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

# Scaling between Normal and Spin-Density-Wave Conductivity in $(TMTSF)_2AsF_6$.

G. KRIZA(*), G. QUIRION, O. TRAETTEBERG and D. JÉROME

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

(received 24 May 1991; accepted in final form 22 August 1991)

PACS. 72.15N – Collective modes; low-dimensional conductors (inc. synthetic metals).

PACS. 75.30F – Spin-density waves.

**Abstract.** – The nonlinear d.c. conduction in the spin-density-wave state of $(TMTSF)_2AsF_6$ (TMTSF is tetramethyltetraselenafulvalene) is investigated as a function of temperature (2 to 10 K) and magnetic field (12 T, parallel to the c*-axis). The total nonlinear conductivity follows the variation of the low-field linear conductivity both as a function of temperature and magnetic field in a wide though limited temperature range. The implications on the origin of spin-density-wave damping are discussed.

Recent electrical transport measurements [1] in the spin-density-wave (SDW) state of $(TMTSF)_2X$ (TMTSF is tetramethyltetraselenafulvalene, X is a monovalent anion) revealed a close analogy with sliding charge-density-wave (CDW) systems. All the main features of CDW conduction [2] have been shown to exist also in SDW conductors. Namely, a nonlinear d.c. conduction above a threshold field [3-7] accompanied by voltage oscillations [5, 7, 8], a pinned mode resonance [9] in the frequency-dependent conductivity, hysteresis [8] and weak relaxation [10] phenomena in the pinned region have been all demonstrated in various SDW systems of the $(TMTSF)_2X$ family.

In most cases the dominant source of the finite threshold field of both CDW and SDW sliding is the interaction of the density wave with crystalline defects. This has been extensively demonstrated for CDW's [2] and also found recently [11] in the SDW state of X-ray irradiated $(TMTSF)_2PF_6$ crystals. An immediate consequence of random pinning is that internal degrees of freedom of the density-wave play an important role in all dynamical phenomena except perhaps in the high-electric-field limit where pinning represents a small perturbation only. A common approach to describe a deformable incommensurate CDW tuned by random defects is to neglect the amplitude fluctuations of the order parameter and treat the fluctuations of the phase $\phi$ in the framework of a Ginzburg-Landau theory. The phase Hamiltonian $H(\phi)$ proposed by Fukuyama, Lee and Rice [12] for CDW's is readily adopted [13] for an incommensurate SDW.

To describe low-frequency dynamical phenomena, the overdamped equation of motion $\gamma\ddot{\phi} + \partial H/\partial\phi = q_e E/Q$ is used where $\gamma$ is a damping constant and the right-hand side describes the coupling to an external electric field $E$, proportional to the condensed charge density $q_e$ ($Q$ is the component of the CDW or SDW wave vector parallel to the conducting chains). As for the origin of the damping, crucial was the observation of Fleming et al. [14] on the temperature dependence of the nonlinear conductivity in various CDW materials. They

<footnote>(*) Permanent address: Central Research Institute for Physics, P.O. Box 49, H-1525 Budapest, Hungary.</footnote>
have found that in constant electric field the non-Ohmic current carried by the CDW shows approximately the same Arrhenius temperature dependence as the Ohmic current carried by the thermally excited normal electrons, i.e. $\gamma \propto \rho_n \propto \exp[\Delta/T]$, where $\rho_n$ is the normal resistance and $2\Delta$ is the CDW gap. The same relation, $\gamma \propto \exp[\Delta/T]$, has been inferred [15, 16] from the temperature dependence of the low-frequency dielectric response.

These findings are in accordance with the theory advanced by Sneddon [17] and further elaborated by Littlewood [18]. They point out that the damping is mainly due to dissipation by normal currents screening the CDW charge fluctuations created by inhomogeneous pinning. An instantaneous screening leads to an enhanced effective damping: $\gamma = \gamma_0 + q_c^2 \rho_n / Q^2$, where $\gamma_0$ is the damping in the absence of screening. Experimental evidence, however, does not rule out the possibility that the origin of similar temperature variation of $\gamma$ and $\rho_n$ is that both quantities reflect temperature activation with similar activation energies. Thermal activation may play a role [19] in CDW depinning, providing a microscopic mechanism for this latter picture. In fact, the activation energy of $\gamma$, deduced from d.c. measurements [14, 20] is always somewhat larger than $\Delta$, so at low temperatures the nonlinearity gradually disappears. Moreover, the relaxation of dielectric polarization follows closely a stretched-exponential time dependence [16] which is usually associated with thermally activated processes [21].

The ideal experiment to choose between the above two pictures would be to change the normal resistance at constant temperature without influencing the CDW or SDW parameters. It is hard to design such an experiment in CDW systems, but the extremely high magnetoresistance [22, 23] of the $(TMSF)_2X$ salts in the insulating SDW state offers a test very close to this ideal.

In this paper we report on measurements of the nonlinear current-voltage characteristics in the spin-density-wave state of $(TMSF)_2AsF_6$ as a function of temperature and magnetic field. We find that the SDW conductivity follows closely the normal conductivity both as a function of temperature and magnetic field in a wide range of these parameters. This result strongly supports the notion that the source of low-frequency SDW damping is the dissipation by normal screening currents.

(TMSF)₂AsF₆ undergoes a metal-insulator transition [24] at $T_c = 11.5$ K, in which SDWs develop over the whole Fermi surface. A recently observed [7] nonlinear conduction accompanied by narrow-band noise and a.c.-d.c. interference effects indicates the existence of a genuine sliding SDW mode in this system.

The single crystals used in this study have been prepared by Lenoir and Batail. For the transport measurements crystals with typical length of 2 to 3 mm in the chain direction and with a homogeneous cross-section (typically (100×50) µm²) have been selected. All measurements have been performed with four stress-free «clamped» contacts [25] on evaporated gold pads.

Figure 1 shows the temperature dependence of the low-electric-field linear resistivity $\rho_n$ in the chain direction in zero magnetic field and in a field of $B = 12$ T parallel to the c*-axis. Without magnetic field the resistivity reaches a constant activation energy curve ($\Delta = 20$ K) close below $T_c$, similarly to $(TMSF)_2PF_6$, but in contrast to CDW systems where the semiconducting gap develops over a wider temperature range. The ratio $\Delta/T_c$ is much smaller than expected for a quasi-one-dimensional system and rather close to the mean-field BCS value. Temperature scans with increasing and decreasing temperature gave identical results; no temperature hysteresis was observed.

The transverse magnetoresistance along the a-axis has been measured up to $B = 12$ T with different field orientations in the (c*, b')-plane. With $B \parallel b'$ the magnetoresistance is smaller than 10% in the whole temperature range investigated. No resistive anomaly has been observed at the spin-flop field [26] $B_{sf} \approx 0.5$ T. In contrast, an extremely high
<figure><img src="image_2.png" /><figcaption>Fig. 1.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2.</figcaption></figure>

Fig. 1. – Temperature dependence of the low-electric-field resistivity of (TMTSF)₂AsF₆ along the *a*-axis in zero magnetic field and in a field of 12 T.

Fig. 2. – Fractional change of the *a*-axis resistivity in a magnetic field at *T* = 6 K. Solid line: the field applied along *c*\* is varied between zero and 12 T. Circles and triangles: the orientation of a constant *B* of 12 T is varied in the (*b*', *c*\*)-plane (*B*<sub>c</sub>*\**<sub>c</sub>* = *B*₀ cos φ), ○ φ < π/2, △ φ > π/2.

magnetoresistance is observed with $B \parallel c^*$. A $\rho_n(B)/\rho_n(0)$ vs. magnetic-field curve taken at 6 K is shown in fig. 2 for this field orientation. This magnetoresistance reproduced within an error bar of about 20% on all three samples investigated. Remarkable is the plateau around $B \approx 5$ T. The position of the plateau does not change with temperature although it is less pronounced at lower temperatures. In fig. 2 we also show the magnetoresistance measured in a constant field $B_0 = 12$ T with various angles $\varphi$ between $B_0$ and $c^*$. Obviously, the resistance is determined by the field component along $c^*$. The values measured with $\varphi$ and $\pi - \varphi$ agree well. All these observations—the good sample-to-sample reproducibility, no significant change upon inverting the field direction, the small magnetoresistance for the $B \parallel b'$ field orientation—indicate that spurious effects due to changes in the current distribution in high magnetic fields have no important influence on the measured magnetoresistance data.

The temperature dependence of the resistance in a field of $B = 12$ T parallel to $c^*$ is shown in fig. 1. The phase transition temperature is 0.4 K higher than without field. The large magnetoresistance develops fast below $T_c$, and below about 7 K ($0.6T_c$) the ratio $\rho_n(12T)/\rho_n(0)$ is about 25 independent of temperature.

The nonlinear conductivity has been measured by a pulsed-current technique above 6 K and by a d.c. lock-in technique at low temperatures. The conductivity becomes strongly nonlinear above a threshold field $E_T$. In fig. 3(a) normalized conductivity curves $\sigma(E)/\sigma_n$ are plotted against electric field for various temperatures. The curves taken between 2.6 K and 4.5 K fall very accurately in top of each other. The normal conductivity changes by about a factor of 20 in this temperature range and the total conductivity $\sigma(E)$ follows the same temperature dependence as $\sigma_n$ at each electric field up to the highest field measured (about $3E_T$). Outside this temperature range the shape of the normalized conductivity curves changes as discussed below.

The same scaling of $\sigma(E)$ with $\sigma_n$ has been found in $B = 12$ T ($B \parallel c^*$) in the temperature range of 2.6 K to 4.2 K (fig. 3(b)). Moreover, for a given temperature $\sigma(E)/\sigma_n$ is practically identical in $B = 12$ T and $B = 0$ T (fig. 3(c)), although $\sigma_n$ changes by a factor of 25.

For a consistent analysis of the strength of the nonlinearity over a wide temperature range and on different samples, we introduce the quantity $\sigma'_{\text{SDW}} = j_{\text{SDW}}(2E_T)/2E_T = \sigma(2E_T) - \sigma_n$,

$$ \frac{J}{2E_T} - \frac{J_n}{2E_T} $$
<figure><img src="image_4.png" /><figcaption>Fig. 3. — Total conductivity $\sigma$ normalized to the low-electric-field conductivity $\sigma_n$ as a function of magnetic field at various temperatures a) in zero magnetic field and b) in a field of 12 T. In panel c) curves taken with (—-) and without (▲) magnetic field are compared.</figcaption></figure>

where $j_{SDW}$ is the nonlinear part of the current density. The temperature dependence of $\sigma'_{SDW}$ normalized to $\sigma_n$ is shown in fig. 4 both for $B=0$ and 12 T. Although the strict scaling between $\sigma'_{SDW}$ and $\sigma_n$ holds only in the temperature range of 2.6 K and 4.2 K, an approximate scaling is valid over a much broader temperature range. The normal conductivity changes by three orders of magnitude between 2 K and 7 K, still the variation of the ratio $\sigma'_{SDW}/\sigma_n$ is smaller by a factor of three. Similarly, $\sigma'_{SDW}/\sigma_n$ values measured with and without magnetic field agree within 20% over the whole temperature range investigated. This error bar is much smaller than the factor of 25 reduction of $\sigma_n$ in a field of 12 T.

<figure><img src="image_5.png" /><figcaption>Fig. 4. — The quantity $\sigma'_{SWD}/\sigma_n$ characterizing the nonlinearity of the conduction as a function of temperature with (○) and without (▲, ▲) magnetic field. The dashed lines are guides to the eye. ●, ○ Sample no. 1; ▲ sample no. 2.</figcaption></figure>

In the discussion we first comment briefly on the effect of a magnetic field on the normal resistivity in the SDW phase. Our measurements as a function of the field orientation indicate that the field acts on the two-dimensional orbitals in the (a, b)-plane. The magnitude of the magnetoresistance for $B\parallel c^*$ agrees well with previous results [23] on $(TMTSF)_2PF_6$. In this work, however, the highest applied field was significantly lower than in our case and the characteristic S-shape of the $\rho_n$ vs. $B$ curve (fig. 2)—reproduced on three different samples—was not detected. Conventionally the high magnetoresistance is explained [23] by the existence of compensated electron and hole bands with high mobilities. It is, however, difficult to understand the nontrivial structure of $\rho_n(B)$ in the framework of a simple semiconductor picture. A further difficulty is the high value of the cyclotron frequency $\omega_c$ in the magnetic fields of our experiments. Using the estimate $\omega_c = \text{veBb}$, with a Fermi velocity in the chain direction $v = 6 \cdot 10^5$ m/s and a lattice constant $b = 0.77$ nm, one gets $\omega_c/2\Delta = 1.5$ in $B=12$ T. With this high value of $\omega_c/2\Delta$, one-band calculations may easily break down. A
systematic study of the magnetoresistance as a function of defect concentration may help to clarify the situation.

The effect of magnetic field on the SDW transition temperature and gap is small in agreement with earlier measurements [22, 23] as well as with expectations for the good nesting situation encountered at ambient pressure. The transition temperature increases by $\delta T_c/T_c \approx 3\%$. We detect no change in the gap parameter $\Delta$; our accuracy, however, allows for a change $\delta\Delta/\Delta \approx 6\%$, which would mean a ratio $\Delta/T_c$ independent of magnetic field.

In a recent communication Bjeliš and Maki [27] examine the SDW amplitude coherence and fluctuations in a magnetic field. The most important effect of the field is the reduction of the transverse coherence length $\xi_\perp$. In the Fukuyama-Lee-Rice weak-pinning theory [12], the decrease of $\xi_\perp$ results in the increase of the threshold electric field: $E_T \propto [\xi_\perp(B)/\xi_\perp(0)]^{-2}$ in three dimensions. Using the numerical results of Bjeliš and Maki [27] for the field dependence of $\xi_\perp$, one gets $E_T(B) = 1.6E_T(0)$ for $\omega_c/2\Delta = 1$ and $E_T(B) = 2.8E_T(0)$ for $\omega_c/2\Delta = 2$. On the other hand, in the strong pinning limit the threshold field is independent of the coherence length [12] and no magnetic field dependence of $E_T$ is expected. Although our measurements suggest a slight increase of the threshold field in $B = 12$ T (see fig. 3c)), this increase is certainly not more than 20%, so the data seem to support a strong-pinning rather than weak-pinning situation. In summary, the magnetic field of our experiments, $B = 12$ T, has a minor effect on the SDW parameters compared to the effect on the normal conductivity.

The main result of this paper is the scaling of the SDW conductivity and normal conductivity, $\sigma'_{SDW}/\sigma_n = \text{const}$, both as a function of temperature and magnetic field. This scaling holds very strictly in the temperature range of 2.6 to 4.2 K. If the normal conductivity changes either due to a change in the number of carriers as a function of temperature, or due to a change in the mobility of carriers as a function of magnetic field, the SDW conductivity follows this change. At these low temperatures the SDW order parameter is saturated, and the parameters appearing in the phase Hamiltonian $H(\phi)$ are temperature independent. Also, as suggested by the above analysis, the magnetic field of 12 T has no major effect on the SDW parameters. Under these circumstances, $1/\sigma'_{SDW}$ is an experimental measure of the damping $\gamma$, and the scaling of $\sigma'_{SDW}$ with $\sigma_n$ strongly suggests that the damping is determined by the normal conductivity. This behaviour is in full agreement with the theory of density-wave damping due to dissipation by normal screening currents [17, 18]. The scaling as a function of magnetic field seems to rule out the models that explain the Arrhenius temperature dependence of $\sigma'_{SDW}$ by the exponential freezing out of SDW charge carriers (such as dislocations). In these models the strong decrease of $\sigma'_{SDW}$ with increasing magnetic fields is unexpected.

The values obtained for $\sigma'_{SDW}/\sigma_n$ on seven different samples are in the range of 0.1 to 1.4 at $T = 4.2$ K, in agreement with previous measurements [28] on $(TMTSF)_2PF_6$. A similar, though somewhat smaller scattering is observed in the values of the threshold field, but no obvious correlation is found between $E_T$ and $\sigma'_{SDW}/\sigma_n$. Based on simple phenomenological arguments, Tomić et al. [28] expect a temperature-independent value $\sigma'_{SDW}/\sigma_n = 1/2$ below $T_c/2$, in rough agreement with the experimental findings.

Outside the temperature range of 2.6 to 4.2 K the shape of $\sigma(E)/\sigma_n$ curves change with temperature and the ratio $\sigma'_{SDW}/\sigma_n$ decreases both with and without magnetic field. Such a decrease is natural close to $T_c$, since $\sigma'_{SDW}$ should disappear at the phase transition [29]. However, the significant decrease in the temperature range of 6 to 8 K is somewhat unexpected in the light of $\rho_n$ vs. T measurements suggesting a temperature-independent order parameter in this region. A possible explanation is that the unrenormalized damping $\gamma_0$ may become comparable to the correction due to screening currents at these temperatures. This hypothesis could be tested on impure samples where $\gamma_0$ is expected to be higher.

More surprising is the quite sudden drop of $\sigma'_{SDW}/\sigma_n$ below 2.6 K. As the screening length of
normal carriers increases exponentially with decreasing temperature, the screening of the SDW charge fluctuations may become incomplete at low temperatures. This effect, however, is expected [30] to lead to an increase rather than decrease of the $\sigma'_{{\text{SDW}}}/\sigma_n$ ratio. The strong decrease of $\sigma'_{{\text{SDW}}}/\sigma_n$ may be related to the appearance of a qualitatively different sliding density-wave mode both in CDW [31] and SDW [32] systems at very low temperatures.

***

We are grateful to P. BATAIL and C. LENOIR for preparing the samples and to J. C. AMELINE for building the sample holder equipped with goniometer. Useful discussions with K. MAKI and S. TOMIĆ are acknowledged. GK acknowledges a visiting researcher position from CNRS. This research was partly supported by ESPRIT fund 3121 DG XIII.

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