In highly anisotropic organic superconductor (TMTSF)$_2$ClO$_4$, superconducting (SC) phase coexists with metallic and spin density wave phases in the form of domains. Using the Maxwell-Garnett approximation (MGA), we calculate the volume ratio and estimate the shape of these embedded SC domains from resistivity data at various temperature and anion disorder, controlled by the cooling rate or annealing time of (TMTSF)$_{2}$ClO$_{4}$ samples. We found that the variation of cooling rate and of annealing time affect differently the shape of SC domains. In all cases the SC domains have oblate shape, being the shortest along the interlayer $z$-axis. This contradicts the widely assumed filamentary superconductivity along $z$-axis, used to explain the anisotropic superconductivity onset. We show that anisotropic resistivity drop at the SC transition can be described by the analytical MGA theory with anisotropic background resistance, while the anisotropic $T_c$ can be explained by considering a finite size and flat shape of the samples. Due to a flat/needle sample shape, the probability of percolation via SC domains is the highest along the shortest sample dimension ($z$-axis), and the lowest along the sample length ($x$-axis). Our theory can be applied to other heterogeneous superconductors, where the size $d$ of SC domains is much larger than the SC coherence length $ξ$, e.g. cuprates, iron based or organic superconductors. It is also applicable when the spin/charge-density wave domains are embedded inside a metallic background, or vice versa.
We calculate angle-dependent heat capacity in a low magnetic field range on the basis of Kramer-Pesch approximation together with an electronic structure obtained by first-principles calculations to determine a superconducting gap function of (TMTSF)2ClO4 through its comparisons with experiments. The present comparative studies reveal that a nodal d-wave gap function consistently explains the experimental results for (TMTSF)2ClO4. Especially, it is emphasized that the observed unusual axis-asymmetry of the angle-dependence eliminates the possibility of s-wave and node-less d-wave functions. It…
Using various transport and magnetotransport probes we study the coexistence of spin-density wave and superconductor states in (TMTSF)2ClO4 at various degrees of ClO4 anions ordering. In the two-phase complex state when both superconductivity and spin-density wave are observed in transport, we find prehistory effects, enhancement of the superconducting critical field, and strong spatial anisotropy of the superconducting state. These features are inconsistent with the conventional model of structural inhomogeneities produced by anion ordering transition. We reveal instead that superconductor an…
We propose a new approach for studying spin density waves (SDW) in the Bechgaard salt (TMTSF)2ClO4 where lattice is dimerized in transverse direction due to anion ordering. The SDW response is calculated in the matrix formulation that rigorously treats the hybridization of inter-band and intra-band SDW correlations. Since the dimerization gap is large, of the order of transverse bandwidth, we also develop an exact treatment of magnetic breakdown in the external magnetic field. The obtained results agree with the experimental data on the fast magneto-resistance oscillations. Experimentally foun…
The noncentrosymmetric ClO$_4$ anions of the organic superconductor (TMTSF)$_2$ClO$_4$ order below 24K. The size of domains where the anions are ordered is substantially dependent on the cooling rate which is a key parameter for the stability of the low temperature electronic ground states. We study the effect of the cooling rate on the SC phase within a self consistent approach in the framework of the time dependent Ginzburg-Landau theory taking into account the superconducting fluctuations. We derive the superconducting transition temperature which is found to decrease with increasing coolin…
Due to the competition of two anion orders, (TMTSF)2ReO4, presents a phase coexistence between semiconducting and metallic (superconducting) regions (filaments or droplets) in a wide range of pressure. In this regime, the superconducting upper critical field for H parallel to both c* and b' axes present a linear part at low fields followed by a divergence above a cross-over field. This cross-over corresponds to the 3D-2D decoupling transition expected in filamentary or granular superconductors. The sharpness of the transition also demonstrates that all filaments are of similar sizes and self o…
We propose a mechanism of nodeless d-wave superconductivity in a quasi-one-dimensional organic superconductor under anion order. It is shown that split of the Fermi surfaces due to the anion order eliminates the line nodes of a d-wave gap function on the Fermi surfaces, and a small finite energy gap appears in the quasi-particle excitations of superconductivity. We discuss that temperature dependences of thermal conductivity and NMR relaxation rate observed in a quasi-one-dimensional organic superconductor (TMTSF)2ClO4 can be explained consistently by the small energy gap.
We report a study of the organic compound $(TMTSF)_2 ClO_4$ in both a sample cooled very slowly through the anion ordering temperature (relaxed state) and a sample cooled more rapidly (intermediate state). For the relaxed state the entire sample is observed to be superconducting below about T_c ~ 1.2 K. The second moment of the internal field distribution was measured for the relaxed state yielding an in-plane penetration depth of ~ 12000 Angstroms. The intermediate state sample entered a mixed phase state, characterized by coexisting macroscopic sized regions of superconducting and spin densi…
The study of the anion ordered (TMTSF)\_2ClO\_4\_(1-x)ReO\_4\_x, solid solution in the limit of a low ReO\_4- substitution level (0<=x<=17%) has revealed a new and interesting phase diagram. Superconductivity is drastically suppressed as the effect of ReO\_4- non magnetic point defects increases following the digamma behaviour for usual superconductors in the presence of paramagnetic impurities. Then, no long range order can be stabilized above 0.1K in a narrow window of substitution. Finally, an insulating SDW ground state in ReO\_4- -rich samples is rapidly stabilized with the decrease of th…
We report comprehensive (magneto)transport studies of the two-phase state in (TMTSF)2ClO4, where superconducting (SC) phase coexists with spin-density wave insulator (SDW). By tuning the degree of ClO4 anion ordering in controlled manner we drive SDW state and study the evolution of the SC spatial texture. We find that as SDW is suppressed, SC regions initially appear inside the SDW insulator in a form of filaments extended in the interlayer direction and further merge into the two-dimensional sheets across the most conducting axis of the crystal. We demonstrate that almost all our results can…
Cantilever magnetometry has been used to measure the upper critical magnetic field $H_{c2}$ of the quasi-one dimensional molecular organic superconductor (TMTSF)$_2$ClO$_4$. From simultaneous resistivity and torque magnetization experiments conducted under precise field alignment, $H_{c2}$ at low temperature is shown to reach 5T, nearly twice the Pauli paramagnetic limit imposed on spin singlet superconductors. These results constitute the first thermodynamic evidence for a large $H_{c2}$ in this system and provide support for spin triplet pairing in this unconventional superconductor
We solve a long-standing problem about a theoretical description of the upper critical magnetic field, parallel to conducting layers and perpendicular to conducting chains, in (TMTSF)$_2$ClO$_4$ superconductor. In particular, we explain why the experimental upper critical field, $H^{b'}_{c2} \simeq 6 \ T$, is higher than both the quasi-classical upper critical field and Clogston paramagnetic limit. We show that this property is due to the coexistence of the hidden Reentrant and Larkin-Ovchinnikov-Fulde-Ferrell phases in a magnetic field in a form of three plane waves with non-zero momenta of t…
We theoretically determine the Ginzburg-Landau slopes of the anisotropic upper critical magnetic field in a quasi-one-dimensional superconductor and correct the previous works on this issue. By using the experimentally measured values of the Ginzburg-Landau slopes in the superconductor (TMTSF)$_2$ClO$_4$, we determine band parameters of its electron spectrum. Our main result is that the so-called quantum dimensional crossover has to happen in this material in magnetic fields, $H = 3-8 \ T$, which are much lower than the previously assumed. We discuss how this fact influences metallic and super…
We present a first-principles band calculation for the quasi-one-dimensional (Q1D) organic superconductor (TMTSF)$_2$ClO$_4$. An effective tight-binding model with the TMTSF molecule to be regarded as the site is derived from a calculation based on maximally localized Wannier orbitals. We apply a two-particle self-consistent (TPSC) analysis by using a four-site Hubbard model, which is composed of the tight-binding model and an on-site (intramolecular) repulsive interaction, which serves as a variable parameter. We assume that the pairing mechanism is mediated by the spin fluctuation, and the s…
Using a proper cooling procedure, a controllable amount of non-magnetic structural disorder can be introduced at low temperature in (TMTSF)2ClO4. Here we performed simultaneous measurements of transport and magnetic properties of (TMTSF)2ClO4 in its normal and superconducting states, while finely covering three orders of magnitude of the cooling rate around the anion ordering temperature. Our result reveals, with increasing density of disorder, the existence of a crossover between homogeneous defect-controlled d-wave superconductivity and granular superconductivity. At slow cooling rates, with…
We present a study of the superconducting properties (Tc and Hc2) in the solid solution (TMTSF)2(ClO4)(1-x)(ReO4)x with a ReO-4 nominal concentration up to x = 6%. The dramatic suppression of Tc when the residual resistivity is increased upon alloying with no modification of the Fermi surface is the signature of non-conventional superconductivity . This behaviour strongly supports p or d wave pairing in quasi one dimensional organic superconductors. The determination of the electron lifetime in the normal state at low temperature confirms that a single particle Drude model is unable to explain…
We suggest theoretical explanation of the high upper critical magnetic field, perpendicular to conducting chains, Hc2, experimentally observed in the superconductor (TMTSF)2ClO4, in terms of singlet superconducting pairing. In particular, we compare the results of d-wave-like nodal, d-wave-like node-less, and s-wave scenarios of superconductivity. We show that, in d-wave-like nodal scenario, superconductivity can naturally exceed both the orbital upper critical magnetic field and Clogston-Shandrasekhar paramagnetic limit as well as reach experimental value, Hc2 = 6T, in contrast to d-wave-like…
We consider a formation of the Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase in a quasi-one-dimensional (Q1D) conductor in a magnetic field, parallel to its conducting chains, where we take into account both the paramagnetic spin-splitting and orbital destructive effects against superconductivity. We show that, due to a relative weakness of the orbital effects in a Q1D case, the LOFF phase appears in (TMTSF)$_2$ClO$_4$ superconductor for real values of its Q1D band parameters. We compare our theoretical calculations with the recent experimental data by Y. Maeno's group [S. Yonezawa et al., Phy…
Recent experiments indicate that the Bechgaard salts (TMTSF)$_2$ ClO$_4$ and (TMTSF)$_2$ PF$_6$ may be unconventional triplet superconductors. The quasiparticle density of states and the uniform spin susceptibility tensor are computed at low temperatures for order parameter symmetries, as an attempt to narrow the number of possibilities based on experimental evidence.
$(TMTSF)_2ClO_4$ is a quasi-one dimensional organic conductor and superconductor with $T_c=1.4$K, and one of at least two Bechgaard salts observed to have upper critical fields far exceeding the paramagnetic limit. Nevertheless, the $^{77}$Se NMR Knight shift at low fields reveals a decrease in spin susceptibility $χ_s$ consistent with singlet spin pairing. The field dependence of the spin-lattice relaxation rate at 100mK exhibits a sharp crossover (or phase transition) at a field $H_s\sim15$kOe, to a regime where $χ_s$ is close to the normal state value, even though \hc2 $\gg H_s$.
We report on dc and microwave experiments of the low-dimensional organic conductors (TMTSF)$_2$PF$_6$ and (TMTSF)$_2$ClO$_4$ along the $a$, $b^{\prime}$, and $c^*$ directions. In the normal state of (TMTSF)$_2$PF$_6$ below T=70 K, the dc resistivity follows a power-law with $ρ_a$ and $ρ_{b^{\prime}}$ proportional to $T^2$ while $ρ_{c^*}\propto T$. Above $T = 100$ K the exponents extracted from the data for the $a$ and $c^*$ axes are consiste1nt with what is to be expected for a system of coupled one-dimensional chains (Luttinger liquid) and a dimensional crossover at a temperature of about 100…
We discuss an analytical model to calculate the superconducting volume ratio. Apart from this, our model can also predict the shape of embedded superconducting domains. We applied our model to calculate the superconducting volume ratios and shape of domains in (TMTSF)$_2$PF$_6$, (TMTSF)$_2$ClO$_4$, YBa$_2$Cu$_4$O$_8$, $β$-(BEDT)TTF$_2$I$_3$ and FeSe. Usually in layered superconductors resistivity drops anisotropically. Our analysis also explains that, this behaviour is due to flat or needle shape of the superconducting samples.
The quasi-1D organic Bechgaard salt (TMTSF)$_2$PF$_6$ displays spin-density-wave (SDW) order and superconductivity in close proximity in the temperature-pressure phase diagram. We have measured its normal-state electrical resistivity $ρ_a(T)$ as a function of temperature and pressure, in the $T \to 0$ limit. At the critical pressure where SDW order disappears, $ρ_a(T) \propto T$ down to the lowest measured temperature (0.1 K). With increasing pressure, $ρ_a(T)$ acquires a curvature that is well described by $ρ_a(T) = ρ_0 + AT + BT^2$, where the strength of the linear term, measured by the $A$ …
There are strong experimental evidences of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state formation in layered organic superconductors in parallel magnetic field. We study theoretically the interplay between the orbital effect and the FFLO modulation in this case and demonstrate that the in-plane critical field anisotropy drastically changes at the transition to the FFLO state. The very peculiar angular dependence of the superconducting onset temperature which is predicted may serve for unambiguous identification of the FFLO modulation. The obtained results permit us to suggest the modulate…
A linear temperature dependence of the electrical resistivity as T -> 0 is the hallmark of quantum criticality in heavy-fermion metals and the archetypal normal-state property of high-Tc superconductors, yet in both cases it remains unexplained. We report a linear resistivity on the border of spin-density-wave order in the organic superconductor (TMTSF)2X (X = PF6, ClO4), whose strength scales with the superconducting temperature Tc. This scaling, also present in the pnictide superconductors, reveals an intimate connection between linear-T scattering and pairing, shown by renormalization group…
Using the quasi-classical formalism we provide the description of the temperature and field-direction dependence of the in-plane upper critical field in layered superconductors, taking into account the interlayer Josephson coupling and the paramagnetic spin splitting. We generalize the Lawrence-Doniach model for the case of high magnetic fields and show that the re-entrant superconductivity is naturally described by our formalism when neglecting the Pauli pair breaking effect. We demonstrate that in layered superconductors the in-plane anisotropy of the onset of superconductivity exhibits four…
We show that there are strong similarities in the spin lattice relaxation of non-magnetic organic charge transfer salts, and that these similarities can be understood in terms of spin fluctuations. Further, we show that, in all of the kappa-phase organic superconductors for which there is nuclear magnetic resonance data, the energy scale for the spin fluctuations coincides with the energy scale for the pseudogap. This suggests that the pseudogap is caused by short-range spin correlations. In the weakly frustrated metals k-(BEDT-TTF)_2Cu[N(CN)_2]Br, k-(BEDT-TTF)_2Cu(NCS)_2, and k-(BEDT-TTF)_2Cu…
The results of 13C NMR spectra and nuclear spin-lattice relaxation rate 1/T_1 for the quasi-one-dimensional quarter-filled organic material (TMTTF)2SCN are presented. A new inhomogeneous line appears below the anion ordering temperature (T_AO), below which the intensity of the distinct line owing to the inner carbon site from the inversion center is halved. This indicates that the charge transfer (disproportionation) occurs from one inner carbon site to the other inner site exclusively within one unit cell triggered by anion ordering. The 1/T_1 for the charge-donating inner site shows activate…
S-band metals such as alkali and alkaline earth metals do not undergo a superconducting transition (SCT) at an ambient pressure, but their high-pressure phases do. In contrast, room temperature stable electride electride (C12A7:e-) in which anionic electrons in the crystallographic sub-nanometer-size cages have high s-character exhibits SCT at 0.2-0.4K at an ambient pressure. In this paper we report that crystal and electronic structure of C12A7:e- are close to those of the high pressure superconducting phase of alkali and alkaline earth metals and the SCT of both materials is induced when ele…
