Organic Superconductivity

• Prof. Denis Jerome, Laboratoire de Physique des Solides, Orsay, France, France

Organic superconductivity refers to the regular phenomenon of superconductivity as it is observed in some metals and metallic inorganic compounds. However, what makes organic superconductivity so distinctive is that conduction in organic molecular conductors is linked to the transport of free charges (electrons or holes) between $\pi$-like molecular orbitals of neighbouring open shell molecules. In addition, these new materials reveal quasi one dimensional features of their electronic transport properties due to their peculiar crystal structure. They are considered as textbook examples for low dimensional physics. They have also been stimulating for the research on high $T_{c}$ superconductors and pnictides.

Some definitions

Superconductivity

Figure 1: The figure displays the evolution of $T_{c}$ in materials according to the date of the discovery of their superconducting properties. It clearly shows a renewed effort in superconductivity at the beginning of the eighties which triggered the research of novel materials with the discovery of organic superconductors in 1980 and high $T_{c}$ cuprates in 1986.

Superconductivity (a new state of matter) is one of the major discovery in physics for the twentieth century given its experimental and theoretical impact on science. From an experimental point of view the finding by G. Holst in the laboratory of H. Kamerlingh Onnes [1] in 1911 of a current travelling without resistance through a metal cooled at very low temperature paved the way to a very large number of industrial applications. This experimental discovery happened long before the proposal of a satisfactory theoretical framework by Bardeen, Cooper and Schrieffer (BCS) [2] in 1957 explaining the two basic properties of the superconducting ground state of a metal namely, the zero resistance state and the magnetic flux expulsion.

Superconductivity requires low temperature conditions to be observed in metals and intermetallic compounds, typically the helium temperature range as can be seen on fig(1). Until the years in the sixties, the critical temperature for superconductivity had been steadily rising as new intermetallic compounds were synthesized according to the paradigm of the BCS theory predicting an isotope effect namely, an electron-electron pairing in the superconducting ground state mediated by the electron-phonon interaction with the ionic mass dependence of the critical temperature, $T_{c}$ $\alpha \,M^{-1/2}$ [2].

Searching for new materials exhibiting the highest possible values of superconducting $T_{c}$ was already a strong motivation in materials sciences in the early 70's, and the term high temperature superconductor was already commonly used referring to the intermetallic compounds of the A15 structure, namely, (Nb₃Sn or V₃Si) [3].

Expending the successful idea of the isotope effect of the BCS theory other models were proposed in which excitations of the lattice responsible for the electron pairing had been replaced by higher energy excitations namely, electronic excitations, with the hope of new materials with $T_{c}$ higher than those explained by the BCS theory. Consequently, the small electronic mass $m_e$ of the polarizable medium would lead to an enhancement of $T_{c}$ of the order of ($M/m_e)^{1/2}$ times the value which is observed in a conventional superconductor, admittedly a huge factor. V.L.Ginzburg [4,5] considered in 1964 the possibility for the pairing of electrons in metal layers sandwiched between polarizable dielectrics through virtual excitations at high energy.

An even more provocative suggestion came from W.A.Little [6, 7] namely, a new mechanism for superconductivity which could lead to a drastic enhancement of the superconducting $T_{c}$ to be observed in especially designed macromolecules. The idea of Little was still rooted in the extension of the isotope effect proposed by BCS replacing the mediating phonon by an electronic excitation which especially designed macromolecules could reveal. However a prerequisite to the model of Little was the achievement of metallic conduction in organic molecular crystals, not a trivial problem in the sixties.

Figure 2: Schematic picture for the Little's suggestion. Charge carriers a and b forming a Cooper pair in the conducting spine are bound via a virtual electronic excitation of the polarizable side groups.

Organic conductors

Electronic conduction proceeds in organic molecular conductors through the transport of free charges (electrons or holes) between $\pi$ -like molecular orbitals of neighbouring open shell molecules rather than between atomic orbitals in regular metallic conductors. One cannot find organic conductors in nature, unlike the metallic ones. The former must always be synthesized in a chemistry laboratory.

The first successful attempt to promote metal-like conduction between open shell molecular species came out in 1954 with the synthesis of the molecular salt of perylene oxidized with bromine [8] although this salt was unstable in air.

History

The idea of Little was based on the use of a long conjugated polymer such as a polyene molecule grafted by polarizable side groups [9]. Admittedly, this formidable task in synthetic chemistry did not reach its initial goal but the idea to link organic metallicity and one dimensionality was launched and turned out to be a very strong stimulant for the development of organic superconductors.

A major step was accomplished in 1970 towards the discovery of new materials for superconductivity with the synthesis by F.Wudl [10] of the new molecule tetrathiafulvalene, (TTF). This molecule containing four sulfur heteroatoms in the fulvalene skeleton can easily donate electrons when it is combined to electron accepting species.

From the charge transfer period to the first organic conductor

A short time later the synthesis of the first stable organic metal, the charge transfer complex TTF-TCNQ came out. This compound is made up of two kinds of flat molecules each forming segregated parallel conducting stacks. It fulfills the conditions for an organic conductor as the orbitals involved in the conduction ($\pi$-HOMO, highest occupied molecular orbital and $\pi$-LUMO, lowest unoccupied molecular orbitals for TTF and TCNQ respectively) are associated with the molecule as a whole rather than with a particular atom. Free carriers within each stacks are given by an interstack charge transfer at variance with other organic conductors such as the conducting polymers in which charges are provided by doping [11].

The announcement of a large and metal-like conduction in TTF-TCNQ was made in 1973, simultaneously by the Baltimore [12] and Pennsylvania groups [13]. The Pennsylvania group made a provocating claim announcing a giant conductivity peak of the order of $10^5$ $(\Omega cm)^{-1}$ at 60 K arising just above a very sharp transition towards an insulating ground state at low temperature. This conductivity peak was attributed by their authors to precursor signs of an incipient superconductor.

Figure 3: Isothermal pressure dependence of the longitudinal conduction of $\mathrm{TTF-TCNQ}$, showing a drop of the conduction in the commensurability (x3) domain of the fluctuating Fröehlich contribution due to pinning by the lattice [15]. The ratio $\sigma (80 K)/\sigma (290 K)$ is evolving from 18 to 3 from ambient pressure up to 18 kbar and rising again at higher pressures.

Thanks to pressure studies it has been shown that the very large conductivity peak exhibited by TTF-TCNQ at 59K is actually the contribution of collective Fröehlich fluctuations in a 1D regime adding to the conduction coming from single particles [14]. Driving the charge transfer through a commensurate value $\rho= 2/3$ under the pressure of $\approx$ 18 kbar provided in turn the proof for collective Fröehlich fluctuations existing in the 1D regime under ambient pressure, leading to a large enhancement of the ordinary single particle conduction since the wave length of these CDW fluctuations is not commensurate with the underlying lattice, in particular under ambient pressure [14], see, fig(3). Furthermore high pressure experiments performed recently in Japan managed to partly suppress the onset of the Peierls instability in TTF-TCNQ [16] after an initial increase up to 30 kbar confirming the early Orsay work [14] but failed to reveal any sign of superconductivity which had been anticipated by Horovitz et-al [17] when phonons become soft in the vicinity of the Peierls transition, hence enhancing $T_{c}$ according to the traditional electron-phonon mechanism.

Furthermore, the $4k_F$ signature in X-ray diffuse scattering experiments [18] have pointed out that 1D electron-electron repulsive interactions cannot be ignored in these kinds of molecular compounds.

Hence, new charge transfer compounds with the acceptor TCNQ were synthesized using other heteroatoms for the donor molecule, i.e. substituting sulfur for selenium in the TTF skeleton thus leading to the TSF molecule where S stands for selenium. Much attention was devoted to the tetramethylated derivative of the TSF molecule which, when combined to the dimethylated TCNQ gave rise to TMTSF-DMTCNQ [19].The outcome of the high pressure study of this latter compound has been truly decisive for the quest of organic superconductivity [19, 20, 21]. First, the metal insulator transition located at 42 K has been identified by X-ray diffuse scattering experiments [22] as a Peierls transition driven by the TMTSF chain and a quarter filling of both donor and acceptor bands of this charge transfer compound has been derived from the measurement of the wave vector $2k_F$ in the Peierls state. Second, for the first time the metallic state of an organic compound could be stabilized down to 1.2 K under pressure albeit without superconductivity [23]. Although the reason for the stability of a metallic state in TMTSF-TCNQ above the pressure of 9 kbar is not yet understood, emphasizing the role of the donor stack (TMTSF) for conduction has been a strong motivation for the synthesis of new organic conductors with structures even simpler than those of the two stacks charge transfer compounds. This is how (TMTSF)₂PF₆, the so-called Bechgaard salt [24], came out of the Copenhagen chemistry laboratory in 1979 inspired by a series of isomorphous radical cationic conductors based on TMTTF previously synthesized at Montpellier [25].

Organic superconductors part of a generic phase diagram

The Copenhagen group led by Klaus Bechgaard and experienced with the chemistry of selenium succeeded in the synthesis of a new series of conducting salts all based on the TMTSF molecule namely, (TMTSF)₂X where X is an inorganic mono-anion with various possible symmetry, spherical (PF₆, AsF₆, SbF₆, TaF₆), tetrahedral (BF₄, ClO₄, ReO₄) or triangular (NO₃) [24],see fig(4). All these compounds but the one with X=ClO₄ did reveal an insulating ground state under ambient pressure.

Figure 4: Side view of the TMTSF molecule (yellow and red dots are selenium and carbon atoms respectively, hydrogens not shown) and (TMTSF)₂PF₆ quasi 1D structure along the $b$ axis, courtesy of J.Ch. Ricquier, IMN, Nantes (left) . First observation of superconductivity in (TMTSF)₂PF₆ under a pressure of 9 kbar [29]. The resistance of two samples is normalized to the 4.5 K value (right).

What is so special with (TMTSF)₂PF₆, the prototype of the so-called Bechgaard salts? Unlike previously investigated TTF-TCNQ, it is the magnetic origin of the ambient pressure insulating state [26] contrasting with the Peierls-like ground states discovered previously in charge transfer compounds. The ground state of (TMTSF)₂PF₆ turned out to be a spin density wave state similar to the predictions made by Lomer [27] in 1962 and by Overhauser [28] for metals. It is the onset of itinerant antiferromagnetism which opens a gap at Fermi level. Since the Fermi surface is nearly planar, the exchange gap develops over the whole surface (although the SDW phase still retains small pockets of carriers, see [30] for references). The magnetic origin of the insulating ground state of (TMTSF)₂PF₆ was thus the first experimental hint for the prominent role played by correlations in these organic conductors.

Superconductivity of (TMTSF)₂PF₆ occurred at 1K as shown by transport, fig(4), once the SDW insulating ground state could be suppressed increasing the transverse overlap between molecular stacks under a pressure of about 9 kbar [29]. AC susceptibility measurements performed on the same material detected an anomaly at T_c indicative of a transition into a diamagnetic state [31]. A confirmation of the bulk nature of organic superconductivity has been given subsequently by the observation of shielding and Meissner signals in (TMTSF)₂PF₆ [32].

The synthesis of the superconducting compound (TMTSF)₂PF₆ allowed to establish a link between this compound and the isostructural family comprising the sulfur molecule with the same series of monoanions discovered earlier. Thanks to an intensive study under pressure, it was realized that (TMTTF)₂X and (TMTSF)₂X salts both belong to a common class of materials forming the generic (TM)₂X phase diagram [33], fig(5).

Figure 5: Generic phase diagram for the (TM)₂X family [33] based on the sulfur compound (TMTTF)₂PF₆ under ambient pressure taken as the origin for the pressure scale. Phases in yellow, green, light green and red colors are long range ordered, Charge Order (CO), Spin Peierls (SP), Magnetic phases Neel antiferromagnet and Spin Density Wave phase and Superconductivity respectively. There exists a small pressure window around 45 kbar in this generic diagram where SC coexists with SDW according to reference [36, 37, 38]. $T_\rho$ marks the onset of 1D charge localization which ends around 15 kbar at the dashed double-dotted line. The (TL)1D to 2D deconfinement occurs at $T^\star$. The dashed line between 2D and 3D regimes defines the upper limit for low temperature 3D coherent domain.

In particular, (TMTTF)₂PF₆ , although the most insulating compound of the phase diagram can be made superconducting at low temperatures under a pressure of 45 kbar [34, 35]. The study of this generic phase diagram has in turn greatly contributed to the experimental exploration of 1D physics and to the comparison with the theory.

(TM)₂X conductors exhibit a band filling commensurate with the underlying 1D lattice due to the 2:1 stoichiometry imposing half a hole per TM molecule. However, non-uniformity of the intermolecular spacing had been noticed from the early structural studies of (TMTTF)₂PF₆ crystal [39] due to the periodicity of the anion packing being twice the periodicity of the molecular packing. This non-uniformity provides a dimerization of the intrastack overlap quite significant in the sulfur series (prominent 1/2 filling Umklapp scattering) and still present in the members of the (TMTSF)₂PF₆ series. An important consequence of the commensurate situation for (TM)₂X materials is the existence of two localization channels either due to electron-electron Umklapp scattering with momentum transfer $4k_F$ (two particles scattering) or to $8k_F$ (four particles scattering) corresponding to half or quarter-filled bands respectively [40, 41]. Consequently, both Umklapp mechanisms compete for the establishment of the 1D charge gap $\Delta_{\rho}$, 1/4 and 1/2 filling Umpklapp scatterings leading to charge ordered and Mott insulators respectively [42].

On the left side of the generic diagram on fig(5) a phase transition towards a long range charge ordered (CO) insulating phase has been observed. This phase at low temperature has been ascribed, according to NMR data, to the onset of a charge disproportionation between molecules on the molecular chains [43]. Increasing pressure, spins localized on dimers of molecules couple to the lattice and give rise to a spin-Peierls ground state evolving through a quantum critical point around 10 kbar [44] into a Néel antiferromagnetic phase and subsequently a spin density wave state under pressure which have been extensively studied by transport [45] and AFMR [46].

Moving toward the right side of the diagram, the onset of charge localization below $T_{\rho}$ (i.e, 1D confinement) decreases under pressure as a result of an interplay between the Mott localization and the interchain coupling (increasing under pressure) which tends to bypass the correlation-induced localization. Once the localization gap vanishes, 1D confinement ceases and the existence of a quasi-1D Fermi surface becomes meaningful[41, 47, 48]. In some respects the deconfinement observed under pressure around 15 kbar on fig(5) is the signature for a crossover from strong to weak coupling in the generic phase diagram. The Néel phase turns into an ordered weak antiferromagnetic spin density wave phase (SDW) above the deconfinement pressure of about 15 kbar.

The temperature $T^\star$ where the $c^\star$-axis transport switches from an insulating to a metallic temperature dependence[49] corresponds to a cross-over between two regimes, see fig(5); a high temperature regime in which the finite QP weight at Fermi energy is smaller than the regular metallic value (possibly a TL liquid in the 1D case described below). $T^\star$ is highly pressure dependent and can be as small as 20 K or so, a value much smaller than the one expected for the bare $t_{\perp}$ of order 200 K. $T^\star$ would recover the value of the bare coupling only under infinite pressure.

Regarding the upper part of the diagram on fig(5) transport in (TM)₂X has been interpreted in terms of the Tomonaga-Luttinger liquid model for a commensurate conductor. A metal-like behaviour of the longitudinal resistance is still observed as long as $T$ is larger than $\Delta_{\rho}$ leading to a resistance displaying a power law $\rho (T) \approx T^{\theta}$ ($\theta >0$). Experimental studies have revealed such a metallic behaviour for the resistance either in (TMTTF)₂X under pressure [49, 50] or in (TMTSF)₂X even at ambient pressure [15]. What has been found is a sublinear exponent, namely $\theta = 0.93$ for the constant volume temperature dependence of the (TMTSF)₂PF₆ resistance [49, 50]. In the vicinity of $T^\star$, this power law evolves toward a constant value in line with a Fermi liquid picture.

When quarter-filled Umklapp scattering prevails at high temperature $\theta = 16K_{\rho}-3$ and consequently $K_{\rho} = 0.23$ according to the data of (TMTSF)₂PF₆ [50].

The temperature $T^{\star}$ where the $c^{\star}$-axis transport switches from an insulating to a metallic temperature dependence [49] corresponds to a cross-over between two regimes, see fig(5); a high temperature regime with no QP weight at Fermi energy (possibly a TL liquid in the 1D case described above) in which the QP weight increases with decreasing temperature.

A three dimensional anisotropic coherent picture prevails at low temperature in (TMTSF)₂X compounds according to the Kohler's rule [51] and angular magnetoresistance oscillations observed in (TMTSF)₂ClO₄ and (TMTSF)₂PF₆ under pressure [52, 53, 54, 55]. In addition, optical reflectance data of light polarised along $c^{\star}$ support the existence for (TMTSF)₂ClO₄ of a weak Drude behaviour in the liquid helium temperature domain when $k_{B}T<t_{\perp c}$ [56]. However, the upper limit for the temperature domain of 3D coherence has been established comparing the temperature dependence of the resistivity along $a$ and $c^{\star}$. This 2D to 3D crossover domain is defined by the temperature above which the temperature dependencies of both components of transport are no longer identical [57]. The 3D coherence regime is displayed on the generic diagram of fig(5). The low temperature limit of the metallic phase will be discussed in section(Organic Superconductivity#Antiferromagnetic fluctuations, spin and charge sectors).

Organic Superconductivity

Figure 6: Electronic contribution to the specific heat of (TMTSF)₂ClO₄, plotted as $C_e/T$ versus $T$, according to ref[59].

Although superconductivity in organic conductors has been first stabilized under pressure, more detailed investigations of this phenomenon have been conducted in (TMTSF)₂ClO₄, the only compound of the 1D-Bechgaard salts series exhibiting superconductivity at ambient pressure below 1.2 K. Evidences for superconductivity in (TMTSF)₂ClO₄ came out from transport [58], specific heat measurements [59] and Meissner flux expulsion [60].

The electronic contribution to the specific heat of (TMTSF)₂ClO₄ in a $C_e/T$ vs $T$ plot, fig(6), displays a very large anomaly around 1.2 K [59]. The total specific heat obeys the classical relation in metals $C/T = \gamma + \beta T^2$, where the Sommerfeld constant for electrons $\gamma$ = 10.5 mJ mol$^{-1}$K$^{-2}$, corresponding to a density of states at the Fermi level $N(E_F)$ = 2.1 states eV$^{-1}$ mol$^{-1}$ for the two spin directions [59]. The specific heat jump at the transition amounts then to $\Delta C_e /\gamma T_c = 1.67$, i.e. only slightly larger than the BCS ratio for a s-wave superconductor. Therefore, the specific heat data and the comparison between the value of the density of states derived specific heat Pauli susceptibility [61] lend support to a weak coupling Fermi liquid picture (at least in the low temperature range) [47]. The first critical field $H_{c1}$ obtained from the Meissner magnetization curves at low temperature read, 0.2,1 and 10 Oe along axes $a$, $b'$and $c^{\star}$ respectively. Following the values for the second critical fields $H_{c2}$ derived either from the Meissner experiments and the knowledge of the thermodynamical field [60] or from a direct measurements of transport, superconductivity can be considered to be in the extreme type II limit. The Ginzburg-Landau parameter $\kappa$ can even overcome 1000 when the field is along $a$ due to the weak interchain coupling in these Q1D conductors making the field penetration very easy for the parallel configuration of the external field. An interpretation for the critical fields assuming the pure limit of type II superconductors has been suggested in 1985 [62]. This proposal was based on the calculation of the microscopic expressions for the effective mass tensor in the Ginzburg-Landau equation near $T_c$ [63].

Figure 7: Diamagnetic shielding of (TMTSF)₂ClO₄ at T=0.05K for magnetic fields oriented along the three crystallographic axes, from ref [60].

Figure 8: Critical fields of (TMTSF)₂ClO₄ determined from $T_{c,onset}$ for the three principal axes with an indication for the Pauli limit at low temperature after [64, 65].

Given the experimental determination of the critical field derivatives near $T_{c}$ of $\mathrm{(TMTSF)_{2}ClO_{4}}$ [64, 65], $dH_{c2}^a/dT$ = 67.5kOe/K, $dH_{c2}^{b\prime}/dT$ = 36.3kOe/K, and $dH_{c2}^c/dT$ = 1.39kOe/K, Eqs.(1-3) lead to band parameters $t_a : t_b : t_c$ = 1200, 300 and 6K respectively are derived when critical fields derivatives are determined from $T_{c,onset}$ for the three principal axes after [64, 65].

Pairing mechanism in organic superconductivity

For several reasons one is entitled to believe that the pairing mechanism in organic superconductivity may differ from the regular electron-phonon driven pairing in traditional superconductors. First, superconductivity of quasi one dimensional Bechgaard salt shares a common border with magnetism as displayed on the generic diagram, see fig(5). Second, strong antiferromagnetic fluctuations in the normal state above $T_{c}$ in the vicinity of the SDW phase provide the dominant contribution to the nuclear hyperfine relaxation and are also controlling the linear temperature dependence of electronic transport. Some experimental results point to the existence of a non conventional pairing mechanism. They will be summarized below.

Non magnetic defects

A basic property of the BCS superconducting s-wave function is the time reversal symmetry of the Cooper pairs. Hence no pair breaking is expected from the scattering of electrons against spinless impurities [66]. Experimentally, this property has been verified in non-magnetic dilute alloys of s-wave superconductors and brought a strong support for the BCS model of conventional s-wave superconductors. However, the condition for time reversal symmetry is no longer met for the case of p-wave pairing. Consequently, $T_c$ for these superconductors should be strongly affected by any non-magnetic scattering. It is actually the extreme dependence of the critical temperature of Sr₂RuO₄ [67] on non-magnetic disorder which has provided a strong support in favour of triplet superconductivity in this compound. It is also the remarkable sensitivity of organic superconductivity to irradiation detected in the early years [68, 69] which led Abrikosov to suggest the possibility of triplet pairing in these materials [70].

A more recent investigation of the influence of non magnetic defects on organic superconductivity has been conducted following a procedure which rules out the addition of possible magnetic impurities as it can be the case for X-ray irradiated samples. Non magnetic defects can be introduced in a controlled way either by fast cooling preventing the complete ordering of the tetrahedral ClO₄ anions or by slow cooling the solid solution (TMTSF)₂(ClO4)_(1-x)ReO_4x.

Figure 9: Phase diagram of (TMTSF)₂(ClO4)_(1-x)ReO_4x governed by non magnetic disorder according to ref[71]. All circles refer to very slowly cooled samples in the R-state (the so-called relaxed state) with different $\mathrm{ReO_{4}^{-}}$ contents. The sample with $\rho_0=0.27(\Omega cm)^{-1}$ is metallic down to the lowest temperature of the experiment. The continuous line is a fit of the data with the Digamma function and $T_{c}{^0}=1.23K$.

As displayed on fig(9), $T_c$ in the solid solution is suppressed and the suppression can be related to the enhancement of the elastic scattering in the normal state. The data on fig9 show that the relation $T_{c}$ versus $\rho_0$ follows the theory proposed for the pair breaking by magnetic impurities in conventional superconductors, namely \[ln\Big(\frac{T_{c}^0}{T_{c}}\Big)=\psi\Big(\frac{1}{2}+\frac{\alpha T_{c}^0}{2\pi T_{c}}\Big)-\psi\Big(\frac{1}{2}\Big),\tag{1}\] with $\psi(x)$ being the Digamma function, $\alpha = \hbar /2 \tau k_{B}T_{c}^0$ the depairing parameter $\tau$. The experimental data follow the latter law with a good accuracy with $T_{c}{^0}=1.23K$.

The conventional pair breaking theory for magnetic impurities in usual superconductors has been generalized to the case of non-magnetic impurities in unconventional materials and the correction to $T_c$ obeys the above relation [72, 73]. Since the additional scattering cannot be ascribed to magnetic scattering according to the EPR checks showing no additional traces of localized spins in the solid solution, the data in fig(9) cannot be reconciled with the picture of a superconducting gap keeping a constant sign over the whole $(\pm k_F)$ Fermi surface. They require a picture of pair breaking in a superconductor with some unconventional gap symmetry.

Figure 10: Possible gap symmetries agreeing with the different experimental results. The d-wave (or g-wave) symmetry is the only symmetry agreeing with all experiments, (yellow column on line).

The influence of non magnetic impurities on the superconducting phase implies the existence of positive as well as negative values for the SC order parameter. It precludes the usual case of s-symmetry but is still unable to discriminate between two possible options namely, singlet-d (g) or triplet-p (f) [74], see fig(10).

Spin susceptibility in the superconducting phase

Regarding the spin part of the $SC$ wave function, a triplet pairing was first claimed in (TMTSF)₂PF₆ from a divergence of the critical field $H_{c2}$ exceeding the Pauli limiting value reported at low temperature in (TMTSF)₂PF₆ under pressure when $H$ is applied along the b' or a axes [75] and from the absence of any change in the $^{77}\mathrm{Se}$ Knight shift at $T_{c}$ [76].

Figure 11: $^{77}Se$ Knight shift vs $T$ for (TMTSF)₂ClO₄, for $H// b'$ and $a$, according to reference [77]. The sign of the variation of Knight shift at $T_{c}$ depends on the sign of the hyperfine field. A variation is observed only under low magnetic field. The difference of $T_{c}$ for the two field orientations is related to the anisotropy of $H_{c2}$. The field of $ 4T$ is still lower than the critical field derived from the onset of the resistive transition ($H_{c2}$$\approx 5T)$.

However more recent experiments performed at fields lower than those used in reference [76] for the work on (TMTSF)₂PF₆ did reveal a clear drop of the $^{77}\mathrm{Se}$ Knight shift below $T_{c}$in the compound (TMTSF)₂ClO₄ [77]. These new data provided a conclusive evidence in favour of singlet pairing, fig(11). In addition, a steep increase of the spin lattice relaxation rate versus magnetic field for both field orientations $\parallel a$ and b' provided evidence for a sharp cross-over or even a phase transition occurring at low temperature under magnetic field between the low field d-wave singlet phase and a high field regime exceeding the paramagnetic limit $H_p$ being either a triplet-paired state [78, 79] or an inhomogenous Fulde-Ferrell-Larkin-Ovchinnikov state [80, 73].

Magneto calorimetric studies

Figure 12: a- Fermi surface of (TMTSF)₂ClO₄ at low temperature in the presence of anion ordering [81]. b- Angular dependence of $C(\phi)$ at 0.14 K and 0.3 T according to ref[82]. c- First and second derivatives of the data displayed on a. d- Sketch of the influence of a magnetic field on the quasi particle DOS. The contribution of Doppler-shifted quasi-particles to the DOS is minimum when the magnetic field is parallel to v$_F$ at one of the nodes (left or right) . e and f- Simulations of the specific heat in the vicinity of the angle $\phi=0$ using the band structure on fig(12a). Arrows correspond to the different orientations of the magnetic field as shown on fig(12).

The response of superconductivity to non magnetic impurities and the loss of spin susceptibility at $T_{c}$ should be sufficient to qualify the $d$-wave alternative as the likely one among the various possibilities displayed on fig(10). In such a case, nodes of the gap should exist on the Fermi surface and could be located. This has been made possible via a measurement of the quasi particle density of states in the superconducting phase performed in an oriented magnetic field [82], see figs (12 b and c). This experiment is based on a Volovik's important remark [83] that for superconductors with line nodes, most of the density of states in their mixed state under magnetic field comes from the superfluid density far outside the vortex cores. The energy of QP's in the superfluid density rotating around the vortex core with the velocity v$_s(\bot H)$ is Doppler-shifted by the magnetic field and reads, \[\delta\omega\propto \textbf{v}_s\cdot \textbf{v}_F(\textbf{k})\tag{2}\] where v$_F$(k) is the Fermi velocity at the k point on the Fermi surface. The Doppler shift can contribute to the DOS in the superconducting state as long as it remains smaller than the superconducting energy gap, i.e only for those k states located in the vicinity of gap nodes on the Fermi surface. Hence, the contribution of the Doppler shift is minimum when the Fermi velocity at a node and the magnetic field are parallel according to eq (2), see fig(12d), and should contribute to a kink in the rotation pattern of the electronic specific heat $C(\phi)$.

The Doppler shift has already been used extensively in the case of 2D superconductors expected to reveal line nodes probing the thermal conduction or the specific heat while the magnetic field is rotated in the basal plane [84,85]. For such 2D conductors v$_F$(k) is usually colinear with k. Therefore, the angular resolved specific heat (or thermal conductivity) enables to reveal the positions of the gap nodes according to the angles corresponding to minima of specific heat (thermal conductivity) [84].

The situation is somewhat peculiar for Q1D Fermi surface. There, anomalies in the rotation pattern of $C_v$ are expected at angles $\phi_n$ between $H$ and v$_F$(k$_n$) where k$_n$ corresponds to a nodal position on the Fermi surface.

A simple model has been used to understand the data of angular resolved $C_v$ of (TMTSF)₂ClO₄ [82]. The rotation pattern has been modeled by, \[N(\phi)\propto \sqrt{H/H_{c2}(\phi)}\Sigma_{n}A_{n}\mid sin(\phi - \phi_{n})\mid\tag{3}\] where the first factor reproduces the anisotropic character of the critical field in the basal plane while the weighted summation over angles accounts for the existence of nodes at angles $\phi_n$ between the magnetic field and the special k$_n$ points where the Fermi velocity is parallel to H. The experimental observation of a rotation pattern at low field and low temperature which is non symmetrical with respect to the inversion of $\phi$ and the existence of kinks for $C(\phi)$ at $\phi=\pm 10^o$ on fig (12b) have been taken as the signature of line nodes [82]. According to the band structure calculation the location of the nodes on the Fermi surface becomes $k_y\sim\pm0.25b^*$. As far as the Pauli limitation is concerned, the superconducting phase diagram of (TMTSF)₂ClO₄ established by thermodynamic experiments indicates a value for the thermodynamic $H_{c2}$ along a of 2.5 T for $H\parallel a$ much smaller than the expected value of 7.7 T for the orbital limitation derived from the measurement of the temperature derivative of $H_{c2}$ close to $T_{c}$for a clean type II superconductor [62]. On the other hand, along the two transverse directions orbital limitation is at work [82]. The domain of field above the paramagnetic limit up to $H_{c2}$ derived from resistivity data where the specific heat recovers its normal state value requires further experimental investigations but it might be the signature of some FFLO phase as suggested by A. Lebed recently [86, 87, 88].

Antiferromagnetic fluctuations, spin and charge sectors

Figure 13: Plot of the nuclear relaxation versus temperature according to the data of ref[89]. A Korringa regime, $T_{1}T$ = const is observed down to 25K. The 2D AF regime is observed below $\approx 15 K$ and the small Curie-Weiss temperature of the 9 kbar run is the signature of the contribution of quantum critical fluctuations to the nuclear relaxation. The Curie-Weiss temperature becomes zero at the QCP. These data show that the QCP should be slightly below 9 kbar with the present pressure scale. The inset shows that the organic superconductor (TMTSF)₂ClO₄ at ambient pressure is very close to fulfill quantum critical conditions.

The metallic phase of (TMTSF)₂PF₆ in the 3D coherent regime when pressure is in the neighbourhood of the critical pressure $P_{c}$ behaves in way far from what is expected for a Fermi liquid.

The canonical Korringa law , $1/T_{1}T \propto {\chi^{2} (q=0,T)} $, is well obeyed at high temperature say, above 25 K, but the low temperature behaviour deviates strongly from the standard relaxation in paramagnetic metals. As shown on fig (13) an additional contribution to the relaxation rate emerges in top of the usual Korringa relaxation. This additional contribution rising at low temperature has been attributed to the onset of antiferromagnetic fluctuations in the vicinity of $P_{c}$ [90, 91, 92]. On the other hand in the low temperature regime the relaxation rate follows a law such as $T_{1}T=C(T+\Theta)$ as shown on fig(13). This is the Curie-Weiss behaviour for the relaxation which is to be observed in a 2D fluctuating antiferromagnet [93, 94, 95, 96]. The positive Curie-Weiss temperature $\Theta$ which provides the energy scale of the fluctuations is zero when pressure is equal to $P_{c}$ (the quantum critical conditions). When $\Theta$ becomes large comparable to T, the standard relaxation mechanism is recovered even at low temperature in agreement with the very high pressure results [97].

The inelastic scattering in transport reveals at once a strong linear term at low temperature evolving to quadratic in the high temperature regime with a general tendency to become quadratic at all temperatures when pressure is way above the critical pressure $P_{c}$ [90], see fig(5).

Figure 14: A log-log plot of the inelastic longitudinal resistivity of (TMTSF)₂PF₆ below $20$ K, according to ref[90].

Figure 15: Coefficient $A$ of linear resistivity as a function of $T_c$ plotted versus $T_{c}/T_{c0}$ for (TMTSF)₂PF₆. $T_c$ is defined as the midpoint of the transition and the error bars come from the 10$\%$ and 90$\%$ points with $T_{c0}=1.23K$ under the pressure of 8 kbar which provides the maximum $T_{c}$ in the $SDW$/$SC$ coexistence regime.The dashed line is a linear fit to all data points except that at $T_c$=0.87K, according to ref[90].

The existence of a linear temperature dependence of the resistivity is at variance with the sole $T^2$ dependence expected from the electron-electron scattering in a conventional Fermi liquid. This is clearly seen on a log-log plot of the resistivity versus T, see fig(14). Furthermore, the investigation of both transport and superconductivity under pressure in (TMTSF)₂PF₆ has established a correlation between the amplitude of the linear in temperature dependence of the resistivity and the value of $T_{c}$ as displayed on fig(15) suggesting a common origin for the inelastic scattering of the metallic phase and pairing in the $SC$ phase (TMTSF)₂PF₆ [90] as discussed in the next section.

Discussions and perspectives

The close proximity between antiferromagnetism and superconducting ground states of (TM)₂X superconductors and the deviation of the metallic phase from the traditional Fermi liquid behaviour have been recognized as early as the beginning eighties. The possibility for a pairing mechanism involving carriers on neighbouring chains in these quasi 1D conductors avoiding the Coulomb repulsion has been proposed by V. Emery in the context of the exchange phonon mechanism [98]. Shortly after, Emery and coworkers introduced the possibility that antiferromagnetic fluctuations play a role in the pairing mechanism [99, 100] but concluded that superconductivity could not emerge from pairing on the same organic chain. The exchange of spin fluctuations between carriers on neighbouring chains was thus proposed [99] to provide the necessary glue for pairing in analogy with the exchange of charge density waves proposed by Kohn and Luttinger [101] in the context of a new pairing mechanism in low dimensional conductors.

L. Caron and C. Bourbonnais [102, 103] extending their theory for the generic (TM)₂X phase diagram to the metallic domain made the proposal for a gap equation with singlet superconductivity based on an interchain magnetic coupling with an attraction deriving from an interchain exchange interaction overcoming the on-stack Coulomb repulsion. Taking into account the interference between the diverging Cooper and Peierls channels the renormalization treatment of Q-1-D conductors received more recently a significant improvement [104].

Within the framework of a weak-coupling limit the problem of the interplay between antiferromagnetism and superconductivity in the Bechgaard salts has been worked out using the renormalization group (RG) approach [74, 96] as summarized below taking into account only the 2D problem. The RG integration of high energy electronic degrees of freedom was carried out down to the Fermi level, and leads to a renormalization of the couplings at the temperature T [96, 104, 105]. The RG flow superimposes the $2k_F$ electron-hole (density-wave) and Cooper pairing many-body processes which combine and interfere at every order of perturbation. As a function of the `pressure' parameter $t_\perp'$ i.e the unnesting interstacks coupling, a singularity in the scattering amplitudes signals an instability of the metallic state toward the formation of an ordered state at some characteristic temperature scale. At low $t_\perp'$, nesting is sufficiently strong to induce a SDW instability in the temperature range of experimental $T_{\rm SDW}\sim 10-20$K. When the antinesting parameter approaches the threshold $t_\perp'^*$ from below ($t_\perp'^* \approx 25.4~{\rm K}$, using the above parameters), $T_{\rm SDW}$ sharply decreases and as a result of interference, SDW correlations ensure Cooper pairing attraction in the superconducting d-wave (SCd) channel. This gives rise to an instability of the normal state for the onset of SCd order at the temperature $T_c$ with pairing coming from antiferromagnetic spin fluctuations between carriers of neighbouring chains. Such a pairing model is actually supporting the conjecture of interchain pairing in order for the electrons to avoid the Coulomb repulsion made by V. Emery in 1983 and 1986 [98, 99].

Figure 16: Calculated phase diagram of the quasi-one dimensional electron gas model from the renormalization group method at the one-loop level [96]. $\Theta$ and the dash-dotted line defines the temperature region of the CW behavior for the inverse normalized SDW response function

The calculated phase diagram fig(16) with reasonable parameters taking $g_1=g_2/2 \approx 0.32$ for the backward and forward scattering amplitudes respectively and $g_3\approx 0.02$ for the longitudinal Umklapp scattering term captures the essential features of the experimentally-determined phase diagram of (TMTSF)₂PF₆ [92, 96], fig(5).

Sedeki et-al [106] have proceeded to an evaluation of the imaginary part of the one-particle self-energy which provides access to the electron-electron scattering rate $\tau^{-1}$. In addition to the regular Fermi-liquid component which goes as $T^2$ low frequency spin fluctuations yield $\tau^{-1} = aT\xi $, where a is constant and the antiferromagnetic correlation length $\xi(T)$ increases according to $\xi = c(T + \Theta)^{-1/2}$ as $T \rightarrow T_c$, where $\Theta$ is the temperature scale for spin fluctuations [106]. It is then natural to expect the Umklapp resistivity to contain (in the limit $T \ll \Theta$) a linear term AT besides the regular $BT^2$, whose magnitude would presumably be correlated with $T_c$, as both scattering and pairing are caused by the same antiferromagnetic correlations. The observation of a T-linear law for the resistivity up to 8K in (TMTSF)₂PF₆ under a pressure of 11.8 kbar as displayed on fig(14) is therefore consistent with the value of $\Theta=8K$ determined from NMR relaxation at 11 kbar on fig(13).

To conclude, both experimental and theoretical views point the contribution of electron correlations to the superconducting pairing problem. The extensive experimental evidence in favor of the emergence of superconductivity in the (TM)₂X family next to the stability threshold for antiferromagnetism has shown the need for a unified description of all electronic excitation that lies at the core of both density-wave and superconducting correlations. In this matter, the recent progresses achieved by the renormalization group method for the 1D-2D electron gas model have resulted in predictions about the possible symmetries of the superconducting order parameter when a purely electronic mechanism is involved, predictions that often differ from phenomenologically based approaches to superconductivity but are in fair agreement with the recent experimental findings. In this respect reexaminations of the superconducting state, specific heat, thermal conductivity,etc,...are highly demanded.

What is emerging from the work on these prototype 1D organic superconductors is their very simple electronic nature with only a single band at Fermi level, no prominent spin orbit coupling and extremely high chemical purity and stability.

They should be considered in several respects as model systems to inspire the physics of the more complex high $T_{c}$ superconductors, especially for pnictides and electron-doped cuprates. Most concepts discovered in these simple low dimensional conductors should be very useful for the study of other low dimensional systems such as carbon nanotubes or artificial 1D structures, etc,.....

The pairing mechanism behind organic superconductivity is likely different from the proposal made by Little but it is nevertheless a phonon-less mechanism, at least in (TM)₂X superconductors.

In conclusion, organic superconductivity has been discovered in the prototype Bechgaard salt (TMTSF)₂PF₆ but several other fascinating aspects of the physics of 1D conductors are worth mentioning namely, the magnetic field confinement discovered in the 1D's leading to the phenomenon of field induced spin density wave phases [107, 108, 109] and the quantization of the Hall effect in these phases [110, 111, 112]. Furthermore, the angular dependent magnetoresistance closely related to these anisotropic conductors [53, 113], (see [114] for a recent survey), the so-called Lebed-Osada-Danner-Chaikin oscillations in (TMTSF)₂X provide a nice illustration for the new features of the quasi-1D conductors [115].

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Related Readings

Organic superconductor http://en.wikipedia.org/wiki/Organic_superconductor

Unconventional superconductivity http://en.wikipedia.org/wiki/Unconventional_superconductor

Conventional superconductivity http://en.wikipedia.org/wiki/Conventional_superconductor

Original article for the discovery of Organic Superconductivity https://www.equipes.lps.u-psud.fr/JEROME/dossier%20complet/Jerome80a.pdf

The physics of organic superconductivity https://www.equipes.lps.u-psud.fr/JEROME/dossier%20complet/Jerome91.pdf

Organic conductors and superconductors https://www.equipes.lps.u-psud.fr/JEROME/dossier%20complet/Jerome82Schulz.pdf

Superconductivity without metal http://www.supraconductivite.fr/en/index.php?p=recherche-nouveaux-moleculaires

Sponsored by: Marcelo Rozenberg, Laboratoire de Physique des Solides, CNRS - Univeristé Paris Sud 11, Orsay, France

Reviewed by: Prof. Claude Bourbonnais, University of Sherbrooke, Sherbrooke, Canada

Reviewed by: Anonymous (via Marcelo Rozenberg, Laboratoire de Physique des Solides, CNRS - Univeristé Paris Sud 11, Orsay, France)

Accepted on: 2014-12-12 06:23:15 GMT