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

**Shusaku Imajo \* and Koichi Kindo**

Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan; kindo@issp.u-tokyo.ac.jp **\*** Correspondence: imajo@issp.u-tokyo.ac.jp

**Abstract:** The superconducting phase diagram for a quasi-two-dimensional organic superconductor, κ-(BEDT-TTF)2Cu[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*<sup>P</sup> 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. Additionally, we discuss the phase diagram and physical parameters of the transition by comparing them with other FFLO candidates.

**Keywords:** FFLO; organic superconductor; penetration depth measurement

#### **1. 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(i*qr*) 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 [1–5]. 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*<sup>P</sup> [6], the BCS superconductivity is destroyed. This is known as the paramagnetic pair-breaking effect. However, the FFLO state can be favorable even above *<sup>H</sup>*<sup>P</sup> by pairing on the split Fermi surfaces owing to the finite *<sup>q</sup>*. Thus, the FFLO

**Citation:** Imajo, S.; Kindo, K. The FFLO State in the Dimer Mott Organic Superconductor κ-(BEDT-TTF)2Cu[N(CN)2]Br. *Crystals* **2021**, *11*, 1358. https:// doi.org/10.3390/cryst11111358

Academic Editor: Andrej Pustogow

Received: 25 October 2021 Accepted: 5 November 2021 Published: 8 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

state can appear only at high fields above the *H*P. In higher magnetic fields, the FFLO state is also suppressed, and many theories [7–9] 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*<sup>P</sup> 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, 21/2*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)2Cu[N(CN)2]Br (hereafter, abbreviated as κ-Br) is known as a high-*T*<sup>c</sup> (~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 [17–20], 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)2Cu(NCS)2 (κ-NCS), which is one of the prime FFLO candidates [22–25]. 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)2GaCl4 (λ-GaCl4), β"-(BEDT-TTF)2SF5CH2CF2SO3 (β"-SF5), and β"-(BEDT-TTF)4[(H3O)Ga(C2O4)3]PhNO2 (β"-GaPhNO2), 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.

**Figure 1.** Crystal structure of κ-(BEDT-TTF)2Cu[N(CN)2]Br. 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.

#### **2. 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 Δ*F* originated only from the penetration depth of the superconductivity. These measurements were performed in a 4He cryostat placed in a 60 T pulse magnet, installed at the International MegaGauss Science Laboratory, Institute for Solid State Physics, The University of Tokyo.

#### **3. Results**

#### *3.1. 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.

**Figure 2.** (**a**) Magnetoresistance at 2.2 K in a perpendicular field (*θ* = 90◦). The inset is the enlarged plot above 40 T to make the quantum oscillation clearer. (**b**) Shift of the penetration depth Δ*F* at various temperatures as a function of field when *θ* = 90◦. For clarity, the datasets include offsets. The inset is the temperature dependence of the penetration depth obtained by the Equations (1) and (2). The blue dashed curve indicates behavior for simple *d*-wave superconductivity.

In Figure 2b, we present the field dependence of Δ*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 Δ*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 Δ*F* between 0 *T* and 20 *T* was directly related to the shift of the penetration depth of the superconductivity Δ*λ*, as shown in the following equation:

$$[\Delta F(0 \ T) - \Delta F(20 \ T)]/F = \ge \Delta \lambda / r,\tag{1}$$

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*) = Δ*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],

$$16\pi^2 \Delta C\_p(T\_c)\lambda(0) = q\_0 T\_c \text{d}H\_{c2'}(T\_c)\rho'(T\_c),\tag{2}$$

leads to *λ*(0) = 0.6 ± 0.2 μm with the reported parameters, heat capacity jump Δ*Cp*(*T*c) = 0.6– 0.7 J/Kmol [18–21] 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 [35–37]. 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 Δ*F* reflects the change in penetration depth in the superconducting state.

#### *3.2. Magnetic-Field Dependence of* Δ*F and d2(*Δ*F)/dH<sup>2</sup> in Nearly Parallel Fields*

In Figure 3a, we present the Δ*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 Δ*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 Δ*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 Δ*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].

**Figure 3.** (**a**) Magnetic-field dependence of Δ*F* at 1.4 K with changing field angle *θ*. (**b**) Second derivative of (**a**) d2*F*/d*H*<sup>2</sup> as a function of field. The black solid lines and the red dots show the background of the normal state and *H*c2. The blue triangles and green squares indicate the anomalous fields of d2*F*/d*H*2. The black dotted, blue dashed, and green dashed curves are eye guides.

#### *3.3. Temperature Dependence of* Δ*F and d2(IF)/dH2 in Perfectly Parallel Fields (θ = 0*◦*)*

To discuss the stability of the FFLO state against temperature, in Figure 4, we present the field-dependent Δ*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].

**Figure 4.** (**a**) Δ*F* (*θ* = 0◦) as a function of field at various temperatures. (**b**) Magnetic-field dependence of d2*F*/d*H*2. The symbols and curves are described using the same definitions as those used in Figure 3b.

#### **4. 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*<sup>⊥</sup> 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*<sup>P</sup> = 1.76*k*B*T*c/(21/2*μ*B)~1.84(*T*c[K])[T] from the balance between the superconducting energy gap Δ<sup>0</sup> = 1.76 *k*B*T*<sup>c</sup> 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 [19–21]. In Agosta's papers [16,23], to discuss the relation between *H*<sup>P</sup> and *H*FFLO more precisely, the following formula:

$$
\mu\_0 H\_\text{P} = a k\_\text{B} T\_\text{c} / \langle 2^{1/2} (\text{g}^\*/\text{g}) \mu\_\text{B} \rangle\_\text{\textdegree} \tag{3}
$$

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*<sup>w</sup> [42]. For κ-Br, this relation led to *μ*0*H*<sup>P</sup> = 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].

**Figure 5.** (**a**) *H*-*T* superconducting phase diagram of κ-Br in parallel fields. The gray boxes are the reported *H*c2 data [21]. The solid line indicates the slope of the *H*c2 curve near *T*<sup>c</sup> at 0 T. The light green region above *H*<sup>P</sup> is the FFLO state. (**b**) Superconducting phase diagram scaled by *H*<sup>P</sup> and *T*c, *H*/*H*<sup>P</sup> vs. *T*/*T*c. The circles and triangles represent *H*c2 and *H*FFLO, respectively. The color of the symbols denotes the material. The filled symbols are taken from the TDO measurements [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 Δ(*T*)/Δ0.

For comparison with other organic FFLO candidates, the *H*-*T* phase diagram in a parallel field was reduced by *H*<sup>P</sup> and *T*c, as shown in Figure 5b. The *H*c2 and *H*FFLO data for other salts are also shown [23,25–27,39,44,45]. 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*<sup>c</sup> 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-GaCl4 and β"-SF5 have significantly different Fermi surfaces [46,47], their parameters were almost identical and gave a similar *H*FFLO. For β"-GaPhNO2, 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*<sup>P</sup> 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*<sup>P</sup> 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)2ClO4, 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.

**Table 1.** Reported *H*FFLO and calculated *H*<sup>P</sup> with parameters characterizing the FFLO state. The abbreviations of the material names are described in the main text. The shown *H*<sup>P</sup> is estimated by the Equation (3) and the parameters shown here, which are typical values taking into account sample dependence, etc. For the estimation of the values of *g\**/*g*, the Wilson' ratio *R*W, calculated by *γ* and *χ*P, is also used to compare with *g\**/*g* determined by angle-dependent quantum oscillations. The values of *α* are taken from heat capacity measurements.


#### **5. 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)2Cu[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*<sup>c</sup> superconducting phase diagrams suggest that *H*<sup>P</sup> 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*<sup>P</sup> is enhanced by the large superconducting gap originating from the strong electron correlations growing in proximity to the Mott metal-insulator boundary.

**Author Contributions:** Conceptualization, S.I.; methodology, S.I. and K.K.; investigation, S.I. writing, S.I.; funding acquisition, S.I. and K.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially supported by the Japan Society for the Promotion of Science KAKENHI Grant No. 20K14406.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data are available from the corresponding author upon reasonable request.

**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.

#### **References**


#### *Article* **Fermi Surface Structure and Isotropic Stability of Fulde-Ferrell-Larkin-Ovchinnikov Phase in Layered Organic Superconductor** *β--***-(BEDT-TTF)2SF5CH2CF2SO3**

**Shiori Sugiura 1,\*,†, Hiroki Akutsu 2, Yasuhiro Nakazawa 2, Taichi Terashima 3, Syuma Yasuzuka 4, John A. Schlueter 5,6 and Shinya Uji 3,\*,†**


**Abstract:** The Fermi surface structure of a layered organic superconductor *β*---(BEDT-TTF)2SF5CH2CF2SO3 was determined by angular-dependent magnetoresistance oscillations measurements and band-structure calculations. This salt was found to have two small pockets with the same area: a deformed square hole pocket and an elliptic electron pocket. Characteristic corrugations in the field dependence of the interlayer resistance in the superconducting phase were observed at any in-plane field directions. The features were ascribed to the commensurability (CM) effect between the Josephson vortex lattice and the periodic nodal structure of the superconducting gap in the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase. The CM effect was observed in a similar field region for various in-plane field directions, in spite of the anisotropic nature of the Fermi surface. The results clearly showed that the FFLO phase stability is insensitive to the in-plane field directions.

**Keywords:** organic superconductor; resistance; FFLO phase; vortex dynamics

#### **1. Introduction**

The discovery of superconductivity has led to breakthroughs in a wide range of fields from fundamental research and applications [1]. In particular, since the discovery of high-temperature superconducting cuprates in 1980s, the search for new superconducting mechanisms has been one of the major trends in superconductivity basic research. Among the various superconductors, organic superconductors in the vicinity of metal-insulator transitions have brought about significant progress in basic research.

Organic conductors based on BEDT-TTF molecules are characterized by a stacked structure with anion molecule (insulating) layers and BEDT-TTF molecule (conducting) layers. These conductors have attracted significant interest because of the presence of various ground states, a dimer-Mott insulating phase, a charge-ordered phase, a density wave phase, and a superconducting phase, where the degree of dimerization of the BEDT-TTF molecules, the Fermi surface instability, and the strong electron correlation play important roles. In particular, the possibility of unconventional superconductivity, mediated by antiferromagnetic spin and/or charge fluctuations, is a central concern.

When the orbital effect is suppressed and the critical field (*H*c2) is Pauli-limited, a unique superconducting phase, namely, the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO)

**Citation:** Sugiura, S.; Akutsu, H.; Nakazawa, Y.; Terashima, T.; Yasuzuka, S.; Schlueter, J.A.; Uji, S. Fermi Surface Structure and Isotropic Stability of Fulde-Ferrell-Larkin-Ovchinnikov Phase in Layered Organic Superconductor *β*---(BEDT-TTF)2SF5CH2CF2SO3. *Crystals* **2021**, *11*, 1525. https:// doi.org/10.3390/cryst11121525

Academic Editor: Andrej Pustogow

Received: 20 November 2021 Accepted: 1 December 2021 Published: 7 December 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

superconducting phase is expected to emerge at high fields [2,3]. In conventional superconductors, the spin-singlet Cooper pairs formed by up and down spins are destroyed in a magnetic field by the Zeeman effect. This pair-breaking effect gives the Pauli limit, *H*Pauli = Δ0/ <sup>√</sup>2*μ*<sup>B</sup> <sup>=</sup> 1.86*T*c, where <sup>Δ</sup><sup>0</sup> is the superconducting energy gap at 0 K, and *<sup>μ</sup>*<sup>B</sup> is the Bohr magneton [4]. In the FFLO phase, the Cooper pairs are formed between up and down spins on the polarized Fermi surface. Therefore, the Cooper pairs have a finite center-of-mass momentum *q* and show a spatial modulation of the order parameter in real space; <sup>Δ</sup>(*r*) = <sup>Δ</sup>cos(*qr*), as shown in Figure 1a. As a result, the superconductivity can be stabilized even above *H*Pauli. In recent years, experimental results suggesting its existence have been obtained in heavy fermion superconductors [5,6], oxide layered superconductors [7], ion-based superconductors [8], and organic superconductors [9–16]. In organic superconductors, the FFLO phase transition was first observed by a tuned-circuit differential susceptometer experiment for *κ*-(BEDT-TTF)2Cu(NCS)2 [10], and, since then, various measurements [17–24] have been performed to confirm the FFLO transition.

**Figure 1.** (**a**) Schematic illustration of the order parameter oscillation <sup>Δ</sup>(*r*) for a single *q* case in a FFLO phase and JV lattice in a layered superconductor. The JVs are easily driven by the Lorentz force in an interlayer current *I*, leading to nonzero interlayer resistance even in the superconducting phase. (**b**) Temperature dependence of the interlayer resistance of the *β*---SF5 salt. The onset of the superconducting transition can be defined as *T*<sup>c</sup> ≈ 4.8 K, consistent with the specific heat measurement [23]. Inset: crystal structure of *β*---SF5 salt.

Highly two-dimensional (2D) layered superconductors can be modeled as Josephsoncoupled multi-layer systems. In such superconductors, magnetic flux lines penetrating the sample can be decomposed into two parts; the pancake vortices (PVs) penetrating the superconducting layers and the Josephson vortices (JVs) penetrating the insulating layers. The JVs are pinned more loosely than the PVs, since the order parameter vanishes in the insulating layers. Therefore, the JVs are easily driven by the Lorentz force in an interlayer current, and, consequently, nonzero interlayer resistance is observed even in the superconducting phase. In the FFLO phase, periodic nodal lines of the order parameter are formed by the finite center-of-mass momentum *q* of the Cooper pairs as depicted in Figure 1a. When the nodal lines are parallel to the JVs, they will work as pinning sites of the JVs. The wavelength of the order parameter oscillation is given by *λ*FFLO = 2*π*/*q*, and the JV lattice spacing is *l* = Φ0/*sH*, where *s* is the interlayer spacing and Φ<sup>0</sup> is the flux quantum. It is expected that JVs are relatively strongly pinned by the nodal line structure for a commensurate condition *l*/*λ*FFLO = *N* (*N* : integer). Since *λ*FFLO is also expected to decrease with an increasing field [25,26], the commensurate condition in the FFLO phase will be periodically satisfied, leading to fine structures in the interlayer resistance curves. This commensurability (CM) effect was first predicted by Bulaevskii et al. [27]. Thus far, the CM effect has been observed in the FFLO phases for various organic superconductors, *λ*-(BETS)2FeCl4 [15], *β*---(BEDT-TTF)4[(H3O)Ga(C2O4)3]C6H5NO2 (*β*---Ga salt) [28], and *β*---(BEDT-TTF)2SF5CH2CF2SO3 (*β*---SF5 salt) [29]. Among them, the highly 2D nature of

the *β*---Ga and *β*---SF5 salts with the large anion layers would provide an excellent platform for the FFLO studies, since JV dynamics play an essential role in the CM effects.

The *β*---SF5 salt with *T*<sup>c</sup> ≈ 4.8 K is composed of the conducting BEDT-TTF molecular layer and the large insulating SF5CH2CF2SO3 layer (inset of Figure 1b). A highly 2D electronic state has been realized, which is characterized by a large ratio of the intralayer to that of interlayer critical fields *H* c2/*H*<sup>⊥</sup> c2 ≈ 11.5. The specific-heat measurements show strongly coupled BCS-like behavior with a full gap given by Δ0/*k*B*T* = 2.18 [30]. In a magnetic field parallel to the conducting layers, the critical field *H*c2 significantly exceeded the Pauli limit *H*Pauli ≈ 10 T, above which the FFLO superconductivity is realized [31–35]. Optical measurements of isostructural *β*---(BEDT-TTF)2SF5*R*SO3 (*R* = CH2, CHFCF2, CH2CF2, and CHF) compounds revealed that the superconducting phase is adjacent to a chargeordered insulating phase [36]. This could indicate superconductivity mediated by charge fluctuations, which is another reason for the interest in the *β*-salts.

In our previous studies, we clarified the FFLO phase boundary in terms of the magnetocaloric effect, torque [35], and resistance measurements [29]. We also observed the CM effect in fields almost parallel to the *a*-axis in the FFLO phase, above ∼9 T. The *λ*FFLO values, ranging from ∼40 nm to ∼210 nm, were obtained under the assumption of a single *q* vector perpendicular to the field. The stability of the FFLO phase is closely related to the nesting instability of the Fermi surface, and the *q* vector leading to a large nesting part is favorable for the FFLO phase. Therefore, the optimum *q* vector depends on the anisotropic structure of the Fermi surface. Meanwhile, the orbital effect stabilizes the *q* vector parallel to the field. This situation can lead to complicated field-direction dependencies of the optimum *q* vector. Even multi-*q*-vector phases are theoretically predicted depending on the field strength and temperature [37].

In this study, we focused on the stability of the FFLO phase in the *β*---SF5 salt, with an anisotropic Fermi surface. Firstly, we clarify the Fermi surface structure from the measurements of angular-dependent magnetoresistance oscillations (AMROs), and then we report the CM effect in various in-plane field directions. The CM effect was surprisingly observed in a similar field region for various in-plane field directions, despite the anisotropic Fermi surface structure. Possible scenarios for explaining these results are presented.

#### **2. Materials and Methods**

Single crystals of the *β*---SF5 salt were synthesized using a standard electrochemical method [38]. Two gold wires of 10 μm diameter were attached to both sides of the single crystal using carbon paste. The interlayer resistance with an electric current perpendicular to the superconducting layers was measured using a conventional four-probe AC technique. The single crystals were mounted on a two-axis rotator in a 3He cryostat with a 15 T superconducting magnet and cooled down to ∼0.5 K at a rate of ∼1 K/min. All measurements were performed at the Tsukuba Magnet Laboratories, NIMS.

#### **3. Results**

Figure 1b shows the temperature dependence of the interlayer resistance for the *β*---SF5 salt. The resistance decreased monotonically with decreasing temperature. At ∼4.8 K, a sudden drop in resistance was observed due to the superconducting transition. Below ∼4 K, the resistance was zero within the noise level. To investigate the 2D Fermi surface structure, we first measured the AMROs in various rotation planes. Typical AMRO data are presented in Figure 2a. The angles *θ* and *ϕ* are defined in the inset. The characteristic *θ* dependence of the interlayer resistance is shown in the upper part of Figure 2a. In the negative second derivative curves, we can observe AMROs, which are periodic with tan(*θ*), as shown in the lower part of Figure 2a. The AMRO period *δ* directly yields the reciprocal lattice vector *<sup>k</sup>*, *<sup>δ</sup>*(*r*) = *<sup>π</sup>*/*sk*(*ϕ*) values. Figure 2b shows the polar plot of *<sup>k</sup>* obtained from the AMRO measurements at various *ϕ*. We can draw the cross-section of the 2D Fermi surface, inscribed in the *k*(*ϕ*) curves, by a solid curve, assuming an elliptical shape. The cross-section of the Fermi surface is very elongated, whose area was ∼6% of the first Brillouin zone. The AMRO results were almost consistent with those of previous reports [39].

**Figure 2.** (**a**) Typical AMRO data and their negative second derivative curves at 1.5 K for 14 T. (**b**) Polar plot of *<sup>k</sup>* obtained from the AMRO measurements. The red solid curve shows the 2D Fermi surface obtained from the AMRO measurements.

In Figure 3a, we present the band calculations by an extended Huckel method [40], using lattice parameters obtained from X-ray crystallography [38]. The calculated 2D Fermi surface is depicted in Figure 3b. The results are different from the reported Fermi surface structure, with a pair of 1D Fermi surface and a closed Fermi surface [38,41], in which the Brillouin zone is apparently wrong. In our calculations, there were two pockets with different carriers: a deformed square electron pocket and an elliptic hole pocket. The areas were equal to each other, and a compensated metal was formed. This is consistent with a single frequency of the quantum oscillation [42–44]. For comparison, the Fermi surface obtained from the AMRO measurements is indicated by a red dotted curve, which is almost consistent with the electron pocket. In the AMRO measurements, the hole pocket was not observed. The reason for this is not clear at present.

**Figure 3.** (**a**) Band structure by an extended Huckel method and (**b**) 2D Fermi surface structure for the *β*---SF5 salt. Deformed square hole and elliptic electron pockets were formed. The red dotted curve indicates the 2D Fermi surface determined by the AMRO measurements.

Figure 4a shows the magnetic field dependence of the interlayer resistance at various temperatures. The field was applied parallel to the *b* axis, in the superconducting *a*- –*b* plane, within the accuracy of 0.1°. At 0.5 K, the resistance increased with the field above 6.5 T, defined as *H*onset. Characteristic corrugation was evident. The corrugation was reduced with increasing temperature. The critical field was determined as *H*c2 ≈ 13 T at 0.7 K from the specific-heat measurements [23]. The finite resistance in the wide field region

below *H*c2 can be ascribed to the motion of the JVs in the insulating layers, as has been observed in various organic superconductors [15,28,29]. To clearly see the corrugations in detail, negative second-derivative curves of the resistance are plotted in Figure 4b. At 0.5 K, we see a broad dip at ∼6.5 T, corresponding to the resistance increase from the noise level. Above ∼8 T, we see a quasi-periodic dip structure, which is most pronounced at ∼10 T. This structure can be ascribed to the CM effect, which is observable only in the FFLO phase, as discussed in the previous reports [29]. At higher fields, the CM effect is reduced and vanishes above ∼12 T, which corresponds to the melting transition of the JV lattice. As temperature increases, the dip structure is suppressed and shifts to a lower field region. Above 2.1 K, such astructure is not evident. As has been discussed [27,29], the dips mean relatively strong pinning of the JV lattice, ascribed to the CM effect between the JV lattice and the periodic nodal structure of the gap <sup>Δ</sup>(*r*).

**Figure 4.** (**a**) Magnetic field dependence of the interlayer resistance at various temperatures. The field was applied parallel to the *b* axis in the superconducting *a*- –*b* plane within an accuracy of 0.1°. (**b**) Negative second-derivative curves of the resistance. Each curve was shifted for clarity.

From the above results, we obtained the temperature–field phase diagram shown in Figure 5. The blue squares indicate the dip fields *H*dip, and the solid curve indicates *H*c2, which was determined from specific-heat measurements [23]. The FFLO phase appears in a wide region above ∼8 T and below ∼2 K. The phase diagram is very similar to that for the *H a*-axis [29], although the Fermi surface was anisotropic.

Figure 6a shows the magnetic field dependence of the resistance at various field angles *θ*. For *θ* = 0°(*H b*-axis), the resistance increased with the field above *μ*0*H*onset = 6 T, which is indicated by an arrow. Figure 6b shows the negative second-derivative curves of the resistance. The low *H*onset value for *θ* = 0°, denoted by an arrow, indicates that only JVs were formed (no PVs), which were pinned very weakly in the insulating layers. When the field was tilted from the superconducting layer, *H*onset increased. This behavior is explained by the stronger pinning of the flux lines in the superconducting layers, where PVs are formed. As the field was further tilted, *H*c2 was steeply reduced, leading to a decrease in *H*onset. For *θ* = 0°, small dips due to the CM effect can be seen above ∼9.5 T. As the field was tilted from the layer, the CM effect was suppressed, and no CM effect was observed for |*θ*| - 0.6°. The stability of the FFLO phase in such a small angle region is consistent with the results for the field almost parallel to the *a* axis [29].

**Figure 5.** Temperature–field phase diagram for *H b* axis. *H*c2 determined by the specific-heat measurements is indicated by a solid curve [23].

**Figure 6.** (**a**) Magnetic field dependence of the resistance at various field angles *θ*. Definition of *θ* is in the inset. (**b**) Negative second-derivative curves of the resistance. Each curve is shifted for clarity.

Figure 7a shows the field dependence of the negative second-derivative curves at various in-plane field directions *ϕ*, as shown in the inset of Figure 7b. For *ϕ* = 0°, we see the onset field *μ*0*H*onset = 6 T (black arrow). The *H*onset value had a nonmonotonic *ϕ* dependence. It should be noted that *H*onset is the depinning field of the JV lattice [29], determined by the pinning strength at the sample edges and/or some other (impurity or defect) pinning sites, which is not related to the FFLO phase transition. The anisotropic behavior of *H*onset is possibly due to the shape effect of the sample. In contrast, we observed many dips above ∼9 T, owing to the CM effect in a similar field region at any *ϕ*. This suggests that the FFLO phase stability was insensitive to the in-plane field direction. An important feature is that the largest dip was evident at *μ*0*H*∗ dip ≈ 9 T (red arrow) in a wideangle region, except for *ϕ* = 0°–45°. This *ϕ* dependence of the dip amplitude suggests some differences in the JV dynamics in the FFLO phase. The largest dip field *H*∗ dip corresponds to the strongest CM effect and is plotted as a function of *ϕ* in Figure 7b. We note that *H*∗ dip is almost isotropic, despite the anisotropic Fermi surface structure as presented in Figure 3b.

**Figure 7.** (**a**) Magnetic field dependence of the negative second-derivative curves at various in-plane field directions *ϕ* defined in the inset of (**b**). Each curve is shifted for clarity. The onset field *H*onset and the largest dip field *H*∗ dip are indicated by black and red arrows, respectively. (**b**) *H*dip as a function of *ϕ*. The dotted curve indicates the expected value for the CM condition *H*dip ∝ 1/cos(*ϕ*) with a single *q* vector.

#### **4. Discussion**

We observed the CM effect in various in-plane field directions, which is recognized as strong evidence of the FFLO phase characterized by the *q* vector. As pointed out, the *q* vector leading to a large nesting part is favorable for the FFLO phase, as schematically depicted in Figure 8a, where the largest number of Cooper pairs can be formed by the *q* vector, perpendicular to the flat part of the Fermi surface. On the other hand, in the presence of the orbital effect, the *q* vector parallel to the field is favorable for an isotropic Fermi surface, leading to no CM effect. The observation of the CM effect at any *ϕ* indicates that the *q* vector is not parallel to the field.

**Figure 8.** (**a**) Spin-polarized 2D Fermi surface in a magnetic field. Up and down Fermi surfaces are indicated by red and blue curves, respectively. (**b**) Schematic nodal line structure of a FFLO phase and JV lattice in a tilted field for the CM condition *l*/*λ*FFLO = 1.

The largest dip at *H*∗ dip in Figure 7a suggest the strongest CM effect, *l*/*λ*FFLO = 1, where all the flux lines can fit into the nodal lines completely. Assuming that the *<sup>q</sup>* vector is fixed to a certain direction (for instance, *a*- -axis), the JV lattice spacing is given by *l* = Φ0/*sH*cos(*ϕ*) as depicted in Figure 8b. This leads to large *ϕ* dependence of *H*∗ dip as indicated by the dotted curve in Figure 7b, which is inconsistent with the experimental result. This inconsistency requires another factor on the stability of the FFLO phase. At the nodal lines, the order

parameter vanishes in the superconducting layers. Therefore, the most stable condition of the vortex structure will be that all the flux lines are almost parallel to the nodal lines; the *q* vector is almost perpendicular to the field. This suggests that the direction of the *q* vector changes with the field direction. In the anisotropic Fermi surface, the *q* vector length should also be anisotropic, depending on the energy dispersion. Although it is difficult to know the *<sup>ϕ</sup>* dependence of the *q* vector, this scenario could explain the lack of significant *<sup>ϕ</sup>* dependence of *H*∗ dip in Figure 7b.

Recent specific-heat measurements show that *H*c2 is the same at a few different inplane field directions in a low temperature range [23]. The fact shows that the FFLO stability is independent of *ϕ*, and seems to be consistent with our results: no significant *ϕ* dependence of *H*∗ dip. Theoretically, the in-plane anisotropies of *H*c2 are led by Fermi surface structure and orbital effects in the FFLO phase [45]. Therefore, no in-plane anisotropy of *H*c2 in the specific-heat measurements suggests that the orbital effect is almost negligible, and the *q* vector is most likely pinned to an optimal direction independent of the in-plane field direction. The inconsistency with our results remains an open question.

Another possible scenario that could explain our results is that multi-*q* vectors [37] are formed in the *β*---SF5 salt, since the two different Fermi pockets are present as shown in Figure 3b. In this case, the *q*-dependent anisotropic stability of the FFLO phase could be smeared out; the FFLO phase may appear in a similar field range, independent of the in-plane field direction. This scenario may also explain the lack of a significant in-plane anisotropy of *H*∗ dip and *H*c2.

Finally, we briefly mention the results for another FFLO superconductor *λ*-(BETS)2FeCl4, which had a pair of 1D and a 2D Fermi surfaces [15]. In this salt, the CM effect was first observed in the field-induced superconducting phase. The CM effect was clearly observed for *<sup>H</sup> <sup>c</sup>* but not for *<sup>H</sup> <sup>a</sup>*. The results show that a single *q* vector was fixed to the *<sup>a</sup>*-axis in the whole FFLO phase. The different behavior of the CM effect between *β*---SF5 salt and *λ*-(BETS)2FeCl4 will be closely related to the Fermi surface structure. More detailed measurements of the CM effect in other FFLO superconductors will be required to clarify the correlation between the Fermi surface structure and the *q* vector.

#### **5. Conclusions**

The AMRO measurements and band-structure calculations in the *β*---SF5 salt show that the Fermi surface is composed of two small pockets, a deformed square electron pocket, and an elliptic hole pocket, which are different from the previous report. The CM effect in the interlayer resistance was observed in a similar field region at any in-plane field directions. This indicates that the stability of the FFLO phase is almost isotropic, which is consistent with the observations of precious specific-heat measurements. Two possible scenarios are proposed: (1) a single center-of-mass momentum *q* of the Cooper pairs, which changes with the in-plane field direction, and (2) multi-*q* vectors, originating from the two anisotropic Fermi surfaces.

**Author Contributions:** S.S., T.T. and S.U. designed the experiments. S.S. and S.Y. mainly performed the resistance measurements and analyzed the data. H.A. and Y.N. performed the band calculation. J.A.S. synthesized the single crystals. S.U. supervised the project. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by KAKENHI 17H01144 and 20K14400.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author on reasonable request.

**Acknowledgments:** Work at ANL was supported by U. Chicago Argonne, LLC, Operator of Argonne National Laboratory ("Argonne"). Argonne, a U.S. Department of Energy Office of Science labor s operated under Contract No. DE-AC02-06CH11357. J.A.S. acknowledges support from the Independent Research = Development program while serving at the National Science Foundation.

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

#### **References**

