# Spin density wave state of (TMTSF)2PF6: pressure and magnetic-field effects

N. Biskup, S. Tomic, Denis Jerome

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N. Biskup, S. Tomic, Denis Jerome. Spin density wave state of (TMTSF)2PF6: pressure and magnetic-field effects. Physical Review B: Condensed Matter (1978-1997), 1995. hal-04988905

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# Spin-density-wave state of tetramethyltetraselenafulvalinium phosphate (TMTSF)₂PF₆: Pressure and magnetic-field effects

Neven Biškup

Laboratoire de Physique des Solides, Université de Paris-Sud, 91405 Orsay, France
and Institute of Physics of the University, P.O. Box 304, 41001 Zagreb, Croatia

Silvia Tomić

Institute of Physics of the University, P.O. Box 304, 41001 Zagreb, Croatia

Denis Jérôme

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

(Received 3 January 1995)

We have used electrical transport measurements to investigate the pressure-temperature phase diagram in the spin-density-wave phase of the organic conductor tetramethyltetraselenafulvalinium phosphate. The transition temperature and the activation energy both decrease with increasing pressure and increase in magnetic field. This behavior results from a change of the degree of nesting and is well described within the imperfect nesting theory.

Low-dimensional organic conductors have attracted significant attention from the first synthesis in the 1970s up to the present. The most investigated materials are members of the Bechgaard family (TMTSF)₂X, where the TMTSF indicates organic molecule tetramethyltetraselenafulvalinium and X represents an inorganic anion: X=PF₆, AsF₆, ClO₄, NO₃, etc.¹,² The large variety of the observed physical phenomena such as the antiferromagnetic spin-density wave (SDW) or superconducting ground state, nonlinear conductivity, field-induced SDW (FISDW), Shubnikov-de Haas and Lebed oscillations keep these materials of interest at present. In particular, there is a large theoretical interest in the above-mentioned phenomena, also in light of the possible connections with high-temperature superconductors.

(TMTSF)₂PF₆ is the prototypical member of this family. It exhibits the SDW transition at 12 K and becomes semiconducting due to the almost perfect nesting of the Fermi surface. The resistance displays activated behavior with the activation energy close to 20 K, which is in good agreement with the BCS-theory prediction. In the SDW state this material shows a large magnetoresistance when the magnetic field is parallel to the least-conducting (c*) crystal axis. An applied pressure decreases SDW transition temperature, and finally, at pressures higher than 8 kbar, the PF₆ becomes a superconductor below a temperature of about 1 K. Once in the superconducting state, a finite magnetic field destroys the superconductivity and large magnetoresistance appears again. At moderate fields, jumps in magnetoresistance indicate concomitant transitions into FISDW states due to the improved nesting. Finally, at high enough fields, the system comes to a

final FISDW state similar to the normal SDW.

The standard mean-field theory for a quasi-one-dimensional (quasi-1D) system with an open Fermi surface provides a description of the SDW state.³ A perfect nesting of the Fermi surface stabilizes the SDW phase. Deviations from the perfect nesting lead to a gradual destruction of the SDW phase. The important point is that the suppression of the SDW phase can be described by the unique parameter which is the characteristic energy of the deviation from perfect nesting. The SDW ground state is completely suppressed once this energy becomes equal to the SDW order parameter in the perfect nesting case. The degree of nesting can be experimentally controlled by the application of hydrostatic pressure and/or magnetic field. The former increases the deviation from perfect nesting, while the latter makes it better. Until now, a complete experimental verification on a quantitative level of the theoretically predicted phase diagram has not yet been performed. Kwak et al.⁴ have performed experiments in magnetic field, but only close to the critical pressure (at 0.87p<sub>C</sub> and 0.99p<sub>C</sub>) to test a possible link between the FISDW states and the ambient-pressure zero-field SDW state. Recently, Nagasawa et al.⁵ have investigated the influence of pressure on the SDW transition temperature and the activation energy. They analyze their results in light of Yamaji's prediction of the existence of the semimetallic SDW state which should appear above the metal-to-insulator SDW phase transition.

We have performed our experiments with the aim of verifying the validity of the mean-field description of the ambient-pressure SDW phase of (TMTSF)₂PF₆. We have used pressure and magnetic field to be able to vary a de-
gree of nesting in such a manner as to cover the whole phase diagram.

Single crystals of $(TMTSF)_2PF_6$ were grown by the standard electrochemical method. Six different samples were measured at different pressures form 1 bar to 8.2 kbar. Typical cross sections were $2 \times 10^{-4} \text{ cm}^2$ and lengths 2–3 mm. The electrical leads were annealed gold wires of 17 mm diameter. They were attached with a silver paste to the golden pads evaporated on the sample. Attention has been paid to the orientation of the samples so that the magnetic field was parallel to the $c^*$ axis inside 5°. Contact resistances were smaller than 10 Ω for all measured samples. Samples were mounted inside a beryllium clamp with silicone oil as a pressure medium. The pressure was measured at room temperature (RT) and is assumed to be the same down to liquid-helium temperature. During the cooling no typical ambient pressure cracks were detected on the samples. The resistance was measured with the standard ac "lock-in" method. Resistance ratios (ratio between RT and minimum resistance just above $T_C$) were quite small and typically of the order of 10, depending on samples and pressures.

The pressure-temperature ($p-T$) phase diagram is shown in Fig. 1. The SDW transition temperature $T_C$ decreases with increasing pressure, as already found by other groups.⁶,⁷ This decrease is rather small up to 4–5 kbar and becomes quite sharp above 7.5 kbar. The critical pressure is found to be 8 kbar, above which the superconducting transition at 1 K is observed. We derive $T_C$ from the peak in $d(\ln R)/d(1/T)$ plots (see the inset of Fig. 1). The peaks are sharp enough at all pressures up to 7.5 kbar, where the broadness of peak increased several times.

The SDW order parameter is the quasiparticle energy gap opened at the Fermi surface. In the electrical trans-

port measurements we consider the activation energy as the best measure of the SDW gap. Therefore, throughout the paper we assume that the activation energy equals the SDW gap and label both of them with the same symbol $\Delta$. Figure 2 shows pressure dependence of the activation energy $\Delta$ without and with a magnetic field of 12 T. We derive $\Delta$ from Arrhenius plots (see the inset) and the obtained values correspond well to the ones we can get from $d(\ln R)/d(1/T)$ plots (see the inset of Fig. 1). We note that, at all pressures, the values of $\Delta$ are well defined in a sufficiently broad temperature range. This behavior is not affected by the external magnetic field. Furthermore, in zero magnetic field, the decrease of $\Delta$ is almost linear up to 6 kbar. Above 6 kbar, the decrease becomes faster and $\Delta$ falls to zero at $p_C$. We also note that the BCS-like ratio at 1 bar between the activation energy and transition temperature also diminishes under pressure. On the other hand, the application of a 12 T magnetic field preserves the ambient-pressure value of $\Delta$ up to nearly 4 kbar, above which it starts to decrease slowly. We note that the activation energy in $H=12$ T normalized to its zero-field value increases with pressure almost linearly up to 6 kbar, above which it starts to increase more rapidly. As $\Delta$ (0 T) vanishes at critical pressure, we expect that the ratio $\Delta(12\text{T})/\Delta(0\text{T})$ disappears at $p_C$. This behavior indicates that although ambient pressure SDW is suppressed at $p_C$, there is a field-induced SDW phase which appears at pressures $p > p_C$.

We have also measured the change of $T_C$ in magnetic field at various pressures [Fig. 3(a)]. The highest pressure presented is 6 kbar. At 7.5 kbar the peak in the $d(\ln R)/d(1/T)$ plot is too broad to detect small changes of a few percent. The influence of magnetic field becomes important above 6 T. We have plotted the increase of $T_C$ in 12 T normalized to its zero-field value as a function of pressure in Fig. 3(b). The increase is approximately

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

FIG. 1. SDW transition temperature ($T_C$) as a function of pressure ($p$). The full line is a guide to the eye. The arrow indicates critical pressure $p_c=8$ kbar. Inset: derivative $d(\ln R)/d(1/T)$ vs $1/T$: open circles, full circles, and triangles for $p=1$ bar, 2 kbar, and 6 kbar, respectively.

<figure><img src="image_3.png" /></figure>

FIG. 2. Activation energy ($\Delta$) as a function of pressure at $H=0$ T (circles) and $H=12$ T (diamonds). Full lines are guides to the eye. Inset: $\log R$ vs $1/T$ for $p=2$ kbar (circles) and $p=6$ kbar (triangles). Open signs are for $H=0$ and full signs for $H=12$ T.
linear with a slope of 0.54 kbar⁻¹.

In what follows, we discuss our experimental results in the framework of the mean-field theory for the spin-density wave state of a quasi-1D system with an open Fermi surface, developed independently by Huang and Maki⁸ and Montambaux.⁹ First, we address the question of the influence of an external pressure on the SDW order parameter Δ and the transition temperature $T_C$. The deviation from perfect nesting is described by a parameter $ε_0$ given by $ε_0=t_b^2/2t_a$, where $t_b$ and $t_a$ are tight-binding transfer integrals for the b and a axes, respectively. For the perfect nesting case, i.e., $ε_0=0$, $t_b$ is negligibly small compared to $t_a$. The transition temperature and the order parameter at $T=0$ K are denoted as $T_{C0}$ and $Δ_0$, respectively. Under pressure, both $t_b$ and $t_a$ increase, and $ε_0$ increases also. In other words, pressure increases dimensionality of the system and warps the Fermi surface in the b direction. When $ε_0$ becomes equal to $Δ_0$ (i.e., $ε_0/Δ_0=1$), the nesting of the Fermi surface is so bad that the SDW state is suppressed. This theoretically predicted behavior is shown as a full line in Fig. 4. For the imper-

fect nesting case and in the limit when magnetic field goes to zero, Huang and Maki derive the following expression for the SDW gap:¹⁰

$$Δ^2 = Δ_0^2 - ε_0^2. \qquad (1)$$

Here we recall that the SDW gap for the imperfect nesting case is not uniquely defined and varies along the Fermi surface. Then one would expect that the activation energy is simply given by the minimum value of the gap.³ However, according to Eq. (1), the SDW gap is better defined as the geometric average of the extreme values. We have used the latter expression in order to analyze our data for the activation energy. We start with the usual assumption that the ambient-pressure SDW phase in the PF₆ compound corresponds to the perfect nesting case. At 1 bar, our measurements show a SDW transition temperature of 12.5 K and an activation energy of 19.4 K. We put values of Δ from our experiments [Figs. 1 and 2(a)] into Eq. (1) to calculate $ε_0$ at each pressure. We take also experimentally obtained values of $T_C$ (normalized to $T_{C0}$) and plot them as a function of $ε_0$, normalized to $Δ_0$ ($β=ε_0/Δ_0$ shown as points in Fig. 4). The overall agreement with the theoretical prediction is quite good. However, there is a slight discrepancy at low pressures (1-4 kbar), while points at high pressures fit well the theoretical curve. This behavior indicates that $T_C$ might not be so pressure sensitive, at least up to 0.5$p_C$, as predicted by theory.

Second, we discuss the variation of the SDW parameters in a magnetic field. For the perfect nesting case, the applied magnetic field does not change the SDW transition temperature, i.e., $T_{C0}(H)=T_{C0}(H=0)$. On the other hand, for the imperfect nesting, i.e., $ε_0>0$, $T_C$ increases in magnetic field like⁹

$$T_C(H)=T_C(H=0)+f(β)\frac{ω_C^2}{t_b^{t^*}}, \qquad (2)$$

where $ω_C=vebH/2$ is the cyclotron frequency and $v, e, b$, and $H$ are Fermi velocity, electron charge, lattice param-

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

<figure><img src="image_5.png" /><figcaption>FIG. 3. (a) The increase of $T_C$ in magnetic field (H) normalized to the value at H=0 T, $T_C(H)/T_C(H=0)$ at five different pressures. (b) The relative increase of $T_C$ at H=12 T [$T_C(12 T)/T_C(0 T)$]-1 as a function of pressure. The full line is a linear fit. Inset: function $f(β)$ vs degree of nesting (β) from Ref. 9. Open circles are values from the fit to Eq. (3) (see text).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 4. $T_C/T_{C0}$ vs $β$ ($β=ε_0/Δ_0$) assuming that $β=0$ at 1 bar. $ε_0$ is calculated from Eq. (1) (see text). The full line is the theoretical curve from Ref. 8. Inset: $T_C/T_{C0}$ vs $β$ using $Δ_0=20.6$ K and $T_{C0}=13$ K (see text).</figcaption></figure>
<table><caption>TABLE I. Fitting parameters from Eq. (3) (see text) at five different pressures. $f(\beta)$ is calculated using $\Delta_0=19.4$ K.</caption><thead><tr><th>p (kbar)</th><th>f(β)/Δ₀ (10<sup>-3</sup> K<sup>-1</sup>)</th><th>f(β)</th><th>β</th></tr></thead><tbody><tr><td>1</td><td>1.28</td><td>0.025</td><td>0.455</td></tr><tr><td>2</td><td>1.68</td><td>0.033</td><td>0.510</td></tr><tr><td>3</td><td>1.84</td><td>0.036</td><td>0.530</td></tr><tr><td>4</td><td>2.55</td><td>0.049</td><td>0.605</td></tr><tr><td>6</td><td>2.86</td><td>0.055</td><td>0.630</td></tr></tbody></table>

eter along the b axis, and magnetic field, respectively. $f(b)$ is the universal function of the degree of nesting $\beta=t'_b/t_b^* \approx \epsilon_0/\Delta_0$ [inset of Fig. 3(b)]. We rewrite Eq. (2) as follows:

$$T_C(H)=T_C(H=0)+\text{const}\times f(\beta)\frac{\omega_C^2}{\Delta_0}, \quad (3)$$

where $\text{const}=\Delta_0/t_b^*=(\Delta_0/T_{C0})1.12=1.96$ in the mean-field approximation. We have fitted our data to Eq. (3) taking $v=1.5 \times 10^5$ m/s and $b=7.7 \times 10^{-10}$ m. Values of $f(\beta)/\Delta_0$ obtained from the fit are listed in Table I. As before, we assume that $\Delta_0$ corresponds to the activation energy at ambient pressure; i.e., we take $\Delta_0=19.4$ K. Obtained values for $f(\beta)$ are also listed in Table I and are shown as points in the inset of Fig. 3(b). Therefore, the degree of nesting changes from 0.45 to 0.63 in the pressure range between 1 and 6 kbar, i.e., between 0.125$P_C$ and 0.75$P_C$. Note that this is in very good agreement with $\beta$ values obtained from Eq. 1 (Fig. 4).

In the imperfect nesting case, Maki also predicts the increase of $\Delta$ in magnetic field which, in the first approximation, is given by$^{10,11}$

$$\frac{\Delta(H)-\Delta(H=0)}{\Delta(H=0)} = \frac{1}{3} \frac{\omega_b^2}{\Delta_0^2} \left[ \frac{\epsilon_0}{\Delta_0} \right]^2, \quad (4)$$

where $\omega_b=2\omega_C$. The best fit to our experimental data taken from Figs. 2 and 4 is obtained for $v=3.2 \times 10^5$ m/s (Fig. 5).

Finally, we address the question as to whether the ambient-pressure SDW phase in the $(TMTSF)_2PF_6$ system corresponds to the perfect nesting case. The extrapolation of linear fit to zero pressure [Fig. 3(b)] shows a small but finite increase of $T_C$ of the order of 1% in magnetic field of 12 T. This finding is in agreement with independent measurements by Auban-Senzier.$^{12}$ Eq. (2) then gives $f(\beta)/t_b^*=8.13 \times 10^{-4}$. Using Eq. (2), the theoretical curve in Fig. 4 and Eq. (1), we get, in two iteration steps, $\beta=\epsilon_0/\Delta_0=0.34$ and $\Delta_0=20.6$ K. There-

<figure><img src="image_7.png" /><figcaption>FIG. 5. Increase of $\Delta$ at $H=12$ T normalized to value at $H=0$ T [$\Delta(12\text{T})-\Delta(0\text{T})]/\Delta(0\text{T})$ as a function of $\beta$ (open circles). Full line is a fit to Eq. (4).</figcaption></figure>

fore, we find that the ambient-pressure SDW phase is not perfectly nested. This finding is also in agreement with the prediction of Montambaux's theory. Taking into account that $\Delta_0=20.6$ K, we get the corrected phase diagram shown in the inset of Fig. 4. Note that the transition temperature for the perfectly nested SDW should then become 13 K.

In conclusion, we have verified the general pressure-temperature SDW phase diagram, originally predicted by Yamaji. The experimentally observed behavior of the SDW transition temperature and the order parameter as a function of pressure and/or magnetic field is well described by imperfect nesting theories, assuming that the activation energy is a good measure of the SDW gap. Moreover, our results reveal that the Fermi surface nesting in $(TMTSF)_2PF_6$ deviates from perfect nesting conditions already at ambient pressure. This finding is in good agreement with our present understanding of the general $(TMTTF)_2X-(TMTSF)_2X$ phase diagram as the SDW phase transition arises (at ambient pressure) at a temperature which is already lowered as compared to the maximum SDW transition temperature [$\approx 16$ K for $(TMTTF)_2Br$ at 5 kbar] of the $(TMTTF)_2X-(TMTSF)_2X$ diagram.$^{13}$

We acknowledge fruitful discussions with G. Montambaux and K. Maki. We thank L. Balicas for constant collaboration and P. Auban-Senzier for technical assistance.

<footnote n="1">¹K. Yamaji, in *Organic Superconductors*, edited by T. Ishiguro and K. Yamaji (Springer-Verlag, Berlin, 1990).</footnote>

<footnote n="2">²D. Jérome, in *Organic Conductors*, edited by J. P. Farges (Dekker, New York, 1994).</footnote>

<footnote n="3">³K. Yamaji, J. Phys. Soc. Jpn. **51**, 2787 (1982).</footnote>

<footnote n="4">⁴J. F. Kwak, J. E. Schriber, P. M. Chaikin, J. M. Williams, H.-H. Wang, and L. Y. Chiang, Phys. Rev. Lett. **56**, 972 (1986).</footnote>

<footnote n="5">⁵M. Nagasawa, T. Sambongi, K. Nomura, and H. Anzai, J. Phys. (Paris) Colloq. **3**, C2-49 (1993).</footnote>

<footnote n="6">⁶D. Jérome and H. Schulz, Adv. Phys. **31**, 299 (1982).</footnote>

<footnote n="7">⁷T. Takahashi, H. Kawamura, T. Ohyama, Y. Maniwa, K. Mu-</footnote>

<footnote>rata, and G. Saito, J. Phys. Soc. Jpn. **58**, 703 (1989).</footnote>

<footnote n="8">⁸X. Huang and K. Maki, Phys. Rev. B **46**, 162 (1992).</footnote>

<footnote n="9">⁹G. Montambaux, Phys. Rev. B **38**, 4788 (1988).</footnote>

<footnote n="10">¹⁰K. Maki, Phys. Rev. B **47**, 11506 (1993).</footnote>

<footnote n="11">¹¹M. Basletić, N. Biškup, B. Korin-Hamzić, S. Tomić, A. Hamzić, K. Bechgaard, and J. M. Fabre, Europhys. Lett. **22**, 279 (1993).</footnote>

<footnote n="12">¹²O. Traetteberg, Ph.D. thesis, Université Paris-Sud, 1993.</footnote>

<footnote n="13">¹³S. S. P. Parkin, F. Creuzet, D. Jérome, J. M. Fabre, and K. Bechgaard, J. Phys. (Paris) **44**, 975 (1983).</footnote>