Universal heat conduction in ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$: Evidence for robust nodal $d$-wave superconducting gap

In the heavy-fermion superconductor ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$, Yb doping was reported to cause a possible change from nodal $d$-wave superconductivity to a fully gapped $d$-wave molecular superfluid of composite pairs near $x\approx 0.07$ (nominal value $x_{\mathrm{nom}}=0.2$). Here we present systematic thermal conductivity measurements on ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$ ($x=0.013$, 0.084, and 0.163) single crystals. The observed finite residual linear term ${\kappa }_0/T$ is insensitive to Yb doping, verifying the universal heat conduction of the nodal $d$-wave superconducting gap in ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$. Similar universal heat conduction is also observed in the $\mathrm{CeCo}({\mathrm{In}}_{1-y}{\mathrm{Cd}}_y$)${}_5$ system. These results reveal a robust nodal $d$-wave gap in ${\mathrm{CeCoIn}}_5$ upon Yb or Cd doping.

Figure 1

(a) Low-temperature resistivity of ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$ single crystals. Here $x$ is the actual Yb concentration. The dashed lines are linear extrapolations to get the residual resistivity ${\rho }_0$. (b) and (c) Doping dependence of ${\rho }_0$ and normalized $T_c/T_{c0}$ ($T_{c0}=2.3$ K for pure ${\mathrm{CeCoIn}}_5$) for our single crystals and the thin films (from Ref. [17]). The shadowed narrow bar marks $x\approx$ 0.07, corresponding to $x_{\mathrm{nom}}\approx 0.2$, where several anomalies were reported, including the Yb valence transition.

Figure 2

(a)–(c) Temperature dependence of resistivity for ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$ single crystals at various magnetic fields $H\parallel c$ up to 5.5 T. (d) Temperature dependence of the upper critical field $H_{c2}\left(T\right)$ for all three samples.

Figure 3

(a)–(c) Temperature dependence of the thermal conductivity divided by temperature $\kappa \left(T\right)/T$ of ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$ under various magnetic fields. The solid lines are linear fits to extrapolate the residual linear term. The dashed lines indicate the normal-state Wiedemann-Franz law expectations $L_0/{\rho }_0$($H_{c2}$), with the Lorenz number $L_0=2.45\times {10}^{-8} \mathrm{W}\mathrm{\Omega }{\mathrm{K}}^{-2}$. (d) Field dependence of ${\kappa }_0/T$. The field is normalized by the $H_{c2}$ of the three samples

Figure 4

(a) Low-temperature resistivity of $\mathrm{CeCo}({\mathrm{In}}_{1-y}{\mathrm{Cd}}_y$)${}_5$ single crystals. Here $y$ is the actual Cd concentration. (b) Temperature dependence of zero-field thermal conductivity divided by temperature $\kappa \left(T\right)/T$ for $\mathrm{CeCo}({\mathrm{In}}_{1-y}{\mathrm{Cd}}_y$)${}_5$. The solid lines are linear fits to extrapolate ${\kappa }_0/T$. (c) ${\kappa }_0/T$ vs doping for all ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$ and $\mathrm{CeCo}({\mathrm{In}}_{1-y}{\mathrm{Cd}}_y$)${}_5$ samples at zero field. The error bar is determined from uncertainties in the geometric factor and the fit. The horizontal dashed line is the theoretical universal value for ${\mathrm{CeCoIn}}_5$ with a 2D nodal $d$-wave superconducting gap [8].