Temperature dependent anisotropy and two-band superconductivity revealed by lower critical field in organic superconductor $\kappa$-(BEDT-TTF)₂Cu[N(CN)₂]Br

Fig. 1 shows the normalized in-plane resistivity $\rho(T)/\rho$(300 K) of $\kappa$-(ET)₂ABr at temperatures from 2 K to 300 K at ambient pressure. The normalized $\rho(T)$ curve displays four distinct subdivisions, which is in agreement with the previous reportsPhysRevLett.104.217003 ; PhysRevB.90.195150 ; PhysRevLett.114.216403 ; PhysRevB.60.9309 ; kamiya2002magnetic ; kund1993anomalous ; watanabe1991lattice ; yu1991anisotropic . The arrow in the inset points to the onset of $T_{c}$ ($T_{c}^{onset}$) of 12.2 K. Below $T_{c}^{onset}$, $\kappa$-(ET)₂ABr is in the superconducting state. The transition width measured from 90$\%$ to 1$\%$ $\rho_{n}$ is only about 0.4 K. The resistivity turns into the behavior of $\rho=\rho_{0}+AT^{2}$ in the temperature region $T_{c}^{onset}<T<$ 30 K, which is consistent with the expectation of the Fermi liquid theory for a metal. As temperature increases further, the normalized $\rho(T)$ curve starts to deviate from the $T^{2}$ dependence behavior and displays a dramatic increase followed by a pronounced maximum around 100 K. In the high-temperature region $T>$ 100 K, the resistivity decreases with increasing temperature, showing a semiconducting behavior. This rapid rising of resistivity above about 40 K and the metal-insulator transition was reported in previous literaturesstrack2005resistivity ; su1998structural ; PhysRevB.60.574 and was attributed to the temperature induced evolution of the correlation effect and density of states near Fermi energyPRLLime .

In order to intuitively demonstrate the anisotropy of this layered organic superconductor, the temperature dependence of in-plane resistivity was measured with the magnetic field oriented perpendicular ($H\perp ac$) and parallel ($H\parallel ac$) to the ac-planes. The temperature dependence of magnetic susceptibility measurement of the same single crystal was carried out before measuring the resistivity. As shown in Fig. 2(c), the magnetic susceptibility curves show the significant diamagnetism signal and a sharp superconducting transition, indicating that the single crystal used for the measurements is of high quality. In the configuration of $H\perp ac$ shown in Fig. 2(a), the superconducting transition widens significantly as the magnetic field increases. However, when $H\parallel ac$, the field induced broadening becomes very weak, the sample $\kappa$-(ET)₂ABr is still in superconducting state at $\mu_{0}H_{\parallel}$ = 9 T. The slight broadening of the superconducting transition indicates that the parallel upper critical field is quite high. The upper critical field $\mu_{0}H_{c2}$ is then determined by using a criterion of 90$\%\rho_{n}(T)$, where $\rho_{n}(T)$ is determined by a fit to the quadratic temperature dependence $\rho=\rho_{0}+AT^{2}$ in the normal state resistivity between 13 K and 20 K. The field-temperature phase diagram $\mu_{0}H_{c2}-T$ is depicted in Fig. 2(d). The slopes of $\mu_{0}H_{c2}-T$ near $T_{c}$ are -1.65 T/K and -19.2 T/K for $H\perp ac$ and $H\parallel ac$, respectively. According to the Werthamer-Helfand-Hohenberg (WHH) formulabrison1995anisotropy

the extrapolated orbital critical fields are $\mu_{0}H_{c2}^{\perp}(0)=13.9$ T and $\mu_{0}H_{c2}^{\parallel}(0)=161.6$ T. The coherence length $\xi$ is given by $H_{c2}=\Phi_{0}/2\pi\xi^{2}$, where $\Phi_{0}$ is the flux quantum. Therefore, it is estimated that the in-plane coherence length $\xi_{\parallel}(0)$ is 4.9 nm and the out-of-plane coherence length $\xi_{\perp}(0)$ is 1.4 nm. The anisotropy $\Gamma$ derived from the formula of $\Gamma=H_{c2}^{\parallel}(0)/H_{c2}^{\perp}(0)$ is 11.6, which indicates highly anisotropic feature of the superconducting state in this organic superconductor.

In fact, the anisotropy $\Gamma$ is not a fixed value at different temperatures. Therefore, exploring the temperature dependence of $\Gamma$ is necessary for a thorough study of the anisotropic properties of this layered organic superconductor. For anisotropic materials, according to the anisotropic GL theory, the angular dependence of $H_{c2}$ can be obtained by following formula

By selecting an appropriate value for $\Gamma$, the angle-resolved in-plane resistivity measured under different fields at a fixed temperature can be scaled onto one curve by using the scaling variable $\tilde{H}=H\sqrt{cos^{2}\theta+\Gamma^{-2}sin^{2}\theta}$PhysRevLett.68.875 . Fig. 3 displays the angle-resolved in-plane resistivity at six measured temperatures. Here $\theta=0^{\circ}$ and $\theta=180^{\circ}$ correspond to the cases with magnetic field perpendicular to the ac-planes, and $\theta=90^{\circ}$ corresponds to the magnetic field parallel to the ac-planes. The scaling results of $\rho$ versus $\tilde{H}$ are shown in Fig. 4(a), and the appropriate value of $\Gamma$ selected for each temperature are shown in Fig. 4(d). One can see that the resistivity curves show good scaling behavior at 2 K and 4 K. However, at higher temperatures, the scaling is not very successful. We extracted $T_{c}$ at different angles $\theta$ under fixed magnetic fields, and adopted the two theoretical models to fit the angle dependence of $T_{c}$, which are shown in Fig. 4(b). The solid line is the fit of $T_{c}(\theta)$ by the 2D-Tinkham modelPhysRevB.40.5263 ; tinkham1996introduction , which is described by,

Here, $T_{c0}$ is the superconducting transition temperature in zero-field. $T_{c}^{\perp}(H)$ and $T_{c}^{\parallel}(H)$ are the superconducting transition temperatures for $H\perp ac$ and $H\parallel ac$, respectively. The dashed line is the fit of $T_{c}(\theta)$ based on anisotropic 3D-GL theoryPhysRevB.40.5263 , which is given by,

Here $H_{0}$ is the applied magnetic field. The ratio of the effective masses is calculated from the formula of $m_{\parallel}/m_{\perp}=(H_{c2}^{\perp}/H_{c2}^{\parallel})^{2}$. It can be seen from our model that the 2D-Tinkham model fits better than that of the anisotropic 3D-GL model. Fig. 4(c) displays the temperature dependence of critical fields at both magnetic field directions defined by three different criteria $90\%\rho_{n}(T)$, $50\%\rho_{n}(T)$ and $5\%\rho_{n}(T)$. The anisotropy $\Gamma$ determined by the ratio of parallel critical field to perpendicular critical field is shown in Fig. 4(d). It is noteworthy that $\Gamma$ defined by the criterion of $5\%\rho_{n}(T)$ has approximately the same temperature dependence as the value determined from the scaling of angular resistivity versus $\tilde{H}$. The anisotropy $\Gamma$ changes a lot from above 20 near $T_{c}$ to a small value at low temperatures, indicating a crossover from the orbital depairing mechanism in high-temperature and low-field region to the paramagnetic depairing mechanism in the high-field and low-temperature region.

The low-field magnetization was measured to explore the magnetic field penetration behavior of $\kappa$-(ET)₂ABr. Now the magnetic field was applied perpendicular to the ac-planes. The initial magnetization curves in Fig. 5(a) exhibit a linear behavior, which shows the Meissner state at low magnetic fields. As the applied magnetic field increases further, the diamagnetization curves deviate from linearity, indicating that the magnetic field starts to penetrate into the sample in the form of vortices. The slope of the magnetization curve of the perfect Meissner state should be $-4\pi M/H=-1$, while the slope of the initial linear part in Fig. 5(a) is -2.82, which indicates that the influence of the demagnetization factor $N$ cannot be ignored. Considering the influence of $N$, the magnetization relationship in the Meissner state is $-4\pi M=H/(1-N)$, where $N$ takes 0.65. The magnetic field that deviates from the linear Meissner shielding of the magnetization curve is defined as the lower critical field $H_{c1}$. The open symbols in Fig. 5(b) show the deviations of the experimental magnetization data from that of the Meissner state: $\bigtriangleup M=M-M_{Meissner}$. Due to the small value of $H_{c1}$ for $\kappa$-(ET)₂ABr, especially at a temperature close to $T_{c}$, only experimental data within the measurement accuracy range of SQUID can be provided. Therefore, in order to determine a more accurate $H_{c1}$ by using a lower criterion, we used the relationship $\bigtriangleup M=a(H-b)^{c}$ to fit the experimental $\bigtriangleup M(H)$ curves in the low-field region, which are shown in the form of solid curves in Fig. 5(b). The fitting parameters at different temperatures are listed in Table 1. With $\triangle M=0.05$ $\rm emu/cm^{3}$ as the criterion, we also extracted the lower critical fields $H_{c1}(T)$ from the experimental data, the results are shown in Fig. 5(c). The temperature dependence of $H_{c1}$ extracted from the fitting curves of $\bigtriangleup M(H)$ are shown in the inset, where the determination criterion of Fit 1 (red symbols) is $\triangle M$ = 0.05 $\rm emu/cm^{3}$ and that of Fit 2 (green symbols) is $\triangle M$ = 0.01 $\rm emu/cm^{3}$. The final values of $H_{c1}$ are obtained by multiplying the deviation field data with $1/(1-N)$. All the $H_{c1}(T)$ curves show the concave shape in wide temperature region.

For a single-band superconductor, the relationship between $H_{c1}$ and the normalized superfluid density $\widetilde{\rho}_{s}$ is given byCARRINGTON2003205 ; PhysRevLett.94.027001 ,

Here, $H_{c1}(0)$ and $\rho_{s}(0)$ are the lower critical field and superfluid density in zero-temperature limit, $\bigtriangleup(T)$ is the superconducting gap function, $f(E,T)$ is the Fermi function, and $E=\sqrt{\epsilon^{2}+\bigtriangleup^{2}}$ is the total energy, where $\epsilon$ is the single-particle energy counting from the Fermi energy. As shown by the fitting curves, neither a single $s$-wave gap nor a single $d$-wave gap can fit $H_{c1}(T)$ well over a wide temperature region. However, $H_{c1}(T)$ can be well fitted using a two-gap model containing an $s$-wave gap and a $d$-wave gap. The normalized superfluid density $\widetilde{\rho}_{s}$ of the two-gap model can be expressed as $\widetilde{\rho}_{s}=x\widetilde{\rho}_{s}^{s}+(1-x)\widetilde{\rho}_{s}^{d}$, where x is the proportion of $\widetilde{\rho}_{s}$ from the $s$-wave gap. The gap values we use to fit $H_{c1}(T)$ are $\bigtriangleup_{s}=0.5$ meV and $\bigtriangleup_{d}=2.2$ meV. The proportion x of the $s$-wave gap is 0.78. The value of $\bigtriangleup_{d}$ used in the above fitting is consistent with the gap value observed in the STM experimentdoi:10.1143/JPSJ.77.114707 . The existence of $d$-wave component indicates a sign change of the superconducting gap cross the Fermi surface, implying a strong coupling superconductivity in this organic superconductor, and the gap ratio of $2\bigtriangleup_{d}/k_{B}T_{c}$ = 4.18 being larger than the expected value (3.53) of the weak-coupling BCS theory. Similar to our fitting result, $Pinteri\acute{c}$ et al. used an $s+d$-wave gap to well describe the temperature dependence of superfluid density obtained by the ac susceptibility techniquerefId0 . In their two-gap model, the proportion of the $s$-wave component is also quite large (about 0.7). Some theoretical calculations point to an eight-node superconductivity with a pairing mechanism of $s_{\pm}+d_{x^{2}-y^{2}}$-wave symmetry in $\kappa$-(ET)₂ABrPhysRevB.97.014530 ; PhysRevLett.116.237001 , our experimental results for the necessity of nodal gaps may partially support to this theoretical prediction. In the $\kappa$-(ET)₂X superconductors, it was predicted that the Fermi surface splits into two parts, one is open (electron-like) and 1D running down two of the Brillouin-zone edges, and another one is a closed ‘quasi-two-dimensional’ hole pocketsingleton2001band . Thus the discovery of two gaps and two bands from our experiment is understandable. But a momentum resolved gap structure is highly desired. Anyway, the pairing symmetry of $\kappa$-(ET)₂X is still under debate, and more experimental and theoretical research are needed to provide more explicit conclusions. In Fig. 5(d), we present the penetration depth $\lambda$ calculated by $\lambda=\sqrt{\frac{\Phi_{0}ln\kappa}{4\pi H_{c1}}}$, where $\kappa$ is the GL parameter. The penetration depth $\lambda$ shows smooth temperature dependence in the low-temperature region, and it increases sharply at a temperature close to $T_{c}$. However, we need to emphasize that some previous reports concluded that $\lambda$ in zero-temperature limit is about 500-800 nmPhysRevLett.83.4172 ; PhysRevResearch.2.043008 ; doi:10.7566/JPSJ.82.064711 , reflecting a much small superfluid density. In the wide temperature region, the overall values of $\lambda$ we obtained are smaller than the values reported in the literature. It was reported once that the penetration depth at zero temperature is even smaller than the value we obtainedCARRINGTON2003205 . We interpret this discrepancy as a consequence of different sample status from different groups. This is reasonable since the effective DOS near the Fermi energy is heavily influenced by the very narrow and shallow bands crossing the Fermi energy. A slight change of the sample status may greatly modify the superfluid density of the system. Our latest experiments and refined analysis reveal a relatively small penetration depth $\lambda(0)\sim$ 200 nm, implying a moderate superfluid density in the system.