$_2$ unit) than in the 'd$_z^2$' metal NbS$_2$. This observation might suggest that the value of $N(E_F)$ is of less significance for superconductivity than the electron-phonon coupling.
Antonova *et al.* (1970) have studied the effect of excess Nb in 2H NbSe₂ and found both $T_c$ and $\delta H_{c2}/\delta T$ to be depressed with increasing Nb concentration (see table). This implies an increase in the coherence length in spite of the disorder introduced by the excess Nb. This effect may be explained by an increase in $v_F$ due to increased filling of the 'd_z²' band and the rise of $E_F$ to a position of greater band curvature. The value for the coherence length normal to the layers given in the table is in accord with the work of Frindt (1972), who found $T_c$ in 2H NbSe₂ to be depressed upon thinning the crystal below 40 Å.

## § 5. CONCLUSIONS

Measurements of the superconducting transition temperature $T_c$ as a function of pressure in the layer crystals 2H NbS₂ and 2H NbSe₂ have indicated that the crystal-polytype does not primarily determine the pressure coefficient of $T_c$. It is suggested that the pressure-enhanced $T_c$ for 2H NbSe₂ below 35 kbar is associated with a superlattice phase at low temperatures. Measurements of the Hall coefficient under pressure would be of value to determine the proposed connection between the Hall sign change at 26 K at 1 bar, and the appearance of the structural transformation. The pressure independence of $T_c$ above 35 kbar in 2H NbSe₂ resembles the behaviour in both 2H and 3R NbS₂, and would rule out the model of pressure-dependent Josephson tunnelling between layers suggested by previous authors.

The measurements of $H_{c2}$ as a function of temperature and pressure have provided values for the effective mass ratio parallel and perpendicular to the c axis, $m_c/m_\perp$, for 2H NbS₂ and 2H NbSe₂. Values for the electron Fermi velocity in the layer planes have been calculated and are given in the table. The change under pressure in the coefficient $\alpha=1/T_c \cdot dH_{c2}/dT$ has been interpreted in terms of a decrease in the effective mass in 2H NbSe₂.

## ACKNOWLEDGMENTS

The experimental work described in this paper was carried out at Orsay. We are extremely grateful to Dr. E. M. Forgan of Birmingham University, England, who supplied the crystals of 2H NbS₂, and to Dr. A. M. Glazer of the Cavendish Laboratory, England, for X-ray polytype determinations. We thank a number of colleagues for valuable discussions, particularly J. Friedel, A. D. Yoffe, P. B. Allen, E. M. Forgan, G. Waysand and R. S. Title. We thank G. Delplanque and G. Malfait for technical assistance, and the S.R.C. for a Fellowship to A.J.G.

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