The FFLO State in the Dimer Mott Organic Superconductor κ -(BEDT-TTF) 2 Cu[N(CN) 2 ]Br

: The superconducting phase diagram for a quasi-two-dimensional organic superconductor, κ -(BEDT-TTF) 2 Cu[N(CN) 2 ]Br, was studied using pulsed magnetic ﬁeld penetration depth measurements under rotating magnetic ﬁelds. At low temperatures, H c2 was abruptly suppressed even by small tilts of the applied ﬁelds owing to the orbital pair-breaking effect. In magnetic ﬁelds parallel to the conducting plane, the temperature dependence of the upper critical ﬁeld H c2 exhibited an upturn and exceeded the Pauli limit ﬁeld H P in the lower temperature region. Further analyses with the second derivative of the penetration depth showed an anomaly at 31–32 T, which roughly corresponded to H P . The origin of the anomaly should not be related to the orbital effect, but the paramagnetic effect, which is almost isotropic in organic salts, because it barely depends on the ﬁeld angle. Based on these results, the observed anomaly is most likely due to the transition between the Bardeen-Cooper-Schrieffer (BCS) and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states. Addition-ally, we discuss the phase diagram and physical parameters of the transition by comparing them with other FFLO candidates.


Introduction
Superconductivity is one of the most intriguing topics in material science, both in terms of basic research and applications. Superconductivity appears when electron pairs are formed and condense in metals. The BCS theory explains the conventional superconductivity that appears in a variety of simple metals and alloys. However, the details of unconventional superconductivity are yet to be elucidated. Unconventional superconductivity is commonly realized nearby metal-insulator transitions, where electron correlations are enhanced. Even in unconventional superconductivity, the electrons are paired by attraction, as suggested by the BCS theory; thus, the details of the pairing mechanism are one of the main topics for unconventional superconductivity studies. The FFLO state, which is one of the unconventional pairings, was proposed by Fulde and Ferrell [1] as well as Larkin and Ovchinnikov [2] in 1964. In the FFLO state, the electrons in a pair have unbalanced momenta, and their total center-of-mass momentum q is finite. The finite q, which modifies the superconducting order parameter with the additional term, exp(iqr) for the FF state [1] and cos(qr) for the LO state [2], induces the spatial modulation of the superconductivity. The superconducting region and the normal-state region appear alternately in real space because the normal state appears at the node positions where the additional term becomes zero. The FFLO state is regarded as an inhomogeneous state, which breaks the rotational symmetry [1][2][3][4][5]. At zero field, the uniform BCS-type pairing with q = 0 is more stable than the inhomogeneous FFLO state; however, when applying magnetic fields, the Zeeman effect causes the Fermi surface to split depending on the spin directions. Above the field where the Zeeman splitting is comparable with the condensation energy of the superconductivity, known as the Pauli limit H P [6], the BCS superconductivity is destroyed. This is known as the paramagnetic pair-breaking effect. However, the FFLO state can be favorable even above H P by pairing on the split Fermi surfaces owing to the finite q. Thus, the FFLO state can appear only at high fields above the H P . In higher magnetic fields, the FFLO state is also suppressed, and many theories [7][8][9] predict that the stability in magnetic fields is affected by various parameters. For example, in the case of isotropic three-dimensional superconductivity, the FFLO phase is very small in the H-T phase diagram [1,2]. Moreover, the difference between the FF state and LO state becomes larger in fields sufficiently higher than H P . It is expected that the superconducting symmetry and the strength of orbital pair-breaking effects also play an important role in the stability of the FFLO state. The investigation of H c2 curve above H P is be important to discuss the details of the FFLO state.
To realize an FFLO state, two conditions need to be met: first, the electronic system should be in a clean limit, and second, the orbital pair-breaking effect should be sufficiently suppressed. The FFLO state hosts the spatial modulation in real space owing to the additional vector q, and impurities smear this modulated pattern with scattering. Therefore, a clean electronic system, in which the mean-free path l is sufficiently larger than the coherence length ξ , is typically required [4,10]. Some theories suggest that the FFLO state can survive even in some disordered systems [11,12]. For the orbital effect, the Maki parameter α M , 2 1/2 H orb /H P , where H orb is the orbital limit, must exceed 1.8 [13,14], because the superconductivity gets destroyed at lower fields before the FFLO state appears if the orbital effect is strong. Basically, for the most superconductivity, the orbital effect is so strong that superconductivity does not survive up to H P . The orbital effect is suppressed when the vortices do not penetrate the superconductor, or the coherence length is sufficiently small, because it originates from the kinetic energy of the supercurrent around the vortices by the Lorentz force. Therefore, the FFLO state may be possible when a magnetic field is precisely applied parallel to the conduction plane of the low-dimensional superconductor, to prevent the magnetic flux from penetrating the superconducting plane, or in the case of heavy electron systems. [4,15,16].
The charge-transfer complex κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Br (hereafter, abbreviated as κ-Br) is known as a high-T c (~12 K) organic superconductor. This salt consists of the organic donor BEDT-TTF layer and the counter anion layer Cu[N(CN) 2 ]Br, as shown in Figure 1. κ-Br has intensively been investigated because of its unconventional superconductivity and proximity to the Mott metal-insulator transition [17,18]. The superconductivity is presumably classified in the d-wave symmetry originating from antiferromagnetic spin fluctuations [17][18][19][20], which grow near the antiferromagnetic Mott insulator phase. The superconductivity has a large superconducting energy gap, leading to a large upper critical field H c2 . Due to the experimental difficulty in performing high-field measurements up to H c2 , the superconducting phase diagram in magnetic fields has not been clarified completely until our recent report [21]. The field-temperature superconducting phase diagram exhibited an upturn of H c2 in a low-temperature and high-field region, which may exceed H P . Moreover, this can be scaled with that of κ-(BEDT-TTF) 2 Cu(NCS) 2 (κ-NCS), which is one of the prime FFLO candidates [22][23][24][25]. Basically, the large effective mass and the electronically quasi-two-dimensionality, suppressing the orbital effect and enhancing the nesting of Fermi surfaces, are advantageous for stabilizing the FFLO state. Although this implies that κ-Br also hosts the FFLO state above H P , there have been no reports on the FFLO phase for κ-Br. Since the electronic structures in the other organic FFLO candidates found so far, such as λ-(BETS) 2 GaCl 4 (λ-GaCl 4 ), β"-(BEDT-TTF) 2 SF 5 CH 2 CF 2 SO 3 (β"-SF 5 ), and β"-(BEDT-TTF) 4 , are different in various aspects, it is necessary to consider several factors for discussing the parameters of the FFLO state. The comparison κ-Br with κ-NCS, which has a very similar electronic state, must be useful to discuss common points underlying the FFLO state. To detect the BCS-FFLO transition, a probe that can yield information even in the superconducting state is needed. As is found in a number of previous studies [23,26,27], the penetration depth is a very sensitive and high-resolution probe of the superconducting state even in short-time pulsed magnetic fields. Therefore, we performed penetration depth measurements to identify κ-Br as the FFLO candidate by detecting the phase boundary between the uniform superconductivity and the FFLO state. Additionally, compared with other FFLO candidates, universal features unique to the FFLO state are discussed.
As divided by the red dashed lines, this material has the two-dimensional layered structure. θ represents the angle from a-axis to b-axis, used for the magnetic-field direction applied in this study.

Radiofrequency (rf) Penetration Depth Measurements
The single crystal measured in the present study was synthesized electrochemically. The out-of-plane electrical resistance of the sample we measured in this study has been reported in [21]. For the rf penetration depth measurements with a tunnel diode oscillator (TDO), the sample, whose dimension was approximately 0.5 mm × 0.5 mm × 0.1 mm, was placed in one of two circles of a 0.7 mm-diameter 8-shaped coil, which could cancel out the voltage induced by the field change of pulsed magnetic fields. The direction of the magnetic field was changed by rotating the sample stage. The TDO circuit was operated at F~82 MHz with LC oscillators, similar to the reported design [26,28]. In this setup, the skin depth of the normal state significantly exceeded the sample thickness, and therefore, the change in the frequency ∆F originated only from the penetration depth of the superconductivity. These measurements were performed in a 4 He cryostat placed in a 60 T pulse magnet, installed at the International MegaGauss Science Laboratory, Institute for Solid State Physics, The University of Tokyo.

Characterization of the Measured Sample
To evaluate whether the present sample was enough clean to host the FFLO state or not, we first estimated the mean-free path l from the quantum oscillations in resistivity, as shown in Figure 2a. The low-field behavior was related to the suppression of the superconductivity. The origin of the peak structure at approximately 10 T has been discussed in previous studies [29,30]. Above 40 T, the Shubnikov-de Haas oscillation was observed. The oscillation frequency was approximately 3900 T, which is consistent with the reported value [31]. Using the Lifshitz-Kosevich formula, the mean-free path l was obtained as 130 nm, which was several times larger than the typical values 30-70 nm [31] and 20 times larger than the in-plane coherence length ξ || =6-7 nm [21,32]. This implies that the present sample was sufficiently clean to form a spatially modulated pattern of the FFLO state.  (1) and (2). The blue dashed curve indicates behavior for simple d-wave superconductivity.
In Figure 2b, we present the field dependence of ∆F in the perpendicular fields. At 13 K, which is higher than the critical temperature T c , the field dependence originated from the magnetoresistance of the Cu wires composing the coil. Below T c , a large response of ∆F was observed at low fields owing to the emergence of superconductivity. The data above 15 T indicated that the magnetoresistance of Cu did not have a large temperature dependence in this temperature region. The difference in ∆F between 0 T and 20 T was directly related to the shift of the penetration depth of the superconductivity ∆λ, as shown in the following equation: where r and x represent the effective sample radius and filling factor of the sample in the coil, respectively. The absolute value of the penetration depth λ(T) is given by the sum of the change and zero-temperature value, λ(T) = ∆λ + λ(0). Because the superfluid density ρ(T) is determined by the relation ρ(T) = [λ(0)/λ(T)] 2 , the Rutgers equation [33], leads to λ(0) = 0.6 ± 0.2 µm with the reported parameters, heat capacity jump ∆C p (T c ) = 0.6-0.7 J/Kmol [18][19][20][21] and the slope of H c2 at T c , µ 0 H c2 (T c )~−15 T/K [21,34]. Despite the large error, the value showed a good agreement with the reported values λ(0) = 0.5-0.7 µm [35][36][37].
The inset shows λ(T) with a fit to the d-wave case (blue dashed curve) [38]. Although a large error made the precise determination of the pairing symmetry difficult, the d-wave model was acceptable for the present data. This result indicated that ∆F reflects the change in penetration depth in the superconducting state.

Magnetic-Field Dependence of ∆F and d 2 (∆F)/dH 2 in Nearly Parallel Fields
In Figure 3a, we present the ∆F data at 1.4 K in fields almost parallel to the conducting plane (θ~0 • ), because the FFLO state occurs at lower temperatures when the orbital effect is sufficiently suppressed. At θ = 0 • , the onset of the change in ∆F from the normal state was approximately 37 T, which was almost consistent with the reported value of H c2 [21]. By tilting the angle from θ = 0 • , H c2 was suppressed. Figure 3b shows the second-field derivative of ∆F. The black lines represent the field-independent baseline of the normal state, and the black dotted curves are the eye guides. This plot further indicates that H c2 was approximately 37 T at θ = 0 • . Notably, these curves had some anomalies (green box and blue triangle) below H c2 , which were not clear in the ∆F data in Figure 3a. This behavior was similar to the results reported in earlier rf penetration depth studies for other organic FFLO candidates [23,26,27]. The anomaly at 31-32 T indicated by the blue triangles appeared to have a bare angle dependence, while the anomaly indicated by the green boxes showed the angle dependence. The angle dependence of the transition should not be significant in organics with weak spin-orbit coupling because the phase transition to the FFLO state is determined by the Zeeman effect; therefore, the anomaly indicated by the blue triangles at 31-32 T is considered to be the BCS-FFLO transition, namely H FFLO = 31-32 T. In fact, the angle-independent behavior was observed in other FFLO candidates [23,25,26,39,40].

Temperature Dependence of ∆F and d 2 (IF)/dH 2 in Perfectly Parallel Fields (θ = 0 • )
To discuss the stability of the FFLO state against temperature, in Figure 4, we present the field-dependent ∆F (a) and its second derivative (b) at θ = 0 • . The symbols shown here are the same as those used in Figure 3b. The BCS-FFLO transition (blue triangle) was observed below 4.0 K and showed a slight temperature dependence, as shown by the blue dashed curve. The additional anomaly in the FFLO state (green box) was immediately smeared out with increasing temperature above 1.4 K. Considering the angle-dependent behavior and the observation at low temperatures, the anomaly indicated by the green box may be related to the vortex transitions in the FFLO state [40,41].

Discussion
In Figure 5a, we organized the obtained H c2 and H FFLO in parallel fields as an H-T phase diagram. The H c2 data reported in the previous study [21] were also plotted. The H c2 obtained in this study (red circle) was consistent with the reported data (gray box). The blue triangles denote the fields in which a kink is observed in Figure 4b. From the initial slope of the H c2 curve (solid line) near T c , the orbital limit field H orb and the perpendicular coherence length ξ ⊥ were estimated to be approximately 130 T, larger than H c2 , and 0.3 nm, five times smaller than the interlayer spacing 1.5 nm [31], respectively. These values indicate that the superconductivity was two-dimensional, and the orbital pair-breaking effect was quenched in parallel fields. To discuss the destruction of superconductivity, only the paramagnetic pair-breaking effect was considered. In a simple assumption based on the BCS theory, this effect gave the Pauli limit µ 0 H P = 1.76k B T c /(2 1/2 µ B )~1.84(T c [K])[T] from the balance between the superconducting energy gap ∆ 0 = 1.76 k B T c and the Zeeman energy gµ B H. However, this assumption often does not work for organic superconductors, because the superconductivity in organics is usually strong-coupling and has unconventional pairing symmetry [19][20][21]. In Agosta's papers [16,23], to discuss the relation between H P and H FFLO more precisely, the following formula: which includes the electron correlation and the coupling strength based on McKenzie's paper [42], was employed. Notably, g* is the effective g-factor, including all many-body effects, and α is the coupling constant of the superconducting gap amplitude. The renormalized ratio g*/g is experimentally determined using quantum oscillation measurements. In addition, g*/g can be estimated by the ratio of the electronic heat capacity coefficient γ and the Pauli paramagnetism χ P , because g*/g can be equal to Wilson's ratio R w [42]. For κ-Br, this relation led to µ 0 H P = 31-32 T, which corresponded to the present H FFLO . This coincidence indicates that the anomaly observed in this study was caused by the transition to the FFLO state. Moreover, the Maki parameter α M~5 .7 was sufficiently larger than required. From the angle dependence of H c2 shown in Figure 3, the FFLO state should exist only in the limited region near the parallel direction θ~0 • because of the disappearance by the slight misalignment. This fragility to the orbital effect is also a characteristic of the FFLO state [14,43].  [23,26,27], whereas the unfilled symbols are from other measurements [25,39,44,45]. The dotted gray line represents an example of the temperature dependence of H orb , which is higher than H c2 at low temperatures. The red dotted curve is a simple theoretical calculation [9] of H c2 for the FFLO state with d-wave superconducting symmetry. The black dashed curve (right axis) is the temperature dependence of the reduced BCS-type superconducting gap ∆(T)/∆ 0 .
For comparison with other organic FFLO candidates, the H-T phase diagram in a parallel field was reduced by H P and T c , as shown in Figure 5b. The H c2 and H FFLO data for other salts are also shown [23,[25][26][27]39,44,45]. The parameters, T c , g*/g, and α, which were used for the estimation of H FFLO , are listed in Table 1. We used a typical value by referring to several references, because there is often some sample dependence in these parameters, and T c depends on the measurement method and definition. Despite the large differences in their electronic states, such as the Fermi surface and dimensionality, these superconductors shared similar H/H P -T/T c phase diagrams. In the κ-type dimer-Mott electronic phase diagram, κ-Br and κ-NCS were located near the Mott metal-insulator boundary [17,18], indicating strong electron correlations originating from the large onsite Coulomb repulsion with the growth of the antiferromagnetic fluctuations. This characteristic resulted in a relatively large g*/g and α of κ-Br and κ-NCS. Althoughλ-GaCl 4 and β"-SF 5 have significantly different Fermi surfaces [46,47], their parameters were almost identical and gave a similar H FFLO . For β"-GaPhNO 2 , the electronic state was expected to be near the charge-ordered phase on its electronic phase diagram and had a strong charge instability, which induced strong-coupling superconductivity α~2.5 [48]. Regardless of the variety in these electronic systems, the calculated H P coincided with H FFLO , as listed in Table 1. This fact demonstrated that the paramagnetic effect, which was the factor underlying H P , mainly governed the transition between the BCS and FFLO states, as predicted by a number of theories. Indeed, the dashed curve shown in the right axis in Figure 5b, which is the temperature dependence of the BCS-type superconducting gap, roughly describes the BCS region (light orange). Importantly, the paramagnetic pair-breaking was determined by the competition between the superconducting gap and the Zeeman effect. However, the H c2 curves above H P indicated that there were small differences in the stability of the FFLO state at high fields. The theoretical curve for simple d-wave superconductivity (red dotted curve) [9] was qualitatively similar to the obtained phase diagram. Nevertheless, the data in Figure 5b were not accurate enough to discuss small differences with the simple model, and therefore, it would be necessary to discuss with an appropriate theoretical model for the electronic system of each material rather than the simple model. The parameters related to the stability of the FFLO state are likely the dimensionality and the shape of the Fermi surface as well as the gap symmetry. For example, (TMTSF) 2 ClO 4 , which was expected to exhibit the FFLO state [49], had a quasi-one-dimensional system, and the difference was expected to be significant. Although its superconducting phase showed strange differences in electrical resistivity and specific heat measurements [50,51], it might be interesting to discuss it through H P . As for the research method to discuss the FFLO state in more detail, for example, the in-plane angular dependence from Refs. [51,52] may be useful. Further research should be completed along with theoretical predictions.

Conclusions
We performed high-field rf-penetration depth measurements to determine whether the FFLO state manifested as a high-field superconducting state distinct from the BCS state in the organic superconductor κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Br. From the quantum oscillation and the phase diagram, it was confirmed that the electronic system was sufficiently clean and two-dimensional to stably host the FFLO state. As has been discussed for the FFLO state in other candidates, the transition field between the BCS and FFLO states had no angle dependence, whereas the FFLO state was very sensitive to the field angle and was immediately smeared out by the slight misalignment. Compared to other organic FFLO candidates, their H/H P -T/T c superconducting phase diagrams suggest that H P certainly corresponds to H FFLO , regardless of the electronic states underlying the superconductivity. This verifies that the BCS-FFLO transition is determined by the competition between the Zeeman energy and the superconducting condensation energy. The FFLO state appears at very high fields above 31-32 T, because κ-Br can also be discussed in this framework, and its H P is enhanced by the large superconducting gap originating from the strong electron correlations growing in proximity to the Mott metal-insulator boundary.  Acknowledgments: We thank Y. Kohama (ISSP, the University of Tokyo) for advice on the rf-TDO measurement.

Conflicts of Interest:
The authors declare no conflict of interest.