Quasiparticle heat transport in single-crystalline ${\text{Ba}}_{1-x}{\text{K}}_x{\text{Fe}}_2{\text{As}}_2$: Evidence for a $k$-dependent superconducting gap without nodes

The thermal conductivity $\kappa$ of the iron-arsenide superconductor ${\text{Ba}}_{1-x}{\text{K}}_x{\text{Fe}}_2{\text{As}}_2\left(T_c\simeq 30\text{K}\right)$ was measured in single crystals at temperatures down to $T\simeq 50\text{mK}\left(\simeq T_c/600\right)$ and in magnetic fields up to $H=15\text{T}\left(\simeq H_{c2}/4\right)$. A negligible residual linear term in $\kappa /T$ as $T\to 0$ shows that there are no zero-energy quasiparticles in the superconducting state. This rules out the existence of line and in-plane point nodes in the superconducting gap, imposing strong constraints on the symmetry of the order parameter. It excludes $d$-wave symmetry, drawing a clear distinction between these superconductors and the high-$T_c$ cuprates. However, the fact that a magnetic field much smaller than $H_{c2}$ can induce a residual linear term indicates that the gap must be very small on part of the Fermi surface, whether from strong anisotropy or band dependence, or both.

Figure 1

(Color online) Temperature dependence of the in-plane resistivity $\rho \left(T\right)$ of ${\text{Ba}}_{1-x}{\text{K}}_x{\text{Fe}}_2{\text{As}}_2$, with $x\simeq 0.25$ and $T_c=26\text{K}$. Inset: same data plotted as a function of $T^2$. The line is a linear fit to $\rho \left(T\right)={\rho }_0+A{T}^2$ below 70 K.

Figure 2

(Color online) Temperature dependence of the thermal conductivity $\kappa \left(T\right)$, plotted as $\kappa /T$ vs $T^{1.65}$, in zero magnetic field. The line is a linear fit over the temperature range shown.

Figure 3

(Color online) Left panel: temperature dependence of the thermal conductivity measured in magnetic fields of 0, 1, 4, 10, and 15 T (from bottom to top), plotted as $\kappa /T$ vs $T^2$. Solid lines are a linear fit to each curve in the range shown. Vertical dashed lines indicate $T=0.1\text{K}$ (blue) and $T=0.15\text{K}$ (green). The value of $\kappa /T$ at those two temperatures is plotted vs magnetic field in the right panel. Right panel: isotherms of $\kappa /T$ as a function of magnetic field $H$, for $T\to 0$ (obtained by extrapolating the linear fits in the left panel to $T=0$), 0.1 and 0.15 K. In all three cases, $\kappa /T$ rises approximately linearly with $H$, with the same slope. The solid line is a linear fit to the $T\to 0$ data, also reproduced in Fig. 4.

Figure 4

(Color online) Residual linear term ${\kappa }_0/T$ in the thermal conductivity of various superconductors as a function of magnetic field $H$, presented on normalized scales ${\kappa }_s/{\kappa }_N$ vs $H/H_{c2}$. For ${\text{Ba}}_{1-x}{\text{K}}_x{\text{Fe}}_2{\text{As}}_2$ (red circles), we use ${\kappa }_N/T$ obtained from the Wiedemann-Franz law applied to the extrapolated residual resistivity ${\rho }_0$ (see text) and $H_{c2}=65\text{T}$ (Ref. 31). The data for the clean and dirty isotropic $s$-wave superconductors Nb and InBi, respectively, are reproduced from (Ref. 24). The data for the $d$-wave superconductor Tl-2201 are for a strongly overdoped sample ($T_c=15\text{K}$ and $H_{c2}\simeq 7\text{T}$) (Ref. 32). The data for multiband superconductor ${\text{NbSe}}_2$ ($T_c=7\text{K}$ and $H_{c2}=4.5\text{T}$) are from Ref. 25.