# Hall effect study of the field-induced instabilities in (TMTSF)2 PF6, under pressure

M. Ribault, Brigitte Piveteau, Denis Jerome, R. C Lacoe, L. Brossard, Alec

Moradpour, François Creuzet

► To cite this version:

M. Ribault, Brigitte Piveteau, Denis Jerome, R. C Lacoe, L. Brossard, et al. Hall effect study of the field-induced instabilities in (TMTSF)2 PF6, under pressure. Journal of Physics C: Solid State Physics, 1986. hal-04989833

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# Hall effect study of the field-induced instabilities in $(TMTSF)_2PF_6$ under pressure

B Piveteau, L Brossard, F Creuzet, D Jérome, R C Lacoe, A Moradpour and M Ribault

Laboratoire de Physique des Solides (associé au CNRS), Université Paris-Sud, 91405, Orsay, France

Received 30 September 1985, in final form 29 January 1986

**Abstract.** We report measurements of the Hall effect in $(TMTSF)_2PF_6$ under a pressure of 10.5 kbar performed at temperatures from 0.15 to 77 K and in magnetic fields up to 9 T parallel to the *c*\* axis. The low-temperature data suggest a series of transitions between semimetallic (SDW) sub-phases above a threshold field. The giant oscillations of the Hall voltage observed above the threshold field suggest for $(TMTSF)_2PF_6$ that, unlike in $(TMTSF)_2ClO_4$, electrons and holes play equally important roles. A particularly sharp anomaly of the Hall voltage detected around 7 T may be the signature of a first-order transition between SDW states with transverse and longitudinal nesting, respectively, below and above the transition field.

## 1. Introduction

In the series of organic conductors $(TMTSF)_2X$ (Jérome and Schulz 1982) both $X = ClO_4$ and $X = PF_6$ salts have revealed the phenomenon of superconductivity at low temperature. The latter compound, however, requires the application of a hydrostatic pressure in excess of 9 kbar for the observation of superconductivity (Jérome et al 1980). From atmospheric pressure up to 9 kbar, the ground state of $(TMTSF)_2PF_6$ is of the spin-density wave (SDW) kind. The superconducting state of $(TMTSF)_2PF_6$ is destroyed by the application of a magnetic field of about 0.1 T along the *c*\* direction. At higher fields, the conducting state is characterised by an anomalously high transverse magnetoresistance which, suddenly, above a threshold field of about 6.5 T at 1.1 K, acquires an oscillating character (Kwak et al 1981). Such behaviour has been interpreted by Kwak et al (1981) in terms of Shubnikov-de Haas oscillations in a state implying a two-dimensional Fermi surface oriented along the *c*\* axis. More high-field studies have been undertaken in $(TMTSF)_2ClO_4$ as superconductivity in this compound can be obtained under ambient pressure (Bechgaard et al 1981). Magnetoresistance (Kajimura et al 1982), NMR (Takahashi et al 1982), Hall effect (Ribault et al 1983, Chaikin et al 1983), specific heat (Garoche et al 1982, Perty et al 1985) and magnetisation (Naughton et al 1985) studies have proved that the 1D Fermi surface of $(TMTSF)_2ClO_4$ undergoes an instability under high magnetic field towards the 2D-like surface of a SDW semimetal.

Gork'ov and Lebed (1984) have explained this behaviour by the one-dimensionalisation of quasi-1D electrons by the strong magnetic field leading to a transition
between a 1D non-magnetic conductor and a 2D magnetic semi-metal at a threshold field. Ribault *et al* (1983) have proposed to relate the pseudo-periodic anomalies observed in the magnetoresistance and Hall effect at high fields to a sequence of phase transitions induced by the magnetic field instead of regular quantum oscillations for a rigid (field-independent) 2D Fermi surface.

Héritier *et al* (1984) have interpreted these phase transitions by a quantised nesting vector argument: within each sub-phase the density of unpaired carriers (and thus the nesting vector) adapts to the field in order to maintain completely filled the occupied Landau levels below the Fermi level. This model gives a possible qualitative explanation for the observation of steps and plateaus of the Hall voltage (Ribault *et al* 1984).

The comparison between the behaviours of $(TMTSF)_2ClO_4$ at ambient pressure and $(TMTSF)_2PF_6$ under pressure is not quite straightforward on account of the influence of the anion lattice on the Fermi surface of the 1D conducting state. The non-centrosymmetric $ClO_4$ anions order below 24 K (Pouget *et al* 1983) with a concomitant doubling of the lattice parameter in the *b* direction (the wave-vector of the superlattice is $(4k_F, \frac{1}{2}, 0)$). This potential induces small additional gaps at $k_y = \pm \pi/2b$ on the Fermi surface. The number of Fermi sheets at the Fermi level doubles below 24 K but the density of states is left practically unaffected by the anion ordering (Grant 1983). However, the anion ordering modifies the geometry of the Fermi surface in the vicinity of $k_y = \pm \pi/2b$. Grant (1983) has suggested that the $(4k_F, \frac{1}{2}, 0)$ anion ordering stabilises the conducting state of $(TMTSF)_2ClO_4$ against the SDW formation, leaving open the path to superconductivity. This argument (in particular the ordering of the anions along the stacking direction) is likely to be valid for compounds comprising non centrosymmetric anions, such as $ClO_4$. However, we wish to point out that superlattice formation may not be a pre-requisite for the existence of superconductivity in compounds with centrosymmetric anions as recent structural data obtained on $(TMTSF)_2PF_6$ under high pressure at 1.7 K did not reveal any superlattice formation due to $PF_6$ ordering (Gallois *et al* 1986).

The purpose of this paper is to present an experimental study of the Hall effect and magnetoresistance of $(TMTSF)_2PF_6$ when the conducting state can be stabilised down to low temperature (at $P > 9$ kbar) under high-field conditions similar to those achieved in previous $(TMTSF)_2ClO_4$ works.

## 2. Experimental details

We have used, in this study, four different electrochemically grown $(TMTSF)_2PF_6$ single crystals of typical dimensions $2.5 \times 0.3 \times 0.12 \text{ mm}^3$. The resistance and Hall voltage (along the *b*\* direction) were measured with a six-probe arrangement of gold-evaporated contacts. All data were obtained using an AC lock-in technique with a current up to 300 µA. The Hall voltage was derived by combining measurements for opposite directions of the magnetic field using always a similar range and speed for the field sweep. The magnetic field was aligned along the *c*\* direction within $10^\circ$. The pressure was applied to the sample in a Be–Cu vessel via an isopentane pressure medium. Several experiments have been performed at various pressures above 9 kbar, but the data reported in this paper deal essentially with the pressure of 10.5 kbar. Data at 1.18 K or at higher temperatures have been obtained up to 9 T while for experiments performed in the dilution refrigerator the field was limited to 7.5 T.

At ambient temperature, up to 10.5 kbar, the conductivity increase reaches a factor four. The resistivity ratio $R(B \to 0, T = 0.15)/R(0, 300) = 480$.
### 3. Results

Figure 1 displays the Hall voltage field dependence at 1.18 K under 10.5 kbar. In the low-field regime the Hall voltage is small (in the nanovolt range for a current of 300 µA) with a quasi-linear field dependence up to 5 T. A change of sign is observed at 6.6 T, followed by a very abrupt increase of two orders of magnitude. This behaviour is very similar to what has been observed in $(TMTSF)_2ClO_4$ in the same temperature domain (Ribault *et al* 1983). The value of the low-field Hall coefficient derived from figure 1 and the sample dimensions amounts to $R_H = 1.3 \times 10^{-9} \text{ m}^3 \text{ A}^{-1} \text{ s}^{-1}$ at 1.18 K. Similar measurements performed at higher temperatures show a slight increase of $R_H$ between 1.18 and 4.2 K ($R_H(4.2 \text{ K}) = 1.9 \times 10^{-9} \text{ m}^3 \text{ A}^{-1} \text{ s}^{-1}$). There is only a weak dependence between 4.2 and 77 K ($R_H(77 \text{ K}) = 2.4 \times 10^{-9} \text{ m}^3 \text{ A}^{-1} \text{ s}^{-1}$) which could be due to the uncertainties in measurements.

<figure><img src="image_2.png" /><figcaption>Figure 1. The Hall voltage of $(TMTSF)_2PF_6$ at 1.18 K under a pressure of 10.5 kbar. The inset shows the low-magnetic-field part of this variation. The measuring current is 300 µA.</figcaption></figure>

The derivative of the transverse magnetoresistance with respect to the magnetic field reveals a first peak at the same field where a change of sign of the Hall voltage is observed as well as more anomalies at higher fields.

Going to low temperatures ($T \approx 0.1$ K) the Hall voltage behaves quite differently. Above a threshold field, which is located around 3–3.5 T at 0.15 K, the magnitude of the Hall voltage rises sharply and the Hall voltage becomes oscillatory in a quasi-symmetrical manner around the zero value; see figure 2. The amplitude of the oscillations
<figure><img src="image_3.png" /><figcaption>**Figure 2.** The Hall voltage of (TMTSF)₂PF₆ at 0.15 K under a pressure of 10.5 kbar. The measuring current is 100 µA. Upper inset: the temperature dependence of the biggest anomalies. Lower inset: a plot of 1/B versus n at 0.15 K; n is arbitrarily chosen equal to one for the highest value of B.</figcaption></figure>

is increasing rapidly with magnetic field. At 7 T, for example (figure 2), the modulus of the Hall coefficient amounts to $750 \times 10^{-9} \text{ m}^3 \text{ A}^{-1} \text{ s}^{-1}$, i.e. about 580 times larger than the low-field Hall coefficient measured at 1.18 K. Data quite similar to those presented in figure 2 have been carefully verified on another sample.

A field hysteresis is observed for the steep rise of the Hall coefficient at the highest field of our study. At $T = 0.39$ K the field hysteresis at 7.1 T is about 0.1 T.

At increasing temperatures we have noticed a rapid increase of the threshold field but the positions of the oscillations shift only weakly in magnetic field. However, the amplitude of the oscillations is dependent on temperature in the interval 0.15-0.6 K with a sluggish saturation observed below about 0.3 K; see the inset in figure 2. Finally, we have determined that at 6 T and 0.15 K the carriers are electrons, a result that implies holes at low fields, in agreement with results for (TMTSF)₂ClO₄ and the band-structure calculation (Grant 1983).

In figure 4 we have drawn a phase diagram for the sdw phases of (TMTSF)₂PF₆ at 10.5 kbar, derived from the temperature dependence of Hall voltage extrema and magnetoresistance anomalies.

In the temperature range near 2 K, the threshold field increases as a function of pressure by ≈10% kbar⁻¹. This variation is slightly smaller than that observed in (TMTSF)₂ClO₄ (F Creuzet *et al* 1985, G Creuzet *et al* 1985). This weak difference can be understood in terms of the decrease of the sample compressibility at high pressure as observed by Gallois *et al* (1985).
<figure><img src="image_4.png" /><figcaption>**Figure 3.** The Hall voltage $V_H$ of $(TMTSF)_2ClO_4$ versus magnetic field at 0.1 K $B||c^*$. Cooling time: 70 h from 30 to 4.2 K. (See Ribault 1985.)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 4.** The phase diagram of $(TMTSF)_2PF_6$ at 10.5 kbar with the magnetic field aligned along $c^*$. The shaded region around 7 T shows the hysteresis of the last observed oscillation. Curve (—) represents the threshold field between the quasi-one-dimensional SDW state. ●, the phase transition detected by Hall voltage; □, the phase transition detected by magnetoresistance.</figcaption></figure>
## 4. Discussion

The first reaction to the Hall effect data in (TMTSF)₂PF₆ described in the previous section is to try and compare these results with similar Hall coefficient measurements performed for (TMTSF)₂ClO₄ (Ribault et al 1983). There are some features common to both systems, which we shall discuss first.

The field and temperature dependences of the Hall coefficient are clearly in favour of a field-induced transition from an open-Fermi-surface 1D conductor at low fields to a semimetal above a temperature-dependent threshold field. This high-field state is very probably magnetic (SDW) according to the ¹H NMR measurements (Azevedo et al 1981). In the 1D conductor regime the value of the Hall coefficient for the (4.2–77 K) temperature domain is indeed very close to the value 3.4 × 10⁻⁹ m³ A⁻¹ s⁻¹ for a 1D tight-binding model (Cooper et al 1976) especially considering the uncertainties in sample dimensions. We cannot at present understand the slight temperature dependence of the low-field Hall coefficient that is observed between 4.2 and 1.18 K. One possible explanation is that the resistivity anisotropy becomes strongly dependent on temperature in the vicinity of the superconducting state because of superconductivity precursor phenomena, and thus can alter the current flows in the crystal. A similar feature has been noticed while studying the low-field Hall coefficient of (TMTSF)₂ClO₄ but for T > 4.2 K.

Next, looking at the high-field features of the Hall coefficient we notice some similarities with as well as striking differences from the experimental situation in (TMTSF)₂ClO₄. The data for (TMTSF)₂PF₆ displayed in figure 2, and especially the existence of a threshold field, suggest that the oscillations of the Hall voltage are not related to ordinary quantum oscillations coming from Landau levels crossing a field-independent Fermi energy. In our measurements the Hall voltage has either strong hole- or electron-type variation. Moreover the amplitude of the Hall constant in each magnetic field range is of the same order of magnitude as in very slowly cooled (TMTSF)₂ClO₄ (Ribault 1985). Thus, we are justified in ascribing the Hall oscillations in (TMTSF)₂PF₆ to transitions between various SDW sub-phases as for (TMTSF)₂ClO₄.

The behaviour of (TMTSF)₂ClO₄ is rather well understood, with a picture that relates the Hall voltage steps to the existence of a sequence of transitions between various SDW sub-phases (Héritier et al 1984, Yamaji 1985b, 1986). For each sub-phase the Fermi level stays between the highest full and the lowest empty Landau levels as the magnetic field varies. This situation optimises the gain in diamagnetic energy and therefore stabilises a given SDW sub-phase. A concomitant condition for the stability of the sub-phase labelled by the integer *n* is $S(q_x)/S_0 = n$. $S(q_x) = 2\pi q_x/b$ is the area in momentum space that is enclosed between the original (quasi 1D) Fermi surface and the same surface shifted by the nesting vector $Q = (2k_F + q_x, \pi/b + q_y, \pi/c)$ and $S_0 = 2\pi eH/c$ is the quantum of phase-space area in the presence of a field *H* (Héritier et al 1984). The previous stability condition implies that $q_x$ is a linear function of the magnetic field in each sub-phase characterised by the integer *n*. The quantisation of the Hall resistivity can be easily derived since

$$ \rho_H \approx (S(q_x))^{-1} \approx h/e^2n. $$

The major difference between ClO₄ and PF₆ salts is in the question of the anion ordering, somewhat irrelevant for (TMTSF)₂PF₆ but responsible for the opening of a small gap of amplitude Δ in the planes $k_y = \pm \pi/2b$ in (TMTSF)₂ClO₄ for the so-called 'relaxed' state at helium temperature.
For $(TMSF)_2PF_6$ a nesting vector close to the commensurate transverse nesting vector $Q_0^T = (2k_F, \pi/b, \pi/c)$ will open a gap over almost the entire Fermi surface, leaving only pockets of holes and electrons with extremely small cross sections ($\ll 1\%$ of the first Brillouin zone area; Kwak 1983). However, the effect of the anion ordering in $(TMSF)_2ClO_4$ is precisely to shift the electron states near $k_y = \pm \pi/2b$ by the small energy $\Delta$ (of the order of the Fermi energy for electrons or holes in the sdw state). This new symmetry changes the fs curvature and thereby strongly modifies the nesting properties, related to the vector $Q_0^T$. However, a situation keeping electrons and hole-pockets seems to be recovered even for $(TMSF)_2ClO_4$, with a concomitant alternation of the sign of the Hall voltage versus magnetic field, provided a perfect long-range order of the $ClO_4$ anions is achieved under conditions of extremely slow cooling in 'hyper-relaxed' samples (Ribault 1985); see figure 3. Thus $(TMSF)_2PF_6$ resembles in some respects 'hyper-relaxed' $(TMSF)_2ClO_4$. However, unlike in $(TMSF)_2ClO_4$, no well defined Hall plateaus have been observed in $(TMSF)_2PF_6$. Maybe the failure to observe plateaus can be attributed to the large density of phase transitions in $(TMSF)_2PF_6$ as compared with $(TMSF)_2ClO_4$ in the field range $B < 7$ T. Such an effect is noticeable comparing relaxed and 'hyper-relaxed' $(TMSF)_2ClO_4$.

In the model proposed by Héritier et al (1984) for the field-induced transitions, the values of the magnetic field corresponding to phase transitions should show an approximate periodicity versus $1/B$. The frequency of the transitions (in units of magnetic field) gives an estimate of the Fermi cross sections that are involved in the sdw semimetallic state. For $(TMSF)_2ClO_4$, in the relaxed state, the transitions detected by Hall effect measurements give a frequency of 28.5 T. However, it is remarkable that either a plot of the four negative extrema of the Hall voltage (the inset in figure 2) or the four positive extrema (extrema that, crudely speaking, correspond to the phase transitions) leads to a fairly good linearity of $1/B$ versus $n$; see the inset in figure 2, with a frequency of 21.7 T. This value gives an order of magnitude for the Fermi surface cross section that is not very different from the $(TMSF)_2ClO_4$ data. The single frequency ($\approx 21.7$ T) derived for both hole- and electron-like Hall voltage oscillations is highly suggestive of an electron and hole situation for the Fermi surface of the sdw states in $(TMSF)_2PF_6$. The nesting vectors that are most likely to be at the origin of such a situation with small Fermi surface cross sections are those located in the immediate vicinity of the transverse vector $Q_0^T$. We have seen earlier that instabilities related to the transverse vector $Q_0^T$ are affected by a doubling of the transverse lattice distance as it occurs in $(TMSF)_2ClO_4$ below 24 K. In the latter system, regions of the Fermi surface near to $k_y = \pm \pi/2b$ are modified and so the Q-vector does not have the same alternative possibilities. This results in a situation where the transitions occur in only one type of FS: the electron-like FS (Héritier et al 1984). The data already known for $(TMSF)_2PF_6$ under pressure (Kwak et al 1981) have led in the high-field regime to a frequency of 76 T at $T = 1.1$ K with a threshold field located at around 6.5 T. Such a cross sectional area is about three times the value obtained at lower fields in the present study of $(TMSF)_2PF_6$. Both sets of data may be reconciled if, as suggested recently by Héritier et al (1985), a change of regime occurs in magnetic field between a regime of transverse nesting with Q close to $Q_0^T = (2k_F, \pi/b, \pi/c)$ and small cross sections at moderate fields and a regime of longitudinal nesting where Q is in the vicinity of $Q_0^L = (2k_F, 0, 0)$ (see also Gork'ov and Lebed 1984, 1985). The T-L transition is associated with an increase in the number of filled Landau levels and Fermi surface cross sections. The strong hysteretic behaviour reported at 7 T in the present study could be taken as the signature for the first-order T-L transition.
## 5. Conclusion

In summary, we have presented the first measurements of the Hall effect in $(TMTSF)_2PF_6$ under pressure below 1 K. The oscillatory field dependence of the Hall constant that is observed above a threshold field suggests, as for $(TMTSF)_2ClO_4$, a destabilisation of the open Fermi surface state against a sequence of sdw phases with 2D-like Fermi surfaces due to orbital effects of the magnetic field on the electron motion. The possibility of field-induced sdw states seems therefore to be a general trend in the $(TMTSF)_2X$ family, independently of the anion symmetry. However, whenever anion ordering arises, as in $(TMTSF)_2ClO_4$ below 24 K, it may modify the curvature of the Fermi surface over regions especially appropriate for good nesting. This feature is probably the reason for the different behaviours of the Hall coefficient in the high-field states of $(TMTSF)_2PF_6$ and $(TMTSF)_2ClO_4$. In the former compound hole and electron sdw phases seem to alternate in magnetic field. The comparison between our data (up to 7 T) and those of Kwak *et al* (1981, 1985) obtained at fields higher than 7 T is a fairly good sign in favour of a first-order transverse-to-longitudinal phase transition at 7 T.

Finally, we wish to emphasise that models such as those proposed by Gork'ov and Lebed (1984), Héritier *et al* (1984) and Yamaji (1985, 1986) provide an overall satisfactory framework for the interpretation of the high-field properties of $(TMTSF)_2X$. They must be modified, however, to explain the electron-hole symmetry that our data have shown to prevail in the supposedly simpler Fermi surface of $(TMTSF)_2PF_6$. This seems to be possible following improvements of the theory proposed recently by Montambaux (1985) and Montambaux *et al* (1985). There is also a need for a better theoretical treatment of the transport properties at high fields.

## Acknowledgments

We would like to acknowledge G Montambaux, M Héritier and P Lederer for useful discussions on the theory, J C Ameline for his help with the high-pressure experiments and C Lenoir for the samples' preparation.

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