Renormalizations in unconventional superconducting states of ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$

We have measured the superconducting penetration depth $\mathrm{\Lambda }\left(T\right)$ in the heavy-fermion/intermediate-valent superconducting alloy series ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$ using a transverse-field muon spin relaxation to study the effect of intermediate-valent Yb doping on Fermi-liquid renormalization. From $\mathrm{\Lambda }\left(T\right)$ we determine the superfluid density ${\rho }_s\left(T\right)$ and find that it decreases continuously with increasing nominal Yb concentration $x$, i.e., with increasing intermediate valence. The temperature-dependent renormalization of the “normal” fluid density ${\rho }_N\left(T\right)={\rho }_s\left(0\right)-{\rho }_s\left(T\right)$ in both the heavy-fermion and intermediate valence limits is proportional to the temperature-dependent renormalization of the specific heat. This indicates that the temperature-dependent Fermi-liquid Landau parameters of the superconducting quasiparticles entering the two different physical quantities are the same. These results represent an important advance in understanding of both intermediate valence and heavy-fermion phenomena in superconductors.

Figure 1

$\mu \mathrm{SR}$ asymmetry spectra $A\left(t\right)$ from ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$, ${\mu }_{0}H=30$ mT, $H\parallel c$. (a) $x=0.05$. (b) $x=0.125$. Red squares: normal state. Blue circles: superconducting state. Solid curves: fits to the data (see text).

Figure 2

Temperature dependencies of $\mathrm{TF}\text{−}\mu \mathrm{SR}$ relaxation rate ${\sigma }_s$ in ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$, ${\mu }_{0}H=30\mathrm{mT}\parallel c$. Data for $x=0$ is from Ref. [30]. Solid curves: power-law fits to the data (see text). Arrows: $T_c$ from transport measurements [3, 4]. Inset: flux-line lattice relaxation rate ${\sigma }_{\mathrm{FLL}}$ from Eq. (4) vs normalized temperature $T/T_c$.

Figure 3

Dependence of normalized $T=0$ superfluid densities ${\rho }_0\left(x\right)/{\rho }_0\left(0\right)$ and Yb valence [4, 15] on nominal Yb concentration $x$ in ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$. The curve is a guide to the eye. Inset: Uemura plot of $T_c$ vs ${\mathrm{\Lambda }}^{-2}$ for various superconductors [32, 33, 34, 35, 36, 37] including ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$.

Figure 4

Temperature dependencies of normalized “normal” fluid density ${\rho }_N\left(T\right)/{\rho }_s\left(0\right)=1-{\mathrm{\Lambda }}^2\left(0\right)/{\mathrm{\Lambda }}^2\left(T\right)$ and electronic specific heat coefficient $C_{\mathrm{el}}/T$ in superconducting ${\mathrm{Ce}}_{1-x}{\mathrm{Yb}}_x{\mathrm{CoIn}}_5$. Solid blue triangles: ${\rho }_N$ measured by $\mu \mathrm{SR}$ experiment. Open blue triangles: ${\rho }_N$ measured using the TDO technique [38]. Red circles: $C_{\mathrm{el}}/T$ (Ref. [3] and present measurements). (a) $x=0$. Inset: low-temperature data. Curves: fits of Eqs. (6) and (7) to the data, and the fitting curve of specific heat data is plotted in light green dashed curve for comparison. (b) $x=0.125$. (c) $x=0.3$.