In-plane superfluid density and microwave conductivity of the organic superconductor $\kappa$-(BEDT-TTF)${}_2$Cu[N(CN)${}_2$]Br: Evidence for $d$-wave pairing and resilient quasiparticles

We report the in-plane microwave surface impedance of a high-quality single crystal of $\kappa$-(BEDT-TTF)${}_2$Cu[N(CN)${}_2$]Br. In the superconducting state, we find three independent signatures of $d$-wave pairing: (i) a strong, linear temperature dependence of superfluid density; (ii) deep in the superconducting state the quasiparticle scattering rate $\Gamma \sim T^3$; and (iii) no BCS coherence peak is observed in the quasiparticle conductivity. Above $T_c$, the Kadowaki-Woods ratio and the temperature dependence of the in-plane conductivity show that the normal state is a Fermi liquid below $\simeq 23$ K, yet resilient quasiparticles dominate the transport up to $\simeq 50$ K.

Figure 1

Temperature dependence of the 19.6 GHz surface impedance, $Z_s=R_s+i{X}_s$. The absolute surface reactance is obtained by finding the temperature-independent offset, $X_{s0}$, that makes $X_s\left(T\right)\equiv X_{s0}+\Delta X_s\left(T\right)$ match $R_s\left(T\right)$ in the Hagen-Rubens regime above $T_c$. Here $X_s\left(T\right)$ is plotted on the assumption that residual $R_s$ is zero. Inset: Real part of the microwave conductivity, ${\sigma }_1\left(T\right)$, at 19.6 GHz. The lowest trace (open circles) indicates ${\sigma }_1\left(T\right)$ extracted on the assumption that there is no residual surface resistance. The upper traces (dashed lines) show ${\sigma }_1\left(T\right)$ for residual $R_s$ of 2, 4, and 6 m$\Omega$, respectively. The initial rise in ${\sigma }_1\left(T\right)$ on cooling through $T_c$ is robust in the face of uncertainties in residual $R_s$, and signifies a rapid drop in quasiparticle scattering on entering the superconducting state. Note the absence of a BCS coherence peak below $T_c$, which would cause ${\sigma }_1\left(T\right)$ to rise almost vertically before falling exponentially at low temperatures.

Figure 2

Superfluid density, ${\rho }_s\left(T\right)\equiv 1/{\lambda }_L^2\left(T\right)=\omega {\mu }_0{\sigma }_2\left(T\right)$, obtained from the imaginary part of the microwave conductivity at 19.6 GHz. ${\lambda }_0=3220$ Å is determined from the absolute measurement of $X_s\left(T\right)$ in Fig. 1, leaving little uncertainty in the shape of ${\rho }_s\left(T\right)$. The superfluid density displays a strong, linear temperature dependence over most of the temperature range, indicating an order parameter with nodes. This is strikingly different from the BCS $s$-wave superfluid density (Ref. 45) (dashed curve). At very low temperatures ${\rho }_s\left(T\right)$ starts to flatten: the solid line is a fit to a linear-to-quadratic crossover function, ${\rho }_s\left(T\right)=1/{\lambda }_0^2-a{T}^2/(T+T_0)$, with crossover temperature $T_0=0.52$ K. The dashed line denotes the expected location of the Kosterlitz-Thouless vortex-unbinding transition. This should occur when the 2D superfluid density ${\rho }_{\mathrm{s}}^{2\mathrm{D}}\equiv {\hslash }^{2}d/4k_{\mathrm{B}}{e}^2{\mu }_0{\lambda }_L^2=(2/\pi )T$, where $d=15$ Å is the interlayer spacing (Ref. 46). ${\rho }_s\left(T\right)$ shows the expected upward curvature close to $T_c$ as it passes through the vortex-plasma regime.

Figure 3

In-plane resistivity, ${\rho }_{\parallel }\left(T\right)$, and quasiparticle scattering rate, $\Gamma \left(T\right)$. Above $T_c$, ${\rho }_{\parallel }\left(T\right)$ grows initially as $T^2$, steepening with increasing temperature. $k_F{\ell }_{\parallel }\approx 50$ at $T_c$, falling to 15 by 30 K, where ${\ell }_{\parallel }$ is the in-plane mean free path and $k_F$ is the Fermi wave vector. Upper inset: ${\rho }_{\parallel }\left(T\right)$ vs. $T^2$. Straight line fit is ${\rho }_{\parallel }\left(T\right)={\rho }_0+A_{\parallel }{T}^2$, with ${\rho }_0=43$ $\mu \Omega$cm and $A_{\parallel }=0.211$ $\mu \Omega$cm/K${}^2$. Lower inset: $\Gamma \left(T\right)$ in the superconducting state, fit to $\Gamma \left(T\right)={\Gamma }_0+B{T}^3/{\Delta }^2\left(T\right)$, with ${\Gamma }_0/2\pi =47$ GHz. $\Delta \left(T\right)={\Delta }_0\left(2.7\sqrt{T_c/T-1}\right)$ approximates the temperature dependence of the superconducting gap (Ref. 72).