**1. Introduction**

Physics of organic conductors has provided the fascinating and rich issues associated with the low dimensionality and effect of electron correlation [1]. One of the most interesting systems are the layered organic conductors composed of the ET donor molecule, where ET denotes bis(ethylenedithio)tetrathiafulvalene. The ET donor molecules form two-dimensional (2D) conducting layers that are separated by insulating anion layers with a monovalent ion X−. Figure 1a shows schematic structure of ET (or BETS) donor conducting layers and anion insulating layers in a layered organic conductor, where BETS denotes bis(ethylenedithio)tetraselenafulvalene. The molecular structures of ET and BETS are shown in the upper right side of Figure 1a. In layered organic superconductors, there are several types (labeled by Greek letters) in packed ET donor layers. The K-, β"-, and λ-type arrangements in donor layers are shown in Figure 1b–d. In the K type compounds, K-(ET)2*X*, the measurements of Shubnikov-de Haas (SdH) and de Haas-van Alphen (dHvA) oscillations have elucidated the presence of a well-defined Fermi surface (FS) with simple structures [2]. Moreover. the moderately heavy effective mass revealed by SdH and dHvA experiments suggests that electron correlation plays a significant role on determining the physical properties of the normal state as well as superconducting (SC) state. It was suggested that superconductivity appears in proximity to the antiferromagnetic insulating state in the electronic phase diagram [3,4]. Since some of these unusual physical properties suggest similarities with high-*T*c cuprates, many researchers have pointed out that the spin fluctuations play a vital role for the appearance of SC state [5–8].

**Citation:** Yasuzuka, S. Interplay between Vortex Dynamics and Superconducting Gap Structure in Layered Organic Superconductors. *Crystals* **2021**, *11*, 600. https:// doi.org/10.3390/cryst11060600

Academic Editor: Toshio Naito

Received: 24 March 2021 Accepted: 20 May 2021 Published: 26 May 2021

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**Figure 1.** (**a**) (Upper left side) Schematic structure of donor conducting layers and anion insulating layers in a layered organic conductor. (Upper right side) Structures of ET and BETS donor molecules. (Lower side) Three patterns within packed donor conducting layers for (**b**) K-, (**c**) β"-, and (**d**) λ-type arrangements.

For layered organic superconductors, many experimental studies have pointed out the existence of nodes in the SC gap [9–15]. The SC gap structure in unconventional superconductors mostly have nodes parallel to certain orientations in the momentum space. The confirmation of point or line nodes is often discussed from power-law temperature (*T*) dependences in physical quantities such as the heat capacity or the nuclear spin relaxation rate. However, determination of the SC gap structure is a harder task. For example, the SC gap symmetry of *dxy* (with nodes along vertical and horizontal directions) differs from *dx* 2 −*y* <sup>2</sup> (diagonal direction) only in the location of the line nodes (π/4 rotation of the latter becomes identical with the former). Therefore, direction-sensitive experimental methods are highly needed to distinguish between these gap structures. Since the SC gap structure is intimately associated with the pairing mechanism [16], it is of fundamental importance to elucidate the SC gap structure.

In the vortex state of *d*-wave superconductors, gapless quasiparticles (QPs) extending outside the vortex core play an important role for determining their physical properties at magnetic fields [17,18]. As an effect of applied magnetic field on *d*-wave superconductors, Volovik [17] theoretically introduced a "Doppler shifted state" in the energy spectrum of the QP which is induced by superfluid flow around the vortices. The QPs with low energy are induced by the application of magnetic field, mainly around the nodes where the QP velocity is perpendicular to the magnetic field orientation. The QP density depends on

the relative direction of the magnetic field with respect to the nodes. Derived from the Doppler sifted state that depend on the magnetic field orientation, the dependence of the heat capacity and the thermal conductivity on applied field orientation was demonstrated to be a useful probe for detecting the nodal directions in a SC gap structure [14,15,19].

Up to now, we have focused on the interplay between vortex dynamics and the SC gap structure in layered organic superconductors because the flux-flow resistivity is a measure of QP dissipation in the vortex dynamics [20–24]. To investigate the correlation between vortex dynamics and the SC gap structure, we measured the in-plane anisotropy of the flux-flow resistivity for the layered organic superconductors K-(ET)2Cu(NCS)2, β"-(ET)2SF5CH2CF2SO3, and λ-(BETS)2GaCl4 [25–27]. SC parameters for these superconductors are listed in Table 1. As shown below, we clearly observed the flux-flow resistivity with fourfold-symmetric anisotropy, owning to the *d*-wave SC gap symmetry in K-(ET)2Cu(NCS)2 and β"-(ET)2SF5CH2CF2SO3. On the other hand, twofold symmetric anisotropy was found in λ-(BETS)2GaCl4 although λ-(BETS)2GaCl4 possesses the similar FS and SC gap structures in the former two superconductors. Interplay between in-plane anisotropy of vortex dynamics and nodal SC gap structures for these superconductors are discussed below.

**Table 1.** Superconducting parameters for K-(ET)2Cu(NCS)2 [2], β"-(ET)2SF5CH2CF2SO3 [2,28–30] and λ-(BETS)2GaCl4 [2,31,32].


### **2. Experimental Methods**

Single crystals of K-(ET)2Cu(NCS)2, β"-(ET)2SF5CH2CF2SO3, and λ-(BETS)2GaCl4 were synthesized electrochemically. The interlayer resistance was measured by a fourterminal ac method, where the electric current is parallel to the least conducting direction. To investigate the in-plane angular variation of the flux-flow resistance for layered organic superconductors, it is crucial to rotate applied field parallel to the conducting layers with high accuracy because a slight field-angle misalignment gives rise to twofold anisotropy of the magnetoresistance related to the huge *H*c2 anisotropy [33]. To check the reproducibility, four samples were simultaneously mounted on a two-axis rotator in a 4He cryostat with a 17-T SC magnet. By using the rotator, it is possible to sweep the θ angle with a resolution of Δθ = 0.1◦, where θ is the polar angle between the least conducting axis and magnetic field direction. In addition, we can discretely control the plane of rotation, which is represented by the azimuthal angle φ with intervals of Δφ = 5◦ or 10 within the conducting layers.

To investigate the in-plane anisotropy of the flux-flow resistance at a constant magnetic field, the resistivity as a function of the angle θ at various fixed φ was measured because we can easily find the position of the *H* parallel to the conducting plane at any φ. For example, the interlayer resistivity of K-(ET)2Cu(NCS)2 is shown in Figure 2a [25]. For the magnetic field exactly applied parallel to the conducting plane, the ρ(θ) curve shows a sharp drop due to the SC transition at the lowest current density of 0.5 mA/cm2. It is remarkable that the resistivity at θ = 90◦ depends strongly on the current density. A sharp peak is clearly observed at θ = 90◦ for highest current density of 100 mA/cm2. The resistivity for |<sup>θ</sup> − <sup>90</sup>◦|>5◦ is independent of current density of up to 100 mA/cm2. Thus, the

effect of Joule heating is negligibly small. Instead, the sharp peaks are due to the vortex dynamics [25,33–38]. Similar features are found for β"-(ET)2SF5CH2CF2SO3 in Figure 2b and λ-(BETS)2GaCl4 in Figure 2c [26,27].

**Figure 2.** Interlayer resistance as a function of polar angle θ for κ-(ET)2Cu(NCS)2 (**a**), β"-(ET)2SF5CH2SF2SO3 (**b**), and λ-(BETS)2GaCl4 (**c**). The insets show the definition of θ and φ. Adapted from refs. [25–27].

#### **3. Results and Discussion**

#### *3.1. κ*-(*ET)2Cu(NCS)2*

The nature of the K-type organic superconductor K-(ET)2*X* possesses similar features to that of high-*T*<sup>c</sup> cuprates, such as quasi-2D electronic structure and the competition of its SC phase with the antiferromagnetic insulating phase [2–4]. To elucidate the pairing mechanism in the K-(ET)2*X* system, its SC gap symmetry has been extensively investigated from both experimental and theoretical points of view [9–14]. Various experiments such as NMR [9], heat capacity [10,11], and mm-wave transmission measurements [12] suggested presence of *d*-wave SC gap in K-(ET)2Cu(NCS)2. As for the nodal direction, scanning tunneling microscopy (STM) [13] and thermal conductivity experiments [14] suggested the line node gap rotated 45◦ relative to the *b* and *c* axes (*dx* 2 −*y* <sup>2</sup> symmetry). Malone et al. [15] argued in favor of *dxy* symmetry (the line nodes along the *b* and *c* axes) from the heat capacity measurement. These results have demonstrated the *d*-wave superconductivity in the K-(ET)2*X* system, but the location of gap nodes is still controversial. In this section, we present in-plane anisotropy of upper critical field and vortex flow resistivity for K- (ET)2Cu(NCS)2 and discuss the nodal structure.

### 3.1.1. In-Plane Anisotropy of Upper Critical Field in K-(ET)2Cu(NCS)2

To discuss the in-plane anisotropy of upper critical field, measurements of the interlayer resistance were performed for φ = 0◦ (*H* // *c*), φ = 45◦, φ = 90◦ (*H* // *b*). Figure 3a presents the temperature dependence of interlayer resistance for various values of magnetic field applied along the *b-axis* (φ = 90◦) up to 14.8 T. Due to a broad transition into the SC state, the following four criteria are used here to extract the SC transition temperature at each magnetic field. These are shown in Figure 3a, where the four criteria of the SC transition temperature: a "junction" *T*J, a "midpoint" *T*M, "a zero resistance extrapolation" *T*<sup>X</sup> (ignoring the tail near *R* = 0), and "zero resistance point" *T*<sup>Z</sup> are defined. A similar fashion was confirmed for φ = 45◦ and φ = 0◦ (*H* // *c*).

Figure 2b presents a *T*–*H* phase diagram for three magnetic field directions, using the four criteria. All criteria clearly show highly isotropic behavior within our experimental error, suggesting the lack of in-plane anisotropy of *H*c2 in spite of many suggestions of

*d*-wave pairing symmetry in K-(ET)2Cu(NCS)2. This is consistent with previous magnetoresistance study by Nam et al. [39]. Lower temperatures will be needed for confirmation of an expected *d*-wave nature in *H*c2. In addition, we find change of slope in *H*c2 line near *T*c, suggesting the dimensional crossover from anisotropic 3D SC to Josephson coupled 2D SC state [40–42]. It means that despite short coherence length along the interlayer direction, the SC state can be regarded as anisotropic 3D SC state only near *H*c2 line.

**Figure 3.** (**a**) Temperature dependence of interlayer resistance for various magnetic fields *H* // *b* (φ = 90◦) in κ- (ET)2Cu(NCS)2. (**b**) *H*–*T* phase diagram derived from (**a**), where four criteria, J (junction), M (midpoint), X (*R* → 0), and Z (*R* = 0) are plotted for *T*c(*H*).

3.1.2. In-Plane Anisotropy of Vortex Dynamics in K-(ET)2Cu(NCS)2

Figure 4a presents the interlayer resistivity as a function of polar angle θ at various φ values with an interval of Δφ = 5◦. As seen in Figure 4a, there is the peak structure at θ = 90◦ at all φ values, which confirms the vortex motion for all in-plane field directions. The peak structure gradually changes with φ. To discuss the in-plane anisotropic effect in flux-flow resistance, we display the φ dependence of the interlayer resistivity at θ = 90◦ at various temperatures in Figure 4b. In the entire temperature range measured, the *ρ*(φ,*T*) curves show minima at φ = 0◦ (*H* // *c*) and 90◦ (*H* // *b*). At temperatures beyond 7.8 K, a sinusoidal angular variation was observed. At elevated temperature, the amplitude of oscillatory behavior is weakened but survives even at 18 K, where the SC phase is vanished. At temperatures below 7.8 K, we find a non-sinusoidal fourfold angular-variation: cusp-like minima are found at φ = 0◦ and 90◦ that are independent of temperature.

As seen in Figure 4b, the ρ(φ) curves seem to be described by a fourfold symmetric origin. Takanaka and Kuboya theoretically showed that the angular variation of the upper critical field is described as *H*c2(φ) ∝ cos4φ in the *d*-wave superconductivity [43–45]. The ρ(φ) curves are naively consistent with this theory; the maximum and minimum of *H*c2 give the minimum and maximum resistivities, respectively. When *dx* 2 −*y* <sup>2</sup> pairing is assumed, *H*c2(φ) has maxima at φ = 0◦ (*H* // *c*) and φ = 90◦ (*H* // *b*), and a minimum at φ = 45◦, whereas *dxy* pairing leads to *H*c2 with maxima and minima reversed with respect to *dx* 2 −*y* <sup>2</sup> pairing. Thus, the fourfold symmetry in Figure 3b is not inconsistent with the *H*c2 anisotropy originating from the *dx* 2 −*y* <sup>2</sup> pairing state. However, detailed magnetoresistance studies at 1.5 and 4.2 K [39] as well as Figure 3 showed the lack of in-plane anisotropy of *H*c2. In addition, the cusp-like minima that appear below 7.8 K cannot be represented by a simple cos4φ dependence. Thus, another mechanism to describe these oscillation patterns is necessary.

**Figure 4.** (**a**) Polar angle dependence of interlayer resistance at 5.0 K under a magnetic field of 13 T for various values of φ. The curves are measured in intervals of Δφ = 5◦ between 120◦ (top curve) and −20◦ (bottom). (**b**) Dependence of resistivity at θ = 90◦ at various temperatures. (Reprinted from [25]. Copyright 2013 The Physical Society of Japan.)

As seen in Figure 4b, we observe sharp peaks related to dissipative feature arising from the vortex dynamics at any azimuth direction within the conducting plane. Therefore, the origin of the cusp-like minima is discussed in terms of the anisotropic vortex dynamics. Since the Josephson vortex essentially does not have a normal core, the pinning effect arising from lattice defects and/or impurities is negligibly small for vortex motion parallel to the conducting layers. In this situation, the friction force is largely affected by the damping viscosity, which depends on the energy dissipation processes of QPs. In the simple description of flux-flow behavior, the Lorentz force on a vortex is balanced by a viscous drag force. Since the Lorentz force keeps constant during the field direction changed, the anisotropic flux-flow resistivity observed here suggests a remarkable variation of the QP excitations by the applied field rotating within the conducting layers because flux-flow resistance can be regarded as a measure of QP dissipation in the vortex dynamics [20–24].

Let us discuss the mechanism of the observed fourfold oscillation in terms of the Doppler effect originally proposed by Volovik [17,18]. In the *d*-wave paring state, a nonzero QP density-of-state (DOS) is generated on the Fermi surface by the effect of Doppler shifted state. Theoretical studies [18] clearly show that the DOS possesses a broad maximum for the field parallel to the antinodal direction and sharp minimum for the field along the node. When an applied field is parallel to the antinodal orientation, all four nodes affect the DOS. On the other hand, when the field is parallel to a nodal orientation, QPs at that node do not affect the DOS. Since a damping of the Josephson vortex motion by QPs is expected, the stronger damping will occur when the magnetic field is applied parallel to the antinodal orientations, leading to the suppression of the flux-flow resistivity for the antinodal orientation. When the *dx* 2 −*y* <sup>2</sup> pairing is assumed, ρ(φ) should become minima at φ = 0◦ (*H* // *c*) and φ = 90◦ (*H* // *b*) and maximum at φ = 45◦, which agrees with Figure 4b. Thus, we consider that our result concerned with flux-flow resistance is consistent with *dx* 2 −*y* <sup>2</sup> pairing state, as suggested by STM [13] and thermal conductivity experiments [14].

The above consideration is essentially based on the semiclassical theory [17,18]. Unfortunately, this semiclassical treatment is valid only at low temperatures and low fields and

does not consider the scattering effect between the QPs and moving vortices. For wider temperature and field ranges, Vorontsov and Vekhter [46,47] developed a microscopic theory for the angular dependent properties of thermal conductivity and heat capacity in *d*-wave SC state. They found that application of the magnetic field parallel to the SC gap nodes may result in maxima or minima in their angular dependence, depending upon the location in the *T*–*H* plane. This variation is attributable to the scattering of the QPs on vortices. Such sign reversal oscillation was observed and discussed in the heavy fermion superconductor CeCoIn5 [48]. A similar problem between the thermal conductivity and heat capacity measurements was discussed for K-(ET)2Cu(NCS)2 [14,15]. According to the *T*–*H* diagrams in [46,47], field-angle dependences of heat capacity and thermal conductivity do not show sign reversal pattern at *H*/*H*c2 ~ 0.43 and *T*/*T*<sup>c</sup> > 0.48, where our experimental study was done. Although our results are not inconsistent with no sign reversal oscillation for a wide temperature region, further experimental study such as confirmation of sign reversal pattern at lower fields will be needed to elucidate the effect of the Doppler shifted state on the vortex dynamics.
