FFLO State Research Articles

Interplay of Fulde-Ferrell-Larkin-Ovchinnikov and Vortex states in two-dimensional Superconductors

Clean superconductors with weakly coupled conducting planes have been suggested as promising candidates for observing the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state. We consider here a layered superconductor in a magnetic field of arbitrary orientation with respect to the conducting plane. In this case there is competition of Pauli spin-pair-breaking effects, favoring the FFLO state, and orbital-pair-breaking effects, favoring the Abrikosov vortex state. In previous work, phase transitions to phases with pairing in Landau levels with quantum numbers n > 0 have been predicted. Here, we calculate the actual structure of the stable states below H c2 by minimizing the free energy. We find new order parameter structures differing from both the traditional Abrikosov and FFLO solutions. These include two-dimensional periodic structures with several zeros of the order parameter, as well as quasi-one-dimensional structures consisting of vortex chains separated by FFLO domains. We discuss the limit of high n, where some interesting but yet unsolved questions appear.

Enhancement of the Upper Critical Field Due to a Fermi-Surface Effect in Quasi-Two-Dimensional Superconductors in Parallel Magnetic Fields

It is shown that the upper critical field of the superconductivity can be remarkably enhanced due to the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) effect, even if the system does not have any flat portion of the Fermi-surface (FS). As an example, we examine a two-dimensional tight binding model at various hole concentrations. The optimum electron concentration at which the ratio of the critical field to the Pauli paramagnetic limit of the uniform BCS state takes maximum is found to be far away from the half-filling. Near there, the ratio exceeds 5 times the Pauli paramagnetic limit. We explain a new mechanism by which the critical field is enhanced due to an FS structure. The FFLO state is discussed in connection with the organic superconductors.

Structure of the non-uniform Fulde–Ferrell–Larkin–Ovchinnikov state in 3D superconductors

We demonstrate that in 3D superconductors the transition between a normal phase and a non-uniform superconducting phase is always of the first order in the pure paramagnetic limit. We also determine the transition temperature and the structure of the modulated `lattice' by means of the generalized Ginzburg–Landau functional near the tricritical temperature, and the exact Gorkov equations in the whole temperature interval.

Exactly Aligned Magnetic Field and the Fulde–Ferrell–Larkin–Ovchinnikov State in Quasi-Low-Dimensional Superconductors

In quasi-low-dimensional type II superconductors, the orbital pair-breaking effect is considerably reduced when the magnetic field is exactly aligned in a direction parallel to the highly conducting layer. Then, since the superconducting phase survives up to a very high magnetic field comparable to the Pauli-paramagnetic limit, we could expect some drastic phenomena, such as the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state. In this paper, we discuss the condition of the tilt angle for the appearance of the FFLO state, and also discuss the stable spatial structure of the FFLO state. Further, we show an example of a quasi-two-dimensional system in which the FFLO state is remarkably enhanced, that is, a two-dimensional tight binding system of appropriate hole concentrations.

Stability of Fulde–Ferrell–Larkin–Ovchinnikov state in type-II superconductors against the phase fluctuations

Phase fluctuations of the order parameters are studied in type-II superconductors in the limit of strong spin magnetism on the basis of a generalized Ginzburg–Landau Hamiltonian, where Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states are assumed to occur in the ground state. Stabilities of the FFLO states with various spatial structures are examined in systems with different symmetries, at finite temperatures. We study rotationally symmetric systems in two dimensions and three dimensions, and systems with anisotropy in Fermi-surface and pairing. We consider the Kosterlitz–Thouless transitions to the quasi-long-range order in two dimensions.

Quasi-One-Dimensional Superconductors: From Weak to Strong Magnetic Field

We discuss the possible existence of a superconducting phase at high magnetic field in organic quasi-one-dimensional conductors. We consider in particular (i) the formation of a Larkin–Ovchinnikov–Fulde–Ferrell state, (ii) the role of a temperature-induced dimensional crossover occurring when the transverse coherence length ξz(T) becomes of the order of the lattice spacing, and (iii) the effect of a magnetic-field-induced dimensional crossover resulting from the localization of the wave functions at high magnetic field. In the case of singlet spin pairing, only the combination of (i) and (iii) yields a picture consistent with recent experiments in the Bechgaard salts showing the existence of a high-field superconducting phase. We point out that the vortex lattice is expected to exhibit unusual characteristics at high magnetic field.

Phase Fluctuations and Kosterlitz-Thouless Transition in Two-Dimensional Fulde-Ferrell-Larkin-Ovchinnikov Superconductors

Effects of the phase fluctuations of the order parameter on the stability of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states are examined in exactly two-dimensional (2D) type-II superconductors with cylindrically symmetric Fermi surface on the basis of a generalized Ginzburg-Landau theory. It is found that for the FFLO states with oscillations in a single direction, not only the long-range order but also quasi-long-range order (QLRO), which is characterized by a power law decay of the order parameter correlation function, is suppressed by the phase fluctuations at any finite temperatures. On the other hand, for the FFLO states with order parameter structures such as triangular and square lattices, it is shown that the QLRO is possible as the uniform BCS state. Systems with anisotropy in the Fermi surface and pairing are also discussed.

Nonmonotonic behavior of the superconducting transition temperature in bimetallic ferromagnet-superconductor structures

For layered ferromagnet/superconductor (F/S) structures we develop a theory of the proximity effect. In contrast to previous approaches, this theory allows for a finite transmission coefficient of the interface between the two metals and competition between the diffusion and spin-wave types of quasiparticle motion in the ferromagnet’s strong exchange field. The superconductivity in F/S systems proves to be a superposition of BCS pairing with a constant-sign pair amplitude in the S-layers and Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) pairing with an oscillating wave function in the F-layers. We show that the oscillatory behavior of the superconducting transition temperature T c is due to oscillations of the Cooper pair flux from the S-layer to the F-layer, which are the result of oscillations of the discontinuity (jump) of the pair amplitude at the F/S boundary as the thickness d f of the F-layer increases. The presence of nonmagnetic impurities leads to heavy damping of the oscillations of the LOFF pair amplitude and rapid deterioration of the coherent coupling of the boundaries of the F-layer in which the T s vs. d f dependence reaches a plateau as d f increases. In F/S superlattices, in contrast to F/S double-layer junctions, there are two forms of the LOFF state, the 0-phase and the π-phase, which differ in their symmetry with respect to the center of the F layer. This gives rise to additional oscillations in the T c (d f ) dependence due to the 0-π transitions. As the most vivid manifestation of LOFF states in F/S-systems, we predict the existence of recurrent and periodically recurrent superconductivities. We give a qualitative explanation of the different behavior of the superconducting transition temperature observed by different groups of experimenters dealing with the same ferromagnet-superconductor structures.

Phase Diagram of the Fulde-Ferrell-Larkin-Ovchinnikov State in a Three-Dimensional Superconductor

The phase diagram of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in a 3-dimensional superconductor is discussed. We use the quasi-classical Green's function to calculate the free energy. It is shown that the phase transition from the normal state is of first order at high temperatures but is of second order near T =0. To describe the phase transition in the Ginzburg-Landau theory, one has to take into account up to eighth order term with respect to the order parameter. The phase transition from the FFLO state to the uniform superconducting state is of second order in all the temperature range T <0.561 T c in accordance with Burkhardt and Rainer's result in a 2-dimensional system.

Fulde-Ferrell-Larkin-Ovchinnikov State in a Quasi-Two-Dimensional Organic Superconductor

The possibility of the existence Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in quasi-low-dimensional organic superconductors is discussed. The critical field and the wave number q of the spatial oscillation are calculated on the basis of a tight binding model of κ- [BEDT- TTF] 2 I 3 , as an example. The critical fields H c at T =0 are estimated to be roughly 1.5, 2, and 2.5 times the Pauli paramagnetic limit H P for s -wave, d x y -wave, and d x 2 - y 2 -wave FFLO states, respectively. We discuss which properties of the structure of the Fermi surface enhance the critical field and are relevant to the direction of q . We show that the number of the lines on the displaced Fermi surfaces due to the Zeeman energy which simaltaneously touch by translation p → p + q are important in addition to the curvature of the Fermi surface and the density of states.

Anomalous superconducting mixed state in single-crystalline CeRu 2

The measurements of magnetization and ultrasonic velocity have been performed on single-crystalline CeRu 2 to clarify the anomalous superconducting mixed state on the basis of the FFLO state. Our experimental results suggest that a novel pinning mechanism works in the FFLO state.

Experimental evidence for a generalized FFLO state in clean type-II superconductors with short coherence length and enhanced Pauli susceptibility

We report an investigation of the magnetic and dilatometric properties of single crystals of the superconductors UPd 2Al 3 and CeRu 2, both compounds exhibiting enhanced spin susceptibilities. Our results suggest for both systems a first-order transition between weak and strong pinning at T < 0.9 T c, somewhat below H c2( T). We argue that these observations are compatible with a staggered order parameter due to the formation of a “generalized Fulde-Ferrell-Larkin-Ovchinnikov state”.

A possibility of a new type of the Fulde-Ferrell-Larkin-Ovchinnikov state

We propose a new type of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state with a cylindrical symmetric order-parameter. We study the Ginzburg-Landau (GL) theory of the strongly Pauli limited type II superconductor with a spherical symmetric fermi surface, near the critical magnetic field of the FFLO state in the ground state. We find that the cylindrical state has a lower energy than the stripe state, which has the lowest energy in the states examined so far.

Strong coupling effects on the upper critical field of the heavy-fermion superconductor UBe13

The upper critical fieldH c2 (T) of a high quality single crystal of UBe13 is studied with very low noise resistivity measurements. It shows a large but finite slope of —45 T/K, an unusual temperature dependence with an inflexion point atT/T C ∼ 0.5 and a large saturated limit forT→ 0 ofH c2 (0) = 14 T. The complete temperature dependence ofH c2 (T) can be described by a simple model of strong coupling superconductor, assuming a full Pauli limitation and the occurence of a non-uniform superconducting state (FFLO state) at low temperature.

Superconducting Properties and de Haas-van Alphen Effect in CeCo2Single Crystal

We have succeeded in growing a high-quality single crystal of CeCo 2 which shows a clear de Haas-van Alphen oscillation. Using this single crystal, we have studied the superconducting properties related to the FFLO state.

Anisotropy of the upper critical field in URu 2Si 2 and FFLO state in antiferromagnetic superconductors

Experimental data of the upper critical field of the heavy-fermion superconductor URu 2Si 2 are analysed, in the clean limit, taking account of both orbital and Pauli limits. A discussion on the possible existence of a non-uniform superconducting state expected in clean Pauli-limited superconductors (FFLO state) follows. We also present calculations determining the influence of an AF ground state on the existence of such a non-uniform superconducting state.

Larkin-Ovchinnikov-Fulde-Ferrell state in quasi-one-dimensional superconductors.

The properties of a quasi-one-dimensional (quasi-1D) superconductor with an open Fermi surface are expected to be unusual in a magnetic field. On the one hand, the quasi-1D structure of the Fermi surface strongly favors the formation of a nonuniform state [Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) state] in the presence of a magnetic field acting on the electron spins. On the other hand, a magnetic field acting on an open Fermi surface induces a dimensional crossover by confining the electronic wave functions along the chains of highest conductivity, which results in a divergence of the orbital critical field and in a stabilization at low temperature of a cascade of superconducting phases separated by first-order transitions. In this paper, we study the phase diagram as a function of the anisotropy by taking into account on the same footing the paramagnetic and the orbital effects of the field. We discuss in detail the experimental situation in the quasi-1D organic conductors of the Bechgaard salts family and argue that they appear as good candidates for the observation of the LOFF state, provided that their anisotropy is large enough. Recent experiments on the organic quasi-1D superconductor (TMTSF${)}_{2}$${\mathrm{ClO}}_{4}$ are in agreement with the results obtained in this paper and could be interpreted as a signature of a high-field superconducting phase. We also point out the possibility to observe a LOFF state in some of the recently discovered quasi-2D organic superconductors due to the particular topology of their Fermi surface.

Existence of the FFLO state in superconducting UPd2Al3.

A Comment on the Letter by K. Gloos et al., Phys. Rev. Lett. 70, 501 (1993).Received 8 March 1993DOI:https://doi.org/10.1103/PhysRevLett.71.3391©1993 American Physical Society