*3.2. β"-(ET)2SF5CH2CF2SO3*

The crystal structure of β"-(ET)2SF5CH2CF2SO3 (*T*<sup>c</sup> = 5.2 K) has triclinic symmetry and the ET donor molecules form 2D conduction layers parallel to the *a*-*b* plane [49]. The insulating SF5CH2CF2SO3 − layer is intercalated between the ET donor layers, which makes the *c*\*-axis the least conducting direction. Both a 2D Fermi pocket and a pair of quasi-1D Fermi sheets is predicted from the band-structure calculation [50]. The SdH effect and AMRO (angular-dependent magnetoresistance oscillation) studies clearly show one small 2D FS with an area of 5% of the first Brillouin zone [50,51]. Reflecting the layered structure, its GL coherence length perpendicular to the conducting layers, ξ⊥(0) (~ 7.9 ± 1.5 Å), is shorter than the interlayer spacing *d* of 17.5 Å [28].

There are three intriguing properties in β"-(ET)2SF5CH2CF2SO3. First, high field AMRO showed the nature of incoherent interlayer transport. This means that the FS of β"-(ET)2SF5CH2CF2SO3 is regarded as the highly 2D confined electron system [52]. Second, the FS structure is very similar to that of K-(ET)2Cu(NCS)2 [49]. Third and most importantly, the SC state of β"-(ET)2SF5CH2CF2SO3 may be mediated by charge fluctuations because a pressure-induced charge ordering state is observed at around 1 GPa [53]. For β" type organic conductors, a large intersite Coulomb repulsion has been theoretically predicted [6,8,54]. Moreover, *d*-wave superconductivity mediated by charge fluctuations has been proposed based on above theoretical study [8,55], but not confirmed experimentally. In this section, we discuss the relationship between in-plane angular variation of *H*c2 and vortex dynamics in β"-(ET)2SF5CH2CF2SO3.

## 3.2.1. Anisotropy of Upper Critical Field in β"-(ET)2SF5CH2CF2SO3

To study the upper critical field, *H*c2(φ), within the conducting *a*-*b* plane, the interlayer resistance as a function of magnetic field at various fixed φ was investigated as shown by Figure 5a. The resistance curves were taken in intervals of Δφ = 10◦. At around the SC transition, the resistance gradually reaches zero with decreasing magnetic field. Figure 5b presents the in-plane angular variation of *H*c2. We define *H*c2, when *R*/*Rn* = 0.9, 0.7, and 0.5, where *Rn* is the normal state resistance given by extrapolating *R*(*H*) in the higher field region of 13 *μ*0*H* 14 T. It is clear that the angular variation of *H*c2 exhibits maxima at around φ = ±90◦ and φ = 0◦. Although the values of *H*c2 are changed by the different criteria, the fourfold oscillation pattern of *H*c2 itself remains.

**Figure 5.** (**a**) Magnetic field dependence of interlayer resistance at various fixed φ in the ohmic regime, where applied field is parallel to the conducting plane. The curves are taken from φ = 100◦ (top curve) to −100◦ (bottom) with intervals of Δφ = 10◦. (**b**) In-plane angular variation of *H*c2 determined from the resistive transition. The *H*c2 values are defined as the fields at which the resistance of the measured sample has reached 90%, 70%, and 50% of its normal-state value. (Reprinted from [27]. Copyright 2015 The Physical Society of Japan.)

For an unconventional superconductor, *H*c2(φ) minima occur for applied field parallel to the nodal directions. Thus, it is considered that the SC gap possesses its node (or minimum) at approximately π/4 from the *b*-axis. This result is in favor of a *dx* 2 −*y* <sup>2</sup> gap symmetry [38].

### 3.2.2. In-Plane Anisotropy of Vortex Dynamics in β"-(ET)2SF5CH2CF2SO3

To further discuss observed *d*-wave like anisotropy of *H*c2, the in-plane anisotropy of the vortex dynamics is next shown. Figure 6 presents the polar angle dependence of the interlayer resistance at various φ values. The structures around θ = 90◦ (i.e., dip or peak) depend on φ, showing the anisotropic vortex dynamics within the conducting layers. To see the anisotropic field effect, the φ-dependence of the interlayer resistance at θ = 90◦ for various currents is shown in Figure 7a. At 100 μA, we observe a fourfold angular oscillation: cusp-like minima are observed at φ = 20◦ and ±90◦. With increasing current, the amplitude increases, showing the remarkable non-ohmic transport phenomena. Figure 7b presents φ-dependence of the interlayer resistance at θ = 90◦ for several magnetic fields. As shown by Figure 7b, the effect of magnetic field on the flux-flow resistance is very similar to Figure 7a. With increasing magnetic field, a non-sinusoidal fourfold angular pattern is found. The cusp-like minima are observed at φ = 20◦ and ±90◦ that are the same as Figure 7a. The current dependence of the flux-flow resistance at a fixed field strength (Figure 7a) is very similar to the field dependence of that at a fixed current (Figure 7b), suggesting that combination of high magnetic field and large current, that is, Lorentz force plays an important role for appearance of the fourfold pattern in the flux-flow resistance. Similar fourfold pattern has been found in the flux-flow resistance in K-(ET)2Cu(NCS)2 [25].

**Figure 6.** Polar angle dependence of interlayer resistance for various fixed φ values. The curves are taken in intervals of Δφ = 10◦ between φ = 100◦ (top curve) and −100◦ (bottom). (Reprinted from [27]. Copyright 2015 The Physical Society of Japan.).

**Figure 7.** Azimuth angle dependence of the flux-flow resistance for several values of current (**a**) and of magnetic field (**b**). (Reprinted from [27]. Copyright 2015 The Physical Society of Japan.)

According to theoretical studies [18,46,47], the Doppler effect in the *d*-wave paring state gives rise to remarkable response of QP generation on the Fermi surface at low temperatures. It is known that as a field is parallel to the antinodal orientation, QPs are excited at four nodes that contribute to the DOS. On the other hand, when the field is directed along a nodal orientation, QPs at that node do not contribute to the DOS, and therefore the Doppler shift vanishes at these points. Since the QPs are expected to dampen the vortex motion, the flux-flow resistance associated with the vortex motion will

be suppressed for the magnetic field applied parallel to the antinodal directions, giving rise to the weaker FFR for the antinodal orientation. For β"-(ET)2SF5CH2CF2SO3, when *dx* 2 −*y* <sup>2</sup> pairing, as discussed in Figure 4b, is assumed, R(φ) should have minima at around φ = ±90◦ and φ = 0◦ and it should have maxima at around φ = ±45◦ orientations, and therefore this is consistent with Figure 7a,b.

Finally, a comment is given on the paring mechanism for the SC state in β"- (ET)2SF5CH2CF2SO3. The nodal orientations discussed here are far from the orientations of most effective nesting vector on the Fermi surface predicted from the band-structure calculation, as well as determined from the AMRO experiments [26,50,51]. If a spin fluctuation scenario is assumed, it is natural to expect the nodes that are parallel to the antiferromagnetic nesting vector, to be along the *b*\* and *a*\* (or *a*) orientations. It is intriguing to point out that superconductivity with another *d*-wave symmetry is theoretically suggested based on the charge fluctuation scenario [55]. Further investigation is needed to elucidate which mechanism (i.e., spin fluctuations versus charge fluctuations) is more likely.

#### *3.3. λ-(BETS)2GaCl4*

The layered organic conductor λ-(BETS)2GaCl4 undergoes a SC transition at *T*<sup>c</sup> of ~8 K [56]. The BETS donor molecules are stacked along the *a*- and *c*-axes. The insulating GaCl4 − anion layers are inserted between the BETS conducting layers. Thus, the 2D conducting layers are formed in λ-(BETS)2GaCl4. Reflecting the layered structure, its GL coherence length perpendicular to the layers is shorter than the interlayer spacing of 18.6 Å [57]. λ-(BETS)2GaCl4 is known as a good candidate for realizing the Fulde– Ferrell–Larkin–Ovchinnikov state [57,58]. Another intriguing point is that its isostructural compound λ-(BETS)2FeCl4 shows a field-induced SC transition [59]. Band-structure calculation [56] predicts the existence of one closed 2D Fermi pocket and two 1D Fermi sheets that are topologically the same as K-(ET)2Cu(NCS)2 [50,51]. Measurements of the SdH and AMROs are qualitatively consistent with the band calculation [60].

As for the SC gap structure, a STM experiment suggested *dxy*–wave symmetry with the line nodes along the *a*\*- and *c*\*-axes [61]. From systematic investigations by chemical substitution in the anions [62,63] or by selecting different donor molecules [64], the SC phase is suggested to exist next to the Mott insulating phase, which is similar to the K- (ET)2*X* system [65]. An NMR study showed the development of spin fluctuations beyond the SC phase transition temperature [66]. A heat capacity study showed a *d*-wave pairing state [67], whereas a μSR study clamed a mixture of the extended *s*- and *d*-wave SC gap [68]. In this section, we discuss the interplay between in-plane anisotropy of vortex dynamics and the SC gap structure for λ-(BETS)2GaCl4 [26].

### In-Plane Anisotropy of Vortex Dynamics in λ-(BETS)2GaCl4

Figure 8 shows the polar angle dependence of the interlayer resistance at various fixed φ values. The sharp peak is clearly observed at θ = 90◦ at all φ values, which shows the vortex dynamics for all φ-directions. Figure 9 shows the azimuth angle dependence of the interlayer resistance at θ = 90◦ for various currents. The in-plane angular dependence is mainly represented by the twofold symmetry with the sharp dip at φ = 0◦ (*H* // *c*). This feature differs from the cases of K-(ET)2Cu(NCS)2 [25] and β"-(ET)2SF5CH2CF2SO3 [27] even though the FS geometry of λ-(BETS)2GaCl4 [56] is similar to that of K-(ET)2Cu(NCS)2 [60] and β"-(ET)2SF5CH2CF2SO3 [50,51].

**Figure 8.** Polar angle dependence of interlayer resistance under rotating field of 8.5 T for various fixed φ. The curves are taken in intervals of Δφ = 10◦ between −80◦ (top curve) and 120◦ (bottom). (Reprinted from [26]. Copyright 2014 The Physical Society of Japan.)

**Figure 9.** Azimuth angle dependence of flux-flow resistance in λ−(BETS)2GaCl4 at various currents. A sharp minimum is observed at φ = 0◦ (*H* // *c*). (Reprinted from [26]. Copyright 2014 The Physical Society of Japan.)

In spite of the similar FS geometry, we found an in-plane twofold flux-flow resistance anisotropy in λ-(BETS)2GaCl4 while fourfold-symmetric flux-flow resistance for β"- (ET)2SF5CH2CF2SO3 and K-(ET)2Cu(NCS)2. As a possible explanation, we consider that the different anisotropic feature may be related to the difference of the interlayer coupling strength [31]. The anisotropy parameter Γ, given by (ξ///ξ⊥) 2, for β"-(ET)2SF5CH2CF2SO3 is Γ ~ 330 that is larger the those of K-(ET)2Cu(NCS)2 (Γ ~ 100) and λ-(BETS)2GaCl4 (Γ ~ 60) [27]. Since λ-(BETS)2GaCl4 is more three dimensional than K-(ET)2Cu(NCS)2 and β"-(ET)2SF5CH2CF2SO3, an orbital pair breaking effect in λ-(BETS)2GaCl4 is stronger than in K-(ET)2Cu(NCS)2 and β"-(ET)2SF5CH2CF2SO3.

Due to the orbital pair breaking effect in λ-(BETS)2GaCl4, the superposition of the normal-state magnetoresistance cannot be avoided in the in-plane anisotropy of the fluxflow resistance. Figure 10 presents the in-plane angular dependence of normal-state magnetoresistance in the magnetic field of 14.8 T [26]. The normal-state magnetoresistance possesses the twofold symmetry with broad maximum at around φ = −20◦. The result is consistent with Tanatar et al. [57]. The twofold symmetric normal-state magnetoresistance can be understood in terms of Fermi-surface anisotropy if it mainly originates from the ellipsoidal 2D pocket elongated along the *c*-axis. In the small Γ system, the large twofold component in the normal-state magnetoresistance may mask the fourfold ones.

**Figure 10.** Azimuth angle dependence of normal-state magnetoresistance in λ−(BETS)2GaCl4 within the conducting plane. (Reprinted from [26]. Copyright 2014 The Physical Society of Japan.)

The origins of the sharp dip (φ = −20◦) and the broad minimum (around *φ* = 90◦) in Figure 9 are next discussed. Similar dip structures are found in K-(ET)2Cu(NCS)2 and β"-(ET)2SF5CH2CF2SO3. The origin of the dip in λ-(BETS)2GaCl4 should be the same as that in K-(ET)2Cu(NCS)2 and β"-(ET)2SF5CH2CF2SO3. To theoretically discuss the SC gap structure for λ-(BETS)2GaCl4, Aizawa et al. [69] performed first-principles band calculation. Considering the spin-fluctuation-mediated mechanism, they discussed the SC gap function by applying the random phase approximation. They showed that the obtained SC gap changes its sign four times along the Fermi surface, suggesting a *d*-wave SC gap in λ- (BETS)2GaCl4. Reflecting the low symmetry of the crystal structure in λ-(BETS)2GaCl4, however, the SC gap has only twofold symmetry. It means that the predicted SC gap has a large gap between narrow opening nodes with an acute angle (around the steep node structure) and a small gap between wide opening nodes with an obtuse angle. Recent magnetic-field-angle-resolved heat capacity study [70] is consistent with theoretically predicted SC gap function. The large gap is located along the *c*-axis [69] which agrees with the position of sharp dip [26]. The small gap exists at around *a*-axis [69], where

we observed broad minimum in flux-flow resistance [26]. Thus, experimental results of flux-flow resistance in λ-(BETS)2GaCl4 [26] are consistent with the *d*-wave gap structure theoretically discussed by Aizawa [69].

#### **4. Summary**

In order to discuss the relationship between vortex dynamics and the SC gap structure with *d*-wave paring symmetry, we investigated in-plane angular variation of vortex dynamics for the layered organic superconductors K-(ET)2Cu(NCS)2, β"-(ET)2SF5CH2CF2SO3, and λ-(BETS)2GaCl4. We observed clear fourfold-symmetric anisotropy in the interlayer flux-flow resistance for K-(ET)2Cu(NCS)2 and β"-(ET)2SF5CH2CF2SO3, while only twofold symmetry in λ-(BETS)2GaCl4. For K-(ET)2Cu(NCS)2 and β"-(ET)2SF5CH2CF2SO3, fluxflow resistivity showing fourfold oscillation can be consistently explained by assuming the enhanced viscous motion of vortices by QPs arising from the Doppler effect. Absence of the fourfold anisotropy in λ-(BETS)2GaCl4 is discussed in the two regimes. The first regime is related to the stronger interlayer coupling in the λ-(BETS)2GaCl4 system as compared with K-(ET)2Cu(NCS)2 and β"-(ET)2SF5CH2CF2SO3. The second regime is discussed in terms of recent theoretical study by Aizawa et al. In this scenario, flux-flow resistivity with twofold anisotropy may be associated with the crystal structure with low symmetry, which is rather different from those of K-(ET)2Cu(NCS)2 and β"-(ET)2SF5CH2CF2SO3.

For these three organic superconductors, the origin of the in-plane anisotropy of fluxflow resistance with sharp minima is discussed in terms of the effect of Doppler shifted state. Based on these results, we claim that angular dependence of the vortex dynamics strongly depends on the SC gap structure. We hope that further understanding of the vortex dynamics presented here leads to clarifying the mechanism of the unconventional superconductors in various strongly correlated electron systems.

**Funding:** These works were partly supported by Grants-in-Aid for Scientific Research on Innovative Areas (Grant No. 20110004) from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, and Grants-in-Aid for Scientific Research (C) (Grant No. 25400383) from the Japan Society for the Promotion of Science (JSPS).

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available within the article.

**Acknowledgments:** The author thanks Shinya Uji, Taichi Terashima, Takako Konoike, Satoshi Tsuchiya, Kaori Sugii, Takayuki Isono, Yoritsugu Iida, Kazuya Saito, Yasuhisa Yamamura, Motoi Kimata, Hidetaka Satsukawa, Jun-ichi Yamada, Biao Zhou, Akiko Kobayashi, Hayao Kobayashi, John A. Schlueter, Hirohito Aizawa, Kazuhiko Kuroki, Nobuhiko Hayashi, and Yoichi Higashi for useful discussions and suggestions.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


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