Exploring the relationship between the magnetic frustration and the emergence of FFLO state on a triangular lattice

The Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) unconventional superconductors attract renewed interest in recent years. However, an unambiguous experimental demonstration of the FFLO state is hindered by the stringent requirements for its realization. In this paper, we explore the relationship between the magnetic frustration and the emergence of the FFLO state on an anisotropic triangular lattice, based on the self-consistent calculation of the Bogoliubov-de Gennes equations. We find that the required lower critical magnetic field to enter the FFLO state decreases with the increase of both the anisotropic ratio and the on-site Coulomb repulsive interaction. This demonstrates that it is easier to enter the FFLO state on the structurally frustrated triangular lattice in comparison with the square lattice, and suggests that the layered organic superconductors with a triangular lattice may be good candidates for exploring the FFLO state.     

Evolution of paramagnetic quasiparticle excitations emerged in the high-field superconducting phase of CeCoIn5

We present (115)In NMR measurements in a novel thermodynamic phase of CeCoIn(5) in a high magnetic field, where exotic superconductivity exists with the incommensurate spin-density wave order. We show that the NMR spectra in this phase provide direct evidence for the emergence of the spatially distributed normal quasiparticle regions. The quantitative analysis for the field evolution of the paramagnetic magnetization and newly emerged low-energy quasiparticle density of states is consistent with the nodal plane formation, which is characterized by an order parameter in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. The NMR spectra also suggest that the spatially uniform spin-density wave is induced in the FFLO phase.     

Texture in the superconducting order parameter of CeCoIn5 revealed by nuclear magnetic resonance

We present a 115In NMR study of the quasi-two-dimensional heavy-fermion superconductor CeCoIn5 believed to host a Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state. In the vicinity of the upper critical field and with a magnetic field applied parallel to the ab plane, the NMR spectrum exhibits a dramatic change below T*(H) which well coincides with the position of reported anomalies in specific heat and ultrasound velocity. We argue that our results provide the first microscopic evidence for the occurrence of a spatially modulated superconducting order parameter expected in a FFLO state. The NMR spectrum also implies an anomalous electronic structure of vortex cores.     

Fulde-Ferrell-Larkin-Ovchinnikov state in a perpendicular field of quasi-two-dimensional CeCoIn5

A Fulde-Ferrell-Larkin-Ovchinnkov (FFLO) state was previously reported in the quasi-2D heavy fermion CeCoIn5 when a magnetic field was applied parallel to the ab plane. Here, we conduct 115In NMR studies of this material in a perpendicular field, and provide strong evidence for FFLO in this case as well. Although the topology of the phase transition lines in the H-T phase diagram is identical for both configurations, there are several remarkable differences between them. Compared to H parallelab, the FFLO phase for H perpendicularab is confined in a much narrower region at the low-T-high-H corner in the H-T plane, and the critical field separating the FFLO and non-FFLO superconducting states almost ceases to have a temperature dependence. Moreover, directing H perpendicularab results in a notable change in the quasiparticle excitation spectrum within the planar node associated with the FFLO transition.     

Novel route to a finite center-of-mass momentum pairing state for superconductors: A current-driven Fulde-Ferrell-Larkin-Ovchinnikov state

The previously studied Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is stabilized by a magnetic field via the Zeeman coupling in spin-singlet superconductors. Here we suggest a novel route to achieve nonzero center-of-mass momentum pairing states in superconductors with Fermi surface nesting. We investigate two-dimensional superconductors under a uniform external current, proportional to a finite pair momentum of q(e). We find that an FFLO state with a spontaneous pair momentum of q(s) is stabilized above a certain critical current that depends on the direction of the external current. A finite q(s) arises in order to make the total pair-momentum of q(t)(=q(s) + q(e)) perpendicular to the nesting vector, which is independent of spin states of Cooper pairs. We also discuss experimental signatures of the FFLO state.     

Quantum phase transition of the Fulde–Ferrell–Larkin–Ovchinnikov states in two-dimensional anisotropic triangular system

Based on the Bogoliubov de Gennes (BdG) equations, we study the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states for d-wave superconductor in anisotropic triangular system self-consistently with a strong magnetic field applied parallel to its conducting planes. We find that the two-dimensional FFLO state transforms to one-dimensional FFLO state as the system frustrated. The calculated local density of states are suggested to distinguish these states.     

Vortex dynamics and the Fulde-Ferrell-Larkin-Ovchinnikov state in a magnetic-field-induced organic superconductor

Under special conditions, a superconducting state where the order parameter oscillates in real space, the so-called FFLO state, is theoretically predicted to exist near the upper critical field, as first proposed by Fulde and Ferrell, and Larkin and Ovchinnikov. We report systematic measurements of the interlayer resistance in high magnetic fields to 45 T in the two-dimensional magnetic-field-induced organic superconductor lambda-(BETS)2FeCl4, where BETS is bis(ethylenedithio)tetraselenafulvalene. The resistance is found to show characteristic dip structures in the superconducting state. The results are consistent with pinning interactions between the vortices penetrating the insulating layers and the order parameter of the FFLO state. This gives strong evidence for an oscillating order parameter in real space.     

Gapless Fermi surfaces of anisotropic multiband superconductors in a magnetic field

We propose that a new state with a fully gapless Fermi surface appears in quasi-2D multiband superconductors in magnetic field applied parallel to the plane. It is characterized by a paramagnetic moment caused by a finite density of states on the open Fermi surface. We calculate thermodynamic and magnetic properties of the gapless state for both s-wave and d-wave cases, and discuss the details of the first order metamagnetic phase transition that accompanies the appearance of the new phase in s-wave superconductors. We suggest possible experiments to detect this state both in the s-wave (2-H NbSe2) and d-wave (CeCoIn5) superconductors.     

The Gor'kov and Melik-Barkhudarov Correction to the Mean-Field Critical Field Transition to Fulde–Ferrell–Larkin–Ovchinnikov States

The Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states, characterized by Cooper pairs condensed at finite-momentum are, at the same time, exotic and elusive. It is partially due to the fact that the FFLO states allow superconductivity to survive even in strong magnetic fields at the mean-field level. The effects of induced interactions at zero temperature are calculated in both clean and dirty cases, and it is found that the critical field at which the quantum phase transition to an FFLO state occurs at the mean-field level is strongly suppressed in imbalanced Fermi gases. This strongly shrinks the phase space region where the FFLO state is unstable and more exotic ground state is to be found. In the presence of high level impurities, this shrinkage may destroy the FFLO state completely.     

Competition between BCS and FFLO States in Magnetic Superconductors in a Cryptoferromagnetic Phase

The possibility of appearance of inhomogeneous superconducting Fulde—Ferrell—Larkin—Ovchinnikov (FFLO) states in magnetic superconductors in a cryptoferromagnetic phase with helical magnetic ordering has been analyzed. The dependence of the critical temperature on the angle between the wave vectors of the spatial modulation of the FFLO state and helical magnetic structure has been calculated within the proposed model. It has been shown that their mutually perpendicular orientation corresponds to the most energetically favorable state. The numerical calculations have also shown the existence of a tricritical point on a line separating the Bardeen—Cooper—Schrieffer and FFLO phases on the phase diagram of states. Furthermore, FFLO states can appear in a magnetic superconductor even at fairly strong exchange fields because of the difference between the effective masses of conduction electrons in different spin subbands and the anisotropy of the Fermi surface.

     

Possible Fulde-Ferrell-Larkin-Ovchinnikov superconducting state in CeCoIn5

We report specific heat measurements of the heavy fermion superconductor CeCoIn5 in the vicinity of the superconducting critical field H(c2), with magnetic fields in the [110], [100], and [001] directions, and at temperatures down to 50 mK. The superconducting phase transition changes from second to first order for fields above 10 T for H parallel [110] and H parallel [100]. In the same range of magnetic fields, we observe a second specific heat anomaly within the superconducting state. We interpret this anomaly as a signature of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) inhomogeneous superconducting state. We obtain similar results for H parallel [001], with the FFLO state occupying a smaller part of the phase diagram.     

Spatial line nodes and fractional vortex pairs in the Fulde-Ferrell-Larkin-Ovchinnikov vortex state of spin-singlet superconductors

A Zeeman magnetic field can induce a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in spin-singlet superconductors. Here we argue that there is a nontrivial solution for the FFLO vortex phase that exists near the upper critical field in which the wave function has only spatial line nodes that form intricate and unusual three-dimensional structures. These structures include a crisscrossing lattice of two sets of nonparallel line nodes. We show that these solutions arise from the decay of conventional Abrikosov vortices into pairs of fractional vortices. We propose that neutron scattering studies can observe these fractional vortex pairs through the observation of a lattice of 1/2 flux quanta vortices. We also consider related phases in noncentrosymmetric superconductors.     

Observation of spin susceptibility enhancement in the possible Fulde-Ferrell-Larkin-Ovchinnikov state of CeCoIn(5)

We report (115)In nuclear magnetic resonance measurements of the heavy-fermion superconductor CeCoIn(5) in the vicinity of the superconducting critical field H(c2) for a magnetic field applied perpendicular to the ? axis. A possible inhomogeneous superconducting state, the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, is stabilized in this part of the phase diagram. In an 11 T applied magnetic field, we observe clear signatures of the two phase transitions: the higher temperature one to the homogeneous superconducting state and the lower temperature phase transition to a FFLO state. We find that the spin susceptibility in the putative FFLO state is significantly enhanced as compared to the value in a homogeneous superconducting state. The implications of this finding for the nature of the low temperature phase are discussed.     

Pressure dependence of the Fulde-Ferrell-Larkin-Ovchinnikov state in CeCoIn5

Pressure studies of the thermodynamics of CeCoIn5 under magnetic fields H parallel to c and H parallel to ab have been made up to P = 1.34 GPa. We recorded the signature of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state for all pressures when H parallel to ab. Also remarkably, the FFLO regime suddenly expands for P = 1.34 GPa. With the help of a microscopic theory for d-wave superconductivity, we have extracted the gyromagnetic ratio g and the Fermi velocities nu(a) and nu(c). Our study is the first evidence for the existence of the FFLO state away from the influence of the antiferromagnetic fluctuations. We find a close parallel between the T-P phase diagram of CeCoIn5 and the T-x phase diagram of the high-Tc cuprates, where x is the hole concentration.     

Electron-hole pairing with nonzero momentum in a graphene bilayer

Electron-hole pairing due to the Coulomb interaction in the system of two graphene sheets has been considered. The critical transition temperature has been determined as a function of both the distance between the electron and hole Fermi lines and the triangular distortion of their spectrum. It has been shown that when the distance between Fermi lines is longer than a critical value, the temperature of the transition to a state with nonzero momentum of Cooper pairs (Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state) is higher than the temperature of the transition to the Bardeen-Cooper-Schrieffer state. The Josephson effect for the FFLO state has been analyzed, which is due to the tunneling of charge carriers between the graphene sheets. It has been shown that the spatial structure of the order parameter of the system in this state can be reconstructed, i.e., the FFLO state can be identified from the dependence of the tunneling current on the magnetic field parallel to the graphene sheets. Other experimental methods for studying the phase diagram of the system have been discussed.     

Large triplet pairing mixing in the FFLO state of spin fluctuation mediated superconductivity in Q1D systems

We study the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state of spin fluctuation mediated superconductivity 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.     

Local tunneling spectroscopy as a signature of the Fulde-Ferrell-Larkin-Ovchinnikov state in s- and d-wave superconductors

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states for two-dimensional s- and d-wave superconductors (s- and d-SCs) are self-consistently studied under an in-plane magnetic field. While the stripe solution of the order parameter is found to have lower free energy in s-SCs, a square lattice solution appears to be energetically more favorable in the case of d-SCs. At certain symmetric sites, we find that the features in the local density of states (LDOS) can be ascribed to two types of bound states. We also show that the LDOS maps for d-SCs exhibit bias-energy-dependent checkerboard patterns. These characteristics can serve as signatures of the FFLO states.     

Calorimetric evidence for a Fulde-Ferrell-Larkin-Ovchinnikov superconducting state in the layered organic superconductor kappa-(BEDT-TTF)2Cu(NCS)2

The specific heat of the layered organic superconductor kappa-(BEDT-TTF)(2)Cu(NCS)(2), where BEDT-TTF is bisethylenedithio-tetrathiafulvalene, has been studied in magnetic fields up to 28 T applied perpendicular and parallel to the superconducting layers. In parallel fields above 21 T, the superconducting transition becomes first order, which signals that the Pauli-limiting field is reached. Instead of saturating at this field value, the upper-critical-field increases sharply and a second first-order transition line appears within the superconducting phase. Our results give strong evidence that the phase, which separates the homogeneous superconducting state from the normal state is a realization of a Fulde-Ferrell-Larkin-Ovchinnikov state.     

Abrikosov-to-Josephson vortex lattice crossover in heavy fermion CeCoIn5

We present torque magnetization measurements on the quasi-2D heavy fermion superconductor CeCoIn₅ at temperatures down to 20 mK and magnetic fields up to 18?T. At orientations with the magnetic field perpendicular to the conducting planes, a prominent vortex lattice peak effect is present at around 0.5H _c2. The peak effect gradually disappears upon rotating the field into the plane parallel orientation. We interpret the absence of the peak effect for the plane parallel case as a transformation of the Abrikosov lattice into a Josephson vortex state, favored by the Pauli paramagnetic limit in CeCoIn₅ together with the unusually large condensation energy. Additionally, we do not observe flux avalanches as found in organic superconductors and suggest that the complete absence of vortex activity in the plane parallel field orientation is crucial for the formation of Fulde–Ferrell–Larkin–Ovchinnikov superconductivity in CeCoIn₅.     

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.