# DIFFERENTIAL THERMAL ANALYSIS OF THE FIELD INDUCED PHASE TRANSITIONS OF (TMTSF)₂ClO₄ ABOVE 1.2 K

B. Piveteau, J.R. Cooper* and D. Jérôme

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

*(Received 22 August 1986; in revised form 5 December 1986 by J. Joffrin)*

We present some results of a differential thermal analysis of the magnetic field induced phase transitions in the organic conductor (TMTSF)₂ClO₄ above 1.2 K. This study shows that transitions between different spin density wave states are first order and that the total entropy change involved in the two detected transitions (in the temperature range 1.2-2 K) is close to that of the quasi-one-dimensional electron gas. Above 2 and 4.2 K, only a single transition has been detected in our measurements. The entropy of that transition decreases and extrapolates to zero near 5 K. We present some arguments suggesting that if longitudinal nesting (2k<sub>F</sub>, 0, 0) is to take place in the semi-metallic SDW phase at high fields it exists only above 2 K or so

## INTRODUCTION

IN THE SERIES OF organic conductors (TMTSF)₂X [1], the X = ClO₄ salt has been extensively studied because of its fascinating properties at atmospheric pressure. Its superconducting ground state is destroyed by the application of a magnetic field of about 25 mT, at 0.9 K [2], along the c*-direction. At higher fields, magnetoresistance [3], NMR [4], Hall effect [5, 6], specific heat [7, 8], and magnetisation [9] studies have proved that, after a threshold field, a spin density wave (SDW) ground state is stabilised. Even more, the magnetic field induces a succession of phase transitions between different SDW states. These phase transitions have been interpreted by a quantized nesting vector argument [10]; the nesting vector is a function of temperature and magnetic field and adapts to the field in order to maintain completely filled Landau levels [10, 11]. So, Héritier et al. [10] have suggested, at a given temperature, as the field is increased or decreased, there occurs a second order transition from metal to semi-metallic phase, then a cascade of first order transitions between different SDW phases.

Recently, Naughton et al [9] have evaluated the entropy change at these transitions: using the Clausius Clapeyron equation for a magnetic system (ΔS = -ΔMdH/dT), they measured the change in magnetisation ΔM and deduced dH/dT from the measured slope of the phase diagram. They found the

metal to semi-metallic SDW phase transition (i.e. the first transition encountered on increasing H) to be second order at T < 3 K and first order for T > 3 K. Moreover, most of the electronic entropy is removed by the three or four lowest field transitions.

In this Communication, we report on direct measurements of entropy change from the field-induced transitions in (TMTSF)₂ClO₄ between 1.2 and 4.2 K. These measurements reveal that, up to 2 K, the total entropy loss is comparable to the entropy of the electron gas γT, where γ is the electronic specific heat coefficient. For T > 2 K the transition still has some first order character but the associated entropy change is smaller and extrapolates to zero around 5 K. Also in the present work no sign of a first order transition could be detected at 4.2 K in magnetic fields up to 11.8 T.

Similar measurements have been made by Garoche et al. [14] at temperatures below 1.2 K, and these also provide a direct measurement of the entropy changes associated with the first order phase transitions, between different SDW phases.

## EXPERIMENTAL

The central portion (2 mm) of a 6 mm long single crystal of (TMTSF)₂ClO₄ weighing 3.29 mg was attached to a miniature ruthenium oxide thermometer with Apiezon vacuum grease. Four 4 mm long 20 µm diameter manganin wires were indium soldered to the thermometer to provide electrical connections and to form a thermal link to a constant temperature heat

<footnote>* On leave from the Institute of Physics of the University, PO Box 304, Zagreb, Yugoslavia</footnote>
sink. A second identical thermometer with no sample attached was mounted nearby in the same way. The two thermometers form two arms of an a.c. Wheatstone bridge with lock-in direction of the balance point. The crystal was aligned by eye with $H \parallel c^*$ to within $10^{-6}$.

The output of the lock-in, i.e. the balance of the bridge, was recorded while the magnetic field was swept between 0 and 12 T at maximum rates of 0.13 T per minute (H increasing) and 0.4 T per minute (H decreasing). When heat was given out or absorbed by the sample the balance of the Wheatstone bridge was altered. By carefully measuring the thermal conductance of the link and the sensitivity of the $RuO_2$ thermometers between 4.2 and 1.2 K (they changed from 1540-1950 ohms) as well as the field sweep rate, the latent heats of the first order transitions could be determined.

In order to establish the relaxed state, the sample was cooled slowly, in about two hours from 40 to 4.2 K. This corresponds approximately to the standard slow cooling rate [9, 13], but not to the ultra-slow cooling rate [13] where different Hall effect behaviour has been observed at very low $T$.

Temperature relaxation of the sample/thermometer assembly towards the sink took place through the manganin wires with a time constant of approximately 8 sec. at 1.2 K and 35 sec at 3.7 K.

Finally we made one preliminary experiment on $(TMTSF)_2PF_6$ under pressure, using isopentan as the pressure medium. The measured thermal conductance of the sample to its surroundings was approximately $1.3 \times 10^{-6} \text{ W K}^{-1}$ at both 4.2 and 1.2 K, i.e. 7.24 times larger than the corresponding value for the experiments on $(TMTSF)_2ClO_4$. Thus, similar experiments on the $PF_6$ under pressure are in principle possible but they would require a factor of ten increase in either the field sweep rate or a corresponding decrease in the noise level of the thermometers, which has not yet been achieved.

## RESULTS

Some typical plots of the output of the Wheatstone bridge vs magnetic field are shown in Fig. 1. At low temperatures ($1.2 \text{ K} < T < 2 \text{ K}$) we observed two transitions for both increasing and decreasing magnetic fields, but for temperatures greater than 2 K, there is only one transition (Fig. 1(b)).

So we were able to partially confirm the phase diagram $(T, H)$ of $(TMTSF)_2ClO_4$. Figure 2 shows the different transitions we observed; the slight deviation from those observed by Naughton *et al* [19] may come from a different cooling rate, different purity

<figure><img src="image_1.png" /></figure>

Fig. 1. Typical field sweeps showing temperature changes arising from the release (field increasing) or absorption of latent heat (field decreasing). (a) at 1.2 K showing hysteresis in upper transition Field sweep rates: 0.12 T/minute (up) and 0.35 T/minute (down). The total area of the peaks is larger for H decreasing because of the faster field sweep. Note also that, for H decreasing, most of the heat is absorbed at the lowest transition (b) at 3.7 K where there is only one transition.

levels or better alignment of the sample. Nevertheless, Fig 1 shows a clear hysteresis on the highest field transition.

We have determined that, when the field is increasing, the thermometer gets warmer at every phase transition, i.e the sample loses entropy, which is in agreement with the fact there are less and less carriers as H is increasing [12].

For every transition, $\Delta S$ was obtained from the area of the peak On Fig 3, we have plotted $\Delta S$ vs $T$.

<figure><img src="image_2.png" /></figure>

Fig. 2. Transition fields obtained in this work compared with published phase diagram determined by magnetisation studies [9] $\nabla$, H decreasing; $\Delta$, H increasing, $\circ$, lower transition with negligible hysteresis.
<figure><img src="image_3.png" /></figure>

Fig. 3 Entropy of transitions compared with $γT$ the conduction electron entropy. ○ lower transition, △, ∇ upper (or single) transition, field increasing or decreasing respectively; □, total entropy in the case of two transitions. According to [7], $γ$ is taken equal to $1.05 ± 0.05 \times 10^5 \text{erg mol}^{-1} \text{K}^{-2}$. The dashed lines represent the uncertainty in the value of $S$ derived from the experimental accuracy of $γ$. The two highest points at 2.5 K ($H ≈ 9T$) belong to a different run. As the value of the entropy is markedly different from the rest of the data, this may be an indication that the first order character of the transition depends on the history of the sample. Similar history effects have been reported in [13].

along the two highest field transition lines of the phase diagram. For $T < 2$ K, we have only plotted data for $H$ increasing because, as shown in Fig. 1, when there is hysteresis the two transitions are not well separated for decreasing $H$. At higher temperature, as there is no more hysteresis, we have plotted $\Delta S$ for transitions at $H$ increasing and decreasing.

In Fig. 3, the total entropy loss is also shown vs $T$ and is compared to the electronic entropy $γT$ [7, 8].

DISCUSSION

The main result of this work is that some of the points made in previous studies of magnetisation [6, 15] have been confirmed directly without using the Clausius Clapeyron equation. Firstly, as shown in Fig. 3, below 2 K the total entropy involved in the two first order phase transitions is close to that of the electron gas, $γT$. At 1.2 K the lower transition plays a dominant role but as the temperature rises the upper transition takes over. Secondly, the first order nature of the high field (8–10 T) transition persists above 2 K although the hysteresis disappears, i.e. the latter arises because there is a SDW–SDW transition. Above 2 K the total entropy detected in our measurements decreases smoothly and extrapolates to zero near 5 K. In contrast to the magnetisation measurements [15] we were not able to detect any first order character (latent heat) above 3.7 K. The disappearance

of the first order character above 5 K may well be connected with the significant changes seen in high frequency de Haas–Schubnikov oscillations disappears and no sharp threshold field is detected.

The difficulty in interpreting the present results is that, unlike specific heat and magnetisation data, they are only affected by first order transitions where the entropy changes sharply within a fraction of a $T$. So, more of the latter measurements would be required to test the pictures proposed below.

It has been suggested that, at low temperature, i.e. below 1 K or about, the high field transition (8–10 T) marks a change from “transverse” or “good” nesting with small electron pockets, to “longitudinal” or “bad” nesting with large electron pockets [17]. We do not think that this can be the case below 2 K where the total entropy loss is of the order of $γT$. Namely the latter is determined by the area of the Fermi surface which will remain large for longitudinal nesting and thus the entropy of the semi-metallic state will not be substantially different from the value in the metallic state.

The main experimental evidence in favour of longitudinal nesting is, however, the observation of high frequency de Haas–Schubnikov oscillations in magnetoresistance by several groups above 2 K. Furthermore, recently, other mechanisms have been suggested for these oscillations, for example field dependent electron–phonon coupling [18], magnetic breakdown [19], or even that they are not a Fermi surface effect at all but arise from very low energy closed orbits [20]. If one of these explanations applies, then, there is no need to invoke longitudinal nesting and it is perfectly possible that the total energy loss remains approximately $γT$ up to at least 4.2 K, as proposed on the basis of magnetisation measurements [15], then the fall we see in $\Delta S$ reflects simply the diminishing first order character of the high field transition as the temperature increases.

However, if the concept of longitudinal nesting is retained in order to account for the high frequency oscillations, or for some other reason, then the entropy argument given above requires another line on the phase diagram in high fields somewhere between 2 and 5 K, marking the changeover from longitudinal to transverse nesting at lower temperatures. Such a longitudinal to transverse wave vector transition is conceivable within the framework of a recently elaborated theory for the onset of SDW at low temperature [21]. Depending on the value of renormalised dimensionality cross-over temperature $T_c$, the driving mechanism of the SDW may be the interchain exchange interaction existing between 1-D spin fluctuations (IEX-model) or the 3D nesting of the Fermi
surface (NAF-model). The Fermi surface of the SDW state should contain large (longitudinal Q-vector) and small (transverse Q-vector) pockets for IEX and NAF instabilities respectively. The existence of a IEX to NAF transition has been suggested for the SDW state of (TMTSF)₂PF₆ which arises below 12 K under ambient pressure [22]. Several experimental features, the vanishing of fast oscillations near 4 K [23] and the recent observation of a phase transition below 4 K by ¹H-n.m.r [22] do support the existence of a IEX-NAF transformation in the vicinity of 4 K. Furthermore, the observation of fast oscillations in the SDW state of (TMTSF)₂ClO₄ at high fields [16] suggests the existence of small Fermi surface pockets with large cross-sections perpendicular to c* (namely, the IEX-model). Therefore, the SDW of the NAF-kind at low fields could transform into the IEX-kind above 7 T, the relative stability of NAF or IEX states at low temperature depending on whether Tₛ is smaller or larger. Both Tₛ and Tₛ should be field dependent, Tₛ and Tₛ respectively increasing and decreasing under magnetic field. We may point out that the metal to SDW state (IEX-kind) transition decreasing T (at H > 7 T) would resemble very much the transition observed in (TMTSF)₂PF₆ at 12 K under ambient pressure. This is a second-order transition

The entropy argument given above must be taken with some caution since, as pointed out by Montambaux [25], there could be significant changes of entropy within each subphase. However, Maxwell's thermodynamic equation [26] states that:

$$ (\partial S / \partial H)_T = + (\partial M / \partial T)_H $$

So providing the magnetisation (M) of each subphase falls as T increases, which must surely be the case, in the field, the entropy must fall monotonically with H. Hence, our estimate from the first order transitions is a lower limit to the total entropy change between the low field metallic state and the high field SDW one.

## CONCLUSION

Using a simple differential thermal analysis technique we have verified that the entropy changes determined by magnetisation studies [6] on (TMTSF)₂ClO₄ are essentially correct and are indeed equal to the conduction electron entropy γT below 2 K. However we find less first order character at 4.2 K [15]. We have pointed out that a total entropy change of γT is very difficult to reconcile with the idea of a longitudinal nesting and large electron or hole pockets. Therefore, we suggest that either longitudinal nesting does not occur at all, or at most only in certain temperature range above 2 or 5 K. In the former case, some other explanation must be sought for the high

frequency Schubnikov-de Haas oscillations observed in magnetoresistance [16], and three possibilities have been proposed recently [18–20]. More complete magnetisation studies at closely spaced temperatures or specific heat measurements combined with differential thermal analysis could provide a definitive answer to these questions

We find that the single phase transition observed above 2 K still retains some first order character (latent heat) which gradually disappears and extrapolates to zero near 5 K. It might be possible to understand this in terms of the behaviour of the fourth term in the microscopic Ginzburg-Landau approach [12]. On the experimental side we should point out that we have not studied the influence of ultra-slow cooling rates [13] on the observed latent heats. It is conceivable that this could have an effect by marking the transitions even sharper.

Finally, the preliminary attempt to perform a similar differential thermal analysis of the high field states in (TMTSF)₂PF₆ under pressure has shown that the study is feasible after some improvement in the sensitivity of the thermometer

*Acknowledgements* — We are grateful to K. Bechgaard and A. Moradpour for the large single crystal of (TMTSF)₂ClO₄ used in this work. Helpful discussions with P. Chaikin, P. Garoche and G. Montambaux are also gratefully acknowledged.

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