Full-Gap Superconductivity Robust against Disorder in Heavy-Fermion ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$

A key aspect of unconventional pairing by the antiferromagnetic spin-fluctuation mechanism is that the superconducting energy gap must have the opposite sign on different parts of the Fermi surface. Recent observations of non-nodal gap structure in the heavy-fermion superconductor ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ were then very surprising, given that this material has long been considered a prototypical example of a superconductor where the Cooper pairing is magnetically mediated. Here we present a study of the effect of controlled point defects, introduced by electron irradiation, on the temperature-dependent magnetic penetration depth $\lambda (T)$ in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$. We find that the fully gapped state is robust against disorder, demonstrating that low-energy bound states, expected for sign-changing gap structures, are not induced by nonmagnetic impurities. This provides bulk evidence for $s_{++}$-wave superconductivity without sign reversal.

Figure 1

Superconducting transitions of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ single crystals before and after electron irradiation. (a) Normalized frequency shift in the TDO measurements as a function of temperature below 0.7 K for all measured samples. Samples were measured while warming to minimize ac field self-heating close to $T_c$. (b) Critical temperature $T_c$ defined as the midpoint of transition normalized by the clean-limit value $T_{c0}=0.71\mathrm{K}$ as a function of irradiation dose (upper axis). We also plot $T_c/T_{c0}$ as a function of residual resistivity ${\rho }_0$ taken from the reported resistivity data $\rho (T)$ [1] (lower axis). The upper axis is adjusted to match the linear relation between the dose and ${\rho }_0$ [1].

Figure 2

Temperature dependence of the change in the penetration depth $\mathrm{\Delta }\lambda$ as a function of normalized temperature $T/T_c$. The origin of $\mathrm{\Delta }\lambda$ at $T\to 0\mathrm{K}$ is determined by the power-law fitting. The vertical axis is normalized by each value at $0.3T_c$. Values of $\mathrm{\Delta }\lambda (0.3T_c)$ are 29, 49, 38, and 30 nm for doses of 0, 2.2, 3.7, and $4.8\mathrm{C}/{\mathrm{cm}}^2$, respectively.

Figure 3

Disorder-induced changes of low-temperature penetration depth in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$. (a) Exponent $n$ of a power-law fit of the experimental data up to $T_{\max }/T_c$. The colors for different doses are the same as in Fig. 2. The dotted (dashed) line shows the clean (dirty) limit case of $n=1$ (2) in unconventional superconductors with line nodes. (b) Similar plot for minimum superconducting gap ${\mathrm{\Delta }}_{\min }$ normalized by $k_B{T}_c$ obtained by the exponential fitting. (c) Exponent $n$ as a function of pair-breaking parameter $g=\hslash /{\tau }_{\mathrm{imp}}{k}_B{T}_{c0}$, in comparison with those for ${\mathrm{BaFe}}_2{({\mathrm{As}}_{1-x}{\mathrm{P}}_x)}_2$ [14] and ${\mathrm{Ce}}_{1-x}{\mathrm{La}}_x{\mathrm{CoIn}}_5$ [20]. For ${\mathrm{Ce}}_{1-x}{\mathrm{La}}_x{\mathrm{CoIn}}_5$, we use the values of ${\lambda }_{ab}(0)=200\mathrm{nm}$ and ${\lambda }_c(0)=280\mathrm{nm}$ [21], and ${\rho }_0$ is estimated from Ref. [22]. (d) Normalized gap minima ${\mathrm{\Delta }}_{\min }/k_B{T}_c$ from the fit for $T_{\max }/T_c=0.2$ plotted against $\mathrm{\Delta }{T}_c/T_{c0}$. For comparison, also plotted are the data for minimum gap in neutron-irradiated ${\mathrm{MgB}}_2$ (closed squares and triangles), which change to a single gap (open symbols) for heavily irradiated samples [23, 24].

Figure 4

Temperature dependence of normalized superfluid density ${\rho }_s(T)={\lambda }^2(0)/{\lambda }^2(T)$ in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ before and after irradiation. Black and gray line are the theoretical temperature dependence of ${\rho }_s(T)$ in the conventional $s$-wave (BCS) and clean-limit $d$-wave cases, respectively. The inset shows the measured lower critical field $H_{c1}$ (closed circles) plotted against the inverse of residual resistivity $1/{\rho }_0$. The dashed line evidences a linear relation, from which the $H_{c1}(0)$ value for $4.8\mathrm{C}/{\mathrm{cm}}^2$ (open diamond) and thus the corresponding $\lambda (0)$ value are estimated.