Influence of s± symmetry on unconventional superconductivity in pnictides above the Pauli limit – two-band model study

The theoretical analysis of the Cooper pair susceptibility shows the two-band Fe-based superconductors (FeSC) to support the existence of the phase with nonzero Cooper pair momentum (called the Fulde-Ferrel-Larkin-Ovchinnikov phase or shortly FFLO), regardless of the order parameter symmetry. Moreover this phase for the FeSC model with s _± symmetry is the ground state of the system near the Pauli limit. This article discusses the phase diagram h-T for FeSC in the two-band model and its physical consequences. We compare the results for the superconducting order parameter with s-wave and s _±-wave symmetry – in first case the FFLO phase can occur in both bands, while in second case only in one band. We analyze the resulting order parameter in real space – showing that the FeSC with s _±-wave symmetry in the Pauli limit have typical properties of one-band systems, such as oscillations of the order parameter in real space with constant amplitude, whereas with s-wave symmetry the oscillations have an amplitude modulation. Discussing the free energy in the superconducting state we show that in absence of orbital effects, the phase transition from the BCS to the FFLO state is always first order, whereas from the FFLO phase to normal state is second order.     

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

We present specific heat and thermal conductivity of the heavy fermion superconductor CeCoIn5 in the vicinity of the superconducting critical fieldH _c2, measured with magnetic field in the plane of this quasi-2D compound and at temperatures down to 50 mK. The superconducting phase diagram and the first order nature of the superconducting phase transition at high fields close to a critical fieldH _c2 indicate the importance of the Pauli limiting effect in CeCoIn₅. In the same range of magnetic field we observe a second specific heat anomaly within the superconducting state, and interpret it as a signature of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) inhomogeneous superconducting state. In addition, the thermal conductivity data as a function of field display a kink at a fieldH _k below the superconducting critical field, which closely coincides with the low temperature anomaly in specific heat tentatively identified with the appearance of the FFLO superconducting state. The enhancement of thermal conductivity within the FFLO state calls for further theoretical investigations of the real space structure of the order parameter (and in particular, the structure of vortices) and of the thermal transport within the inhomogeneous FFLO state.     

Josephson effect in fulde-ferrell-larkin-ovchinnikov superconductors

Because of the difference in the momenta of the superconducting order parameters, the Josephson current in a Josephson junction between a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductor and a conventional BCS superconductor is suppressed. We show that the Josephson current may be recovered by applying a magnetic field in the junction. The field strength and direction at which the supercurrent recovery occurs depend upon the momentum and structure of the order parameter in the FFLO state. Thus the Josephson effect provides an unambiguous way to detect the existence of an FFLO state, and to measure the momentum of the order parameter.     

FFLO States in Layered Organic Superconductors

In this short review, the recently found experimental evidence that Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states are realized in quasi‐two‐dimensional (2D) organic superconductors is reported. At low temperatures and when a high magnetic field is aligned parallel to the conducting organic layers, an upturn of the upper critical field much beyond the Pauli limit is observed, as proven by thermodynamic measurements. Under certain conditions, a second thermodynamic transition emerges inside the FFLO state. Nuclear magnetic resonance (NMR) work has added strong microscopic support for the realization of the FFLO state. The NMR spectra in the FFLO phase can very well be explained by a nonuniform one‐dimensionally modulated superconducting order parameter. All these features, appearing only in a very narrow angular region close to parallel‐field orientation, give robust evidence for the realization of the FFLO state in organic superconductors.     

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.     

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.     

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.     

Superconductivity in an organic insulator at very high magnetic fields

We investigate by electrical transport the field-induced superconducting state (FISC) in the organic conductor lambda-(BETS)2FeCl4. Below 4 K, antiferromagnetic-insulator, metallic, and eventually superconducting (FISC) ground states are observed with increasing in-plane magnetic field. The FISC state survives between 18 and 41 T and can be interpreted in terms of the Jaccarino-Peter effect, where the external magnetic field compensates the exchange field of aligned Fe3+ ions. We further argue that the Fe3+ moments are essential to stabilize the resulting singlet, two-dimensional superconducting state.     

The Fulde-Ferrell-Larkin-Ovchinnikov states in s and d-wave superconductor with spin-dependent bandwidth imbalance on square lattice

We study numerically the phase diagram for s and d-wave fermionic superfluidity/superconductivity with spin-dependent bandwidth imbalance on a two-dimensional square-lattice. We also investigate the spontaneous space symmetry breaking states at low temperatures by solving the Bogoliubov-de Gennes equations. It is found that, the spatial configuration of the order parameter,both the uni-directional Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) states and the two-dimensional FFLO state may show up in the presence of finite spin-dependent bandwidth imbalance. Moreover, we calculate the spectra of local density of states, and the experimental proposals of observing such FFLO states are therefore suggested.     

Anisotropic effect of Cd and Hg doping on the Pauli limited superconductor CeCoIn5

We studied the effect of impurity on the first order superconducting (SC) transition and the high field-low temperature (HFLT) SC state of CeCoIn5 by measuring the specific heat of CeCo(In1-xCdx)_{5} with x=0.0011, 0.0022, and 0.0033 and CeCo(In1-xHgx)_{5} with x=0.000 16, 0.000 32, and 0.000 48 at temperatures down to 0.1 K and fields up to 14 T. Cd substitution rapidly suppresses the crossover temperature T0, where the SC transition changes from second to first order, to T=0 K with x=0.0022 for H parallel[100], while it remains roughly constant up to x=0.0033 for H parallel[001]. The associated anomaly of the proposed FFLO state in Hg-doped samples is washed out by x=0.000 48, while remaining at the same temperature, indicating high sensitivity of that state to impurities. We interpret these results as supporting the nonmagnetic, possibly FFLO, origin of the HFLT state in CeCoIn5.     

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.     

Enhancing of the critical temperature of an in-plane FFLO state in heterostructures by the orbital effect of the magnetic field

It is well-known that the orbital effect of the magnetic field suppresses superconducting T _c . We show that for a system, which is in the Larkin-Ovchinnikov-Fulde-Ferrell (FFLO) state at zero external magnetic field, the orbital effect of an applied magnetic field can lead to the enhancement of the critical temperature higher than T _c at zero field. We concentrate on two systems, where the in-plane FFLO state was predicted recently. These are equilibrium S/F bilayers and S/N bilayers under nonequilibrium quasiparticle distribution. However, it is suggested that such an effect can take place for any plane superconducting system, which is in the in-plane FFLO state (or is close enough to it) at zero applied field.     

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.     

Multiple superconducting phases in heavy fermion compounds PrOs4Sb12 and CeCoIn5

In recently discovered heavy fermion compounds, quasi-two-dimensional CeCoIn₅ and skutterudite PrOs₄Sb₁₂, multiple superconducting phases with different symmetries manifest themselves belowT _c. The angle-resolved magnetothermal transport measurements revealed that in PrOs₄Sb₁₂ a novel change in the symmetry of the superconducting gap function occurs deep inside the superconducting state. The ultrasound velocity measurements revealed that in CeCoIn5 the Fulde-Ferrel-Larkin-Ovchinikov (FFLO) phase, in which the order parameter is spatially modulated and has planar nodes aligned perpendicular to the vortices, appears at low temperature and high field. These results open up a new realm for the study of the superconductivity with multiple phases.     

New superconducting phases in field-induced organic superconductor lambda-(BETS)2FeCl4

We derive the parallel upper critical field, Hc2, as a function of the temperature T in quasi-2D organic compound lambda-(BETS)2FeCl4, accounting for the formation of the nonuniform Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) state. To further check the 2D LOFF model, we propose to study the Hc2(T) curve at low T in tilted fields, where the vortex state is described by the high Landau level functions characterized by the index n. We predict a cascade of first-order transitions between vortex phases with different n, between phases with different types of the symmetry at given n and the change of the superconducting transition from the second order to the first order as FeCl4 ions are replaced partly by GaCl4 ions.     

Double-stage continuous-discontinuous superconducting phase transition in the Pauli paramagnetic limit of a 3D superconductor: the URu<Subscript>2</Subscript>Si<Subscript>2</Subscript> case

The sharp suppression of the de-Haas van-Alphen oscillations observed in the mixed superconducting (SC) state of the heavy fermion compound URu₂Si₂ is shown to confirm a theoretical prediction of a narrow double-stage SC phase transition, smeared by fluctuations, in a 3D paramagnetically-limited superconductor. The predicted scenario of a second order transition to a nonuniform (FFLO) state followed by a first order transition to a uniform SC state, obtained by using a non-perturbative approach, is also found to be consistent with recent thermal conductivity measurements performed on this material.     

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.     

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.     

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.

     

Cold attractive spin polarized Fermi lattice gases and the doped positive U Hubbard model

Experiments on polarized fermion gases performed by trapping ultracold atoms in optical lattices allow the study of an attractive Hubbard model for which the strength of the on-site interaction is tuned by means of a Feshbach resonance. Using a well-known particle-hole transformation we discuss how results obtained for this system can be reinterpreted in the context of a doped repulsive Hubbard model. In particular, we show that the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state corresponds to the striped state of the two-dimensional doped positive U Hubbard model. We then use the results of numerical studies of the striped state to relate the periodicity of the FFLO state to the spin polarization. We also comment on the relationship of the d(x(2)-y(2)) superconducting phase of the doped 2D repulsive Hubbard model to a d-wave spin density wave state for the attractive case.