# Organic Superconductors: To Fluctuate or not to Fluctate?

Denis Jerome

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Denis Jerome. Organic Superconductors: To Fluctuate or not to Fluctate?. Molecular Crystals and Liquid Crystals, 1985, 119 (1), pp.1 - 9. 10.1080/00268948508075125. hal-05011635

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# Organic Superconductors: To Fluctuate or not to Fluctate?

D. Jerome<sup>a</sup>

<sup>a</sup> Laboratoire de Physique des Solides (associé au CNRS) Université Paris Sud, 91405, Orsay, France

Published online: 17 Oct 2011.

**To cite this article:** D. Jerome (1985) Organic Superconductors: To Fluctuate or not to Fluctate?, Molecular Crystals and Liquid Crystals, 119:1, 1-9

**To link to this article:** http://dx.doi.org/10.1080/00268948508075125

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# ORGANIC SUPERCONDUCTORS : TO FLUCTUATE OR NOT TO FLUCTUATE?

D.JEROME

Laboratoire de Physique des Solides (associé au CNRS)

Université Paris Sud 91405 Orsay, France.

**Abstract** New experimental data and recent improvements of the theory suggest that the one-dimensional regime could extend much lower in temperature than what can be expected from the knowledge of band parameters. This approach allows the interpretation of the properties of (TMTSF)₂X conductors which are not observed in ordinary metallic conductors and supports the existence of well developed cooperative phenomena up to 30 K.

## INTRODUCTION

Approximately four years have passed since the initial observation of superconduction in a conducting organic solid. This conference comes therefore at a time which is very appropriate to check the results of research efforts developed in the field of organic superconductivity and to evaluate the possibilities for future improvements.

As this field is blooming in various directions, the present account intends to display very briefly some of the new flowers which have emerged since the last ICSM conference held in December 1982.

Superconductivity has been first observed in the single chain organic conducting salt: tetramethyltetraselenafulvalene -hexafluorophosphate ¹, (TMTSF)₂PF₆, a member of the broader family (TMTSF)₂X where X is a mineral counter-anion playing no direct role in the electron conduction which is provided by the organic molecule TMTSF ².

In the PF₆ salt superconductivity is observed below 1 K only under a hydrostatic pressure exceeding ~9 Kbar. Important progresses in material preparation have followed shortly after the initial discovery. First, superconductivity has been stabilized in the perchlorate salt(TMTSF)₂ClO₄³⁻ below 1.2K under atmospheric pressure. Subsequently, superconductivity has been obtained with an other organic molecule, BEDT-TTF, containing sulfur instead of selenium heteroatoms , first under pressure in (BEDT-TTF)₂ReO₄⁴⁻ and subsequently at atmospheric pressure in (BEDT)TTF)₂I₃⁵⁻ below 1.4 K. Recently, new phases of the (BEDT)TTF-iodine conductor have been reported to exhibit superconductivity at atmospheric pressure below 2.5 K⁶.

Besides efforts performed in the synthesis of new organic
molecules, sophisticated experimental and theoretical studies have
led first to a better understanding of the electronic properties
of prototype systems such as (TMTSF)₂ClO₄ and secondly to the ob-
servation of new and remarkable physical properties which are
tightly related to the quasi-one-dimensionality of these conductors.

THE CONDUCTING STATE OF (TMTSF)₂ClO₄

(TMTSF)₂X conductors present only one conducting chain per unit cell. The band structure is, in principle, relatively simple and practically insensitive to the temperature as long as the conducting state is stable. This feature has to be opposed to the case of two-chain conductors such as TTF-TCNQ or HMTSF-TCNQ where cova- lency effects between donor and acceptor chains lead to a tempera- ture dependent band structure.⁷ However, the perfect planarity of the Fermi surface of independent conducting chains is likely to be slightly modified by the existence of the non-negligible overlap between molecular wave functions belonging to molecules on near neighbouring chains. In the (TMTSF)₂X structure, this is the coupling along the transverse b-direction which is relatively strong on account of short Se···Se intermolecular contacts⁸.

At high temperature (T~300~ K) the Fermi surface is thermally smeared over an energy interval given by W/τ_{>//}, where τ_{>//} is the intra-chain electron scattering time. Since DC and optical conductivity data give τ_{>//} ~3x10⁻¹⁵s (W/τ_{>//}) ~0.2 eV) the Fermi surface smearing is larger than measured or calculated values for the largest inter-chain interaction, namely t_b^↓ ~12 meV ⁹. Hence, the motion of electrons along b is diffusive^1 and a metallic-like behaviour with a plasma edge around 10.000 cm⁻¹ is observed only for light polarized along the high-conductivity a-axis¹⁰. In the high T-domain the hopping rate between chains 1/τ_⊥ is smaller than t_b^⊥ since 1/τ_⊥~(t_b^//)^2τ_{>//} and t_b^⊥ τ_{>//} <1, (K = 1).

It is also the hopping rate 1/τ⊥ which establishes a cut-
off for the t⁻¹/² 1-D time decay of the electron spin autocorrela-
tion funciton at large times ¹¹. This property is particularly im-
portant in Q-1-D conductors as it enables a direct determination
of 1/τ⊥ via a study of the NMR relaxation rate versus the applied
magnetic field . When the nuclear spin-lattice relaxation is due
to the hyperfine coupling of the nuclear spins to electrons moving
in a given direction, 1/T₁~H⁻¹/²/τ// as long as ω_e τ⊥ > 1, where
ω_e is the electron Larmor frequency. However at low fields if ω_e
becomes smaller than a cross-over frequency ω_c (ω τ⊥ = 1) the usual
field-independent relaxation of the 3-D regime is recovered. These
properties have been studied extensively in TTF-TCNQ where the
characteristic H⁻¹/² field dependence is observed for the ¹H re-
laxation rate above 8 kOe. A 1-D regime (1/T₁~H⁻¹/²) has been
observed similarly in (TMTSF)₂X conductors at ~300 K but above
20 kOe ¹².
The comparison between NMR data of (TMTSF)₂X and selectively deuterated TTF-TCNQ leads to the following remark: $t_↓$ in (TMTSF)X is about twice that between TNCQ chains in TTF-TCNQ (i.e. $t_↓^b \sim 10$ meV meV in (TMTSF)₂ClO₄). It is interesting to recall that high-field $T_↓$ data of HMTSF-TCNQ have failed to show any field dependence up to 60 kOe¹⁵. This result implies that $t_↓$ of (TMTSF)₂X cannot be much larger than $t_↓$ in HMTSF-TCNQ, a system which undergoes a Peierls transition at 24 K¹⁴ with high temperature precursor effects in spite of a strong corrugation of the Fermi surface. At low temperatures the thermal broadening of the Fermi surface becomes weaker and when $1/t_↓/t_↓^c$ (this condition should not be taken "stricto-sensu") the transverse electron motion becomes coherent. A transverse band-like behaviour is indeed observed below ~100 K in FIR reflectance data of (TMTSF)₂X with light polarized along the b-direction¹⁵. The ratio of $t_↓$ to $t_↓^c$ plasma edges is about 10.

Hall effect experiments performed in the low temperature conducting state of (TMTSF)₂ClO₄¹⁶ reveal a linear field dependence of the Hall voltage up to 30 kOe, also T-independent in the helium temperature range, leading to $R_H = 4 \times 10^{-9} m^3/A \cdot S$. The experimental value agrees fairly well with 1-D tight binding determination in a quarter-filled band corresponding to the 2:1 stoichiometry of (TMTSF)₂X salts¹⁷, namely $R_H^{th} = 3.4 \times 10^{-9} m^3/A \cdot S$. The low field Hall effect data strongly suggest that in spite of a somewhat important $t_↓$ the Fermi surface of (TMTSF)₂X conductors remains quasi planar with a small corrugation in the b-direction even at low temperatures.

Consequently, parallel and transverse polarized light reflection data lead to a band structure anisotropy $t/t_↓ \sim 10^{15}$. The interchain coupling in the c-direction is about one order of magnitude smaller than in the b-direction and $t/t_↓ \sim 300$. There exists much dispersion in the magnitude of $t/t_↓$ but the range $t_↓ \sim 0.12 - 0.2$ eV seems to be a reasonable compromise between experimental data and calculations.

## THE 1-D REGIME IN A Q-1-D CONDUCTOR

The basis of 1-D physics is that electron-electron (Cooper) and electron-hole (Peierls) channels diverge simultaneously because of the Fermi surface invariance by a shift of wave-vector $Q = (2k_F, q_↓)$ independent of the value of $q_↓^{14}$.

The Fermi surface warping in a real Q=1-D conductor can be neglected at high temperature as long as $T \gg t_↓$. The response functions of the 1-D non-interacting electron gas diverge logarithmically, namely $\chi(2k_F, T) \sim \ln(\frac{E_F}{T})$. However, the two-channel mixing of the 1-D regime is stopped when $T < T_X$ where $T_X$ is the cross-over temperature between 1-D and 3-D behaviours for the correlation functions. In a non interacting electron gas $T_X = T_X^0 \sim t_↓ / \pi^{18}$.
Below $T^0$ the Peierls divergence levels off at $\ln \frac{E_F}{t_\perp}$ whereas nothing $x^*$ prevents the Cooper divergence from a possible onset of superconductivity at $T_c$.

With the accepted values of $t_\perp$, $T^0$ should be located around 80 K. Therefore, if the 3-D domain of the correlation functions begins at nitrogen temperature it is difficult to claim the existence of strongly developed precursors of a superconducting transition which occurs at liquid helium temperature only. Superconductivity should thus develop in a quasi-mean field way ¹⁹. However, Bourbonnais ²⁰ has shown that the cross-over temperature may be overestimated by, say a factor 10, not taking into account Coulomb interactions between electrons. A particular consequence of the channel mixing of the 1-D regime is the existence of long wavelength and low energy modes giving rise to:

(i) 1-D short range correlations and (ii) a depression of the single-particle density of states at the Fermi level $N(E_F)$ which reads $\sim (t_F^{-2\alpha})$ where the exponent $0<\alpha<1$ depends on the interactions $g_1$ and $g_2$ in the geology model ²¹. The drop in $N(E_F)$ upon cooling in the 1-D regime is also called the formation of a "pseudo-gap". Its existence has been established in the 1-D regime of 1-D conductors undergoing a Peierls transition ¹⁴.

In the vicinity of $T^0$ the transfer of single particles between neighbouring chains is slowed down by the concomitant opening of the pseudo-gap. This effect has been recognized to stabilize the 1-D regime at temperatures lower than $T^0$. This means that mixed intra-chain correlations persist down to $T^0$ a renormalized cross-over temperature $T_x$ which is determined by the size of the Coulomb interactions. In Bourbonnais' model $^{20}T_x$ reads $\frac{t_F}{\sqrt{2\alpha}}$ ($\frac{t_F}{\sqrt{2\alpha}}$)$^{\alpha/1-\alpha}$. As shown below $\alpha \approx 0.5$ is an acceptable value for $(TMSF)_2$XF. This would extend the 1-D regime down to about 8 K since $t_\perp \sim E_F \approx 1/10$.

## COOPERATIVE PHENOMENA IN $(TMSF)_2$X CONDUCTORS

A major interesting aspect of $(TMSF)_2$X compounds is, besides the existence of superconductivity, to know whether 1-D theories are required for their understanding or whether a band structure anisotropy of 10 is low enough to justify an anisotropic mean-field treatment for the onset of superconductivity ¹⁹. The 3-D anisotropic model for superconductivity implies that the precursor signs of the transition develop in all three directions simultaneously. Therefore the fluctuating regime cannot be spread over more than a few Kelvin above the onset of long-range order at 1.2 K. It is important to see that this picture can be supported by some experimental data if a large amount of other results are left out. This is the case of specific heat data ²² indicating that the thermodynamics of the superconducting transition adheres rather well to a standard BCS-like model. However, the validity of an anisotropic 3-D mean-field model implies that in the T-domain 5 to 30 K the properties of the electron gas in $(TMSF)_2$ClO₄ do not deviate significantly
from those exhibited in a regular metallic conductor at low temperature. The rest of this section is devoted to the demonstration that this is precisely not true and consequently that the conducting state of $(\text{TMTSF})_2\text{ClO}_4$ or other members of the series below 30 K cannot be assimilated to an ordinary metallic state. We shall summarize some recent NMR results which are presented in greater details by F. Creuzet at this conference$^{23}$. The first point we want to emphasize is that the spin susceptibility of $(\text{TMTSF})_2\text{ClO}_4$ is both temperature and field independent below 40 K and up to $^{10}$kOe respectively$^{24}$. Under these conditions the hyperfine spin-lattice relaxation should vary linearly with the temperature and be independent of the magnetic field (following the so-called Korringa law).

Single crystal NMR data$^{23}$ have clearly established that both properties are not followed in the R-state of $(\text{TMTSF})_2\text{ClO}_4$:

(i) Strong enhancements above the Korringa relaxation$^2$ are observed upon cooling below $\sim$25 K.

(ii) A large field dependence of $1/T_1$ is observed in the same T-domain (see figure 1 of reference $^{12,3}$).

A field dependence of $1/T_1$ has also been reported for $^{1}$H, $^{13}$C and $^{77}$Se nuclei in $(\text{TMTSF})_2\text{PF}_6$ under pressure in the limited range of temperatures below $^{4,2}$K$^{25}$. Azevedo$^{25}$ has ascribed the field dependence of $1/T_1$ ($\propto \log H$) to electron spins diffusing in a low dimensional conductor ($2D^0$ motion). However the slope of $1/T_1$ versus $H^{-1/2}$ or $\log H$ is proportional to $1/\tau_1$; the inverse intra-chain electron scattering. Therefore the field dependence of $1/T_1$ should be reduced at low temperature by a factor $\sim 10^3$ below the room temperature value since the ratio of the room temperature to the 10 K resistivity reaches approximately $10^3$ in good quality samples$^{26}$. According to the known field dependence at room temperature $d(1/T_1)/dH^{-1/2} = 2 s^{-1} \cdot \text{kOe}^{1/2}$ it is impossible to attribute the important$^0$ field dependence of $1/T_1$ at low temperature to low dimensional spin dynamics with the assumption of an entirely single-particle conduction below 30 K. An other interpretation has been proposed for the low temperature relaxation data$^{25}$. The non-Korringa behaviour versus temperature can be understood in terms of a strong enhancement of the 2k SDW response at low temperature. The SDW response becomes quite F significant if, for example, the 1-D regime extends down to the 5-8 K domain. As indicated in the previous section the 1-D to 3-D cross over temperature can be low if short-range Coulomb interactions are repulsive ($g_1>0$). This assumption is very plausible for the $(\text{TMTSF})_2X$ series since:

(i) the existence of a SDW (but not CDW) state for $(\text{TMTSF})_2\text{PF}_6$ and other members is the signature of a positive $g_1^{21}$.

(ii) $g_1^{21}$ no gaps are observed in the T-dependence of the susceptibility$^{21}$.

In the presence of interchain hopping the SDW susceptibility follows a power law divergence in the 1-D regime and saturates at
T. In the same regime $1/T_1$ is non-Korringa. Below $T_x$ spin-lattice relaxation tends towards an other, but enhanced, Korringa regime.

The NMR studies suggest that $T_x$ may be located in the vicinity of 5 K. This value is admittedly not far from the estimate obtained in the previous section assuming $\alpha = 0.5$ in agreement with repulsive Coulomb interactions.

As the renormalized value $T_x$ decreases with decreasing $t_1$, one may expect that under large transverse magnetic fields orbital effects begin to play an important role restricting the electronic motion perpendicular to the chains $^{28}$. This argument has been proposed $^{23}$ for the origin of the field dependence of $1/T_1$ in the 1-D regime when cooperative phenomena become predominant.

The NMR properties of the conducting state have revealed the existence of magnetic fluctuations at low temperature in $(\text{TMTSF})_2\text{ClO}_4$, R-state. However, in the 1-D regime, magnetic fluctuations may coexist with superconducting precursor effects since the coupling between electron-hole and pairing correlations is the heart of 1-D physics. We shall briefly recall some experimental data related to transport and density of states properties which are hard(if not impossible) to understand within the framework of a gas of single-particles.

DC conduction becomes extremely large at low temperature ($\alpha(2\text{K})\gtrsim 10^6 (\text{cm})^{-1}$) in $(\text{TMTSF})_2\text{ClO}_4$ with an anomalously strong sensitivity to the application of a transverse magnetic field $^{29}$.

The high conductivity is restricted to a narrow mode at zero frequency. Its width, derived from FIR reflectance studies, is about $<0.5 \text{ cm}^{-1}$ at $2 \text{ K}^{30}$. Furthermore, Timusk et al $^{31}$ have shown that it is impossible to fit the frequency dependence of the conductivity in the FIR regime by a Drude-like behaviour. Finally, as more and more electron states near the Fermi surface couple to build up (in the 1-D regime) the long wavelength low energy collective mode, the density of states at the Fermi level becomes depressed (formation of a pseudo-gap). This phenomenon is probably responsible for the "gap" observed in the FIR conductivity below $30 \text{ cm}^{-1}$ $^{30}$.

The pseudo-gap formation could also explain the behaviour of recent tunneling data of electrons in lead (superconducting)-insulator-$(\text{TMTSF})_2\text{ClO}_4$ junctions between 5 and $1.2\text{K}^{32}$. Junctions of high quality reveal a smearing of the typical BCS singularities at the lead gap edges. This effect is pretty well understood by a tunneling response being the convolution between the ordinary $N(E)$ of superconducting lead and the density of states of $(\text{TMTSF})_2\text{ClO}_4$ showing a pronounced depression over an energy width of about 3.6 - 3.8 meV.

The anomalous drop of the thermal conduction at low temperatures $^{33}$has also been related to the joint formation of a DC collective mode contributing to the electron transport (but not to heat transport) and of the pseudo-gap in the single-particle density of states.
## CONCLUSION

Progresses in the theory have shown how One-Dimensionality can be stabilized at low temperatures by 1-D correlations in a 1-D Coulomb electron gas. This model reconcile the numerous anomalous features of the NMR observed below 30 K with the modest band structure anisotropy of 10.

It is clear that the theory of Q-1-D superconductors will require further improvements, but we feel confident that a Landau-Ginzburg treatment of the order parameter phase fluctuations close enough to $T_c$, when Peierls and Cooper channels become decoupled ³, constitute a reasonable starting point. Such a picture has the merit to reconcile the quasi BCS behaviour of the superconducting transition itself with the existence of a broad fluctuation domain.

Other data require 1-D physics for their interpretation. It is, for instance, the case of the interplay between magnetism and superconductivity observed in all $(TMTSF)_2X$ superconductors ³⁵ and the stabilization of a SDW semimetallic 2D state (in terms of Fermi surface topology) under high transverse fields ³⁶, ³⁷. This latter aspect has led to spectacular developments also related to the Hall quantization which are also presented at this conference ²⁶, ³⁸, ³⁹. A crucial question is still left without answer: it is addressed to the nature of the pairing (spin singlet or triplet) in the 3-D superconducting state as well as in the precursor regime. There are no, so far, direct evidences showing triplet pairing besides the remarkably strong sensitivity of organic superconductivity to traces of non-magnetic impurities and the g-ology model which favours triplet pairing when $g_1$ is positive ³⁵.

Similarly, a correct treatment of the fluctuation conductivity has to be developed; especially under high field where the model of Efetov ⁴⁰ would require some modifications to be adapted to the $g_1>0$ situation.

In conclusion, the results presented at this conference show that on several points the ideas about Q-1-D organic superconductors are converging, for ex, role of Coulomb interactions on the definition of the 1-D regime, SDW state at high field, FIR conductivity, etc... More effort in chemistry is still needed to improve materials properties as far as superconductivity is concerned. This seems to be "conceptually" feasible.

I wish to acknowledge the constant cooperation of all my colleagues at the Laboratoire de Physique des Solides d'Orsay and at the Oersted Institute of Copenhagen.

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