Distorted superconducting nodal line on a single Fermi surface in the anisotropic organic superconductor $\lambda -{\left(\mathrm{BETS}\right)}_2{\mathrm{GaCl}}_4$

The superconducting (SC) gap structure appearing in systems with single Fermi surface (FS) is generally described by the single gap function with $s$- or $d$-wave symmetry. The organic superconductor $\lambda$-(BETS)${}_2{\mathrm{GaCl}}_4$ [BETS = (${\mathrm{CH}}_2{)}_2{\mathrm{S}}_2{\mathrm{Se}}_2{\mathrm{C}}_6{\mathrm{Se}}_2{\mathrm{S}}_2$(${\mathrm{CH}}_2{)}_2$] endeavors to examine a novel SC gap structure on the distorted single cylindrical FS. Here, we show an example of the formation of the distorted SC nodal line by using the positive muon spin rotation (${\mu }^{+}\mathrm{SR}$) spectroscopy on $\lambda$-(BETS)${}_2{\mathrm{GaCl}}_4$. Our analysis method of the ${\mu }^{+}\mathrm{SR}$ data reveals that the nodal line has a narrower width than that of the traditional $d$-wave by the steepness factor of 4.6(2.1), and a flat part with the maximum gap exists. We found that the amplitude of the SC gap is $2\mathrm{\Delta }/k_{\text{B}}{T}_c$ = 3.9(2) and the in-plane penetration depth is ${\lambda }_{\text{ac}}$(0) = 560(5) nm. Our present study gives insight into the relation of the FS distortion and the unusual Cooper pair formation mediated by the anisotropic spin fluctuations.

Figure 1

(a) Crystal structure of $\lambda$-(BETS)${}_2{\mathrm{GaCl}}_4$. Magenta and brown layers are ${\left[{\mathrm{GaCl}}_4\right]}^{-}$ insulating and [([(${\mathrm{CH}}_2{)}_2{\mathrm{S}}_2{\mathrm{Se}}_2{\mathrm{C}}_6{\mathrm{Se}}_2{\mathrm{S}}_2$(${\mathrm{CH}}_2{)}_2{\left]\right)}_2{]}^{+}$ conducting layers, respectively. (b) Conducting plane of $\lambda$-(BETS)${}_2{\mathrm{GaCl}}_4$. Cyan area is the unit cell. BETS molecule 1–2 and 3–4 form a dimer. The blue and red lobes indicate the maximally localized Wannier orbital of the $\pi$ orbital on the BETS dimer (positive: blue, negative: red). In the inset, the first (second) nearest neighbors transfer integrals are indicated by black $t$ (orange $t^{\prime }$), with $t^{\prime }/t=0.8$. Another sublattice with the red $t$, cyan $t^{\prime }$, and $t^{\prime }/t=0.2$ alternates in the $a$ direction. (c) The 3D Fermi surface estimated from the DFT calculation [10, 11, 12, 13, 14].

Figure 2

(a)–(c) Normalized ${\mu }^{+}\mathrm{SR}$ time spectra measured in $\lambda$-(BETS)${}_2{\mathrm{GaCl}}_4$ under the transverse field (TF) of 3, 6, and 15 mT, respectively, at 10 K (cyan) and 0.3 K (black). The red and black solid lines are the best-fitting results using the Gaussian-type-damped cosine equation. The trace of the envelope represents the damping behavior of the muon-spin precession at 10 K (red) and 0.3 K (black). (d) Temperature dependence of the damping rate, $\sigma \left(T\right)$, and that in the SC state, ${\sigma }_{\text{SC}}\left(T\right)$, after subtracting the nuclear dipole moment part above $T_c$ by ${\sigma }^2={\sigma }_{\text{SC}}^2+{\sigma }_{\text{NM}}^2$. The dashed line indicates $T_c=5.3$ K determined from our resistivity and magnetization measurements [34]. (e) Temperature dependence of field shift extracted from the fitting parameter of $B_1$ and $B_2$ in the Eq. (1) which is shifted from those measured at $T=10\mathrm{K}$. (f) Transverse-field dependence of ${\mu }^{+}\mathrm{SR}$ damping rate in the superconducting state, ${\sigma }_{\text{SC}}\left(H\right)$. Magenta circle and solid line are the data and fitting line by using Eq. (3) roughly estimating $H_{\text{c2}}$. Cyan and cyan broken lines are the $H_{\text{c1}}$ and $H_{\text{c2}}$ determined by the SQUID measurement in Fig. 3.

Figure 3

(a) $H\text{−}T$ phase diagram of the superconducting state determined from magnetic susceptibility measurements. Cyan, magenta, and cyan circles show the lower, irreversible, and upper critical fields, respectively ($H_{\text{c1}}, H_{\text{irr}}, H_{\text{c2}}$) determined by the isothermal field dependence of magnetization measurement ($MH$ loop) at several temperatures. Colored solid lines are the fitting lines by using Eq. (4). The inset shows two measured $MH$ loops. (b) Temperature dependence of normalized inverse square London penetration depth obtained from TF-${\mu }^{+}\mathrm{SR}$ and SQUID measurement.

Figure 4

Temperature dependence of the inverse square of the in-plane London penetration depth, ${\lambda }_{\text{ac}}^{-2}={\rho }_{\text{s}}{c}^{-2}$, in $\lambda$-(BETS)${}_2{\mathrm{GaCl}}_4$ fitted by (a) $d$-wave, (b) $s$-wave, (c) multigap $s+d$-wave, and (d) $h$-wave distorted nodal line symmetry for $a=4.6$, respectively. The lower inset in (c) shows the obtain gap structure considering the upper limit of ${\sigma }_{\text{NM}}=0.1131\mu {\text{s}}^{-1}$ [10] for deducing the ${\sigma }_{\text{SC}}$ from $\sigma$. The broken lines in (d) indicate the trajectory for $a=1$. Black solid lines are the best-fitting results for gap symmetries described by each tested gap function, $g\left(\varphi \right)$. Green solid lines are the difference between data and the fit. Blue and red color areas in the insets show changes in the sign of the SC gap phase $\varphi$, and the $\mathrm{\Delta }\left(\varphi \right)$ is in the unit of meV.

Figure 5

(a) The plotting of the $h$-wave model in the case of $a=1$ and ${\mathrm{\Delta }}_{\text{h}}\left(0\right)=0.89$ meV, ${\lambda }_{\text{ac}}^{-2}\left(0\right)=3.16 \mu {\text{m}}^{-2}$, the same parameters obtained in Fig. 4. (b) The fitting result of the $h$-wave model in case we fixed $a=1$ and let other parameters free. The obtained parameters are simulated in the inset with broken magenta and cyan lines working very similar to the one obtained by $d$-wave symmetry plotted as solid lines filled with their red and blue area which is the inset of Fig. 4. The solid green line shows the difference between the fitting result of the black line and the data point.