Triplet superconductivity in an organic superconductor probed by NMR Knight shift

Pairing symmetries in the Zeeman-coupled extended attractive Hubbard model

By introducing the possibility of equal- and opposite-spin pairings concurrently, we show that the ground state of the extended attractive Hubbard model (EAHM) exhibits rich phase diagrams with a variety of singlet, triplet, and mixed parity superconducting orders. We study the competition between these superconducting pairing symmetries invoking an unrestricted Hartree-Fock-Bogoliubov-de Gennes (HFBdG) mean-field approach, and we use the d-vector formalism to characterize the nature of the stabilized superconducting orders. We discover that, while all other types of orders are suppressed, a non-unitary triplet order dominates the phase space in the presence of an in-plane external magnetic field. We also find a transition between a non-unitary to unitary superconducting phase driven by the change in average electron density. Our results serve as a reference for identifying and understanding the nature of superconductivity based on the symmetries of the pairing correlations. The results further highlight that EAHM is a suitable effective model for describing most of the pairing symmetries discovered in different materials.

Modern History of Organic Conductors: An Overview

This short review article provides the reader with a summary of the history of organic conductors. To retain a neutral and objective point of view regarding the history, background, novelty, and details of each research subject within this field, a thousand references have been cited with full titles and arranged in chronological order. Among the research conducted over ~70 years, topics from the last two decades are discussed in more detail than the rest. Unlike other papers in this issue, this review will help readers to understand the origin of each topic within the field of organic conductors and how they have evolved. Due to the advancements achieved over these 70 years, the field is nearing new horizons. As history is often a reflection of the future, this review is expected to show the future directions of this research field.

The Zeeman, spin-orbit, and quantum spin Hall interactions in anisotropic and low-dimensional conductors

To construct a microscopic theory of electrons and holes in anisotropic conductors that self-consistently treats their band effective mass anisotropy with their interactions with applied electric and magnetic fields, the Dirac equation is extended for an electron or hole in an orthorhombically-anisotropic conduction band. Its covariance is established both by a modified version of the Klemm-Clem transformations to a space in which it is isotropic, and also in its fully anisotropic form by making the most general proper and improper Lorentz transformations, proving its validity in both the relativistic and non-relativistic limits. The appropriate Foldy-Wouthuysen transformations are extended to expand about the non-relativistic Hamiltonian limit to fourth order in the inverse of the particle's Einstein rest energy. The results have important consequences for magnetic measurements of many classes of clean anisotropic semiconductors, metals, and superconductors. In all of these cases, the Zeeman interaction is found to depend strongly upon the effective mass anisotropy. When an electron or hole is traveling in an atomically thin one-dimensional conduction band, its Zeeman, spin-orbit, and quantum spin Hall interactions are vanishingly small. Accurate expressions for the Zeeman, spin-orbit and quantum spin Hall interactions for two-dimensional conductors are provided.

Synthetic Weyl Points and Chiral Anomaly in Majorana Devices with Nonstandard Andreev-Bound-State Spectra

We demonstrate how to design various nonstandard types of Andreev-bound-state (ABS) dispersions, via a composite construction relying on Majorana bound states (MBSs). Here, the MBSs appear at the interface of a Josephson junction consisting of two topological superconductors (TSCs). Each TSC harbors multiple MBSs per edge by virtue of a chiral or unitary symmetry. We find that, while the ABS dispersions are 2π periodic, they still contain multiple crossings which are protected by the conservation of fermion parity. A single junction with four interface MBSs and all MBS couplings fully controllable, or networks of such coupled junctions with partial coupling tunability, open the door for topological band structures with Weyl points or nodes in synthetic dimensions, which in turn allow for fermion-parity (FP) pumping with a cycle set by the ABS-dispersion details. In fact, in the case of nodes, the FP pumping is a manifestation of chiral anomaly in 2D synthetic spacetime. The possible experimental demonstration of ABS engineering in these devices further promises to unveil new paths for the detection of MBSs and higher-dimensional chiral anomaly.

Triplet superconductivity in coupled odd-gon rings

Shedding light on the nature of spin-triplet superconductivity has been a long-standing quest in condensed matter physics since the discovery of superfluidity in liquid 3He. Nevertheless, the mechanism of spin-triplet pairing is much less understood than that of spin-singlet pairing explained by the Bardeen-Cooper-Schrieffer theory or even observed in high-temperature superconductors. Here we propose a versatile mechanism for spin-triplet superconductivity which emerges through a melting of macroscopic spin polarization stabilized in weakly coupled odd-gon (e.g., triangle, pentagon, etc) systems. We demonstrate the feasibility of sustaining spin-triplet superconductivity with this mechanism by considering a new class of quasi-one-dimensional superconductors A2Cr3As3 (A = K, Rb, and Cs). Furthermore, we suggest a simple effective model to easily illustrate the adaptability of the mechanism to general systems consisting of odd-gon units. This mechanism provides a rare example of superconductivity from on-site Coulomb repulsion.

Exotic superconducting states in the extended attractive Hubbard model

We show that the extended attractive Hubbard model on a square lattice allows for a variety of superconducting phases, including exotic mixed-symmetry phases with [Formula: see text] and [Formula: see text] symmetries, and a novel [Formula: see text] state. The calculations are performed within the Hartree-Fock Bardeen-Cooper-Schrieffer framework. The ground states of the mean-field Hamiltonian are obtained via a minimization scheme that relaxes the symmetry constraints on the superconducting solutions, hence allowing for a mixing of s-, p- and d-wave order parameters. The results are obtained within the assumption of uniform-density states. Our results show that extended attractive Hubbard model can serve as an effective model for investigating properties of exotic superconductors.

Microscopic model of the Knight shift in anisotropic and correlated metals

We present a microscopic model of nuclear magnetic resonance in metals. The spin-1/2 local nucleus and its surrounding orbital electrons interact with the arbitrary constant B(0) and perpendicular time-oscillatory magnetic inductions B1(t) and with each other via an anisotropic hyperfine interaction. An Anderson-like Hamiltonian describes the excitations of the relevant occupied local orbital electrons into the conduction bands, each band described by an anisotropic effective mass with corresponding Landau orbits and an anisotropic spin g tensor. Local orbital electron correlation effects are included using the mean-field decoupling procedure of Lacroix. The Knight resonance frequency and corresponding linewidth shifts are evaluated to leading orders in the hyperfine and Anderson excitation interactions. While respectively proportional to (B1/B0)2 and a constant for weak B(0) >> B1, both highly anisotropic shifts depend strongly upon B(0) when a Landau level is near the Fermi energy. Electron correlations affect the anisotropy of the linewidth shift. The model is easily generalizable to arbitrary nuclear spin I.

Observation of orbital resonance Hall effect in (TMTSF)2ClO4

We report the observation of a Hall effect driven by orbital resonance in the quasi-1-dimensional (q1D) organic conductor (TMTSF)2ClO4. Although a conventional Hall effect is not expected in this class of materials due to their reduced dimensionality, we observed a prominent Hall response at certain orientations of the magnetic field B corresponding to lattice vectors of the constituent molecular chains, known as the magic angles (MAs). We show that this Hall effect can be understood as the response of conducting planes generated by an effective locking of the orbital motion of the charge carriers to the MA driven by an electron-trajectory resonance. This phenomenon supports a class of theories describing the rich behavior of MA phenomena in q1D materials based on altered dimensionality. Furthermore, we observed that the effective carrier density of the conducting planes is exponentially suppressed in large B, which indicates possible density wave formation.

The Fulde-Ferrell-Larkin-Ovchinnikov state in layered d-wave superconductors: in-plane anisotropy and resonance effects in the angular dependence of the upper critical field

We study the anisotropy of the in-plane upper critical magnetic field coupled to the orbital motion and the spins of electrons in a layered d(x2-y2) organic superconductor in the spatially modulated Fulde-Ferrell-Larkin-Ovchinnikov phase. We show that the interplay between the nodal structure of the order parameter and its spatial modulation results in the very peculiar angular dependence of the onset of superconductivity in the high-field regime. The principal axis of the field-direction dependence of the onset of superconductivity is tilted by π/4 in the temperature range 0.056 < or approximately equal T < 0.56. In some cases the resonance between the modulation wavevector and the vector potential of a parallel magnetic field may lead to anomalous cusps in the temperature and in-plane angular dependences of the onset of superconductivity. The obtained results support the interpretation of the recent experiments as evidence of the FFLO state.

In-plane magnetic field anisotropy of the Fulde-Ferrell-Larkin-Ovchinnikov state in layered superconductors

There is strong experimental evidence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state formation in layered organic superconductors in a 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 modulated phase stabilization as the origin of the magnetic-field angle dependence of the onset of superconductivity experimentally observed in (TMTSF)2ClO4 organic conductors.

Upper critical magnetic field far above the paramagnetic pair-breaking limit of superconducting one-dimensional Li0:9Mo6O17 single crystals

The upper critical field H(c2) of purple bronze Li0:9Mo6O17 is found to exhibit a large anisotropy, in quantitative agreement with that expected from the observed electrical resistivity anisotropy. With the field aligned along the most conducting axis, H(c2) increases monotonically with decreasing temperature to a value 5 times larger than the estimated paramagnetic pair-breaking field. Theories for the enhancement of H(c2) invoking spin-orbit scattering or strong-coupling superconductivity are shown to be inadequate in explaining the observed behavior, suggesting that the pairing state in Li0:9Mo6O17 is unconventional and possibly spin triplet.

Transverse spin current in the s-wave/p-wave Josephson junction

We report a theoretical study on spin transport in the hybrid Josephson junction composed of singlet s-wave and triplet p-wave superconductor. The node of the triplet pair potential is considered perpendicular to the interface of the junction. Based on a symmetry analysis, we predict that there is no net spin density at the interface of the junction but instead a transverse mode-resolved spin density can exist and a nonzero spin current can flow transversely along the interface of the junction. The predictions are numerically demonstrated by means of the lattice Matsubara Green's function method. It is also shown that, when a normal metal is sandwiched in between two superconductors, both spin current and transverse mode-resolved spin density are only residing at two interfaces due to the smearing effect of the multimode transport. Our findings are useful for identifying the pairing symmetry of the p-wave superconductor and generating spin current.

Hidden reentrant and Larkin-Ovchinnikov-Fulde-Ferrell superconducting phases in a magnetic field in a (TMTSF)2ClO4

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 a (TMTSF)(2)ClO(4) superconductor. In particular, we explain why the experimental upper critical field, H(c2)(b')≃6 T, is higher than both the quasiclassical upper critical field and the 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 the form of three plane waves with nonzero momenta of the Cooper pairs. Our results are in good qualitative and quantitative agreement with the recent experimental measurements of H(c2)(b') and support a singlet d-wave-like scenario of superconductivity in (TMTSF)(2)ClO(4).

Strong parity mixing in the Fulde-Ferrell-Larkin-Ovchinnikov superconductivity in systems with coexisting spin and charge fluctuations

We study the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state of spin fluctuation mediated pairing, and focus on the effect of coexisting charge fluctuations. We find that (i) consecutive transitions from singlet pairing to FFLO and further to S_{z}=1 triplet pairing can generally take place upon increasing the magnetic field when strong charge fluctuations coexist with spin fluctuations, and (ii) the enhancement of the charge fluctuations lead to a significant increase of the parity mixing in the FFLO state, where the triplet/singlet component ratio in the gap function can be close to unity. We propose that such consecutive pairing state transition and strong parity mixing in the FFLO state may take place in a quasi-one-dimensional organic superconductor (TMTSF)2X.

Spontaneous spin accumulation in singlet-triplet josephson junctions

We study the Andreev bound states in a Josephson junction between a singlet and a triplet superconductors. Because of the mismatch in the spin symmetries of pairing, the energies of the spin-up and -down quasiparticles are generally different. This results in imbalance of spin populations and net spin accumulation at the junction in equilibrium. This effect can be detected using probes of local magnetic field, such as the scanning SQUID, Hall, and Kerr probes. It may help to identify potential triplet pairing in (TMTSF)2X, Sr2RuO4, and oxypnictides.

Anomalous in-plane anisotropy of the onset of superconductivity in (TMTSF)2ClO4

We report the magnetic-field amplitude and field-angle dependence of the superconducting onset temperature Tconset of the organic superconductor (TMTSF)2ClO4 in magnetic fields H accurately aligned to the conductive ab' plane. We revealed that the rapid increase of the onset fields at low temperatures occurs both for H || b' and H || a, irrespective of the carrier confinement. Moreover, in the vicinity of the Pauli-limiting field, we report a shift of a principal axis of the in-plane field-angle dependence of Tconset. This feature may be related to an occurrence of Fulde-Ferrell-Larkin-Ovchinnikov phases.

Fabrication and efficiency evaluation of a hybrid NiCrAl pressure cell up to 4 GPa

A hybrid NiCrAl pressure cell was fabricated to measure magnetic quantities under high pressure above 3 GPa. A pressure of 4.0 GPa was achieved and the pressure cell was found to be reusable even after a pressurizing trial up to 4.0 GPa. Pressure was monitored using (63)Cu nuclear quadrupole resonance of Cu(2)O and ruby fluorescence. The pressure efficiency of a fresh cell was maintained at 96%, and no appreciable deformation was observed at pressures below 3 GPa; on the other hand, the efficiency after pressurizing trials decreased gradually and reached 75% at 4 GPa accompanied by a maximum expansion inside the cylinder of 2%.

Superconducting state of the organic conductor (TMTSF)2ClO4

(TMTSF)2ClO4 is a quasi-one-dimensional organic conductor and superconductor with Tc=1.4 K, and one of at least two Bechgaard salts observed to have upper critical fields far exceeding the paramagnetic limit. Nevertheless, the 77Se NMR Knight shift at low fields reveals a decrease in spin susceptibility chi(s) consistent with singlet spin pairing. The field dependence of the spin-lattice relaxation rate at 100 mK exhibits a sharp crossover (or phase transition) at a field Hs approximately 15 kOe, to a regime where chi(s) is close to the normal state value, even though Hc2>> Hs.

Spin Triplet Superconducting State due to Broken Inversion Symmetry in Li(2)Pt(3)B

We report (11)B and (195)Pt NMR measurements in noncentrosymmetric superconductor Li(2)Pt(3)B. We find that the spin susceptibility measured by the Knight shift remains unchanged across the superconducting transition temperature T(c). With decreasing temperature (T) below T(c), the spin-lattice relaxation rate 1/T(1) decreases with no coherence peak and is in proportion to T3. These results indicate that the Cooper pair is in the spin-triplet state and that there exist line nodes in the superconducting gap function. They are in sharp contrast to those in the isostructural Li(2)Pd(3)B which is a spin-singlet, s-wave superconductor, and are ascribed to the enhanced spin-orbit coupling due to the lack of spatial inversion symmetry. Our finding points to a new paradigm where exotic superconductivity arises in the absence of electron-electron correlations.

Effect of the vortices on the nuclear spin relaxation rate in the unconventional pairing states of the organic superconductor (TMTSF)2PF6

This Letter theoretically discusses quasiparticle states and nuclear spin relaxation rates T1-1 in the quasi-one-dimensional superconductor (TMTSF)2PF6 under a magnetic field applied parallel to the conduction chains. We study the effects of Josephson-type vortices on T1(-1) by solving the Bogoliubov-de Gennes equation for p-, d- or f-wave pairing interactions. In the presence of line nodes in pairing functions, T1(-1) is proportional to T in sufficiently low temperatures because quasiparticles induced by vortices at the Fermi energy relax spins. We also try to identify the pairing symmetry of (TMTSF)2PF6.

Spin-wave contribution to the nuclear spin-lattice relaxation in triplet superconductors

We discuss collective spin-wave excitations in triplet superconductors with an easy axis anisotropy for the order parameter. Using a microscopic model for interacting electrons, we estimate the frequency of such excitations in Bechgaard salts and ruthenate superconductors to be 1 and 20 GHz, respectively. We introduce an effective bosonic model to describe spin-wave excitations and calculate their contribution to the nuclear spin-lattice relaxation rate. We find that, in the experimentally relevant regime of temperatures, this mechanism leads to the power law scaling of 1/T1 with temperature. For two- and three-dimensional systems, the scaling exponents are 3 and 5, respectively. We discuss experimental manifestations of the spin-wave mechanism of the nuclear spin-lattice relaxation.

Triplet superconducting pairing and density-wave instabilities in organic conductors

Using a renormalization group approach, we determine the phase diagram of an extended quasi-one-dimensional electron gas model that includes interchain hopping, nesting deviations, and both intrachain and interchain repulsive interactions. We find a close proximity of spin-density- and charge-density-wave phases and singlet d-wave and triplet f-wave superconducting phases. There is a striking correspondence between our results and recent puzzling experimental findings in the Bechgaard salts, including the coexistence of spin-density-wave and charge-density-wave phases and the possibility of a triplet pairing in the superconducting phase.

Resonant nernst effect in the metallic and field-induced spin density wave States of (TMTSF)2ClO4

We examine an unusual phenomenon where, in tilted magnetic fields near magic angles parallel to crystallographic planes, a "giant" resonant Nernst signal has been observed by Wu et al. [Phys. Rev. Lett. 91, 056601 (2003)] in the metallic state of an organic conducting Bechgaard salt. We show that this effect appears to be a general feature of these materials and is also present in the field-induced spin density wave phase with even larger amplitude. Our results place conditions on any model that treats the metallic state as a state with finite Cooper pairing.

Coexistence of superconductivity and antiferromagnetism probed by simultaneous nuclear magnetic resonance and electrical transport in (TMTSF)2PF6 system

We report simultaneous NMR and electrical transport experiments in the pressure range near the boundary of the antiferromagnetic spin density wave (SDW) insulator and the metallic/superconducting (SC) phase in (TMTSF)2PF6. Measurements indicate a tricritical point separating a line of second-order SDW/metal transitions from a line of first-order SDW/metal(SC) transitions with coexistence of macroscopic regions of SDW and metal(SC) order, with little mutual interaction but strong hysteretic effects. NMR results quantify the fraction of each phase.

77Se NMR probe of magnetic excitations of the magic angle effect in (TMTSF)2PF6

We report 77Se spin-lattice relaxation rates for (TMTSF)2PF6, carried out in the regime where a set of spectacular transport anomalies known as the "magic angle effects" are observed. In situ resistance measurements (R(zz)) were used to verify the experimental conditions and give precise sample alignment information. We found that the 77Se T-11 exhibits no significant changes as the magnetic-field orientation is rotated through the magic angles, and conclude that there is no evidence for either a single-particle gap or a spin gap. The clearly observed field-induced spin-density wave transition temperature is also, unexpectedly, not enhanced at the magic angles.

Evidence of gap anisotropy in superconducting YNi2B2C using directional point-contact spectroscopy

We present a study of the anisotropy in the superconducting energy gap in a single crystal of YNi2B2C (T(c) approximately 14.6 K) using directional point-contact spectroscopy. The superconducting energy gap at 2.7 K, when measured for I||c, is 4.5 times larger than that for I||a. The energy gaps in the two directions also have different temperature dependences. Our results support a scenario with s + g like symmetry.

Point-contact tunneling involving low-dimensional spin-triplet superconductors

We modify and extend previous microscopic calculations of tunneling in superconducting junctions based on a nonequilibrium Green function formalism to include the case of spin-triplet pairing. We show that distinctive features are present in the I-V characteristics of different kinds of junctions, in particular, when the effects of magnetic fields are taken into account, that permit to identify the type of pairing. We discuss the relevance of these results in the context of quasi-one-dimensional organic superconductors such as (TMTSF)2PF6 and layered compounds like Sr2RuO(4).

Magnetic determination of H(c2) under accurate alignment in (TMTSF)2ClO4

Cantilever magnetometry has been used to measure the upper critical magnetic field H(c2) of the quasi-one-dimensional molecular organic superconductor (TMTSF)2ClO4. From simultaneous resistivity and torque magnetization experiments conducted under precise field alignment, H(c2) at low temperature is shown to reach 5 T, 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.

Giant Nernst effect and lock-in currents at magic angles in (TMTSF)2PF6

We have measured the thermoelectric signal along the a axis in (TMTSF)2PF6 at 10 kbar as a function of the orientation of the applied magnetic field. Resonantlike Nernst signals were found with a dramatic sign change as the field was rotated through the "Lebed magic angles." The sign change indicates that the electrical current is "locked in" to the magic angle (interchain) directions for field alignment close to, but on either side of, the magic angles. The amplitude of signals near these angles is many orders of magnitude larger than expected from conventional Boltzmann transport theory.

Polarization of interacting bosons with spin

We prove that in the absence of explicit spin-dependent forces one of the ground states of interacting bosons with spin is always fully polarized. Generally, this state is degenerate with other states, but one can specify the exact degeneracy. For T>0, the magnetization and zero-field susceptibility exceed that of a pure paramagnet. The results are relevant to experimental work on triplet superconductivity and condensation of atoms with spin. They eliminate the possibility, raised in some theoretical speculations, that the ground state or positive temperature state might be antiferromagnetic.

Critical field enhancement near a superconductor-insulator transition

We have discovered a phenomenon where the orbital pair breaking effect is reduced, if not eliminated. It appears as a striking enhancement in the upper critical field H(c2) for (TMTSF)2PF6 and a strong upward curvature in the critical field versus temperature in the region of pressure-temperature phase space near the superconductor-spin density wave insulator boundary. A simple model based on self-consistently dividing the superconductor into layers explains the observations remarkably well and provides a unique way around orbital frustration and toward higher critical fields.