Introduction

As theorized by Bardeen, Cooper, and Schrieffer (BCS), superconductivity occurs when itinerant electrons form pairs, so-called Cooper pairs, via an attractive force. Although many superconducting properties are well described by BCS theory, long-standing studies have found various superconductors beyond the BCS framework and many intriguing open questions. One of the exotic unconventional superconducting states, the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state, was independently theorized by Fulde & Ferrell1 and Larkin & Ovchinnikov2 in 1964. An up-spin electron with momentum k is coupled with a down-spin electron with momentum −k + q in an FFLO pairing, leading to a finite center-of-mass momentum of Cooper pairs q ≠ 0, whereas an ordinary superconducting state is formed by electrons whose momenta are k and −k, as illustrated in Fig. 1a. For ordinary superconductivity, in which spins in paired electrons are antiparallel to each other, a magnetic field destabilizes the superconductivity through the orbital effect and the Zeeman effect. In most superconductors, superconductivity is suppressed by the orbital effect caused by the Lorentz force on vortices, which mainly determines the upper critical field Hc2. However, when the orbital effect is quenched, the Zeeman effect governs Hc2. In this case, the ordinary spin-singlet superconducting state is destroyed at the field where the Zeeman splitting energy reaches the superconducting energy gap Δ, known as the Pauli paramagnetic limit HP. In contrast, the FFLO state is stable even above HP due to a gain in spin polarization energy of the nonzero q. This finite q adds a term, cos(qr), to the order parameter of the superconductivity Δ. The modified gap function Δcos(qr) indicates that the order parameter spatially oscillates in real space, as shown in Fig. 1b3,4,5,6,7,8. The oscillatory pattern composed of the normal state and the superconductivity endows the FFLO state with emergent anisotropy depending on the q vector. Since disorder stunts the formation of the spatial modulation, the FFLO state appears only in the clean limit4. Besides, as mentioned-above, the emergence of the FFLO state is allowed when the orbital effect is sufficiently weaker than the paramagnetic effect, as characterized by the Maki parameter αM > 1.85. Consequently, these restrictions narrow the candidate materials in the search for FFLO superconductivity7,8,9,10,11,12,13,14 and have disturbed experimental examination of FFLO physics despite numerous theoretical studies1,2,3,4,5,6,15,16,17,18,19. In particular, spatial anisotropy, one of the main features of the FFLO state, has never been experimentally observed.

Fig. 1: Schematics of the FFLO state, crystal structure, and experimental setup.
figure 1

a Schematics of ordinary spin-singlet pairing and FFLO pairing. k and σ represent the momentum and spin of the electrons. b Spatial modulation of order parameter Δ(r) in the FFLO state (dashed curve). The normal state (blue) appears at nodes of the superconducting order parameter Δcos(qr). c Quasi-two-dimensional crystal structure of κ-(BEDT-TTF)2Cu(NCS)2. d Fermi surface of κ-(BEDT-TTF)2Cu(NCS)2. The solid and dashed black lines are the first Brillouin zone and the extended Brillouin zone, respectively. The blue and red curves show the Fermi surfaces, and the green arrow indicates the most predominant nesting vector. e Experimental setup for the multidirectional ultrasound measurements using longitudinal sound waves. The magnetic field was rotated from the c-axis to the a*-axis. θ is the polar angle from the c-axis. f Relative change in the sound velocity Δv/v (red, left axis) and attenuation coefficient Δα (blue, right axis) at T = 1.6 K and θ =90° as a function of magnetic field. The black arrow indicates the upper critical field of the superconductivity Hc2.

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The organic superconductor κ-(BEDT-TTF)2Cu(NCS)2 (BEDT-TTF is an abbreviation of bis(ethyleneditio)tetrathiafulvalene) is known as the prime candidate for exhibiting the FFLO state and has been examined by various measurements9,20,21,22,23,24,25,26. As displayed in Fig. 1c, the layered structure formed by alternating stacking of conducting and insulating layers provides a quasi-two-dimensional (quasi-2D) electronic structure. This compound undergoes a superconducting transition at ~9.5 K and changes into a d-wave superconductivity27,28, which is theoretically expected to manifest using nesting vectors including the predominant nesting vector Q1 shown in Fig. 1d29,30. When a magnetic field is parallel to the conducting plane, emergence of FFLO pairing is highly expected in κ-(BEDT-TTF)2Cu(NCS)2 because of the large Maki parameter, relatively long mean-free path9, and quasi-2D Fermi surface. The heat capacity data21 show a 1st-order transition at HFFLO~21 T. Tunnel diode oscillator (TDO) measurement22 and torque magnetometry23,24 also detect this anomaly, which is smeared out by a slight field misalignment. Based on the existence of an additional superconducting phase with an upturn in its field-temperature phase diagram, these works suggest that a putative FFLO phase appears in the high-field region. Moreover, a NMR study26 detects the formation of Andreev bound states related to a phase twist of the order parameter, which strongly indicates the presence of the FFLO phase31. A slight in-plane anisotropy in Hc223 implies that the FFLO state has a d-wave-like fourfold gap symmetry in momentum space. However, conclusive evidence for anisotropy related to spatial modulation in real space is still missing. Employing multidirectional ultrasound propagation, we examine the plausible FFLO state of κ-(BEDT-TTF)2Cu(NCS)2 and establish that this state certainly exhibits an anisotropic response, which is a hallmark of the FFLO state.

Results

To discuss the anisotropy in the FFLO state, we arranged two pairs of transducers, generating and detecting longitudinal ultrasonic waves, on all sides of a cuboid-shaped single crystal, as shown in Fig. 1e. First, in Fig. 1f, we show the relative change in sound velocity Δv/v and ultrasonic attenuation Δα at 1.6 K in magnetic fields perpendicular to the conducting plane θ = 90°. The polarization vector u (parallel to the ultrasound propagation vector for longitudinal waves) is along the b-axis. At low fields, the obtained data reproduce the reported behavior32. As indicated by the arrow, Δv/v exhibits an anomaly accompanied by suppression of the superconductivity at 3 T ( = Hc2(90°)). The lattice softening in most of its superconducting region in a perpendicular field (Fig. 1f) agrees with the fact that the vortex lattice melts at a much lower field (<0.5 T) when H | | a*,33. From the equation Hc2(90°)=φ0/(2πξ||2), where φ0 is the flux quantum, an in-plane coherence length of ξ||~10 nm is determined. The gradual increase in Δv/v between 3 T and 7 T reflects the suppression of fluctuating superconductivity above Hc234 since Δv/v is a sensitive indicator of fluctuations of the superconducting order parameter in organic superconductors32,35. At higher fields, both properties exhibit the acoustic de Haas–van Alphen (acoustic dHvA) oscillations mainly composed of two orbits, whose frequencies are estimated to be approximately 610 T and 3300 T. The obtained frequencies well coincide with the reported values of the α orbit (blue area in Fig. 1d) and the β − α orbit36,37. Detailed analyses and discussions are described in the Supplementary Materials. For the α orbit, an estimation of the mean-free path l from a fit to the typical Lifshitz-Kosevich formula leads to l~90 nm. This value sufficiently larger than ξ||~10 nm indicates that the electronic system is in the clean limit, which meets one of the requirements for the emergence of the FFLO state. For the Maki parameter, the phase diagram, which is consistent with our results discussed later, indicates αM~89, which is approximately 4 times larger than required5. Thus, the present sample satisfies the conditions required to form the FFLO state.

Since the FFLO state appears at low temperatures when the orbital effect is sufficiently suppressed, in Fig. 2, we show the magnetic field dependence of the elastic properties (u | |b) at 2.1 K near the parallel direction, with θ < 1.2°. Note that no clear hysteresis was observed in our present measurements (see Supplementary Fig. 4). At θ = 0°, the field dependence of Δv/v has two dips at ~21.3 T (blue circle) and ~24.5 T (black triangle), as indicated by the symbols in Fig. 2a. These anomalies are observed as peaks in Δα in Fig. 2b. Based on the results of previous studies9,20,21,22,23,24,25,26, these characteristic fields correspond to HFFLO and Hc2, respectively, and the FFLO state appears between HFFLO and Hc2. Upon tilting the sample away from 0°, Hc2 abruptly decreases, whereas HFFLO shows barely any change. Since these two anomalies finally merge into one sharp anomaly at θ = 1.2°, the FFLO state is completely suppressed by this slight tilt. This result is perfectly consistent with the report that the FFLO state at 2.0 K only exists for θ < 1.2° 23. To closely examine the ultrasonic properties of the FFLO state, the datasets of θ = 0° and θ = 1.2° are enlarged in Fig. 2c, d. The light green area corresponds to the contribution of the FFLO state. This result indicates that the formation of the FFLO state leads to the lattice hardening in the u | |b direction. For Δα in Fig. 2d, attenuation of the sound wave propagation by FFLO formation is natural because of the spatially inhomogeneity. Near Hc2, a flux flow gives excess attenuation appearing as a peak in Δα38,39, and therefore, the combination of the two peaks at HFFLO and Hc2 produces the observed behavior above HFFLO.　Note that the difference below HFFLO (gray area) originates from perpendicular components of the applied fields because it appears when a field is tilted away from θ = 0°. The perpendicular component, which penetrates the conducting plane and forms pancake vortices, induces excess dynamics of the pancake vortices. Therefore, when θ ≠ 0°, the lattice is softened and Δα is enhanced, leading to the difference highlighted by the gray area.

Fig. 2: Low-temperature and high-field ultrasonic properties when θ~0°.
figure 2

a, b Magnetic field dependence of Δv/v a and Δα b at 2.1 K. The blue and black symbols indicate the dips in Δv/v and the peaks in Δα. The black line represents offset for each dataset. c, d Enlarged plot of the datasets at 0° and 1.2°. The green area indicates the difference between the 0° and 1.2° data, which reflects the contribution of the FFLO state.

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In Fig. 3a, b, we show the HFFLO and Hc2 of the detected anomalies at 2.1 K and 6.3 K as an H vs. θ plot. The cusp-like angular dependence of Hc2 at 6.3 K can be described by the Tinkham 2D model40. Indeed, the interlayer coherence length ξ⊥~1.4 nm (=φ0/2πξ||Hc2(0°)) is smaller than the interlayer distance of 1.5 nm, indicating that the interlayer coupling of the superconductivity is weak. This anisotropic behavior and the values of Hc2 agree well with the results reported in refs. 22,24,33. However, this model cannot reproduce Hc2 at 2.1 K because of the emergence of the FFLO state. The Hc2 determined by the resistivity (see Supplementary Materials) at 1.6 K also exhibits similar behavior, as shown by the pink triangles (right axis) in Fig. 3a. The abrupt suppression of Hc2 when moving away from θ = 0° means that the FFLO state is easily destabilized even by the small orbital effect induced by the slight tilt. This fragility to the orbital effect is also a well-known characteristic peculiar to the FFLO state5,18,41. In contrast, the angle dependence of HFFLO is not significant. As HFFLO corresponds to HP determined by the paramagnetic effect, the angle-insensitive behavior is suggestive of isotropic Pauli paramagnetism. This fact is also consistent with the almost isotropic g-factor in the organic compounds composed of light atoms with weak spin-orbit coupling. In Fig. 3c, we organize the present results as the obtained H-T superconducting phase diagram at θ = 0°. For comparison, we additionally show the data of earlier reports (blank symbols)9,20,21,22,23,24,25,26. Our results are in good agreement with the reported data. In addition, the temperature dependence of the reduced superconducting gap amplitude Δ(T)/Δ(0 K) calculated by the basic BCS theory is also shown on the right axis. Since the α model, a simple extension of the BCS theory, well describes the thermal variation in | Δ(T)/Δ(0 K) | 42, the behavior roughly reconciling with the temperature dependence of HP for the homogeneous superconducting state is reasonable. Above we assumed that the FFLO state would appear between HFFLO and Hc2 according to the results of previous studies, this consistency certainly confirms that the high-field phase is non-BCS superconductivity emerging above HP.

Fig. 3: Superconducting phase diagrams.
figure 3

a, b Upper critical field Hc2 (black triangles) as a function of the absolute value of the field angle | θ| at a 2.1 K and b 6.3 K. The blue circles in a indicate transition fields to the FFLO state HFFLO. The green and red areas represent the regions for the FFLO phase and ordinary superconducting phase, respectively. The dotted curves in a, b are fits to the Tinkham 2D model, |Hc2(θ)cosθ/Hc2(90°)|+(Hc2(θ)sinθ/Hc2(0°))2 = 1. c Field-temperature superconducting phase diagram obtained in this work (filled symbols) and previous reports (blank symbols)9,21,22,26. The dashed curve (right axis) is the temperature dependence of the reduced superconducting gap amplitude Δ(T)/Δ(0 K) calculated by the basic BCS theory.

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Apart from the phase diagram, examination in further detail of the pinning effect enhanced in the FFLO state is interesting. In Fig. 4a, we compare the field dependence of Δv/v (θ = 0°) taken for the parallel (u | |c) and perpendicular (u | |b) configurations under the same conditions. There is only a small difference depending on the sound wave direction below HFFLO. The difference becomes significantly larger in the FFLO region HFFLO < H < Hc2 (green area). Since the acoustic response for θ = 90° is almost isotropic in the whole field region, as shown in Fig. 4b, the behavior is clear evidence of the emergent anisotropy of the FFLO state.

Fig. 4: Emergent anisotropy in the FFLO state.
figure 4

a, b Δv/v at 1.6 K for u | |b (blue) and u | |c (red) in magnetic fields a parallel (θ = 0°) and b perpendicular (θ = 90°) to the 2D conducting plane. Direction dependence of the longitudinal sound wave propagation is observed only in the FFLO state (green area), whereas the direction dependence of Δv/v in the BCS state and the normal state is not significant. The inset schematics show the electronic states (red: superconductivity, blue: normal state), directions of applied fields (light green arrows), and sound polarization vectors (striped arrows).

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Discussion

Since the anisotropy appears only in the reported field-temperature region of the FFLO state, the present results demonstrate that the emergent anisotropy originates from the formation of the FFLO state. Given an additional periodicity of the spatial modulation, it seems natural that the stiffness of the lattice in the direction across that modulation pattern should increase. Thus, sound velocity measures the stiffness of crystal lattice. Namely, the q vector is oriented along the b-axis, perpendicular to the field direction in the present setup. Nevertheless, how this manifests as a change in Δv/v must be discussed. The most likely possibility is flux pinning because vortices have a strong influence on elastic properties in the superconducting state38,39,43. Typically, suppression of vortex motion results in lattice hardening through increase in spring constant. As shown in Fig. 2a, the enhancement of Δv/v with increasing magnetic field in the lower-field region indicates compression of the flux-line lattice, which reduces the vortex dynamics. When the vortex lattice melts, Δv/v decreases and shows a minimum. In Figs. 2c and 4a, the lattice hardening observed in u | |b suggests that the spatial modulation when H | | c reinforces pinning of the flux lines. When this spatial modulation traps the flux lines, the Josephson vortices should be pinned at nodes of the spatial modulation of the FFLO state because the Josephson supercurrent is absent at the node positions. Therefore, for the strong pinning effect, the wavelength of the order parameter oscillation 2π/q (q = |q | ) should be comparable to dJV/n, using simple integers n and the Josephson vortex lattice constant dJV (dJV = φ0/sH, where s is the layer spacing)41,44,45. Using the a*-axis length as the layer spacing, 1.5 nm, dJV at 21 T is estimated to be approximately 60 nm. This estimation means that pinning by the FFLO formation requires q~n*108 m−1. In the FFLO state, the energy gain by the momentum q is larger than the Zeeman splitting energy of the up- and down-spin Fermi surfaces. This energy balance gives the relation qħvF = gμBHFFLO. Here, vF, g, and μB are the Fermi velocity, the g-value, and the Bohr magneton, respectively. This rough approximation leads to q = 6*107 m−1 (2π/q~100 nm) at 21 T. This estimation yields n~0.6 at 21 T, which indicates the less strong pinning effect at 21 T. With increasing the field up to Hc2, the size of 2π/q abruptly decreases down to πξ||~30 nm45,46, and thus, n should pass through 1 and reaches about 2 near Hc2. The change in the pinning effect with the variation in n results in the observed peak structure in the field dependence of Δv/v in the FFLO state. The commensurability effect on the Josephson vortices41,44,45 can be confirmed by the anomalies in the field dependence of the out-of-plane resistivity (see Supplementary Materials). Thus, the lattice hardening in u | |b in the FFLO state may originate from the strong pinning effect of the flux-line lattice. Nevertheless, we should take into account other effects leading to the lattice hardening because the coupling between sound waves and Josephson-vortex lattice has not been well studied yet. For example, the change in the electron-lattice coupling and/or the local density of states related to the Andreev bound states may also modify the elastic properties. The present results certainly demonstrate the fact that the FFLO state yields the emergent anisotropy with the q vector along the b-axis; however, the detailed mechanism should be clarified by further measurements in the future.

Next, we need to consider the question of why the orientation of the q vector is mainly along the b-axis, perpendicular to the field direction in the present configuration. Note, here, that the effect of the spin-orbit interaction on the FFLO state in the present salt is negligible because the organic salt is composed of light elements. In the case of ideal isotropic 3D superconductors, the q vector always points in the field direction1,2. Since the q vector can be oriented in any direction in 3D, the anisotropy can be discussed in the Heisenberg-type model. According to this framework, in the present measurement with H | | c, the q vector should be parallel to the c-axis, not the b-axis. However, the present superconductivity is described by the 2D model (Fig. 3b). For 2D superconductors, the better nesting vectors on the Fermi surface make the FFLO state more stable, and the anisotropy of the Fermi surface often locks the direction of the q vector according to the predominant nesting vectors6,7,8,45. Namely, the FFLO state in the anisotropic 2D superconductor is expected to show Ising-type behavior. Indeed, theoretical studies29,30 suggest that the nesting vector Q1, parallel to the kb-axis (green arrow in Fig. 1d), always strongly relates to the Cooper pairing in κ-type organic salts regardless of the emergent pairing symmetry. Thus, the q vector parallel to the b-axis in the FFLO state should be reasonable for the present 2D superconductor. This result suggests that the direction of the emergent anisotropy and the model describing it, such as Ising, XY, and Heisenberg, can be controlled by changing the shape of the Fermi surface and dimensionality. Future studies of the in-plane field angle dependence of the emergent anisotropy will allow for further detailed discussions of the relation between the q vector and nesting vector. Furthermore, similar measurements in other FFLO candidates would also be interesting, as different anisotropies using other nesting vectors should occur in other FFLO candidates.

The present multidirectional ultrasound measurements demonstrate the emergent anisotropy of the FFLO state induced by the spatial modulation of the order parameter. Since κ-(BEDT-TTF)2Cu(NCS)2 is a 2D clean superconductor, the FFLO state shows Ising anisotropy originating from the anisotropic Fermi surface. Further studies of other FFLO candidates with various features, such as 3D and slight dirtiness, will facilitate a deeper understanding of the FFLO state.

Methods

Sample preparation

Single crystals of κ-(BEDT-TTF)2Cu(NCS)2 measured in this study were synthesized by typical electrochemical process and crystallized as black hexagonally-shaped blocks. The shape of the crystals used in the ultrasonic measurements was modified as described in the Supplementary Materials. Ultrasonic measurements: Using the typical pulse-echo methods, the ultrasonic properties were measured. Longitudinal ultrasound waves, whose frequencies were in the range of 37–39 MHz, were generated and detected by LiNbO3 piezoelectric transducers (90 μm thickness) attached on side surfaces of the crystals. Further details of the setup are presented in the Supplementary Materials.