Universal Heat Conduction in the Iron Arsenide Superconductor ${\mathrm{KFe}}_2{\mathrm{As}}_2$: Evidence of a $d$-Wave State

The thermal conductivity $\kappa$ of the iron arsenide superconductor ${\mathrm{KFe}}_2{\mathrm{As}}_2$ was measured down to 50 mK for a heat current parallel and perpendicular to the tetragonal $c$ axis. A residual linear term at $T\to 0$, ${\kappa }_0/T$ is observed for both current directions, confirming the presence of nodes in the superconducting gap. Our value of ${\kappa }_0/T$ in the plane is equal to that reported by Dong et al. [Phys. Rev. Lett. 104, 087005 (2010)] for a sample whose residual resistivity ${\rho }_0$ was 10 times larger. This independence of ${\kappa }_0/T$ on impurity scattering is the signature of universal heat transport, a property of superconducting states with symmetry-imposed line nodes. This argues against an $s$-wave state with accidental nodes. It favors instead a $d$-wave state, an assignment consistent with five additional properties: the magnitude of the critical scattering rate ${\Gamma }_c$ for suppressing $T_c$ to zero; the magnitude of ${\kappa }_0/T$, and its dependence on current direction and on magnetic field; the temperature dependence of $\kappa (T)$.

Figure 1

(a) Electrical resistivity of the two samples of ${\mathrm{KFe}}_2{\mathrm{As}}_2$ studied here, with $J\parallel a$ (full red circles, left axis) and $J\parallel c$ (full blue squares, right axis). Our $a$-axis data are compared to those of Dong et al. [7] (open circles, left axis), normalized here to have the same value at $T=300\mathrm{K}$ (see text). The lines are a fit to $\rho ={\rho }_0+a{T}^{\alpha }$ from which we extrapolate ${\rho }_0$ at $T=0$. (b) Same data for the two $a$-axis samples, up to 300 K. (c) Abrikosov-Gorkov formula for the decrease of $T_c$ with scattering rate $\Gamma$ (line), used to obtain a value of $\Gamma /{\Gamma }_c$ for the three samples of ${\mathrm{KFe}}_2{\mathrm{As}}_2$, given their $T_c$ values and the factor 10 in ${\rho }_0$ between the two $a$-axis samples (circles), assuming a disorder-free value of $T_{c0}=3.95\mathrm{K}$.

Figure 2

Thermal conductivity of ${\mathrm{KFe}}_2{\mathrm{As}}_2$, plotted as $\kappa /T$ vs $T^2$, for $J\parallel a$ (${\kappa }_a$, circles, left axis) and $J\parallel c$ (${\kappa }_c$, squares, right axis), for a magnetic field $H\parallel c$ as indicated. Our $a$-axis data are compared to those of Dong et al. [7] (open circles, left axis), normalized by the same factor as in Fig. 1 (see text). Lines are a fit to $\kappa /T=a+b{T}^{\alpha }$, used to extrapolate the residual linear term $a\equiv {\kappa }_0/T$ at $T=0$. For our $a$-axis sample (full red circles), $\alpha =2.0$, while for others $\alpha <2$.

Figure 3

Field dependence of ${\kappa }_0/T$ obtained as in Fig. 2 (with corresponding symbols). The dashed line is a theoretical calculation for a $d$-wave superconductor with $\hslash \Gamma /{\Delta }_0=0.1$ [38]. Inset: Zoom at low field. Lines are a power-law fit to extract the value of ${\kappa }_0/T$ at $H=0$.

Figure 4

Dependence of ${\kappa }_0/T$ (a) and $T_c$ (b) on impurity scattering rate $\Gamma$, normalized by $T_{c0}$, the disorder-free superconducting temperature. (a) ${\kappa }_0/T$ for ${\mathrm{KFe}}_2{\mathrm{As}}_2$ (red circles; see text) and the cuprate ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ (blue squares; from Ref. 30), normalized by the theoretically expected value for a $d$-wave superconductor, ${\kappa }_{00}/T=3.3$ and $0.16\mathrm{mW}/{\mathrm{K}}^2\mathrm{cm}$, respectively (see text). The typical dependence expected of an $s$-wave state with accidental nodes is also shown, from a calculation applied to pnictides (black line; from Ref. 31). (b) $T_c$ for ${\mathrm{KFe}}_2{\mathrm{As}}_2$ [red circles; from Fig. 1c] and for the pnictides ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ and ${\mathrm{SrFe}}_2{\mathrm{As}}_2$ at optimal doping (from Ref. 20).